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Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory
Antonio Acuaviva, Pablo Acuaviva
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Summary
This paper investigates the capacity of current language models to contribute to mathematical research in Banach space theory. The authors developed an automated system that searches literature for open problems and attempts solutions at scale. AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. The study highlights the potential of language models for mathematical discovery while emphasizing the continuing importance of expert verification.
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Antonio Acuaviva → affiliatedwith → Lancaster University
confidence 95% · Antonio Acuaviva School of Mathematical Sciences, Fylde College, Lancaster University
Pablo Acuaviva → affiliatedwith → University of Bern
confidence 95% · Pablo Acuaviva Institute of Computer Science, University of Bern
Tomasz Kania → proposedproblem → Toroidal Elton–Odell theorem
confidence 95% · The first two problems were proposed by Tomasz Kania... Problem 1. Toroidal Elton–Odell theorem
Tomasz Kania → proposedproblem → Non-Calkin unital Banach algebras
confidence 95% · The first two problems were proposed by Tomasz Kania... Problem 2. Non-Calkin unital Banach algebras
Kevin Beanland → proposedproblem → Strict cosingularity and adjoints
confidence 95% · the third and fourth problems were proposed by Kevin Beanland... Problem 3. Strict cosingularity and adjoints
Kevin Beanland → proposedproblem → Weakly compact basis factorization
confidence 95% · the third and fourth problems were proposed by Kevin Beanland... Problem 4. Weakly compact basis factorization
Automated System → searchesliteraturein → arXiv
confidence 90% · developed an automated system that searches the literature for open problems... search the arXiv literature
Language Models → →
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Abstract
Abstract:We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification.
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- Source: https://arxiv.org/abs/2607.17388v1
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Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory Antonio Acuaviva School of Mathematical Sciences, Fylde College, Lancaster University, LA1 4YF, United Kingdom ahacua@gmail.com and Pablo Acuaviva Institute of Computer Science, University of Bern, Neubrückstrasse 10, 3012 Bern, Switzerland pablohacuaviva@gmail.com (Date: 19 July 2026) Abstract. We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification. Key words and phrases: AI-assisted mathematics, large language models, mathematical discovery, proof search, proof verification, Banach space theory 2020 Mathematics Subject Classification: Primary 46B20; Secondary 46B25, 46B28, 47B01, 68T20, 68T42, 68T50 “C’est par la logique qu’on démontre, c’est par l’intuition qu’on invente.” Henri Poincaré, Science et méthode Contents I Survey and Discussion 1 Introduction and discussion 2 Selected problems and provenance 3 Implications for mathematical exploration 4 Limitations, credit, and responsibility I Selected Problems 5 Problem 1. Toroidal Elton–Odell theorem 6 Problem 2. Non-Calkin unital Banach algebras 7 Problem 3. Strict cosingularity and adjoints 8 Problem 4. Weakly compact basis factorization 9 Problem 5. Primariness of Lp(L1)L_p(L_1) I The Automated Pipeline 10 Technical and methodological considerations 11 Selected results from the automated pipeline References Part I Survey and Discussion 1. Introduction and discussion The main purpose of this article is to draw attention to a development which, in our view, the mathematical community can no longer regard as merely speculative. Current language model systems, when placed in suitable workflows and checked by experts, can already do useful work on live research level mathematics. The claim is not merely that such systems can solve polished exercises, reproduce known arguments, or assist with exposition, but that they can sometimes generate serious proof candidates for questions that arise naturally in the research literature. This point is made concrete in Part˜I, where we present selected Banach space problems whose proofs were generated essentially by the model before human verification, editing, and integration; the provenance of the individual problem papers is discussed more carefully in Section˜2. A second purpose is to begin raising the questions that follow from this fact. Autonomous or semi-autonomous proof search with general-purpose models is no longer only a matter for benchmark design or future speculation. We place this development in recent and historical context in Section˜1.2. The important point is that these systems are beginning to affect how mathematical problems can be found, attempted, checked, organized, and written up. If this continues, the mathematical community will need to think carefully about its consequences: standards of verification, attribution and credit, the role of human judgement, the value of open problem lists, pressure on peer-review, and the ways in which research training and collaboration may change. We return to these broader issues in Section˜3 and 4. 1.1. Origin and scope of the project This article grew out of an automatic-search project in Banach space theory. The original objective was to search the arXiv literature for questions, conjectures, and open problems that appeared sufficiently local to be attacked, and then to try to settle them with minimal human guidance. The motivation was to work systematically through unresolved points left in papers. Such questions are often not the central open problems of a field, but answering them can still clarify the literature: a proposed problem may have a proof, a counterexample, an answer already implicit in known results, or an obstruction explaining why it remains open. In this sense, the project was concerned with accelerating the gradual closure of mathematical knowledge. As the project developed, the model-generated outputs were stronger than expected. In particular, many of the claims presented by the model as full solutions survived subsequent mathematical checking. This changed the scope of the experiment. We continued the automatic search, but also began to test the same style of automatic proof search on harder problems selected by humans: problems suggested in discussions with other mathematicians, or chosen from the authors’ knowledge of the field. Thus the project came to have two distinct components. Part˜I contains the human-selected component. The problems treated there were not discovered by the automatic pipeline. They were chosen precisely because they seemed to be substantial, non-routine research problems: difficult enough that a solution would, in our judgement, constitute a meaningful advance in Banach-space theory. Their purpose is therefore not to exhibit model performance on routine exercises, but to test autonomous proof search on problems that specialists would regard as serious mathematical targets. Thus, the selection was human and deliberately demanding, but the proof search itself was fully model-driven. The arguments printed in Part˜I were autonomously produced by the model before human verification, editing, and integration into self-contained mathematical notes. Part˜I records the automatic-search component. Here, both the selection of targets and the attempted solutions came from the pipeline. The purpose of this part is not to present polished, standalone papers, but to document the wider machine-selected, machine-attempted exploration from which the project began. The choice of field should not be interpreted as a claim that the phenomenon is specific to Banach space theory. It reflects a practical constraint on this kind of work: generated mathematics requires verification by someone who understands the relevant definitions, the cited results, and the local literature. In both parts of the article, human verification enters after generation. The authors formulate or select problems in the human-chosen cases, run the exploratory process, identify outputs worth checking, verify the mathematics, assess significance, and edit the final exposition, fixing some minor mistakes whenever necessary. 1.2. Recent context Several recent projects evaluate AI on advanced or research-level mathematics. FrontierMath currently has two components: Tiers 1–4, a benchmark of unpublished problems authored and peer-reviewed by expert mathematicians, and Open Problems, a separate collection of unsolved research problems [41, 33]. RealMath derives automatically verifiable mathematical tasks from research papers and mathematical forums and is designed as a continually refreshable benchmark [74]. LemmaBench similarly constructs a live research-level benchmark by extracting recent lemmas from arXiv and rewriting them as self-contained statements [63]. IMProofBench is a private benchmark of peer-reviewed proof problems developed by expert mathematicians; models operate in an agentic environment with tools such as web search and SageMath, while complete proofs are graded by human experts [67]. First Proof focuses more specifically on questions that arose in the work of professional mathematicians, had known solutions, and had not previously appeared online. In its first, informal batch, the organisers released ten such questions and temporarily withheld the answers. In the organisers’ preliminary tests, each query was run once, without iterative interaction or rerunning [1]. The second batch was a formal benchmark using ten new questions and four systems: ChatGPT 5.5 Pro and three academic harnesses. Each system received the problems under a fixed one-shot protocol and returned solutions without further interaction from the organisers; the problems, human-generated and AI-generated solutions, costs, evaluations including referee reports, logs, and harness code were subsequently released [2]. The aim of the present paper is different. We do not seek to compare systems under a fixed protocol or compute budget, or to produce a leaderboard result. Instead, we ask what current models can contribute when used for repeated exploration of live problems, with final human verification of claims that may be genuinely new. A complementary collaborative paradigm is represented by the AI co-mathematician, an interactive and stateful workbench through which mathematicians guide agents across ideation, literature search, computational exploration, theorem proving, and theory building [75]. The process studied here was not that kind of specialist-guided collaboration: a mathematician did not supply the next lemma or choose how to use each intermediate observation. Problems could be revisited, and partial outputs could inform later attempts, but the mathematical content of the promoted proofs was generated in the model attempts rather than supplied by a mathematician; see Section˜2. A related but methodologically distinct line of work uses automated evaluators to guide discovery. FunSearch combines an LLM with evolutionary search over programs and an automated evaluator; it produced new cap-set constructions and improved online bin-packing heuristics [64]. Subsequent work made this style of search more accessible to working mathematicians [31]. AlphaEvolve generalizes evaluator-guided evolutionary search to direct modification of algorithmic code and a broader range of scientific and computational tasks [60]. Georgiev, Gómez-Serrano, Tao, and Wagner applied AlphaEvolve to 67 mathematical problems, recovering best-known solutions in most cases and improving them in several others [39]. Such methods apply most directly in settings where candidate objects can be represented computationally and scored automatically. The problems considered here generally do not have that form. The outputs are natural-language proof candidates for questions extracted from papers or selected by mathematical judgement. Their correctness is not fully captured by a numerical objective or a short verification computation. They must instead be checked by reading the proof, verifying the hypotheses of cited results, comparing the claim with the original problem, and rewriting the argument in a clear mathematical form. Other recent work has developed longer-horizon scientific and mathematical agents with memory, tool use, literature search, theorem retrieval, and verification. The AI Scientist and The AI Scientist v2 automate stages of machine-learning research including idea or hypothesis generation, experiment execution, analysis, manuscript writing, and automated or simulated review [53, 73]. Kosmos coordinates literature search, hypothesis generation, and data analysis through a structured world model [54], while Arbor uses Hypothesis-Tree Refinement to preserve hypotheses, artifact versions, evidence, and distilled insights across long research runs [46]. Iteris uses an explore–plan–execute loop for numerical experimentation, construction, and proof development on open problems in computational mathematics; in its case studies, the resulting artifacts were followed by expert review and correction [24]. In mathematics more specifically, Aletheia iteratively generates, verifies, and revises natural-language solutions and has also been evaluated on the inaugural First Proof problems [34, 35]. Rethlas combines informal proof search with the Matlas theorem-search engine, while Archon uses LeanSearch to formalize and verify the resulting arguments in Lean 4 [48]. Recent AxiomProver-backed papers report a Lean formalization of the combinatorial identity used in one proof and autonomous Lean formalizations and machine-checkable proofs of six conjectures in another [23, 25]. Harmonic’s Aristotle combines informal reasoning with Lean proof search and a dedicated geometry solver [5]. The setup used in the present project is much lighter: direct interaction with ChatGPT 5.5 Pro and Codex agents operating over a filesystem. An automatic pipeline extracts open-problem signals from arXiv source files and organizes them into candidate questions; the agents then attempt the problems, and their outputs are submitted for human review; see Section˜10. We do not claim that this is an optimal architecture for automated mathematics. The point is that a relatively ordinary setup can already produce useful proof candidates in both settings considered here: difficult problems selected by humans and literature questions selected by machines; see Part˜I and Table˜2. 1.3. Contribution and organization The paper has two main contributions. The first is to help raise awareness within the mathematical community that current language-model systems can already play a substantive role in research-level mathematical exploration, and to open a discussion about how such tools should be understood, used, disclosed, verified, and credited. The second contribution is mathematical. The paper presents selected results in Banach space theory for human-selected problems. These results are not included merely as demonstrations of model behaviour. In our view, they are meaningful mathematical contributions in their own right, and several of them could plausibly have supported independent papers. At the same time, their provenance gives them an additional role: they provide concrete evidence that current systems can sometimes engage with serious research-level problems. The project was not fully autonomous in the strongest sense. The authors chose which outputs to promote, verified the mathematical content, checked references, assessed significance, and edited the exposition; they take responsibility for the final text and the mathematical correctness. Nor are the proofs formally verified. They are ordinary mathematical proofs, checked by human mathematical reading. At the same time, the model’s role was not confined to exposition or stylistic assistance. It generated proof ideas, proof structures, and in several cases, complete or essentially complete arguments for problems drawn from the literature. The paper is organised as follows. Part˜I gives the survey-level discussion, including the list of selected problems and comments on verification, disclosure, credit, and responsibility. Part˜I contains the selected problem papers. These are intended to be readable as self-contained mathematical notes. Part˜I records both technical details of the automatic-search pipeline and selected outputs from that pipeline. 2. Selected problems and provenance The project is problem-driven. The problems listed below are not toy examples or prompt demonstrations, but mathematical problems which could reasonably be considered serious research questions in Banach space theory. The provenance of the proofs is therefore central. For the selected problems, the proofs arose primarily from model-generated proof search, followed by human verification, editing, and rewriting. We discuss this in more detail in Subsection 2.1. The original AI outputs can be found on the project website. Thus, the problem papers should be read neither as conventional unaided proofs nor as unverified model transcripts, but as AI-assisted mathematical arguments for which the authors take responsibility after human verification and editing. The first two problems were proposed by Tomasz Kania, the third and fourth problems were proposed by Kevin Beanland, and the last problem was proposed by the first-named author. Since the individual problem papers are written mainly as proof notes rather than as fully introduced articles, we give here some brief background on each problem and its significance. When available, we also include comments supplied by the proposer. Problem summaries and background P1. Toroidal Elton–Odell theorem. Proposed by Tomasz Kania. The following background and comments were given by Tomasz Kania and are lightly edited here for style. Kottman’s theorem asserts that the unit sphere of every infinite-dimensional normed space contains a sequence whose mutual distances are strictly greater than one. The classical Elton–Odell theorem strengthens this by providing a uniform margin: there are ε>0 >0 and a sequence in the unit sphere whose mutual distances are all at least 1+ε1+ . Over the complex field, the natural projective analogue identifies vectors which differ by multiplication by a unimodular scalar. Accordingly, for x,y∈SXx,y∈ S_X, one considers the toroidal distance d(x,y)=infθ∈‖x−θy‖.d_T(x,y)= _θ \|x-θ y\|. The toroidal Elton–Odell problem asks whether every infinite-dimensional complex normed space admits ε>0 >0 and a sequence (xn)⊂SX(x_n)⊂ S_X such that d(xn,xm)≥1+ε(n≠m).d_T(x_n,x_m)≥ 1+ (n≠ m). This problem was raised explicitly in [43, §5.3.1]. A strict but non-uniform analogue holds in complete generality: every infinite-dimensional complex normed space contains a sequence (xn)⊂SX(x_n)⊂ S_X such that d(xn,xm)>1(n≠m).d_T(x_n,x_m)>1 (n≠ m). This does not settle the uniform problem, since the excess over one may depend on the pair and tend to zero. Uniform positive results were also known under additional geometric hypotheses, including finite cotype, lower q-estimates, and asymptotic uniform convexity, together with computations of the toroidal separation constant for several classical spaces [50]. The remaining question was whether the additional hypotheses could be removed and a single positive margin obtained in every infinite-dimensional complex normed space. The proof included here gives a complete solution to this problem. P2. Non-Calkin unital Banach algebras. Proposed by Tomasz Kania. This problem concerns the realisation problem for Calkin algebras. For a Banach space X, its Calkin algebra is the quotient ℬ(X)/(X)B(X)/K(X) of the bounded operators on X by the compact operators. The corresponding realisation problem asks which unital Banach algebras can occur, up to Banach-algebra isomorphism, as Calkin algebras. This question was recorded in Tarbard’s thesis and subsequently studied by Horváth and Kania [71, 44]. The scalar-plus-compact theorem of Argyros and Haydon provided a striking early solution to a particular realisation problem [10]. They constructed an infinite-dimensional Banach space X on which every bounded operator is a scalar multiple of the identity plus a compact operator. Consequently, ℬ(X)/(X)≅ℂ.B(X)/K(X) . Thus even the scalar field can occur as the Calkin algebra of an infinite-dimensional Banach space. Subsequent constructions demonstrated considerable flexibility in the class of algebras which can be realised as Calkin algebras. Motakis, Puglisi, and Zisimopoulou realised C(K)C(K) for every countable compact metric space K [59], and Motakis later extended this to every compact metric space [55]. Further examples include broad classes of diagonal scalar-plus-compact algebras [58], infinite-dimensional reflexive Calkin algebras [56], and the unitisation of the noncommutative algebra (c0)K(c_0) [57]. Despite these positive realisation results, it remained unknown whether every unital Banach algebra is isomorphic to the Calkin algebra of some Banach space. Horváth and Kania obtained a partial negative result: they constructed simple unital AF C∗C^*-algebras which are not isomorphic to the Calkin algebra of any separable Banach space [44]. Their argument did not, however, rule out a representation over a nonseparable Banach space. This left open whether there exists a unital Banach algebra A such that A≇ℬ(X)/(X)A (X)/K(X) for every Banach space X. The proof included here answers this question affirmatively. The principal difficulty is the nonseparable case, where density alone provides no obstruction. The constructions combine large density and topological simplicity with carefully chosen matrix-unit or shift relations. If either algebra were the Calkin algebra of a nonseparable Banach space, topological simplicity would force every separable-range operator to be compact. The additional algebraic relations are then used to construct a noncompact operator with separable range, giving the required contradiction. P3. Strict cosingularity and adjoints. Proposed by Kevin Beanland. The following background and comments were given by Kevin Beanland and are lightly edited here for style. This problem concerns the relationship between strictly singular and strictly cosingular operators. Pełczyński introduced and studied strictly cosingular operators as a dual counterpart to strictly singular operators, and proved several duality results between the two ideals. In particular, if T:X→YT X→ Y is strictly cosingular and weakly compact, then T∗T^* is strictly singular; conversely, if T is strictly singular and X is reflexive, then T∗T^* is strictly cosingular. He also showed that the weak compactness assumption in the first implication cannot be omitted in general: the canonical inclusion c0↪ℓ∞c_0 _∞ is strictly cosingular, while its adjoint is not strictly singular. The separable-range case arose from Beanland’s 2008 paper with George Androulakis on descriptive set theoretic methods for strictly singular and strictly cosingular operators [8]. Earlier work had introduced an ordinal hierarchy of αS_α-strictly singular operators using the transfinite Schreier families, and it was natural to ask whether a corresponding hierarchy could be obtained for strictly cosingular operators. A natural route was to define such classes through duality, but this required understanding whether Pełczyński’s implication remains true when the range space is separable and the weak compactness hypothesis is omitted. The main obstruction was the following question: if T∗T^* restricts to an isomorphism on an infinite-dimensional subspace of Y∗Y^*, must there be an infinite-dimensional weak-star closed subspace of Y∗Y^* on which T∗T^* is still an isomorphism? Beanland later posted this question on MathOverflow [17]. The proof included here gives an affirmative answer in the required form and proves that, when Y is separable, T is strictly cosingular if and only if T∗:Y∗→X∗T^*:Y^*→ X^* is strictly singular. P4. Weakly compact basis factorization. Proposed by Kevin Beanland. The following background and comments were given by Kevin Beanland and are lightly edited here for style. Davis, Figiel, Johnson, and Pełczyński, in their seminal paper introducing the interpolation method used to prove the weakly compact factorization theorem, also established conditions under which the DFJP interpolation space admits a Schauder basis [26]. Specifically, they showed that if T:X→YT X→ Y is weakly compact and the range space Y has a shrinking Schauder basis, then the interpolation space may be constructed so that it also has a Schauder basis. This hypothesis cannot simply be omitted. Not all separable Banach spaces have the bounded approximation property, while a separable Banach space has the bounded approximation property if and only if its identity operator factors through a Banach space with a Schauder basis [62]. Consequently, one cannot expect the DFJP interpolation space to admit a basis in complete generality. Obtaining a basis is not immediate from the definition of the DFJP interpolation space. If one simply constructs the interpolation space from the relatively weakly compact set T(BX)T(B_X), there is no evident basis structure. Instead, Davis, Figiel, Johnson, and Pełczyński enlarge the generating set by adjoining the images of T(BX)T(B_X) under the coordinate projections associated with the shrinking basis of Y. The shrinking property is then used to show that this enlarged set remains relatively weakly compact. This argument is highly specific to shrinking bases; in general, the same enlargement process need not preserve weak compactness, even when T itself is weakly compact. The corresponding result in the unconditional case was subsequently established by Figiel, Johnson, and Tzafriri [36]. Later, Ghoussoub, Maurey, and Schachermayer proved that the DFJP interpolation space may likewise be constructed with a Schauder basis whenever the range space is isomorphic to C[0,1]C[0,1] [40]. Beanland’s interest in these questions arose from attempts to combine the DFJP interpolation method with the descriptive set theoretic framework developed by Dodos and others [28], in order to obtain uniform factorization theorems for classes of weakly compact operators. A principal motivation is the theorem of Dodos and Ferenczi [29], who proved that the class of separable reflexive Banach spaces is strongly bounded. Thus, although no universal separable reflexive Banach space exists, every analytic family of separable reflexive Banach spaces embeds isomorphically into a single separable reflexive Banach space. Moreover, when each member of the family has a Schauder basis, these embeddings may be chosen to be complemented. Passing from Banach spaces to weakly compact operators naturally suggests an analogous theory in which uniform embedding theorems are replaced by uniform factorization theorems. The principal remaining obstacle is that, even for analytic classes of weakly compact operators whose range spaces possess Schauder bases, one needs the associated DFJP interpolation spaces to admit Schauder bases in order to apply the existing descriptive set theoretic machinery. Prior to the present work, this was known only in the special situations described above. In June 2016, Beanland asked on MathOverflow whether the DFJP interpolation space could always be chosen to have a Schauder basis when the range space is isomorphic to L1[0,1]L_1[0,1] [16]. Bill Johnson gave an elegant argument establishing this case. Using that observation, the descriptive set theoretic machinery was extended to this setting. In particular, it was proved that the collection of all weakly compact operators T:X→L1T X→ L_1, where X is separable, is analytic, and consequently there exists a single separable reflexive Banach space through which every such operator factors. The more general question of whether the DFJP interpolation space could always be chosen to admit a Schauder basis whenever the range space has a Schauder basis remained open until the proof presented here. P5. Primariness of Lp(L1)L_p(L_1). Proposed by the first named author. A Banach space is primary if, whenever it is decomposed as a direct sum of two complemented subspaces, one of those subspaces is isomorphic to the whole space. The problem considered here asks whether the mixed norm space Lp(L1)L_p(L_1) is primary for 1<p<∞1<p<∞. This question belongs to the broader programme of understanding primariness for classical Banach spaces and their mixed norm variants. Lechner, Motakis, Müller, and Schlumprecht proved that L1(Lp)L_1(L_p) is primary for 1<p<∞1<p<∞ [51]. In the same paper, they identified the primariness of Lp(L1)L_p(L_1) as one of the prominent remaining open cases. They subsequently studied this problem further in their work on multipliers on bi-parameter Haar system Hardy spaces [52], where they established factorization results constituting a first step towards a proof of primariness, while leaving the full question unresolved. The first named author’s interest in this question grew out of this line of work, which led him to study primariness for biparameter constructions involving C(K)C(K) spaces [6]. From that perspective, Lp(L1)L_p(L_1) was a natural remaining case, although one somewhat outside his original background. When the problem was proposed to the model, it was included with little expectation that it would actually be solved. It was meant partly as a technical test of whether the exploratory process could handle a long and highly structured problem in Banach space theory. In hindsight, this makes the outcome especially interesting, while the provenance discussion below explains why the original raw output should still not be confused with a finished proof. 2.1. Proof provenance and AI writing This subsection aims to record more precise comments on the AI outputs, the final proofs, and the human intervention and modification that connected them. We begin with some general comments about the outputs produced by the AI. For Problems 1, 2, 3, and 4, the original AI outputs could be regarded as essentially correct, up to verification, small modifications, and editing. Somewhat surprisingly, most of the required work was more a matter of form than substance: the issues were often in the writing, presentation, and organisation of the arguments rather than in the underlying mathematics. The main proof was already present. In some places, however, the arguments were written in a nonstandard or unnatural mathematical style. In a few other places, the model misattributed a theorem, cited a result imprecisely, or made a small error. These issues did not require substantial mathematical repair. Another recurring feature was that the model sometimes left parts of an argument not fully fleshed out, presenting them as “immediate” even when some verification was still needed. Problem 5 was different, largely because of its length and complexity. The raw output could not be regarded as a complete proof as it stood. It contained the right proof architecture, and most of the technical details and mathematical content were already present, but the assembly left much to be desired. Some parts of the proof were only indicated rather than fully written out, some terminology was used before being defined, and the exposition was at times difficult to follow. In this case, the human intervention was not merely local editing. It involved reorganising the proof, making the indicated arguments precise, and turning the architecture provided by the model into a complete written argument. 3. Implications for mathematical exploration The examples in this paper come from Banach space theory merely from practical limitations when verifying the significance and correctness of the solutions. The broader lesson is not that AI-assisted proof discovery is particular to this field, but that similar modes of exploration may become available in many areas where there is enough human expertise to verify, repair, and contextualise the generated arguments. 3.1. From solving to exploring AI systems may make it cheaper to generate many possible routes through a problem. The value of this is not only that a model may occasionally produce a complete proof. It may also suggest intermediate lemmas, illuminate alternative decompositions of a problem, or produce failed attempts whose failure is itself informative. In this sense, AI may shift part of mathematical research from solving isolated problems to exploring large families of possible approaches. Many generated ideas will fail, some will require substantial repair, and only a few may ultimately become complete mathematical arguments. Nevertheless, the ability to explore a much larger mathematical search space may itself become a valuable research capability. 3.2. Acceleration rather than replacement Current AI systems do not solve every mathematical problem, and many difficult questions remain well beyond their present capabilities. Rather than replacing mathematical research, they may accelerate it. A mathematician can formulate promising questions, provide context, reject unproductive directions, and repeatedly redirect the exploration towards approaches that appear mathematically meaningful. Even when a complete proof is not obtained, the generated intermediate ideas may shorten the path to a human solution. We therefore view AI not as a replacement for mathematical research, but as a tool that may substantially accelerate parts of the discovery process when guided by human expertise. 3.3. Human mathematical judgement If proof generation becomes easier, human judgement does not disappear. It shifts. Choosing worthwhile problems, recognizing genuinely interesting ideas, identifying hidden gaps, deciding which proof strategies deserve further development, connecting new results to existing theory, and judging whether a theorem is mathematically important may become increasingly central tasks. In particular, the ability to distinguish between mathematically significant discoveries and technically correct but uninteresting statements may become one of the defining skills of future mathematical research. 3.4. Verification and formal mathematics One of the principal bottlenecks encountered during this project was not proof generation itself, but verification. Producing plausible mathematical arguments became substantially easier than establishing, with confidence, that every step was correct and every cited ingredient was applicable. Formal proof assistants such as Lean may therefore represent a natural next step. If AI-generated proofs can eventually be translated into formally verified mathematics with relatively little additional effort, then the verification bottleneck may be substantially reduced. Such developments would not diminish the importance of human mathematicians. Rather, they would further shift human effort towards selecting important problems, evaluating ideas, organizing mathematical knowledge, and deciding which directions are worth pursuing. 3.5. Distillation as a mathematical skill As the volume of mathematically plausible text continues to grow, the ability to distill ideas into clear, elegant, and verifiable mathematics may become increasingly valuable. A proof that is technically correct but opaque is less useful than one that reveals the underlying mechanism, exposes the essential ideas, and can be readily understood and reused by other mathematicians. Clear exposition has always been a central mathematical skill. If proof generation becomes substantially cheaper, its importance may only increase. One part of the mathematician’s role may increasingly involve recognising important ideas, verifying them, organising them into coherent theories, and communicating them in a form that advances mathematical understanding. 4. Limitations, credit, and responsibility The examples in this article should be read as evidence that current AI systems can contribute substantially to proof discovery in suitable mathematical workflows. They should not be read as evidence that raw model outputs can be trusted without expert checking, or that the present project gives a general measurement of AI reliability in mathematics. The central lesson is more specific: when combined with human problem selection, verification, repair, and exposition, model generated proof search can already produce mathematically serious results. Precisely because this contribution can be substantial, it also raises a series of challenges about verification, credit, attribution, disclosure, and responsibility. The rest of this section is devoted to those challenges. 4.1. Disclosure and incentives The possibility of substantial AI assistance raises a delicate question about disclosure. If theorems obtained with such assistance are automatically devalued, while undisclosed AI assistance remains difficult to detect, then the mathematical community risks creating a perverse incentive: honest disclosure may be penalized, whereas nondisclosure may be rewarded. This concern is especially serious for early-career researchers, for whom a small number of results can have a large professional impact. The issue is not merely hypothetical. Some of the results recorded in the selected problem papers are, in our judgement, substantial enough to stand as standalone mathematical contributions. Such results can affect very concrete forms of professional evaluation: hiring, fellowships, grants, invitations, collaborations, and related opportunities. It is therefore important to address the incentive problem directly. If mathematically valuable work receives less professional recognition solely because AI assistance played a substantial, or even decisive, role in its discovery, then authors may have an incentive to present the history of the work selectively rather than transparently. One possible way forward is to keep the emphasis primarily on the usual mathematical standards: correctness, novelty, significance, clarity of presentation, and responsibility for verification. There is also a second, less visible incentive problem. If substantial AI assistance is taken to diminish the professional value of a result, researchers may become reluctant to use these systems on their most promising ideas. They may reserve their best questions, conjectures, and proof strategies for unaided work, not because AI assistance would be mathematically inappropriate, but because they fear “losing” the result in professional terms if the model contributes decisively. This would be an especially unfortunate outcome. Mathematical progress depends on testing ideas with the strongest available tools, and a norm that discourages researchers from exploring their best ideas with AI could slow discovery, delay the resolution of important problems, and reduce the collective value produced by the field. Transparency should not require researchers to choose between using an effective method of exploration and receiving appropriate recognition for the mathematics that results. In short, the lasting value of a theorem should ultimately depend on its mathematical content: the strength of its statement and proof, the clarity and depth of the ideas, the quality of the exposition, and the new avenues it opens. There is something personally and culturally meaningful about a theorem being discovered by a particular mathematician through their own insight, and that aspect of mathematics is real. Nevertheless, once a theorem has been stated, proved, and responsibly verified, its primary mathematical value should not depend on whether the path to it involved pencil and paper, conversation with colleagues, computer search, formal verification, or interaction with an AI system. This distinction also matters for incentives: if researchers, especially early-career applicants, expect less credit for results obtained with substantial AI assistance, they may be discouraged from disclosing that assistance. A healthier norm would evaluate the theorem and proof on their mathematical merits, independently of how the result came to be, while still requiring transparency about the process and full responsibility for the correctness of the final written work. 4.2. Understanding and stewardship Mathematics is not merely the production of correct theorem statements and proofs. A result becomes part of mathematics through understanding: one must know why the statement is natural, which ideas make the proof work, how it relates to the surrounding theory, and what new questions it makes possible. An automatically generated proof candidate should therefore normally be regarded as the beginning of a mathematical process rather than its end. Even when it contains the decisive idea or an essentially complete argument, turning it into durable mathematical knowledge requires people to verify it, identify its conceptual content, place it in context, and explain it in a form that others can use and develop further. There is a useful analogy with software. Software developed by, or in close collaboration with, people who understand its architecture is generally easier to inspect, maintain, repair, and extend. By contrast, code generated largely by AI can be difficult to work with when no person understands the reasons for its design choices or can explain how its components fit together. The problem is not that people have moved on from a system that they once understood, but that the system may never have been properly understood by anyone in the first place. Open-source access to the code does not by itself solve this problem: a system can be completely visible and yet remain practically opaque. A similar danger arises in mathematics. A discovery pipeline may generate correct arguments while producing a body of work whose structure and significance have not been absorbed by the mathematical community. It would be irresponsible to build systems for producing mathematics while accepting that neither the systems nor their outputs need to be understood by the people presenting and relying on them. Scale must therefore be accompanied by distillation and stewardship. The aim should not be to accumulate an ever larger collection of claims, but to transform selected outputs into mathematics that can be checked, taught, reused, and extended. People consequently remain part of the system at every stage. Mathematical problems are embedded in communities and histories, and the production of a final proof is only one contribution among many. Formulating a fruitful question, developing the surrounding theory, identifying the relevant literature, recognising a useful idea, verifying and repairing an argument, and presenting it clearly are all substantive parts of mathematical work. The role of an AI system should be disclosed accurately, but authorship and responsibility must remain with people who understand the result sufficiently well to explain it, defend it, and correct it when necessary. The same principles should govern any public platform for generated attempts. Unverified material should be unmistakably labelled as unverified, changes and corrections should be recorded, the origins of the questions should be credited, and experts should be encouraged to inspect and improve the arguments. Such a platform should support collaboration and reduce duplicated effort, rather than overwhelm the literature with unfiltered claims or create the impression that generated outputs can replace the community whose accumulated knowledge made them possible. If proof generation becomes cheaper, human understanding, verification, organisation, and communication become more important, not less. 4.3. Responsibility for correctness AI-assisted mathematics does not remove responsibility from the authors. On the contrary, it increases the burden of responsibility. The authors of an AI-assisted paper remain responsible for the correctness of the statements, the validity of the proofs, the accuracy of the references, and the clarity of the exposition. AI output should be treated as a source of possible ideas, not as a certificate of truth. 4.4. Pressure on peer-review There is also a practical danger. If AI systems make it easier to generate large numbers of plausible proofs, then the already strained peer-review system may face an additional burden. Referees should not be expected to clean up unclear AI-generated arguments or to discover basic gaps that the authors could have found themselves. For this reason, AI-assisted work should be held to a high standard of exposition and verification. Authors using such tools should make special efforts to write arguments in a form that is easy to check, to separate standard ingredients from new ones, to give precise references, and to indicate where human verification has entered. In this possible future, clear distillation may become one of the central skills of the working mathematician. As proof search becomes cheaper, the ability to recognize the useful idea, discard misleading fragments, repair gaps, organize the argument, and present it in a readable and verifiable form may become even more important than it already is. 4.5. A cautious optimism Despite these challenges, our view of this direction is ultimately optimistic. Used carefully, these systems need not diminish mathematical research. They may enlarge the range of problems we can seriously explore, help us follow long technical paths, test approaches that would otherwise seem too costly, and reach parts of the mathematical landscape that might otherwise remain out of practical reach. This optimism does not lessen the need for verification, attribution, and responsibility. Rather, it places those obligations in service of a positive goal: accelerating discovery while preserving the standards of the subject. In this sense, we are sympathetic to Hilbert’s maxim, “Wir müssen wissen. Wir werden wissen.” Properly used, AI systems may become one of the tools that helps us move further toward that ideal. 4.6. A moving target. One final caution is in order. All of the exploratory work underlying the results discussed here was carried out with ChatGPT 5.5 Pro, before the release of the newer generation of models. These results should therefore be read as a snapshot of what was possible with that earlier generation, rather than as an estimate of the present frontier. The release of the new models shortly before the completion of this manuscript, together with our preliminary and necessarily anecdotal experiments with them, has only strengthened this impression. The improvements are perhaps most immediately apparent in matters of form (for example, in the clarity, organization, and overall quality of the mathematical writing) but our early experience suggests that they extend to matters of mathematical substance as well. Thus, if anything, the examples in this paper may understate what can now be achieved with rapidly improving AI systems and further reinforce our conviction that the questions raised here deserve serious attention. Part I Selected Problems 5. Problem 1. Toroidal Elton–Odell theorem Abstract. We prove a toroidal form of the Elton–Odell theorem. More precisely, every infinite-dimensional complex normed space contains a sequence of unit vectors which are separated by a distance strictly larger than one, even after multiplication by unimodular scalars. Contents of this problem paper Run LaTeX again to generate this local table of contents. 1. Statement and notation Throughout, all spaces are complex. For unit vectors x,yx,y in a normed space, set d(x,y)=inf|θ|=1‖x−θy‖.d_T(x,y)= _|θ|=1\|x-θ y\|. We will prove the following statement. Theorem 5.1 (Toroidal Elton–Odell). Let X be an infinite-dimensional complex normed space. Then there are ε>0 >0 and unit vectors (xn)n=1∞⊂X(x_n)_n=1^∞⊂ X such that ‖xn−θxm‖≥1+ε(n≠m,|θ|=1).\|x_n-θ x_m\|≥ 1+ (n≠ m,\ |θ|=1). Equivalently, d(xn,xm)≥1+εd_T(x_n,x_m)≥ 1+ for all n≠mn≠ m. The model produced two different proofs of the theorem. This is itself noteworthy: although the proofs share some techniques, their central arguments are sufficiently distinct to be regarded as different proof methods rather than variations of a single argument. We include only the first proof, which has been manually verified and edited. The second has not yet undergone the level of manual verification required for inclusion in the paper, but is made publicly available on the project website as part of the record of the exploratory process. 2. Proof strategy and organisation The proof is organised as a sequence of reductions. In Section˜5, we treat the two classical easy cases. If X contains an isomorphic copy of c0c_0 or ℓ1 _1, then the desired toroidally separated sequence is obtained from explicit sequences in c0c_0 and ℓ1 _1, and then transferred to X by James’ distortion theorem. In Section˜5, we introduce asymptotically monotone bases and flat blocks. In Section˜5, we prove the main combinatorial ingredient, the rapid flat-block alternative, Section˜5. Roughly speaking, it says that, inside the closed span of an asymptotically monotone basic sequence, failure of toroidal separation forces a rigid obstruction: there must exist successive almost-flat blocks d1<d2<…d_1<d_2<… and phases αj∈ _j such that supN∈ℕ‖∑j=1Nαjdj‖<∞. _N \| _j=1^N _jd_j \|<∞. In Section˜5, we show that this obstruction cannot occur in the two remaining structural cases needed later. First, if X contains a boundedly complete basic sequence, we pass to an asymptotically monotone boundedly complete block basis and apply Section˜5; the bounded twisted partial sums produced by the proposition contradict bounded completeness. Second, if X contains a non-weakly-convergent strongly summing basic sequence, we form normalised differences of far-apart terms. These differences are weakly null, so we pass to an asymptotically monotone basic subsequence and again apply Section˜5. The resulting bounded flat-block sums are then expanded back in the original strongly summing sequence, forcing convergence of a scalar series whose terms do not tend to zero. Finally, in Section˜5, we assemble the cases. If X contains c0c_0 or ℓ1 _1, Section˜5 applies. If X is reflexive, then it contains a boundedly complete basic sequence, so Section˜5 applies. If X is nonreflexive and contains neither c0c_0 nor ℓ1 _1, Rosenthal’s ℓ1 _1 theorem gives a non-weakly-convergent weak-Cauchy sequence, and Rosenthal’s c0c_0 theorem gives a strongly summing subsequence; hence Section˜5 applies again. This exhausts all infinite-dimensional Banach spaces. 3. The easy c0c_0 and ℓ1 _1 cases We first treat the simple case where the space X contains an isomorphic copy of either c0c_0 or ℓ1 _1. We shall use the following elementary transfer estimate. Lemma 5.2 (Almost isometric transfer). Let E and X be normed spaces, and let (un)n=1∞⊂SE(u_n)_n=1^∞⊂ S_E satisfy infn,m∈ℕn≠md(un,um)≥σ>1. _ subarraycn,m \\ n≠ m subarrayd_T(u_n,u_m)≥σ>1. Suppose that T:E→XT E→ X is linear and that, for some 0<δ<10<δ<1, (1−δ)‖u‖≤‖Tu‖≤(1+δ)‖u‖(u∈E).(1-δ)\|u\|≤\|Tu\|≤(1+δ)\|u\| (u∈ E). For n∈ℕn , set xn=Tun‖Tun‖.x_n= Tu_n\|Tu_n\|. Then d(xn,xm)≥(1−δ)σ−2δ1+δ(n,m∈ℕ,n≠m).d_T(x_n,x_m)≥ (1-δ)σ-2δ1+δ (n,m ,\ n≠ m). In particular, if δ<(σ−1)/(σ+3)δ<(σ-1)/(σ+3), then the right-hand side is strictly larger than 11. Proof. For each k∈ℕk , put ak=‖Tuk‖.a_k=\|Tu_k\|. Since ‖uk‖=1\|u_k\|=1, the hypothesis on T gives 1−δ≤ak≤1+δ(k∈ℕ).1-δ≤ a_k≤ 1+δ (k ). In particular, ak>0a_k>0, and Tuk=akxkTu_k=a_kx_k. Fix distinct indices n,m∈ℕn,m and fix θ∈θ . On the one hand, ‖Tun−θTum‖≥(1−δ)‖un−θum‖≥(1−δ)σ.\|Tu_n-θ Tu_m\|≥(1-δ)\|u_n-θ u_m\|≥(1-δ)σ. On the other hand, ‖Tun−θTum‖ \|Tu_n-θ Tu_m\| =‖anxn−θamxm‖ =\|a_nx_n-θ a_mx_m\| ≤an‖xn−θxm‖+|an−am| ≤ a_n\|x_n-θ x_m\|+|a_n-a_m| ≤(1+δ)‖xn−θxm‖+2δ. ≤(1+δ)\|x_n-θ x_m\|+2δ. Combining the two estimates gives ‖xn−θxm‖≥(1−δ)σ−2δ1+δ.\|x_n-θ x_m\|≥ (1-δ)σ-2δ1+δ. Taking the infimum over θ∈θ proves the asserted estimate. ∎ As an easy consequence, we get the following. Lemma 5.3. If a Banach space X contains an isomorphic copy of c0c_0 or an isomorphic copy of ℓ1 _1, then X satisfies Theorem˜5.1. Proof. In ℓ1 _1, the unit vector basis (en)n=1∞(e_n)_n=1^∞ satisfies ∥en−θem∥1=2(n,m∈ℕ,n≠m,|θ|=1).\|e_n-θ e_m\|_1=2 (n,m ,\ n≠ m,\ |θ|=1). Hence the conclusion follows for every Banach space containing ℓ1 _1 from James’ distortion theorem [45] and Section˜5. In c0c_0, define un=e1+…+en−en+1(n∈ℕ).u_n=e_1+…+e_n-e_n+1 (n ). Then (un)n=1∞⊂Sc0(u_n)_n=1^∞⊂ S_c_0. If n<mn<m, then ‖un−θum‖∞≥max|1−θ|,|1+θ|(|θ|=1).\|u_n-θ u_m\|_∞≥ \|1-θ|,|1+θ|\ (|θ|=1). Since inf|θ|=1max|1−θ|,|1+θ|=2, _|θ|=1 \|1-θ|,|1+θ|\= 2, we have d(un,um)≥2d_T(u_n,u_m)≥ 2 whenever n≠mn≠ m. Again, James’ distortion theorem [45] and Section˜5 give the conclusion for every Banach space containing c0c_0. ∎ 4. Asymptotically monotone bases and flat blocks We shall need the following definitions. Definition 5.4. Let (en)n=1∞(e_n)_n=1^∞ be a normalized basic sequence. For N∈ℕN , set MN=sup‖∑i=pqaiei‖∑i=praiei‖:N≤p≤q≤r,(ai)i=pr⊂ℂ,∑i=praiei≠0.M_N= \ \| _i=p^qa_ie_i \| \| _i=p^ra_ie_i \|:N≤ p≤ q≤ r,\ (a_i)_i=p^r ,\ _i=p^ra_ie_i≠ 0 \. We say that (en)n=1∞(e_n)_n=1^∞ is asymptotically monotone if MN⟶1(N→∞).M_N 1 (N→∞). Equivalently, for every γ>0γ>0 there is N0∈ℕN_0 such that, whenever N0≤p≤q≤rN_0≤ p≤ q≤ r and (ai)i=pr⊂ℂ(a_i)_i=p^r , ‖∑i=pqaiei‖≤(1+γ)‖∑i=praiei‖. \| _i=p^qa_ie_i \|≤(1+γ) \| _i=p^ra_ie_i \|. This tail-initial form of asymptotic monotonicity is the only projection property used below. Definition 5.5 (Flat blocks). Let (en)n=1∞(e_n)_n=1^∞ be a basic sequence. A flat block of (en)n=1∞(e_n)_n=1^∞ is a nonzero vector of the form d=∑i∈Fλiei,d= _i∈ F _ie_i, where F⊂ℕF is finite and nonempty, and |λi|=1| _i|=1 for every i∈Fi∈ F. If 0<η<10<η<1, an η-flat unit block is a unit vector of the form b=d‖d‖,b= d\|d\|, where d is a flat block satisfying |‖d‖−1|≤η.|\|d\|-1|≤η. For two finite-support vectors u and v, write u<vu<v if maxsuppu<minsuppv < . Observe that, in the previous definition, the finite set F need not be an interval of integers and thus gaps are allowed. 5. The rapid flat-block alternative The next proposition is the main combinatorial step in the proof. It says that, if toroidal separation fails inside the closed span of an asymptotically monotone basic sequence, then this failure forces a very rigid alternative: one can find successive flat blocks d1<d2<…d_1<d_2<…, almost of norm one, and phases αj∈ _j whose partial sums are uniformly bounded. The idea is that failure of separation gives, on each sufficiently far-out tail, a finite cover of the almost-flat unit blocks, up to multiplication by unimodular scalars. These finite covers are then chained backwards: starting from a far-out block, one repeatedly chooses an earlier cover element close to the normalised block already built and subtracts a suitable phase multiple of it. This produces finite phased sums with uniformly bounded initial pieces; a diagonal argument then gives the infinite sequence. The rest of the proof is devoted to showing that this alternative is impossible in the cases we need. Proposition 5.6 (Rapid flat-block alternative). Let (en)n=1∞(e_n)_n=1^∞ be a normalized asymptotically monotone basic sequence. Suppose that, for every ε>0 >0, there is no sequence (xn)n=1∞⊂span¯en:n∈ℕ(x_n)_n=1^∞⊂ span\e_n:n \ of unit vectors satisfying d(xn,xm)>1+ε(n,m∈ℕ,n≠m).d_T(x_n,x_m)>1+ (n,m ,\ n≠ m). Then there are successive flat blocks (dj)j=1∞(d_j)_j=1^∞ and phases (αj)j=1∞⊂( _j)_j=1^∞ such that ‖dj‖⟶1\|d_j\| 1 and supN∈ℕ‖∑j=1Nαjdj‖<∞. _N \| _j=1^N _jd_j \|<∞. Proof. Choose two positive null sequences (ηj)j=1∞( _j)_j=1^∞ and (γj)j=1∞( _j)_j=1^∞ such that 0<ηj<2−j−10and0<γj<2−j−10(j∈ℕ).0< _j<2^-j-10 0< _j<2^-j-10 (j ). For j∈ℕj , set ρj=∑k=j∞(2ηk+γk). _j= _k=j^∞(2 _k+ _k). By choosing the terms recursively and sufficiently small at each step, we may assume that ρj+1≤ηj(j∈ℕ). _j+1≤ _j (j ). We first construct, by induction on j∈ℕj , integers mjm_j and finite families Bj=bj,1,…,bj,rjB_j=\b_j,1,…,b_j,r_j\ of ηj _j-flat unit blocks, satisfying the following properties: (i) For every j∈ℕj , the tail after mjm_j is (1+γj)(1+ _j)-monotone: whenever mj≤p≤q≤rm_j≤ p≤ q≤ r and ap,…,ar∈ℂa_p,…,a_r , ‖∑i=pqaiei‖≤(1+γj)‖∑i=praiei‖. \| _i=p^qa_ie_i \|≤(1+ _j) \| _i=p^ra_ie_i \|. (1) (i) For every j∈ℕj , the family BjB_j is a finite maximal family of ηj _j-flat unit blocks supported after mjm_j such that d(bj,i,bj,ℓ)>1+ηj(1≤i<ℓ≤rj).d_T(b_j,i,b_j, )>1+ _j (1≤ i< ≤ r_j). (i) The families are successive: every member of BjB_j is supported before mj+1m_j+1. Here, a vector is supported after mjm_j if its support is contained in mj+1,mj+2,…\m_j+1,m_j+2,…\. Maximality in (i) is meant with respect to inclusion among families satisfying the displayed separation condition. Thus, once BjB_j has been chosen, no further ηj _j-flat unit block supported after mjm_j can be added to BjB_j while preserving pairwise toroidal separation greater than 1+ηj1+ _j. Equivalently, every ηj _j-flat unit block supported after mjm_j lies within distance at most 1+ηj1+ _j, up to multiplication by a unimodular scalar, of some member of BjB_j. We now carry out the construction. By asymptotic monotonicity, for each j∈ℕj there is an integer NjN_j such that (1) holds whenever Nj≤p≤q≤rN_j≤ p≤ q≤ r. Choose m1≥N1m_1≥ N_1. Having chosen mjm_j, consider the class of all ηj _j-flat unit blocks supported after mjm_j. By the standing assumption, this class contains no infinite subfamily whose distinct members are pairwise at d_T-distance greater than 1+ηj1+ _j. Hence a greedy selection process gives a finite maximal family Bj=bj,1,…,bj,rjB_j=\b_j,1,…,b_j,r_j\ with the separation property in (i). For each 1≤i≤rj1≤ i≤ r_j, choose a flat block dj,id_j,i such that bj,i=dj,i‖dj,i‖,|‖dj,i‖−1|≤ηj.b_j,i= d_j,i\|d_j,i\|, |\|d_j,i\|-1|≤ _j. (2) Since BjB_j is finite, we may choose mj+1≥Nj+1m_j+1≥ N_j+1 larger than every index appearing in the supports of the vectors in BjB_j. This defines BjB_j and mj+1m_j+1, and hence completes the recursive construction. The maximality of BjB_j gives the following covering property. If u is any ηj _j-flat unit block supported after mjm_j, then there are 1≤i≤rj1≤ i≤ r_j and θ∈θ such that ‖bj,i−θu‖≤1+ηj.\|b_j,i-θ u\|≤ 1+ _j. (3) Indeed, otherwise u could be added to BjB_j, contradicting maximality. This finishes the construction of the finite covers BjB_j. We now use them to build finite flat block combinations with uniformly bounded partial sums. The construction is done first at a fixed finite length L, and then a diagonal argument will let L→∞L→∞. Fix L∈ℕL . We shall construct, by backward induction on j=L,L−1,…,1j=L,L-1,…,1, indices ιj(L)∈1,…,rj, _j^(L)∈\1,…,r_j\, phases θj(L)∈ _j^(L) for 1≤j<L1≤ j<L, and flat blocks zL(L),zL−1(L),…,z1(L)z_L^(L),z_L-1^(L),…,z_1^(L) such that the following hold. (a) At the last level, zL(L)=dL,ιL(L).z_L^(L)=d_L, _L^(L). (b) For every 1≤j<L1≤ j<L, if uj+1(L)=zj+1(L)‖zj+1(L)‖,u_j+1^(L)= z_j+1^(L)\|z_j+1^(L)\|, then ‖bj,ιj(L)−θj(L)uj+1(L)‖≤1+ηj, \|b_j, _j^(L)- _j^(L)u_j+1^(L) \|≤ 1+ _j, (4) and zj(L)=dj,ιj(L)−θj(L)zj+1(L).z_j^(L)=d_j, _j^(L)- _j^(L)z_j+1^(L). (5) (c) For every 1≤j≤L1≤ j≤ L, |‖zj(L)‖−1|≤ρj.|\|z_j^(L)\|-1|≤ _j. (6) We start at level L. Choose any ιL(L)∈1,…,rL _L^(L)∈\1,…,r_L\ and set zL(L)=dL,ιL(L).z_L^(L)=d_L, _L^(L). This gives (a). Moreover, by (2), |‖zL(L)‖−1|≤ηL≤ρL,|\|z_L^(L)\|-1|≤ _L≤ _L, so (c) holds at level L. Now suppose that 1≤j<L1≤ j<L and that the construction has been completed at level j+1j+1. In particular, (c) gives |‖zj+1(L)‖−1|≤ρj+1≤ηj.|\|z_j+1^(L)\|-1|≤ _j+1≤ _j. Since the families B1,B2,…B_1,B_2,… are successive, zj+1(L)z_j+1^(L) is supported after mjm_j. Hence the normalized vector uj+1(L)=zj+1(L)‖zj+1(L)‖u_j+1^(L)= z_j+1^(L)\|z_j+1^(L)\| is an ηj _j-flat unit block supported after mjm_j. By the covering property (3), there are ιj(L)∈1,…,rj _j^(L)∈\1,…,r_j\ and θj(L)∈ _j^(L) such that (4) holds. We then define zj(L)=dj,ιj(L)−θj(L)zj+1(L).z_j^(L)=d_j, _j^(L)- _j^(L)z_j+1^(L). Thus (b) holds at level j. Since the two terms have successive disjoint supports and unimodular coefficients on their supports, zj(L)z_j^(L) is again a flat block. It remains to verify (c) at level j. Put sj(L)=‖dj,ιj(L)‖,tj+1(L)=‖zj+1(L)‖.s_j^(L)=\|d_j, _j^(L)\|, t_j+1^(L)=\|z_j+1^(L)\|. By (2) and the induction hypothesis, |sj(L)−1|≤ηj,|tj+1(L)−1|≤ρj+1.|s_j^(L)-1|≤ _j, |t_j+1^(L)-1|≤ _j+1. By (b), and by the definitions of sj(L)s_j^(L), tj+1(L)t_j+1^(L) and uj+1(L)u_j+1^(L), we have zj(L)=sj(L)bj,ιj(L)−θj(L)tj+1(L)uj+1(L).z_j^(L)=s_j^(L)b_j, _j^(L)- _j^(L)t_j+1^(L)u_j+1^(L). Therefore, ‖zj(L)‖ \|z_j^(L)\| =‖sj(L)bj,ιj(L)−θj(L)tj+1(L)uj+1(L)‖ = \|s_j^(L)b_j, _j^(L)- _j^(L)t_j+1^(L)u_j+1^(L) \| (7) ≤‖bj,ιj(L)−θj(L)uj+1(L)‖+|sj(L)−1|+|tj+1(L)−1| ≤ \|b_j, _j^(L)- _j^(L)u_j+1^(L) \|+|s_j^(L)-1|+|t_j+1^(L)-1| ≤1+ηj+ηj+ρj+1 ≤ 1+ _j+ _j+ _j+1 =1+2ηj+ρj+1. =1+2 _j+ _j+1. For the lower estimate, dj,ιj(L)d_j, _j^(L) is an initial interval projection of zj(L)z_j^(L). Since mj≥Njm_j≥ N_j, (1) gives ‖dj,ιj(L)‖≤(1+γj)‖zj(L)‖.\|d_j, _j^(L)\|≤(1+ _j)\|z_j^(L)\|. Therefore ‖zj(L)‖≥1−ηj1+γj≥1−ηj−γj,\|z_j^(L)\|≥ 1- _j1+ _j≥ 1- _j- _j, (8) where in the last inequality we used the elementary estimate 1−a1+b≥1−a−b(a,b>0). 1-a1+b≥ 1-a-b (a,b>0). Combining (7) and (8), we obtain |‖zj(L)‖−1|≤2ηj+γj+ρj+1=ρj.|\|z_j^(L)\|-1|≤ 2 _j+ _j+ _j+1= _j. Thus (c) holds at level j. This proves the induction step and hence completes the backward construction. Unwinding (5), and setting α1(L)=1,αj(L)=(−1)j−1θ1(L)⋯θj−1(L)(2≤j≤L), _1^(L)=1, _j^(L)=(-1)^j-1 _1^(L)·s _j-1^(L) (2≤ j≤ L), we obtain z1(L)=∑j=1Lαj(L)dj,ιj(L).z_1^(L)= _j=1^L _j^(L)d_j, _j^(L). (9) In particular, by (6), ‖z1(L)‖≤1+ρ1(L∈ℕ).\|z_1^(L)\|≤ 1+ _1 (L ). (10) Thus, for each fixed L∈ℕL , the construction gives a finite sequence d1,ι1(L)<d2,ι2(L)<…<dL,ιL(L)d_1, _1^(L)<d_2, _2^(L)<…<d_L, _L^(L) and phases α1(L),…,αL(L)∈ _1^(L),…, _L^(L) . If N≤LN≤ L, then the partial sum over the first N levels is an initial interval projection of z1(L)z_1^(L). Applying (1) at level 11, we get ∥∑j=1Nαj(L)dj,ιj(L)∥≤(1+γ1)(1+ρ1)=:C. \| _j=1^N _j^(L)d_j, _j^(L) \|≤(1+ _1)(1+ _1)=:C. (11) Finally, we pass from finite lengths to an infinite sequence. For each fixed j∈ℕj and L≥jL≥ j, the indices ιj(L) _j^(L) take values in the finite set 1,…,rj\1,…,r_j\, while the phases αj(L) _j^(L) lie in the compact set T. A diagonal compactness argument gives a subsequence Lk→∞L_k→∞ such that, for each fixed j∈ℕj , ιj(Lk)=ιj eventually,αj(Lk)→αj∈. _j^(L_k)= _j eventually, _j^(L_k)→ _j . Set dj=dj,ιj(j∈ℕ).d_j=d_j, _j (j ). Then d1<d2<…d_1<d_2<…, and (2) gives ‖dj‖→1\|d_j\|→ 1. Fixing N∈ℕN in (11) and passing to the limit along LkL_k yields ‖∑j=1Nαjdj‖≤C. \| _j=1^N _jd_j \|≤ C. Since N∈ℕN was arbitrary, supN∈ℕ‖∑j=1Nαjdj‖<∞. _N \| _j=1^N _jd_j \|<∞. This proves the proposition. ∎ 6. Excluding the bad alternative The rapid flat-block alternative leaves one obstruction: bounded twisted partial sums of successive almost-flat blocks. We now record the selection principle that allows us to arrange asymptotic monotonicity without losing the structural properties needed later. Lemma 5.7 (Asymptotically monotone selection). The following holds. (i) Every seminormalized weakly null sequence has a subsequence whose termwise normalization is an asymptotically monotone basic sequence. (i) Every boundedly complete basic sequence has a normalised asymptotically monotone block basis. Moreover, the block basis is boundedly complete. Proof. (i) is precisely the asymptotically monotone selection principle; see [12, Lemma 3.1]. (i) follows from [43, Lemma 2.4]. Finally, normalising a block basis does not change its canonical projections, and bounded completeness is preserved under such normalisation. ∎ Using this, we are ready to prove the boundedly complete case. Lemma 5.8 (Boundedly complete case). Let X contain a boundedly complete basic sequence. Then X satisfies Theorem˜5.1. Proof. Assume toward a contradiction that no sequence of unit vectors in X is toroidally (1+ε)(1+ )-separated for any ε>0 >0. Choose a boundedly complete basic sequence in X. By Section˜5(i), pass to a normalised asymptotically monotone boundedly complete block basis (en)n=1∞(e_n)_n=1^∞. Apply Section˜5 to (en)n=1∞(e_n)_n=1^∞. We obtain successive flat blocks (dj)j=1∞(d_j)_j=1^∞ with ‖dj‖→1\|d_j\|→ 1 and phases (αj)j=1∞⊂( _j)_j=1^∞ such that supN∈ℕ‖∑j=1Nαjdj‖<∞. _N \| _j=1^N _jd_j \|<∞. Since (dj)j=1∞(d_j)_j=1^∞ is a seminormalised block basic sequence of the boundedly complete basis (en)n=1∞(e_n)_n=1^∞, it is boundedly complete. Hence the series ∑j=1∞αjdj _j=1^∞ _jd_j converges. Its terms must then tend to 0, contradicting ‖αjdj‖=‖dj‖⟶1(j→∞).\| _jd_j\|=\|d_j\| 1 (j→∞). Thus, the desired toroidally separated sequence must exist. ∎ We shall need the following definition. Definition 5.9. A basic sequence (sn)n=1∞(s_n)_n=1^∞ is called strongly summing if it is weak-Cauchy and, whenever a scalar sequence (cn)n=1∞(c_n)_n=1^∞ satisfies supN∈ℕ‖∑n=1Ncnsn‖<∞, _N \| _n=1^Nc_ns_n \|<∞, the scalar series ∑n=1∞cn _n=1^∞c_n converges. We now treat the case in which the Banach space X contains a nontrivial strongly summing basic sequence. Lemma 5.10 (Strongly summing case). Suppose that a Banach space X contains a strongly summing basic sequence (sn)n=1∞(s_n)_n=1^∞ which is not weakly convergent. Then X satisfies Theorem˜5.1. Proof. Assume, toward a contradiction, that no sequence of unit vectors in X is toroidally (1+ε)(1+ )-separated for any ε>0 >0. Since (sn)n=1∞(s_n)_n=1^∞ is weak-Cauchy and not weakly convergent, it is not norm Cauchy. Hence there are r>0r>0 and indices p1<q1<p2<q2<…p_1<q_1<p_2<q_2<… such that ‖spn−sqn‖≥r(n∈ℕ).\|s_p_n-s_q_n\|≥ r (n ). (12) Define yn=spn−sqn‖spn−sqn‖(n∈ℕ).y_n= s_p_n-s_q_n\|s_p_n-s_q_n\| (n ). Then (yn)n=1∞(y_n)_n=1^∞ is normalized and weakly null. Indeed, fix f∈X∗f∈ X^*. Since (sn)n=1∞(s_n)_n=1^∞ is weak-Cauchy, the scalar sequence (f(sn))n=1∞(f(s_n))_n=1^∞ is Cauchy. As pn,qn→∞p_n,q_n→∞, it follows that f(spn)−f(sqn)⟶0.f(s_p_n)-f(s_q_n) 0. Since the denominators in the definition of yny_n are bounded below by r, we get f(yn)→0f(y_n)→ 0. Thus (yn)n=1∞(y_n)_n=1^∞ is weakly null. By Section˜5 (i), pass to a normalized asymptotically monotone basic subsequence of (yn)n=1∞(y_n)_n=1^∞. Relabelling this subsequence and the corresponding pairs, we may assume that (yn)n=1∞(y_n)_n=1^∞ itself is asymptotically monotone. Apply Section˜5 to (yn)n=1∞(y_n)_n=1^∞. We obtain successive flat blocks dj=∑n∈Fjλj,nyn,|λj,n|=1(n∈Fj),F1<F2<…,d_j= _n∈ F_j _j,ny_n, | _j,n|=1 (n∈ F_j), F_1<F_2<…, and phases (αj)j=1∞⊂( _j)_j=1^∞ such that A:=supN∈ℕ‖∑j=1Nαjdj‖<∞.A:= _N \| _j=1^N _jd_j \|<∞. (13) Define scalars (an)n=1∞(a_n)_n=1^∞ by an=αjλj,n(n∈Fj),an=0(n∉⋃j=1∞Fj).a_n= _j _j,n (n∈ F_j), a_n=0 (n∉ _j=1^∞F_j ). This is well defined because the sets FjF_j are successive, hence disjoint. Moreover, every nonzero ana_n has modulus 11, and there are infinitely many nonzero ana_n. Let K be the basis constant of (yn)n=1∞(y_n)_n=1^∞. If PmP_m denotes the m-th coordinate projection relative to (yn)n=1∞(y_n)_n=1^∞, then ‖Pm‖≤K\|P_m\|≤ K for every m∈ℕm . For each m∈ℕm , choose N large enough that m≤maxFNm≤ F_N. Then ∑n=1manyn=Pm(∑j=1Nαjdj). _n=1^ma_ny_n=P_m ( _j=1^N _jd_j ). Thus (13) gives B:=supm∈ℕ‖∑n=1manyn‖≤KA<∞.B:= _m \| _n=1^ma_ny_n \|≤ KA<∞. (14) We now expand these sums in the original strongly summing sequence. For each n∈ℕn , anyn=an‖spn−sqn‖spn−an‖spn−sqn‖sqn.a_ny_n= a_n\|s_p_n-s_q_n\|s_p_n- a_n\|s_p_n-s_q_n\|s_q_n. Define scalar coefficients (ck)k=1∞(c_k)_k=1^∞ by cpn=an‖spn−sqn‖,cqn=−an‖spn−sqn‖(n∈ℕ),c_p_n= a_n\|s_p_n-s_q_n\|, c_q_n=- a_n\|s_p_n-s_q_n\| (n ), and set ck=0c_k=0 whenever k∉pn,qn:n∈ℕk∉\p_n,q_n:n \. If an=0a_n=0, then the two displayed coefficients are also 0. We claim that the vector partial sums ∑k=1mcksk _k=1^mc_ks_k are bounded. Since p1<q1<p2<q2<…,p_1<q_1<p_2<q_2<…, every partial sum stops either after a completed pair or halfway through the next pair. More precisely, suppose first that, for some n∈ℕn , qn≤m<pn+1.q_n≤ m<p_n+1. Then the pairs (pi,qi)(p_i,q_i) have been completely included in the partial sum for 1≤i≤n1≤ i≤ n, and no later pair has begun. Hence ∑k=1mcksk=∑i=1n(cpispi+cqisqi)=∑i=1naiyi, _k=1^mc_ks_k= _i=1^n (c_p_is_p_i+c_q_is_q_i )= _i=1^na_iy_i, so the norm is at most B by (14). Suppose instead that, for some n∈ℕn , pn≤m<qn.p_n≤ m<q_n. Then the pairs (pi,qi)(p_i,q_i) have been completely included for 1≤i<n1≤ i<n, while the n-th pair has contributed only its first term. Therefore ∑k=1mcksk=∑i=1n−1aiyi+an‖spn−sqn‖spn. _k=1^mc_ks_k= _i=1^n-1a_iy_i+ a_n\|s_p_n-s_q_n\|s_p_n. Since (sn)n=1∞(s_n)_n=1^∞ is weak-Cauchy, it is bounded. Writing M=supn∈ℕ‖sn‖<∞M= _n \|s_n\|<∞, the second term has norm at most M/rM/r by (12). Thus every partial sum has norm at most B+M/rB+M/r, and hence supm∈ℕ‖∑k=1mcksk‖<∞. _m \| _k=1^mc_ks_k \|<∞. (15) Strong summability of (sn)n=1∞(s_n)_n=1^∞ implies that the scalar series ∑k=1∞ck _k=1^∞c_k converges. This is impossible. Indeed, for every n∈ℕn such that an≠0a_n≠ 0, |cpn|=1‖spn−sqn‖≥12M.|c_p_n|= 1\|s_p_n-s_q_n\|≥ 12M. There are infinitely many such n, so the scalar sequence (ck)k=1∞(c_k)_k=1^∞ does not converge to 0. This contradicts the convergence of ∑k=1∞ck _k=1^∞c_k. Therefore, the assumed failure of toroidal separation is impossible. ∎ 7. Proof of the main result We can finally obtain the proof of Theorem˜5.1. Proof of Theorem˜5.1. Assume first that X is an infinite-dimensional Banach space. If X contains an isomorphic copy of c0c_0 or ℓ1 _1, then Section˜5 applies. Hence assume that X contains neither c0c_0 nor ℓ1 _1. If X is reflexive, then, by the basic sequence theorem, X contains a basic sequence. Its closed linear span is reflexive, and therefore the basis is boundedly complete. Section˜5 applies. It remains to consider the case when X is nonreflexive and contains neither c0c_0 nor ℓ1 _1. Since X is nonreflexive, its closed unit ball is not weakly compact. By the Eberlein–Šmulian theorem, the closed unit ball is therefore not weakly sequentially compact. Hence there is a bounded sequence (xn)n=1∞⊂X(x_n)_n=1^∞⊂ X with no weakly convergent subsequence. Since X contains no copy of ℓ1 _1, the complex version of Rosenthal’s ℓ1 _1 theorem due to Dor [30], gives a weak-Cauchy subsequence of (xn)n=1∞(x_n)_n=1^∞. This weak-Cauchy subsequence is not weakly convergent, because otherwise (xn)n=1∞(x_n)_n=1^∞ would have a weakly convergent subsequence. Since X contains no copy of c0c_0, Rosenthal’s c0c_0 theorem [66] gives a strongly summing subsequence. Section˜5 applies. A standard completion-and-density argument now yields the result for an arbitrary infinite-dimensional, not necessarily complete, normed space. ∎ 6. Problem 2. Non-Calkin unital Banach algebras Abstract. We construct unital complex Banach algebras which are not isomorphic, as Banach algebras, to ℬ(X)/(X)B(X)/K(X) for any Banach space X. We give two constructions. The first uses a large Leavitt-type quotient and an ℓ1 _1 matrix-unit obstruction. The second uses a large shift algebra on c0(Γ<ω)c_0( ^<ω) and forces a copy of c0c_0 inside a nonseparable space. Contents of this problem paper Run LaTeX again to generate this local table of contents. 1. Statement and notation All Banach spaces and Banach algebras are complex. For a Banach space X, write ℬ(X)=T:X→X:T is bounded and linear,B(X)=\T X→ X:T is bounded and linear\, and let (X)K(X) be the closed ideal of compact operators on X. The Calkin algebra of X is (X)=ℬ(X)/(X).Q(X)=B(X)/K(X). A Banach-algebra isomorphism means a bounded complex-linear algebra isomorphism; by the open mapping theorem, its inverse is then bounded. Theorem 6.1. There is a unital Banach algebra A which is not isomorphic, as a Banach algebra, to ℬ(X)/(X)B(X)/K(X) for any Banach space X. We present two proofs. Both constructions follow the same strategy. We first build a Banach algebra which is large enough to rule out being the Calkin algebra of a separable Banach space, and which is topologically simple. If such an algebra were the Calkin algebra of a nonseparable Banach space, topological simplicity would force the ideal of separable-range operators to coincide with the compact operators. The two proofs then use different algebraic structures to contradict this conclusion: in the first, a matrix-unit configuration forces a noncompact separable-range operator through a map into ℓ1 _1; in the second, the structure forces a copy of c0c_0 to appear in the underlying Banach space. 2. First proof 2.1. Proof strategy and organisation The first proof starts from a large Leavitt-type algebra. In Section˜6, we record two elementary facts about quotient norms and about absorbing countably many compact errors into a separable invariant subspace. In Section˜6, we construct a topologically simple Banach algebra AκA_κ from the one-vertex graph with κ=(2ℵ0)+κ=(2 _0)^+ loops. In Section˜6, we compute its density character. The obstruction to being a Calkin algebra is proved in Section˜6. If X is nonseparable and (X)Q(X) is topologically simple, then separable-range operators on X must coincide with compact operators. We then show that this is incompatible with a countable matrix-unit system whose first row has uniformly bounded c0c_0-type finite sums. Finally, Section˜6 shows that AκA_κ contains precisely such a matrix-unit system, and Section˜6 completes the proof. 2.2. Quotient-norm preliminaries We shall use the following elementary quotient-norm observation to pass from estimates on a corrected operator on X to estimates for the induced operator on X/NX/N. The point is that adding an operator whose range lies in the quotient kernel changes the lift but not the quotient action. Lemma 6.2 (Induced operator norm from a corrected lift). Let X be a Banach space, let N⊂XN⊂ X be a closed subspace, and let π:X→X/Nπ X→ X/N be the quotient map. Suppose that U,K∈ℬ(X)U,K (X) satisfy U(N)⊂NU(N)⊂ N and K(X)⊂NK(X)⊂ N. Then U+KU+K also leaves N invariant, the operators induced by U and U+KU+K on X/NX/N are equal, and ‖U¯‖≤‖U+K‖,\| U\|≤\|U+K\|, where U¯ U denotes the operator induced by U on X/NX/N. Proof. Since K(X)⊂NK(X)⊂ N, in particular K(N)⊂NK(N)⊂ N, so U+KU+K leaves N invariant. For every x∈Xx∈ X, π((U+K)x)=π(Ux)+π(Kx)=π(Ux).π((U+K)x)=π(Ux)+π(Kx)=π(Ux). Thus U and U+KU+K induce the same operator on X/NX/N. The norm of an induced operator is bounded above by the norm of any operator inducing it, and the estimate follows. ∎ We shall also need a simple separability device which absorbs countably many compact ranges while remaining invariant under a prescribed countable family of operators. Lemma 6.3 (Absorbing countably many compact errors). Let X be a Banach space, let (Rn)n=1∞⊂ℬ(X)(R_n)_n=1^∞ (X), and let (Km)m=1∞⊂(X)(K_m)_m=1^∞ (X). Then there is a separable closed subspace N⊂XN⊂ X such that Km(X)⊂N(m∈ℕ),K_m(X)⊂ N (m ), and Rn(N)⊂N(n∈ℕ).R_n(N)⊂ N (n ). Proof. For each compact operator KmK_m, the closed linear span of Km(X)K_m(X) is separable. Indeed, Km(BX)K_m(B_X) has compact norm closure, hence is separable, and Km(X)=⋃r=1∞rKm(BX).K_m(X)= _r=1^∞rK_m(B_X). Let N0N_0 be the closed linear span of ⋃mKm(X) _mK_m(X). Then N0N_0 is separable. Let N be the closed linear span of all vectors of the form Ri1Ri2…Rikx,R_i_1R_i_2… R_i_kx, where k≥0k≥ 0, i1,…,ik∈ℕi_1,…,i_k , and x∈N0x∈ N_0. There are only countably many finite words in the countable family (Rn)n∈ℕ(R_n)_n , and the image of a separable space under a bounded operator is separable. Hence N is separable. By construction, N contains every Km(X)K_m(X) and is invariant under every RnR_n. ∎ 2.3. The Leavitt-type quotient Let =2ℵ0,κ=+. c=2 _0, κ= c^+. We shall use a very special Leavitt path algebra. The general theory may be found in [4, Chapter 1] and in the original papers [3, 9]; for arbitrary graphs and infinite emitters, see also [42, 72]. In the present case, however, all the needed features are quite elementary, and we record the details. Let EκE_κ be the directed graph with one vertex v and κ loops. Informally, the associated Leavitt path algebra is the algebra generated by the loops, together with formal reverse loops, subject to the relations saying that a reverse loop cancels the corresponding loop and annihilates the others. We first describe the algebra explicitly. Let LκL_κ be the unital complex algebra generated by symbols sα,tα(α<κ),s_α,\ t_α (α<κ), subject to the relations tαsβ=δαβ1(α,β<κ).t_αs_β= _αβ1 (α,β<κ). (16) Equivalently, LκL_κ is the quotient of the free unital complex algebra on the symbols sα,tαs_α,t_α, α<κα<κ, by the two-sided ideal generated by the elements tαsβ−δαβ1(α,β<κ).t_αs_β- _αβ1 (α,β<κ). Thus tαt_α is a left inverse for sαs_α, while tαt_α annihilates sβs_β whenever β≠αβ≠α. This is precisely the Leavitt path algebra Lℂ(Eκ)L_C(E_κ) of the graph EκE_κ with one vertex v and κ loops. In that graph-theoretic language, the loop corresponding to α<κα<κ is sαs_α, and its formal reverse edge is tαt_α. The vertex v gives the identity element, which we denote by 11. Since there is only one vertex, all paths begin and end at v, so the usual source and range relations simply say that 1sα=sα1=sα,1tα=tα1=tα(α<κ).1s_α=s_α1=s_α, 1t_α=t_α1=t_α (α<κ). The Cuntz–Krieger cancellation relation is exactly tαsβ=δαβ1(α,β<κ).t_αs_β= _αβ1 (α,β<κ). The only possible additional Cuntz–Krieger relation would be the finite-emitter relation 1=∑s(e)=vee∗.1= _s(e)=ve^*. However, the unique vertex emits infinitely many loops. In the definition of a Leavitt path algebra, this relation is imposed only at vertices emitting a finite nonzero number of edges. Therefore no relation of the form 1=∑α∈Fsαtα1= _α∈ Fs_αt_α is imposed for a finite set F⊂κF⊂κ. We shall use the following notation for words. Let WκW_κ denote the set of all finite sequences μ=(α1,…,αn)μ=( _1,…, _n) with entries in κ, together with the empty word ∅ . We write such a word as μ=α1…αnμ= _1… _n, and define its length by |μ|=n|μ|=n. For μ=α1…αnμ= _1… _n, set sμ=sα1…sαn,tμ=tαn…tα1.s_μ=s_ _1… s_ _n, t_μ=t_ _n… t_ _1. For the empty word, set s∅=t∅=1.s_ =t_ =1. The order in the definition of tμt_μ is reversed because tμt_μ represents the formal inverse path to μ. We shall use the standard spanning description of Leavitt path algebras; see [4, Lemma 1.2.12(i),(i)]. In the present one-vertex case, it says that LκL_κ is linearly spanned by the elements sμtν(μ,ν∈Wκ).s_μt_ν (μ,ν∈ W_κ). Concretely, products of these monomials are computed by repeatedly replacing adjacent subwords tαsβt_αs_β with tαsβ=δαβ1.t_αs_β= _αβ1. Thus every product in the generators sα,tαs_α,t_α reduces to a finite linear combination of terms of the form sα1…sαmtβn…tβ1=sμtν,s_ _1… s_ _mt_ _n… t_ _1=s_μt_ν, where μ=α1…αmμ= _1… _m and ν=β1…βnν= _1… _n. Lemma 6.4. The algebra LκL_κ is algebraically simple. Proof. This also follows immediately from the simplicity criterion for Leavitt path algebras of arbitrary graphs [4, Theorem 2.9.1]: since Eκ0=vE_κ^0=\v\, its only hereditary saturated subsets are ∅ and Eκ0E_κ^0, and every cycle has an exit, because it uses only finitely many of the κ loops and hence there is another loop available. We give the direct argument in this special case. Let I be a nonzero two-sided ideal of LκL_κ, and choose 0≠a∈I0≠ a∈ I. Write a in normal form, a=∑r=1ncrsμrtνr,a= _r=1^nc_rs_ _rt_ _r, where cr≠0c_r≠ 0 and the pairs (μr,νr)( _r, _r) are distinct. Choose an index r0r_0 such that |μr0|+|νr0|| _r_0|+| _r_0| is minimal among the pairs appearing in this expression. Put μ=μr0μ= _r_0 and ν=νr0ν= _r_0. Then tμasν=cr01+b,t_μas_ν=c_r_01+b, where b is a finite linear combination of monomials sσtτs_σt_τ with (σ,τ)≠(∅,∅)(σ,τ)≠( , ). Indeed, the chosen term gives cr01c_r_01, and the minimality of |μ|+|ν||μ|+|ν| prevents any other term from contributing another scalar term. Only finitely many non-empty words occur in the monomials appearing in b. Since κ is infinite, choose a letter λ<κλ<κ which is not the first letter of any of these non-empty words. Then every non-scalar term in b is killed by multiplying on the left by tλt_λ and on the right by sλs_λ. Hence tλ(tμasν)sλ=cr0tλsλ=cr01.t_λ(t_μas_ν)s_λ=c_r_0t_λs_λ=c_r_01. Thus 1∈I1∈ I, and so I=LκI=L_κ. Therefore LκL_κ is algebraically simple. ∎ Now put Yκ=ℓ1(Wκ).Y_κ= _1(W_κ). For α<κα<κ, define operators Sα,Tα∈ℬ(Yκ)S_α,T_α (Y_κ) on the unit vector basis by Sαδw=δαw(w∈Wκ),S_α _w= _α w (w∈ W_κ), and Tαδβw=δw,β=α,0,β≠α,Tαδ∅=0.T_α _β w= cases _w,&β=α,\\ 0,&β≠α, cases T_α _ =0. Thus SαS_α prefixes a word by the letter α, while TαT_α removes an initial α when it is present and sends the vector to zero otherwise. The operator SαS_α is an isometry on ℓ1(Wκ) _1(W_κ). The operator TαT_α is contractive, since it keeps only the coordinates indexed by words beginning with α and then relabels them. Since TαSα=IT_αS_α=I, both operators have norm one: ‖Sα‖=‖Tα‖=1.\|S_α\|=\|T_α\|=1. Moreover, for α,β<κα,β<κ, TαSβ=δαβI.T_αS_β= _αβI. Indeed, if α=βα=β, then SβS_β adds the first letter β and TαT_α removes it; if α≠βα≠β, then TαT_α kills the vector. By the universal property of LκL_κ, these operators induce a unital homomorphism ρ:Lκ→ℬ(Yκ),ρ(sα)=Sα,ρ(tα)=Tα.ρ L_κ (Y_κ), ρ(s_α)=S_α, ρ(t_α)=T_α. Since ρ(1)=I≠0ρ(1)=I≠ 0, the kernel of ρ is a proper two-sided ideal of LκL_κ. By Section˜6, this kernel must be zero. Hence ρ is faithful. Let Bκ=ρ(Lκ)¯⊂ℬ(Yκ)B_κ= ρ(L_κ) (Y_κ) be the norm-closed unital Banach algebra generated by ρ(Lκ)ρ(L_κ). We shall use the standard Zorn-lemma fact that every unital Banach algebra has maximal proper closed two-sided ideals; for completeness we recall the proof. Lemma 6.5. Let A be a unital Banach algebra. Then A has a maximal proper closed two-sided ideal. Proof. Partially order the proper closed two-sided ideals of A by inclusion. Let (Ij)j∈J(I_j)_j∈ J be a chain of proper closed two-sided ideals, and set I=⋃j∈JIj¯.I= _j∈ JI_j. Then I is a closed two-sided ideal of A. We claim that I is proper. If 1∈I1∈ I, then there is some x∈⋃j∈JIjx∈ _j∈ JI_j such that ‖1−x‖<1.\|1-x\|<1. But then x is invertible, so the ideal containing x must be all of A, contradicting the properness of the ideals in the chain. Hence I is proper. Zorn’s lemma gives a maximal proper closed two-sided ideal. ∎ Applying Lemma 6 to A=BκA=B_κ, choose a maximal proper closed two-sided ideal M of BκB_κ and define Aκ=Bκ/M.A_κ=B_κ/M. Let q:Bκ→Aκq B_κ→ A_κ be the quotient map. Then AκA_κ is a unital Banach algebra. Moreover, it has no nonzero proper closed two-sided ideals. Indeed, if J is a closed two-sided ideal of AκA_κ, then q−1(J)q^-1(J) is a closed two-sided ideal of BκB_κ containing M. By maximality of M, we have q−1(J)=Mq^-1(J)=M or q−1(J)=Bκq^-1(J)=B_κ, and hence J=0J=\0\ or J=AκJ=A_κ. We now verify that the maximal ideal chosen above does not meet the embedded Leavitt algebra nontrivially, so the Leavitt algebra survives faithfully in the quotient. Lemma 6.6. We have M∩ρ(Lκ)=0.M∩ρ(L_κ)=\0\. Consequently, q∘ρ:Lκ→Aκq ρ:L_κ→ A_κ is injective. Proof. Suppose that 0≠a∈M∩ρ(Lκ)0≠ a∈ M∩ρ(L_κ). Since ρ is faithful and LκL_κ is algebraically simple, the algebra ρ(Lκ)ρ(L_κ) is algebraically simple. Therefore the algebraic two-sided ideal generated by a inside ρ(Lκ)ρ(L_κ) is all of ρ(Lκ)ρ(L_κ). Hence there exist x1,…,xn,y1,…,yn∈ρ(Lκ)x_1,…,x_n,y_1,…,y_n∈ρ(L_κ) such that 1=∑r=1nxrayr.1= _r=1^nx_ray_r. The right-hand side belongs to M, because a∈Ma∈ M and M is a two-sided ideal of BκB_κ. Thus 1∈M1∈ M, contradicting the properness of M. Hence M∩ρ(Lκ)=0M∩ρ(L_κ)=\0\. Finally, if z∈ker(q∘ρ)z∈ (q ρ), then ρ(z)∈M∩ρ(Lκ)=0ρ(z)∈ M∩ρ(L_κ)=\0\. Since ρ is faithful, z=0z=0. Thus q∘ρq ρ is injective. ∎ 2.4. Density of the quotient algebra We now prove that passing to the quotient has not changed the intended size of the algebra. The next lemma computes the density character of AκA_κ. Lemma 6.7. The Banach algebra AκA_κ has density character dens(Aκ)=κ=+. dens(A_κ)=κ= c^+. Proof. For α<κα<κ, put pα=sαtα∈Lκp_α=s_αt_α∈ L_κ, and use the same notation for its image in AκA_κ. By Section˜6, each pαp_α is nonzero in AκA_κ. The relations give pairwise orthogonal idempotents: pαpβ=δαβpα.p_αp_β= _αβp_α. If p and q are nonzero orthogonal idempotents in a normed algebra, then ‖p−q‖≥1\|p-q\|≥ 1, because p(p−q)=p(p-q)=p. Thus AκA_κ contains a 11-separated set of cardinality κ, and hence dens(Aκ)≥κ dens(A_κ)≥κ. Conversely, LκL_κ has cardinality at most κ: its elements are finite complex linear combinations of finite words in κ many generators, and κ≥|ℂ|κ≥|C|. Since BκB_κ is the norm closure of ρ(Lκ)ρ(L_κ), we have dens(Bκ)≤κ dens(B_κ)≤κ, and therefore dens(Aκ)≤κ dens(A_κ)≤κ. ∎ We shall use the following elementary fact about the Calkin algebra of a separable Banach space. Lemma 6.8. If X is separable, then dens((X))≤. dens(Q(X))≤ c. Proof. A separable Banach space has cardinality at most c. Fix a countable dense subset D⊂XD⊂ X. Every bounded operator T∈ℬ(X)T (X) is determined by its restriction to D, since T is continuous. Therefore |ℬ(X)|≤|X||D|≤ℵ0=.|B(X)|≤|X|^|D|≤ c _0= c. It follows that dens((X))≤|(X)|≤ dens(Q(X))≤|Q(X)|≤ c. ∎ 2.5. The matrix-unit obstruction We isolate the obstruction which will rule out representing the quotient algebra on a nonseparable Calkin algebra. In a topologically simple Calkin algebra, a uniformly controlled system of matrix units would force an impossible separable reduction. Lemma 6.9 (The ℓ1 _1 matrix-unit obstruction). Let X be a nonseparable Banach space. Suppose that (X)Q(X) has no nonzero proper closed two-sided ideals. Then (X)Q(X) cannot contain elements (eij)i,j∈ℕ(e_ij)_i,j satisfying the following conditions: (i) eijekl=δjkeile_ije_kl= _jke_il for all i,j,k,l∈ℕi,j,k,l ; (i) e11≠0e_11≠ 0; (i) there is Cr<∞C_r<∞ such that, for every finite F⊂ℕF and every scalar family (aj)j∈F(a_j)_j∈ F, ‖∑j∈Faje1j‖≤Crsupj∈F|aj|; \| _j∈ Fa_je_1j \|≤ C_r _j∈ F|a_j|; (iv) there is Cc<∞C_c<∞ such that supi∈ℕ‖ei1‖≤Cc _i \|e_i1\|≤ C_c. Proof. We proceed by contradiction, and suppose that such a family (eij)i,j∈ℕ(e_ij)_i,j exists. Let q:ℬ(X)→(X)q (X) (X) be the quotient map. Define (X)=T∈ℬ(X):T(X)¯ is separable.S(X)=\T (X): T(X) is separable\. This is a closed two-sided ideal of ℬ(X)B(X), it contains (X)K(X), and it is proper because X is nonseparable. Hence (X)/(X)S(X)/K(X) is a closed ideal of (X)Q(X). Since (X)Q(X) is topologically simple, this ideal is either 0 or all of (X)Q(X). It is not all of (X)Q(X), because its inverse image is (X)≠ℬ(X)S(X) (X). Therefore (X)=(X).S(X)=K(X). (17) Choose arbitrary lifts Eij∈ℬ(X)E_ij (X) with q(Eij)=eijq(E_ij)=e_ij. By (i), Dij,kl=EijEkl−δjkEil∈(X)D_ij,kl=E_ijE_kl- _jkE_il (X) for all i,j,k,l∈ℕi,j,k,l . For each finite F⊂ℕF and each rational complex tuple a=(aj)j∈F∈(ℚ+iℚ)Fa=(a_j)_j∈ F∈(Q+iQ)^F, set UF,a=∑j∈FajE1j.U_F,a= _j∈ Fa_jE_1j. By (i) and the definition of the quotient norm, choose CF,a∈(X)C_F,a (X) such that ‖UF,a+CF,a‖≤(Cr+1)supj∈F|aj|.\|U_F,a+C_F,a\|≤(C_r+1) _j∈ F|a_j|. (18) Similarly, by (iv), for each i∈ℕi choose Hi∈(X)H_i (X) such that ‖Ei1+Hi‖≤Cc+1.\|E_i1+H_i\|≤ C_c+1. (19) The compact operators Dij,kl(i,j,k,l∈ℕ),CF,a(F⊂ℕ finite,a∈(ℚ+iℚ)F),Hi(i∈ℕ)D_ij,kl\ (i,j,k,l ), C_F,a\ (F finite,\ a∈(Q+iQ)^F), H_i\ (i ) form a countable family. By Section˜6, applied to the countable family (Eij)i,j∈ℕ(E_ij)_i,j and to this countable family of compact operators, there is a separable closed subspace N⊂XN⊂ X such that Eij(N)⊂NE_ij(N)⊂ N for all i,ji,j, and such that the ranges of all compact errors just listed are contained in N. Let Z=X/NZ=X/N, and let π:X→Zπ X→ Z be the quotient map. Since each EijE_ij leaves N invariant, it induces an operator E¯ij∈ℬ(Z) E_ij (Z) defined by E¯ijπx=πEijx(x∈X). E_ijπ x=π E_ijx (x∈ X). Since Dij,kl(X)⊂ND_ij,kl(X)⊂ N, the induced operators satisfy exact matrix-unit relations: E¯ijE¯kl=δjkE¯il. E_ij E_kl= _jk E_il. (20) By Section˜6, (18) implies, first for rational scalar families and then by density, that ‖∑j∈FajE¯1j‖≤(Cr+1)supj∈F|aj| \| _j∈ Fa_j E_1j \|≤(C_r+1) _j∈ F|a_j| (21) for every finite F⊂ℕF and every scalar family (aj)j∈F(a_j)_j∈ F. Likewise, (19) gives supi∈ℕ‖E¯i1‖≤Cc+1. _i \| E_i1\|≤ C_c+1. (22) We next show that E¯11≠0 E_11≠ 0. If E¯11=0 E_11=0, then E11(X)⊂NE_11(X)⊂ N, so E11E_11 has separable range. By (17), E11E_11 is compact, and hence e11=q(E11)=0e_11=q(E_11)=0, contradicting (i). Choose y∈Zy∈ Z such that E¯11y≠0 E_11y≠ 0, and put z0=E¯11yz_0= E_11y. Then z0≠0z_0≠ 0 and E¯11z0=z0 E_11z_0=z_0, by (20). Choose φ∈Z∗ ∈ Z^* with φ(z0)=1 (z_0)=1. We claim that, if we define Θx=(φ(E¯1jπx))j=1∞, x= ( ( E_1jπ x) )_j=1^∞, then Θ is a well-defined bounded operator from X to ℓ1 _1. Indeed, for x∈Xx∈ X and n∈ℕn , (21) gives ∑j=1n|φ(E¯1jπx)| _j=1^n| ( E_1jπ x)| =sup|aj|≤1|φ(∑j=1najE¯1jπx)| = _|a_j|≤ 1 | ( _j=1^na_j E_1jπ x ) | ≤(Cr+1)‖φ‖‖πx‖≤(Cr+1)‖φ‖‖x‖. ≤(C_r+1)\| \|\,\|π x\|≤(C_r+1)\| \|\,\|x\|. Taking the supremum over n shows that Θx∈ℓ1 x∈ _1 and that ‖Θx‖ℓ1≤(Cr+1)‖φ‖‖x‖.\| x\|_ _1≤(C_r+1)\| \|\,\|x\|. Thus Θ:X→ℓ1 X→ _1 is bounded, and linearity is trivial. For i∈ℕi , put zi=E¯i1z0z_i= E_i1z_0. By (22), the sequence (zi)i∈ℕ(z_i)_i is bounded. Choose lifts xi∈Xx_i∈ X with πxi=ziπ x_i=z_i and ‖xi‖≤‖zi‖+1\|x_i\|≤\|z_i\|+1. Then (xi)i∈ℕ(x_i)_i is bounded. For i,k∈ℕi,k , (20) gives φ(E¯1kzi)=φ(E¯1kE¯i1z0)=δkiφ(E¯11z0)=δkiφ(z0)=δki. ( E_1kz_i)= ( E_1k E_i1z_0)= _ki ( E_11z_0)= _ki (z_0)= _ki. Hence Θxi x_i is the i-th unit vector of ℓ1 _1. Since (xi)i∈ℕ(x_i)_i is bounded and the unit vector basis of ℓ1 _1 has no norm-convergent subsequence, Θ is not compact. Choose a bounded sequence (wi)i=1∞⊂X(w_i)_i=1^∞⊂ X with no norm-convergent subsequence, and define R:ℓ1→X,R((ai)i∈ℕ)=∑i=1∞aiwi.R: _1→ X, R((a_i)_i )= _i=1^∞a_iw_i. Then R is bounded. The operator T=RΘT=R has separable range, since T(X)⊂span¯wi:i∈ℕT(X)⊂ span\w_i:i \. However, Txi=wiTx_i=w_i for every i, so T is not compact. This contradicts (17), which finishes the proof. ∎ It remains to connect the concrete algebra AκA_κ with the abstract obstruction above. We do this by extracting a countable system of matrix units from the Leavitt generators. Lemma 6.10. The algebra AκA_κ contains elements (eij)i,j∈ℕ(e_ij)_i,j satisfying the hypotheses of Section˜6. Proof. Choose distinct α1,α2,…∈κ _1, _2,…∈κ, viewed as loop labels in the one-vertex graph defining LκL_κ, and define eij=qρ(sαitαj)∈Aκ(i,j∈ℕ).e_ij=qρ(s_ _it_ _j)∈ A_κ (i,j ). We verify the four hypotheses of Section˜6. Since q∘ρq ρ is injective by Section˜6, we have e11≠0e_11≠ 0, so (i) holds. The Leavitt relations (16) give eijekl=δjkeil(i,j,k,l∈ℕ),e_ije_kl= _jke_il (i,j,k,l ), so (i) holds. We next verify Item˜(i). Fix a finite set F⊂ℕF and scalars (aj)j∈F(a_j)_j∈ F. Consider the concrete operator U=∑j∈FajSα1Tαj∈ℬ(Yκ).U= _j∈ Fa_jS_ _1T_ _j (Y_κ). Since ρ(sα1tαj)=Sα1Tαjρ(s_ _1t_ _j)=S_ _1T_ _j and e1j=qρ(sα1tαj)e_1j=qρ(s_ _1t_ _j), we have q(U)=∑j∈Faje1j.q(U)= _j∈ Fa_je_1j. Thus, it suffices to estimate ‖U‖\|U\|. If x=(cw)w∈Wκ∈ℓ1(Wκ)x=(c_w)_w∈ W_κ∈ _1(W_κ), then ‖Ux‖1≤supj∈F|aj|∑w∈Wκ∑j∈F|cαjw|≤supj∈F|aj|‖x‖1.\|Ux\|_1≤ _j∈ F|a_j| _w∈ W_κ _j∈ F|c_ _jw|≤ _j∈ F|a_j|\,\|x\|_1. Hence ‖∑j∈Faje1j‖≤supj∈F|aj|. \| _j∈ Fa_je_1j \|≤ _j∈ F|a_j|. This proves (i), with Cr=1C_r=1. Finally, since ‖SαiTα1‖≤1\|S_ _iT_ _1\|≤ 1 for every i∈ℕi and ei1=qρ(sαitα1)e_i1=qρ(s_ _it_ _1), we have supi∈ℕ‖ei1‖≤1. _i \|e_i1\|≤ 1. This proves (iv), with Cc=1C_c=1. Thus the elements (eij)i,j∈ℕ(e_ij)_i,j satisfy the hypotheses of Section˜6. ∎ 2.6. Conclusion of the first proof We can now finish the first construction: density rules out separable Calkin algebras, while the matrix-unit obstruction rules out the nonseparable case. First proof of Theorem˜6.1. Let A=AκA=A_κ. Assume, toward a contradiction, that Aκ≅(X)A_κ (X) for some Banach space X. If X is separable, then Section˜6 and Section˜6 give dens((X))≤<+=dens(Aκ), dens(Q(X))≤ c< c^+= dens(A_κ), which is a contradiction. Thus X is nonseparable. Since AκA_κ is topologically simple, so is (X)Q(X). Let Φ:Aκ→(X) A_κ (X) be a Banach-algebra isomorphism. By Section˜6, choose elements (eij)i,j∈ℕ⊂Aκ(e_ij)_i,j ⊂ A_κ satisfying the hypotheses of Section˜6. The elements Φ(eij) (e_ij) are matrix units in (X)Q(X), and Φ(e11)≠0 (e_11)≠ 0. Moreover, the row and column estimates are preserved up to the factor ‖Φ‖\| \|, that is ‖∑j∈FajΦ(e1j)‖≤‖Φ‖supj∈F|aj|, \| _j∈ Fa_j (e_1j) \|≤\| \| _j∈ F|a_j|, and supi∈ℕ‖Φ(ei1)‖≤‖Φ‖. _i \| (e_i1)\|≤\| \|. This contradicts Section˜6 and finishes the proof. ∎ 3. Second proof 3.1. Proof strategy and organisation The second proof is organised as follows. In Section˜6, we avoid the Leavitt-path algebra core and instead construct a very large shift algebra on a space of the form c0(W)c_0(W). In Section˜6, the density of this algebra is chosen so large that, if it were isomorphic to a Calkin algebra (X)Q(X), then the Banach space X would have to have density strictly larger than the continuum. The next step is to exploit the shift relations inside the alleged Calkin algebra. In Section˜6, we remove countably many compact errors so that the relevant relations can be witnessed on a carefully chosen subspace. In Section˜6, these relations are used to force X to contain a copy of c0c_0. Finally, Section˜6 shows that a nonseparable Banach space containing c0c_0 has a nonsimple Calkin algebra: the separable-range operators give a nonzero proper closed ideal in (X)Q(X). This contradicts the topological simplicity of the algebra constructed in Section˜6. The contradiction is assembled in Section˜6. 3.2. The shift algebra Let (ℶn)n=0∞( _n)_n=0^∞ be the Beth sequence, defined by ℶ0=ℵ0,ℶn+1=2ℶn. _0= _0, _n+1=2 _n. Set λ=ℶω=supn<ωℶn.λ= _ω= _n<ω _n. Then λ is a strong-limit cardinal, and in particular λ>2ℵ0.λ>2 _0. Let Γ be a set of cardinality λ. We write W=Γ<ωW= ^<ω for the set of finite words over the alphabet Γ , including the empty word ∅ . Put E=c0(W),E=c_0(W), and denote by ewe_w, w∈Ww∈ W, the canonical unit vector basis of c0(W)c_0(W). For each α∈Γα∈ , define two operators Sα,Tα∈ℬ(E)S_α,T_α (E) as follows. The operator SαS_α prefixes a word by α: Sαew=eαw(w∈W).S_αe_w=e_α w (w∈ W). The operator TαT_α removes an initial α when it is present, and sends the vector to zero otherwise: Tαeβw=ew,β=α,0,β≠α,Tαe∅=0.T_αe_β w= casese_w,&β=α,\\ 0,&β≠α, cases T_αe_ =0. Thus SαS_α is an isometric embedding of c0(W)c_0(W) onto the coordinate subspace supported on the cylinder αWα W, while TαT_α is the corresponding contractive retraction. Hence ‖Sα‖=‖Tα‖=1.\|S_α\|=\|T_α\|=1. Moreover, TαSβ=δαβIE(α,β∈Γ).T_αS_β= _αβI_E (α,β∈ ). We shall use the following simple consequence of the c0c_0 geometry. If F⊂ΓF⊂ is finite and (aα)α∈F⊂ℂ(a_α)_α∈ F , then ‖∑α∈FaαSα‖=maxα∈F|aα|. \| _α∈ Fa_αS_α \|= _α∈ F|a_α|. (23) Indeed, the vectors SαxS_αx, α∈Fα∈ F, have pairwise disjoint supports, since they are supported on the disjoint sets αWα W. Hence, whenever ‖x‖≤1\|x\|≤ 1, ‖∑α∈FaαSαx‖≤maxα∈F|aα|. \| _α∈ Fa_αS_αx \|≤ _α∈ F|a_α|. This gives one inequality. For the reverse inequality, choose α0∈F _0∈ F such that |aα0|=maxα∈F|aα|.|a_ _0|= _α∈ F|a_α|. Then, for any w∈Ww∈ W, ‖∑α∈FaαSαew‖≥|aα0|. \| _α∈ Fa_αS_αe_w \|≥|a_ _0|. Therefore equality holds in (23). We now define the Banach algebra, which will be the counterexample. Let BλB_λ be the closed unital subalgebra of ℬ(E)B(E) generated by IE,Sα,Tα(α∈Γ).I_E, S_α, T_α (α∈ ). Thus BλB_λ is the norm-closed algebra generated by the identity and by the forward and backward shifts attached to the alphabet Γ . By Section˜6, choose a maximal proper closed two-sided ideal M of BλB_λ, and define Aλ=Bλ/M.A_λ=B_λ/M. (24) This quotient is the Banach algebra which will serve as the counterexample. By the maximality of M, the algebra AλA_λ is topologically simple. Lemma 6.11 (Density of the quotient algebra). The Banach algebra AλA_λ has density character dens(Aλ)=λ. dens(A_λ)=λ. Proof. The algebra BλB_λ is generated, as a closed algebra, by λ many elements. Noncommutative polynomials with coefficients in ℚ+iℚQ+iQ and variables from finite subsets of the generators form a dense subset of cardinality at most λ. Hence dens(Bλ)≤λ dens(B_λ)≤λ, and therefore dens(Aλ)≤λ dens(A_λ)≤λ. For the reverse inequality, write σα=Sα+M,τα=Tα+M _α=S_α+M, _α=T_α+M inside AλA_λ. If α≠βα≠β, then τα(σα−σβ)=1Aλ. _α( _α- _β)=1_A_λ. Also ‖τα‖≤1\| _α\|≤ 1 and ‖1Aλ‖=1\|1_A_λ\|=1. Therefore 1≤‖σα−σβ‖.1≤\| _α- _β\|. Thus AλA_λ contains a 11-separated set of cardinality λ, so dens(Aλ)≥λ dens(A_λ)≥λ. ∎ 3.3. A density trap We next record the cardinal estimate which explains the choice of λ=ℶωλ= _ω. A Banach space of density κ has at most 2κ2^κ many bounded operators up to density, and hence its Calkin algebra also has density at most 2κ2^κ. Since λ is a strong-limit cardinal and dens(Aλ)=λ dens(A_λ)=λ, this forces any Banach space X with (X)≅AλQ(X) A_λ to have density at least λ. Lemma 6.12. Let X be a Banach space. If dens(X)=κ dens(X)=κ is infinite, then dens((X))≤2κ. dens(Q(X))≤ 2^κ. Consequently, if (X)Q(X) is Banach-algebra isomorphic to AλA_λ, then dens(X)≥λ>2ℵ0. dens(X)≥λ>2 _0. Proof. A metric space of density κ has cardinality at most κℵ0κ _0. Thus |X|≤κℵ0|X|≤κ _0. Every operator in ℬ(X)B(X) is determined by its values on a fixed dense subset of cardinality κ, and so |ℬ(X)|≤|X|κ≤(κℵ0)κ=κ=2κ.|B(X)|≤|X|^κ≤(κ _0)^κ=κ^κ=2^κ. Therefore dens((X))≤2κ dens(Q(X))≤ 2^κ. Now suppose that (X)Q(X) is Banach-algebra isomorphic to AλA_λ. Since density character is preserved by Banach-space isomorphism, Section˜6 gives dens((X))=dens(Aλ)=λ. dens(Q(X))= dens(A_λ)=λ. On the other hand, the estimate just proved gives λ=dens((X))≤2κ.λ= dens(Q(X))≤ 2^κ. We claim that this forces κ≥λκ≥λ. Indeed, if κ<λκ<λ, then the strong-limit property of λ=ℶωλ= _ω gives 2κ<λ.2^κ<λ. This contradicts λ≤2κλ≤ 2^κ. Therefore κ≥λκ≥λ, as required. ∎ 3.4. Killing countably many compact errors We shall next need a simple consequence of nonseparability at very large density. Compact operators have separable behaviour on bounded sets: the image of the unit ball under a compact operator is norm-compact, hence separable. Therefore, a countable family of compact operators can only see a separable amount of information. If the ambient space has density strictly larger than the continuum, we can choose a unit vector on which all of these compact operators are simultaneously small. Lemma 6.13. Let X be a Banach space with dens(X)>2ℵ0 dens(X)>2 _0. Let (Kn)n=1∞(K_n)_n=1^∞ be a countable family of compact operators on X, and let (εn)n=1∞( _n)_n=1^∞ be a sequence of positive real numbers. Then there is x∈Xx∈ X such that ‖x‖=1\|x\|=1 and ‖Knx‖≤εn(n∈ℕ).\|K_nx\|≤ _n (n ). Proof. Fix a compact operator K∈(X)K (X) and ε>0 >0. By Schauder’s theorem, K∗:X∗→X∗K^*:X^*→ X^* is compact. Hence K∗(BX∗)K^*(B_X^*) is norm-totally bounded. Choose g1,…,gm∈BX∗g_1,…,g_m∈ B_X^* such that for every f∈BX∗f∈ B_X^* there is 1≤r≤m1≤ r≤ m with ‖K∗f−K∗gr‖<ε.\|K^*f-K^*g_r\|< . Set Y=⋂r=1mker(K∗gr)Y= _r=1^m (K^*g_r). Then Y has finite codimension, and for y∈Yy∈ Y, ‖Ky‖=supf∈BX∗|K∗f(y)|≤ε‖y‖.\|Ky\|= _f∈ B_X^*|K^*f(y)|≤ \|y\|. Thus for each KnK_n there is a finite-codimensional closed subspace Yn⊂XY_n⊂ X such that ∥Kn|Yn∥≤εn\|K_n|_Y_n\|≤ _n. Let Y∞=⋂n=1∞YnY_∞= _n=1^∞Y_n. If Y∞=0Y_∞=\0\, then the map X→∏n=1∞X/Yn,x↦(x+Yn)nX→ _n=1^∞X/Y_n, x (x+Y_n)_n is injective. Each quotient X/YnX/Y_n is finite-dimensional over ℂC, so the product has cardinality at most (2ℵ0)ℵ0=2ℵ0(2 _0) _0=2 _0. This contradicts dens(X)>2ℵ0 dens(X)>2 _0. Hence Y∞≠0Y_∞≠\0\. Choose 0≠x∈Y∞0≠ x∈ Y_∞, and normalise it; this gives the result. ∎ 3.5. The column system forces c0c_0 We now turn the shift relations in the Calkin algebra back into geometry inside the Banach space X. The elements sjs_j should be thought of as the first column of a system of matrix units, while the elements tjt_j are left inverses modulo compact operators. The estimate on the sums of the sjs_j says that this column behaves like the unit vector basis of c0c_0 in the quotient algebra. The point of the next lemma is that, when the density of X is large enough, the compact errors can be made simultaneously small; the quotient-level c0c_0 behaviour then lifts to an actual copy of c0c_0 inside X. Lemma 6.14. Let X be a Banach space with dens(X)>2ℵ0 dens(X)>2 _0. Suppose that there are elements (sj)j=1∞(s_j)_j=1^∞ and (tj)j=1∞(t_j)_j=1^∞ in (X)Q(X) satisfying the following conditions. (i) We have tisj=δij1(X)(i,j∈ℕ).t_is_j= _ij1_Q(X) (i,j ). (i) There is a constant C<∞C<∞ such that ‖∑j=1majsj‖≤Cmax1≤j≤m|aj| \| _j=1^ma_js_j \|≤ C _1≤ j≤ m|a_j| for every m∈ℕm and every choice of scalars a1,…,am∈ℂa_1,…,a_m . (i) We have supj∈ℕ‖tj‖<∞. _j \|t_j\|<∞. Then X contains an isomorphic copy of c0c_0. Proof. Choose lifts Sj∈ℬ(X)S_j (X) of sjs_j. By (i), we may choose lifts Tj∈ℬ(X)T_j (X) of tjt_j such that supj∈ℕ‖Tj‖<∞. _j \|T_j\|<∞. Put D=max1,supj∈ℕ‖Tj‖<∞.D= \1, _j \|T_j\| \<∞. For i,j∈ℕi,j , define Rij=TiSj−δijIX.R_ij=T_iS_j- _ijI_X. By (i), we have Rij∈(X)(i,j∈ℕ).R_ij (X) (i,j ). Let ℚ(i)=ℚ+iℚQ(i)=Q+iQ. For every nonzero finitely supported family a=(a1,…,am,0,0,…)∈c00(ℚ(i)),a=(a_1,…,a_m,0,0,…)∈ c_00(Q(i)), condition (i) gives ‖∑j=1majsj‖(X)≤Cmax1≤j≤m|aj|. \| _j=1^ma_js_j \|_Q(X)≤ C _1≤ j≤ m|a_j|. (25) Since ∑j=1majSj _j=1^ma_jS_j is a lift of ∑j=1majsj _j=1^ma_js_j, the definition of the quotient norm and (25) allow us to choose Ka∈(X)K_a (X) such that ‖∑j=1majSj+Ka‖≤(C+1)max1≤j≤m|aj|. \| _j=1^ma_jS_j+K_a \|≤(C+1) _1≤ j≤ m|a_j|. (26) Apply Section˜6 to the countable family consisting of all operators RijR_ij, i,j∈ℕi,j , and all operators KaK_a, 0≠a∈c00(ℚ(i))0≠ a∈ c_00(Q(i)). For i,j∈ℕi,j , set εij=2−j−1. _ij=2^-j-1. For 0≠a=(a1,…,am,0,0,…)∈c00(ℚ(i))0≠ a=(a_1,…,a_m,0,0,…)∈ c_00(Q(i)), set εa=max1≤j≤m|aj|. _a= _1≤ j≤ m|a_j|. The lemma gives a unit vector x∈Xx∈ X such that ‖Rijx‖≤εij=2−j−1(i,j∈ℕ),\|R_ijx\|≤ _ij=2^-j-1 (i,j ), (27) and ‖Kax‖≤εa=max1≤j≤m|aj|\|K_ax\|≤ _a= _1≤ j≤ m|a_j| (28) whenever 0≠a=(a1,…,am,0,0,…)∈c00(ℚ(i))0≠ a=(a_1,…,a_m,0,0,…)∈ c_00(Q(i)). Set xj=Sjx(j∈ℕ).x_j=S_jx (j ). We claim that (xj)j=1∞(x_j)_j=1^∞ is equivalent to the canonical basis of c0c_0. Let 0≠a=(a1,…,am,0,0,…)∈c00(ℚ(i)).0≠ a=(a_1,…,a_m,0,0,…)∈ c_00(Q(i)). By (26) and (28), ‖∑j=1majxj‖ \| _j=1^ma_jx_j \| =‖∑j=1majSjx‖ = \| _j=1^ma_jS_jx \| ≤‖(∑j=1majSj+Ka)x‖+‖Kax‖ ≤ \| ( _j=1^ma_jS_j+K_a )x \|+\|K_ax\| ≤(C+2)max1≤j≤m|aj|. ≤(C+2) _1≤ j≤ m|a_j|. For the lower estimate, choose i∈1,…,mi∈\1,…,m\ such that |ai|=max1≤j≤m|aj|.|a_i|= _1≤ j≤ m|a_j|. Then Ti(∑j=1majxj)=aix+∑j=1majRijx.T_i ( _j=1^ma_jx_j )=a_ix+ _j=1^ma_jR_ijx. Using (27) and ‖x‖=1\|x\|=1, we get ‖Ti(∑j=1majxj)‖≥max1≤j≤m|aj|−max1≤j≤m|aj|∑j=1m2−j−1. \|T_i ( _j=1^ma_jx_j ) \|≥ _1≤ j≤ m|a_j|- _1≤ j≤ m|a_j| _j=1^m2^-j-1. Since ∑j=1m2−j−1≤∑j=1∞2−j−1=12, _j=1^m2^-j-1≤ _j=1^∞2^-j-1= 12, we obtain ‖Ti(∑j=1majxj)‖≥12max1≤j≤m|aj|. \|T_i ( _j=1^ma_jx_j ) \|≥ 12 _1≤ j≤ m|a_j|. Since ‖Ti‖≤D\|T_i\|≤ D, it follows that ‖∑j=1majxj‖≥12Dmax1≤j≤m|aj|. \| _j=1^ma_jx_j \|≥ 12D _1≤ j≤ m|a_j|. Since the estimates are continuous in the finitely many coefficients involved, they extend from c00(ℚ(i))c_00(Q(i)) to all finitely supported complex scalar families. Hence, for every m∈ℕm and every a1,…,am∈ℂa_1,…,a_m , 12Dmax1≤j≤m|aj|≤‖∑j=1majxj‖≤(C+2)max1≤j≤m|aj|. 12D _1≤ j≤ m|a_j|≤ \| _j=1^ma_jx_j \|≤(C+2) _1≤ j≤ m|a_j|. Define U:c00→XU:c_00→ X by Uej=xj(j∈ℕ).Ue_j=x_j (j ). The upper estimate shows that U is bounded for the c0c_0 norm, so it extends uniquely to a bounded operator U:c0→XU:c_0→ X. The lower estimate shows that U is bounded below, hence injective with closed range. Therefore U(c0)U(c_0) is a subspace of X isomorphic to c0c_0. This finishes the proof. ∎ 3.6. A nonseparable space containing c0c_0 has nonsimple Calkin algebra The previous subsection shows that the shift relations force a copy of c0c_0 inside any sufficiently large Banach space whose Calkin algebra realises the model algebra. We now explain why this is incompatible with topological simplicity of the Calkin algebra. The point is that, in a nonseparable Banach space, the separable-range operators form a proper closed operator ideal larger than the compact operators as soon as X contains a copy of c0c_0. After quotienting by the compact operators, this gives a nonzero proper closed two-sided ideal in the Calkin algebra. Lemma 6.15. Let X be a nonseparable Banach space. If X contains an isomorphic copy of c0c_0, then (X)Q(X) has a nonzero proper closed two-sided ideal. Proof. Let i:c0↪Xi:c_0 X be an isomorphic embedding. By the Josefson-Nissenzweig theorem, choose (fn)n=1∞⊂X∗(f_n)_n=1^∞⊂ X^* such that ‖fn‖=1\|f_n\|=1 for every n and fn(x)→0f_n(x)→ 0 for every x∈Xx∈ X. Define J:X→c0,Jx=(fn(x))n=1∞.J:X→ c_0, Jx=(f_n(x))_n=1^∞. This is bounded. It is not compact: compact subsets of c0c_0 have uniformly vanishing tails, whereas sup‖x‖≤1|(Jx)n|=‖fn‖=1 _\|x\|≤ 1|(Jx)_n|=\|f_n\|=1 for every n∈ℕn . Hence iJ:X→XiJ:X→ X is a noncompact operator with separable range. Let (X)=T∈ℬ(X):T(X)¯ is separable.S(X)=\T (X): T(X) is separable\. Then (X)S(X) is a closed two-sided ideal of ℬ(X)B(X), and (X)⊂(X)K(X) (X). Since X is nonseparable, IX∉(X)I_X (X), so (X)≠ℬ(X)S(X) (X). Since iJ∈(X)∖(X)iJ (X) (X), we have (X)⊊(X)⊊ℬ(X).K(X) (X) (X). Thus (X)/(X)S(X)/K(X) is a nonzero proper closed two-sided ideal of (X)Q(X). ∎ 3.7. Conclusion of the second proof Second proof of Theorem˜6.1. Let A=AλA=A_λ. By Section˜6, dens(A)=λ dens(A)=λ, and by construction A is topologically simple. Suppose, toward a contradiction, that there is a Banach-algebra isomorphism Φ:A→(X) :A (X) for some Banach space X. Then (X)Q(X) is topologically simple. Section˜6 gives dens(X)≥λ>2ℵ0. dens(X)≥λ>2 _0. Choose pairwise distinct elements αj∈Γ _j∈ , j∈ℕj , and define σj=Sαj+M,τj=Tαj+M(j∈ℕ), _j=S_ _j+M, _j=T_ _j+M (j ), as elements of A=Bλ/MA=B_λ/M. Then, for all i,j∈ℕi,j , τiσj=δij1A. _i _j= _ij1_A. Moreover, if m∈ℕm and a1,…,am∈ℂa_1,…,a_m , then (23), followed by passage to the quotient, gives ‖∑j=1majσj‖≤‖∑j=1majSαj‖=max1≤j≤m|aj|. \| _j=1^ma_j _j \|≤ \| _j=1^ma_jS_ _j \|= _1≤ j≤ m|a_j|. Also, supj∈ℕ‖τj‖≤1. _j \| _j\|≤ 1. Set sj=Φ(σj),tj=Φ(τj)(j∈ℕ).s_j= ( _j), t_j= ( _j) (j ). Then, for all i,j∈ℕi,j , tisj=δij1(X).t_is_j= _ij1_Q(X). Furthermore, for every m∈ℕm and every a1,…,am∈ℂa_1,…,a_m , ‖∑j=1majsj‖=‖Φ(∑j=1majσj)‖≤‖Φ‖max1≤j≤m|aj|. \| _j=1^ma_js_j \|= \| ( _j=1^ma_j _j ) \|≤\| \| _1≤ j≤ m|a_j|. Finally, supj∈ℕ‖tj‖≤‖Φ‖. _j \|t_j\|≤\| \|. Section˜6 implies that X contains an isomorphic copy of c0c_0. Since dens(X)>2ℵ0 dens(X)>2 _0, the space X is nonseparable. Therefore Section˜6 implies that (X)Q(X) is not topologically simple. This contradicts the topological simplicity of (X)Q(X), and completes the proof. ∎ 7. Problem 3. Strict cosingularity and adjoints Abstract. Let X and Y be Banach spaces and let T:X→YT X→ Y be bounded. We prove that, when Y is separable, T is strictly cosingular if and only if T∗:Y∗→X∗T^*:Y^*→ X^* is strictly singular. Contents of this problem paper Run LaTeX again to generate this local table of contents. 1. Statement and notation Throughout, we work over the scalar field K, which is either ℝR or ℂC. Unless explicitly stated otherwise, the infinite-dimensional subspaces used in the operator-theoretic definitions and theorem statements are norm closed. If E⊂X∗E⊂ X^* is a subspace, its preannihilator in X is E⟂=x∈X:e(x)=0 for all e∈E.E =\x∈ X:e(x)=0 for all e∈ E\. If M⊂XM⊂ X is a subspace, its annihilator in X∗X^* is M⟂=x∗∈X∗:x∗(m)=0 for all m∈M.M =\x^*∈ X^*:x^*(m)=0 for all m∈ M\. We write σ(X∗,X)σ(X^*,X) for the weak-star topology on X∗X^*. Definition 7.1. Let S:E→FS E→ F be a bounded operator between Banach spaces. We say that S is strictly singular if S is not bounded below on any infinite-dimensional subspace of E. Let T:X→YT X→ Y be bounded. We say that T is strictly cosingular if, for every infinite-codimensional subspace M⊂YM⊂ Y, the operator QMT:X→Y/MQ_MT X→ Y/M is not surjective, where QM:Y→Y/MQ_M:Y→ Y/M denotes the quotient map. The terminology goes back to Pełczyński [61]. We shall use the standard dual formulation of strict cosingularity: T:X→YT X→ Y is strictly cosingular if and only if T∗T^* is not bounded below on any infinite-dimensional weak-star closed subspace of Y∗Y^*. This is recalled, for instance, in [8, Section 3]. In particular, T∗ strictly singular⟹T strictly cosingular.T^* strictly singular T strictly cosingular. The converse is false in general. The classical counterexample, due to Pełczyński, is the canonical inclusion c0↪ℓ∞c_0 _∞: it is strictly cosingular, while its adjoint is not strictly singular; see [61] and also [8, Section 3]. The result below shows that the converse does hold when the range space is separable. Theorem 7.2. Let X be a Banach space, let Y be a separable Banach space, and let T:X→YT X→ Y be bounded. Then T is strictly cosingular if and only if T∗T^* is strictly singular. 2. Proof strategy and organisation The proof is organised around the following idea. Strict cosingularity is naturally detected by the adjoint on weak-star closed subspaces of Y∗Y^*, while strict singularity asks about arbitrary infinite-dimensional subspaces of Y∗Y^*. Thus the difficult implication is to start with an arbitrary subspace on which T∗T^* is bounded below and manufacture a weak-star closed witness. In Section˜7, we record two elementary reductions. First, surjectivity of an operator is equivalent to its adjoint being bounded below. Second, strict cosingularity of T is equivalent to saying that T∗T^* is not bounded below on any infinite-dimensional weak-star closed subspace of Y∗Y^*. In Section˜7, we prove the main technical lemma. It uses separability of Y to diagonalise over a countable dense linear subspace of Y. The conclusion is that, after passing to an infinite-dimensional subspace E0⊂X∗E_0⊂ X^*, a given operator U:E→Y∗U E→ Y^* becomes the restriction of an adjoint A∗A^*, where A:Y→X/(E0)⟂A Y→ X/(E_0) . In Section˜7, we apply this lemma with E=T∗(Z)E=T^*(Z), where T∗T^* is bounded below on an infinite-dimensional subspace Z⊂Y∗Z⊂ Y^*. The extracted adjoint identity gives a quotient factorization AT=q:X→X/(E0)⟂.AT=q:X→ X/(E_0) . Since q is onto an infinite-dimensional quotient, this produces an infinite-codimensional quotient of Y on which T is onto. Hence T is not strictly cosingular. 3. Preliminaries We start recording two standard facts which will be used throughout the proof. Both are classical and elementary consequences of basic Banach space duality; we include the details for completeness and to fix the precise form in which they will be used. Lemma 7.3 (Surjectivity and the adjoint). Let S:X→ZS X→ Z be bounded. Then S is surjective if and only if S∗S^* is bounded below. Proof. Suppose first that S is surjective. By the open mapping theorem, there is C>0C>0 such that, for every z∈Zz∈ Z, one can choose x∈Xx∈ X with Sx=zSx=z and ‖x‖≤C‖z‖\|x\|≤ C\|z\|. Hence, for every z∗∈Z∗z^*∈ Z^*, ‖z∗‖=sup‖z‖≤1|z∗(z)|≤Csup‖x‖≤1|z∗(Sx)|=C‖S∗z∗‖.\|z^*\|= _\|z\|≤ 1|z^*(z)|≤ C _\|x\|≤ 1|z^*(Sx)|=C\|S^*z^*\|. Thus S∗S^* is bounded below. Conversely, suppose that S∗S^* is bounded below. Then S∗S^* is injective and has closed range. By the closed range theorem, ranSranS is closed. Also (ranS)⟂=kerS∗=0.(ranS) = S^*=\0\. Therefore ranS¯=Z ranS=Z, and since ranSranS is closed, S is surjective. ∎ Proposition 7.4 (Weak-star characterization of strict cosingularity). Let T:X→YT X→ Y be bounded. Then the following are equivalent. (i) The operator T is strictly cosingular. (i) The operator T∗T^* is not bounded below on any infinite-dimensional weak-star closed subspace of Y∗Y^*. Proof. Let M⊂YM⊂ Y be closed, and let QM:Y→Y/MQ_M Y→ Y/M be the quotient map. The adjoint QM∗Q_M^* identifies (Y/M)∗(Y/M)^* isometrically and weak-star homeomorphically with M⟂⊂Y∗M ⊂ Y^*. Moreover, (QMT)∗=T∗QM∗.(Q_MT)^*=T^*Q_M^*. By Section˜7, the map QMTQ_MT is surjective if and only if (QMT)∗(Q_MT)^* is bounded below. Through the above identification of (Y/M)∗(Y/M)^* with M⟂M , this is equivalent to T∗T^* being bounded below on M⟂M . It remains only to translate the subspaces. Every σ(Y∗,Y)σ(Y^*,Y)-closed subspace W⊂Y∗W⊂ Y^* has the form W=(W⟂)⟂,W=(W ) , and W is infinite-dimensional if and only if Y/W⟂Y/W is infinite-dimensional. Thus closed subspaces M⊂YM⊂ Y of infinite codimension correspond exactly to infinite-dimensional weak-star closed subspaces of Y∗Y^*. The equivalence follows. ∎ The next lemma is a finite-dimensional form of Goldstine’s theorem. We include a short direct proof. Lemma 7.5 (Finite-dimensional interpolation). Let F⊂X∗F⊂ X^* be finite-dimensional and let φ∈F∗ ∈ F^*. For every η>0η>0, there exists x∈Xx∈ X such that f(x)=φ(f)(f∈F),f(x)= (f) (f∈ F), and ‖x‖≤(1+η)‖φ‖.\|x\|≤(1+η)\| \|. If φ=0 =0, one may take x=0x=0. Proof. Define RF:X→F∗,(RFx)(f)=f(x).R_F X→ F^*, (R_Fx)(f)=f(x). The adjoint RF∗:F∗→X∗R_F^* F^**→ X^* is, under the canonical identification F∗=F^**=F, the inclusion of F into X∗X^*. In particular, RF∗R_F^* is an isometry. First, RFR_F is onto. Indeed, if ranRFranR_F were a proper subspace of the finite-dimensional space F∗F^*, then there would be a nonzero element f∈(F∗)∗=Ff∈(F^*)^*=F which annihilates ranRFranR_F. This would mean that 0=f(RFx)=(RFx)(f)=f(x)(x∈X),0=f(R_Fx)=(R_Fx)(f)=f(x) (x∈ X), which is impossible for a nonzero element f∈F⊂X∗f∈ F⊂ X^*. Equip F∗F^* with the quotient norm ∥ψ∥q=inf∥x∥:RFx=ψ.\|ψ\|_q= \\|x\|:R_Fx=ψ\. Thus ‖ψ‖q\|ψ\|_q is the least norm, up to an arbitrarily small error, of a vector x∈Xx∈ X which represents ψ by evaluation on F. We compare this quotient norm with the original dual norm on F∗F^*. Let f∈Ff∈ F. The dual norm of f, when F∗F^* is equipped with ∥⋅∥q\|·\|_q, is ‖f‖q,∗=sup‖ψ‖q≤1|ψ(f)|.\|f\|_q,*= _\|ψ\|_q≤ 1|ψ(f)|. By the definition of the quotient norm, taking the supremum over ‖ψ‖q≤1\|ψ\|_q≤ 1 is the same as taking the supremum over vectors of the form ψ=RFxψ=R_Fx with ‖x‖≤1\|x\|≤ 1, up to an arbitrarily small enlargement of the unit ball. Therefore ‖f‖q,∗=sup‖x‖≤1|(RFx)(f)|.\|f\|_q,*= _\|x\|≤ 1|(R_Fx)(f)|. Since (RFx)(f)=f(x)(R_Fx)(f)=f(x), we get ‖f‖q,∗=sup‖x‖≤1|f(x)|=‖f‖.\|f\|_q,*= _\|x\|≤ 1|f(x)|=\|f\|. Thus the quotient norm on F∗F^* induces on (F∗)∗=F(F^*)^*=F exactly the original norm inherited from X∗X^*. The original norm on F∗F^* also induces this same dual norm on F. Since F∗F^* is finite-dimensional, a norm is determined by its dual norm. Hence the quotient norm ∥⋅∥q\|·\|_q and the original norm on F∗F^* coincide. Consequently, inf∥x∥:RFx=φ=∥φ∥. \\|x\|:R_Fx= \=\| \|. The desired choice of x follows. ∎ 4. The preadjoint extraction lemma The next lemma is the main point of the proof. It says that, after passing to a further infinite-dimensional subspace E0⊂E_0⊂ E, the operator U:E→Y∗U E→ Y^* behaves as though it had a preadjoint. More precisely, once we quotient X by the common kernel (E0)⟂(E_0) , each functional e∈E0e∈ E_0 becomes a functional JeJe on P=X/(E0)⟂P=X/(E_0) , and the lemma constructs an operator A:Y→PA Y→ P such that A∗Je=Ue(e∈E0).A^*Je=Ue (e∈ E_0). Thus, on E0E_0, the map U is realised by the adjoint of an operator defined on Y. The idea of the proof is a diagonal construction. Since Y is separable, we choose a countable dense linear set y1,y2,…⊂Y\y_1,y_2,…\⊂ Y. At stage n, we choose a vector xn∈Xx_n∈ X which represents the values of UeUe at yny_n for all previously chosen functionals e1,…,en−1e_1,…,e_n-1. We then choose the next functional en∈Ee_n∈ E so that all earlier interpolation identities remain valid. After this recursive construction, the rule Ayn=qxnAy_n=qx_n is well defined on the dense set, because all linear relations among the yny_n are killed by the functionals in E0E_0. The resulting operator A:Y→PA Y→ P has adjoint agreeing with U on E0E_0. Lemma 7.6 (Preadjoint extraction). Let Y be separable, E⊂X∗E⊂ X^* be a closed infinite-dimensional subspace, and U:E→Y∗U E→ Y^* be bounded. Then there are a closed infinite-dimensional subspace E0⊂E_0⊂ E and a bounded operator A:Y→P:=X/(E0)⟂A Y→ P:=X/(E_0) such that ‖A‖≤2‖U‖\|A\|≤ 2\|U\| and A∗Je=Ue(e∈E0),A^*Je=Ue (e∈ E_0), (29) where J:E0→P∗J E_0→ P^* is the canonical map defined by (Je)(qx)=e(x)(e∈E0,x∈X),(Je)(qx)=e(x) (e∈ E_0,\ x∈ X), and q:X→Pq X→ P is the quotient map. Proof. Let =ℚ,=ℝ,ℚ+iℚ,=ℂ.F= casesQ,&K=R,\\ Q+iQ,&K=C. cases Choose a countable dense F-linear subspace D⊂YD⊂ Y, and enumerate it as D=y1,y2,….D=\y_1,y_2,…\. We recursively construct linearly independent unit vectors (en)n=1∞(e_n)_n=1^∞ in E and vectors (xn)n=1∞(x_n)_n=1^∞ in X. Suppose that e1,…,en−1e_1,…,e_n-1 have already been chosen, and put Fn−1=spane1,…,en−1.F_n-1=span\e_1,…,e_n-1\. Define φn∈Fn−1∗ _n∈ F_n-1^* by φn(e)=(Ue)(yn)(e∈Fn−1). _n(e)=(Ue)(y_n) (e∈ F_n-1). Then ‖φn‖≤‖U‖‖yn‖.\| _n\|≤\|U\|\|y_n\|. By Section˜7, choose xn∈Xx_n∈ X such that e(xn)=(Ue)(yn)(e∈Fn−1),e(x_n)=(Ue)(y_n) (e∈ F_n-1), (30) and ‖xn‖≤2‖U‖‖yn‖.\|x_n\|≤ 2\|U\|\|y_n\|. (31) If yn=0y_n=0, we take xn=0x_n=0. Having already defined h1,…,hn−1∈E∗h_1,…,h_n-1∈ E^*, define hn∈E∗h_n∈ E^* by hn(e)=(Ue)(yn)−e(xn)(e∈E).h_n(e)=(Ue)(y_n)-e(x_n) (e∈ E). The subspace Hn=⋂j=1nkerhjH_n= _j=1^n h_j has finite codimension in the infinite-dimensional space E, and is therefore infinite-dimensional. Choose en∈Hn∖Fn−1,‖en‖=1.e_n∈ H_n F_n-1, \|e_n\|=1. This completes the recursion. Set E0=span¯en:n∈ℕ⊂E.E_0= span\e_n:n \⊂ E. Then E0E_0 is closed and infinite-dimensional. We claim that (Ue)(yj)=e(xj)(e∈E0,j∈ℕ).(Ue)(y_j)=e(x_j) (e∈ E_0,\ j ). (32) Equivalently, for each fixed j∈ℕj , we must show that hjh_j vanishes on E0E_0. It is enough first to check this on the vectors eie_i. If i<ji<j, then ei∈Fj−1e_i∈ F_j-1, so (30) gives ei(xj)=(Uei)(yj),e_i(x_j)=(Ue_i)(y_j), that is, hj(ei)=0h_j(e_i)=0. If i≥ji≥ j, then ei∈Hi⊂kerhje_i∈ H_i⊂ h_j, so again hj(ei)=0h_j(e_i)=0. Thus each hjh_j vanishes on every eie_i, and by continuity each hjh_j vanishes on E0=span¯ei:i∈ℕ.E_0= span\e_i:i \. This proves (32). Let P=X/(E0)⟂P=X/(E_0) and let q:X→Pq X→ P be the quotient map. Define A0:D→PA_0 D→ P by A0yn=qxn(n∈ℕ).A_0y_n=qx_n (n ). (33) We verify that this is F-linear. In other words, we must check that every F-linear relation among the vectors yny_n is respected by the images qxnqx_n. Therefore, whenever ∑j=1majynj=0, _j=1^ma_jy_n_j=0, with a1,…,am∈a_1,…,a_m , we must prove that ∑j=1majqxnj=0 _j=1^ma_jqx_n_j=0 in P. Thus, suppose that ∑j=1majynj=0,a1,…,am∈. _j=1^ma_jy_n_j=0, a_1,…,a_m . For every e∈E0e∈ E_0, equation (32) gives e(∑j=1majxnj)=∑j=1maj(Ue)(ynj)=(Ue)(∑j=1majynj)=0.e ( _j=1^ma_jx_n_j )= _j=1^ma_j(Ue)(y_n_j)=(Ue) ( _j=1^ma_jy_n_j )=0. Therefore ∑j=1majxnj∈(E0)⟂ _j=1^ma_jx_n_j∈(E_0) , and hence ∑j=1majqxnj=0. _j=1^ma_jqx_n_j=0. Thus all F-linear relations in D are respected. If y∈Dy∈ D, choose n∈ℕn with y=yny=y_n. By (31), ‖A0y‖=‖qxn‖≤‖xn‖≤2‖U‖‖y‖.\|A_0y\|=\|qx_n\|≤\|x_n\|≤ 2\|U\|\|y\|. Thus A0A_0 extends uniquely to a bounded F-linear map A:Y→PA Y→ P with ‖A‖≤2‖U‖\|A\|≤ 2\|U\|. The extension is K-linear: in the real case this follows by approximating real scalars by rationals, and in the complex case by approximating complex scalars by elements of ℚ+iℚQ+iQ. Recall that J:E0→P∗J E_0→ P^* is defined by (Je)(qx)=e(x)(e∈E0,x∈X).(Je)(qx)=e(x) (e∈ E_0,\ x∈ X). This is well defined: if qx=qx′qx=qx , then x−x′∈(E0)⟂x-x ∈(E_0) , and since e∈E0e∈ E_0 we have e(x−x′)=0e(x-x )=0. Hence e(x)=e(x′)e(x)=e(x ). Now let e∈E0e∈ E_0. For every n∈ℕn , since A extends A0A_0 and A0yn=qxnA_0y_n=qx_n, we have (A∗Je)(yn)=(Je)(Ayn)=(Je)(qxn)=e(xn)=(Ue)(yn).(A^*Je)(y_n)=(Je)(Ay_n)=(Je)(qx_n)=e(x_n)=(Ue)(y_n). The set D=yn:n∈ℕD=\y_n:n \ is dense in Y, so the continuous functionals A∗JeA^*Je and UeUe agree on all of Y. This proves (29). ∎ Remark 7.7. No basis property is required of the sequence (en)n=1∞(e_n)_n=1^∞. Linear independence is enough, because the argument only uses that E0=span¯en:n∈ℕE_0= span\e_n:n \ is infinite-dimensional. 5. Quotient factorization and proof of the theorem We first prove the nontrivial implication in a factorized form. Theorem 7.8 (Quotient factorization). Let Y be separable and let T:X→YT X→ Y be bounded. If T∗T^* is bounded below on some infinite-dimensional subspace of Y∗Y^*, then there exist an infinite-dimensional Banach space P, a quotient map q:X→Pq X→ P, and an operator A:Y→PA Y→ P such that AT=q.AT=q. In particular, T is not strictly cosingular. Proof. Suppose that T∗T^* is bounded below on an infinite-dimensional subspace of Y∗Y^*. Passing to the norm closure of that subspace, choose a closed infinite-dimensional subspace Z⊂Y∗Z⊂ Y^* and a constant c>0c>0 such that ‖T∗z‖≥c‖z‖(z∈Z).\|T^*z\|≥ c\|z\| (z∈ Z). Then E=T∗(Z)⊂X∗E=T^*(Z)⊂ X^* is closed and infinite-dimensional, and U=(T∗|Z)−1:E→Z⊂Y∗U=(T^*|_Z)^-1:E→ Z⊂ Y^* is bounded. By Section˜7, there is a closed infinite-dimensional subspace E0⊂E_0⊂ E such that, if P=X/(E0)⟂P=X/(E_0) and q:X→Pq X→ P denotes the quotient map, then there is an operator A:Y→PA Y→ P satisfying A∗Je=Ue(e∈E0).A^*Je=Ue (e∈ E_0). (34) For e∈E0e∈ E_0, it follows from (34) that (AT)∗Je=T∗A∗Je=T∗Ue=e=q∗Je.(AT)^*Je=T^*A^*Je=T^*Ue=e=q^*Je. (35) Here the last equality follows directly from the definition of J. It remains to extend (35) from E0E_0 to all of P∗P^*. The adjoint q∗q^* is an isometric weak-star homeomorphism from P∗P^* onto q∗(P∗)=((E0)⟂)⟂.q^*(P^*)=((E_0) ) . Since q∗Je=eq^*Je=e for every e∈E0e∈ E_0, we have q∗(J(E0))=E0q^*(J(E_0))=E_0. Also q∗q^* identifies P∗P^* weak-star homeomorphically with ((E0)⟂)⟂((E_0) ) , and by the annihilator bipolar identity this latter space is E0¯σ(X∗,X) E_0^\,σ(X^*,X). Hence q∗(J(E0))q^*(J(E_0)) is weak-star dense in q∗(P∗)q^*(P^*). Pulling back through the weak-star homeomorphism q∗q^*, we conclude that J(E0)J(E_0) is weak-star dense in P∗P^*. The operators (AT)∗:P∗→X∗(AT)^* P^*→ X^* and q∗:P∗→X∗q^* P^*→ X^* are weak-star-to-weak-star continuous. By (35), they agree on J(E0)J(E_0). Since J(E0)J(E_0) is weak-star dense in P∗P^*, they agree on all of P∗P^*. Hence (AT)∗p∗=q∗p∗(p∗∈P∗).(AT)^*p^*=q^*p^* (p^*∈ P^*). Since two bounded operators into P are equal whenever their adjoints agree, it follows that AT=q.AT=q. This proves the first part of the theorem. We now show that this factorization implies that T is not strictly cosingular. The quotient map q is surjective, so AT=qAT=q forces A to be surjective. Put M=kerA.M= A. Then A induces an isomorphism A~:Y/M→P. A Y/M→ P. The space P is infinite-dimensional, since P∗P^* contains the infinite-dimensional subspace J(E0)J(E_0). Hence M has infinite codimension in Y. Let QM:Y→Y/MQ_M Y→ Y/M be the quotient map. Then we have A~QMT=AT=q. AQ_MT=AT=q. It follows that QMT=A~−1q.Q_MT= A^-1q. Since q is surjective and A~−1 A^-1 is onto Y/MY/M, the operator QMTQ_MT is surjective. Thus T is not strictly cosingular. ∎ Using the previous result, we can now prove the main duality result. Proof of Theorem˜7.2. Assume first that T∗T^* is strictly singular. Then T∗T^* is not bounded below on any infinite-dimensional subspace of Y∗Y^*, and in particular it is not bounded below on any infinite-dimensional weak-star closed subspace of Y∗Y^*. By Section˜7, T is strictly cosingular. For the converse, we prove the contrapositive. Suppose that T∗T^* is not strictly singular. Then T∗T^* is bounded below on some infinite-dimensional subspace of Y∗Y^*. By the quotient factorization theorem, Theorem˜7.8, the operator T is not strictly cosingular. Therefore, if T is strictly cosingular, then T∗T^* must be strictly singular. This proves the equivalence. ∎ The same argument gives the following geometric formulation. Corollary 7.9 (Weak-star extraction). Let Y be separable and let T:X→YT X→ Y be bounded. If T∗T^* is bounded below on an infinite-dimensional norm-closed subspace of Y∗Y^*, then T∗T^* is bounded below on an infinite-dimensional weak-star closed subspace of Y∗Y^*. Proof. Use the notation obtained in the proof of Theorem˜7.8. Since A:Y→PA Y→ P is surjective, W=A∗(P∗)=(kerA)⟂W=A^*(P^*)=( A) is an infinite-dimensional weak-star closed subspace of Y∗Y^*. The identity AT=qAT=q gives T∗A∗=q∗.T^*A^*=q^*. For p∗∈P∗p^*∈ P^*, ‖T∗A∗p∗‖=‖q∗p∗‖=‖p∗‖≥1‖A‖‖A∗p∗‖.\|T^*A^*p^*\|=\|q^*p^*\|=\|p^*\|≥ 1\|A\|\|A^*p^*\|. Thus T∗T^* is bounded below on W. ∎ Before we finish, let us point out where the separability hypothesis enters the argument. It is used only in Section˜7, through the choice of a countable dense F-linear subspace of Y. No separability assumption on X is used. This also explains why the classical example of the inclusion c0↪ℓ∞c_0 _∞ is not covered by the theorem: the range space ℓ∞ _∞ is nonseparable, so the countable diagonal construction used above is unavailable. Finally, the proof does not require the weak-star closed witness for T∗T^* to lie inside the original subspace on which T∗T^* is bounded below. Starting from such a subspace, the argument constructs a new weak-star closed subspace W⊂Y∗W⊂ Y^* on which T∗T^* is bounded below. 8. Problem 4. Weakly compact basis factorization Abstract. We prove a basis-valued refinement of the Davis–Figiel–Johnson–Pelczynski factorization theorem. If Y is separable and has the bounded approximation property, then every weakly compact operator T:X→YT X→ Y factors through a reflexive Banach space with a Schauder basis. In particular, every weakly compact operator into a Banach space with a Schauder basis admits such a factorization. Contents of this problem paper Run LaTeX again to generate this local table of contents. 1. Statement and notation Throughout, Banach spaces are over the real or complex scalar field. We write BEB_E for the closed unit ball of a Banach space E, and we identify E with its canonical image in E∗E^**. Our main findings are the following. Theorem 8.1 (Weakly compact basis factorization). Let Y be a separable Banach space with the bounded approximation property. If T:X→YT X→ Y is weakly compact, then T factors through a reflexive Banach space with a Schauder basis. As an automatic consequence we get. Corollary 8.2. Let Y have a Schauder basis. If T:X→YT X→ Y is weakly compact, then T factors through a reflexive Banach space with a Schauder basis. 2. Proof strategy and organisation The proof is organised in two stages, after the preliminary results recorded in Section˜8. In Section˜8, we use weak compactness of T and the bounded approximation property of Y to produce finite-rank operators An:X→YA_n X→ Y which converge to T in two dual senses: An∗y∗→T∗y∗(y∗∈Y∗),An∗x∗→T∗x∗(x∗∈X∗).A_n^*y^*→ T^*y^* (y^*∈ Y^*), A_n^**x^**→ T^**x^** (x^**∈ X^**). The adjoint convergence is the only delicate point. The map T∗:BY∗→X∗T^* B_Y^*→ X^* is Baire-one for the weak-star topology on BY∗B_Y^*, while the finite-rank approximants initially give continuous maps which converge only weakly. A tail-convexification argument, using Mazur’s theorem, converts this weak convergence into pointwise norm convergence. In Section˜8, the sequence (An)n=1∞(A_n)_n=1^∞ is used to encode T as a map into c(Y)=(yn)n=1∞:yn converges in norm in Y.c(Y)=\(y_n)_n=1^∞ y_n converges in norm in Y\. The “freezing” projections Qm(y1,y2,…)=(y1,…,ym,ym,ym,…)Q_m(y_1,y_2,…)=(y_1,…,y_m,y_m,y_m,…) are arranged to be finite-rank on the eventual DFJP interpolation space. They approximate the identity there and determine a finite-dimensional decomposition. The reflexive basisification theorem from Section˜8 then embeds the interpolation space as a complemented subspace of a reflexive space with a Schauder basis. The final assembly is given in Section˜8. 3. Preliminaries We shall use two standard structural results. We state them in the precise forms needed below. The first is the Davis–Figiel–Johnson–Pelczynski interpolation construction; see the construction preceding [26, Lemma 1], together with parts (i), (i), and (iv) of that lemma. Theorem 8.3 (Davis–Figiel–Johnson–Pelczynski interpolation). Let E be a Banach space and let K⊂EK⊂ E be bounded, closed, absolutely convex and weakly compact. For j∈ℤj , set Uj=2jK+2−jBE,U_j=2^jK+2^-jB_E, let |⋅|j|·|_j be the Minkowski functional of UjU_j, and define Δ(K)=z∈E:∑j∈ℤ|z|j2<∞ (K)= \z∈ E _j |z|_j^2<∞ \ with norm ‖z‖Δ(K)=(∑j∈ℤ|z|j2)1/2.\|z\|_ (K)= ( _j |z|_j^2 )^1/2. Then Δ(K) (K) is a reflexive Banach space, the inclusion Δ(K)→E (K)→ E is bounded, and K is bounded as a subset of Δ(K) (K). The second result is a reflexive basisification result for spaces with finite-dimensional decompositions. It follows from the finite-dimensional stabilisation theorem of Johnson–Rosenthal–Zippin; see [47, Corollary 4.12(a)]. It is in the same spirit as Pelczynski’s basisification theorem [62]. Theorem 8.4 (Reflexive basisification from an FDD). Let Z be a reflexive Banach space with a finite-dimensional decomposition. Then Z is isomorphic to a complemented subspace of a reflexive Banach space with a Schauder basis. Proof. Let (En)n=1∞(E_n)_n=1^∞ be a finite-dimensional decomposition of Z, and let (PN)N=1∞(P_N)_N=1^∞ be its natural projections. By [47, Corollary 4.12(a)], there is an absolute constant K such that, for every n∈ℕn , there is a finite-dimensional Banach space GnG_n containing EnE_n as a 11-complemented subspace and having a basis with basis constant at most K. Let πn:Gn→En _n G_n→ E_n be a norm-one projection and put Hn=kerπnH_n= _n, equipped with the norm inherited from GnG_n. Thus Gn=En⊕HnG_n=E_n H_n algebraically. For e∈Ene∈ E_n and h∈Hnh∈ H_n, put ‖(e,h)‖2=(‖e‖2+‖h‖2)1/2\|(e,h)\|_2=(\|e\|^2+\|h\|^2)^1/2. Since ‖πn‖=1\| _n\|=1 and ‖I−πn‖≤2\|I- _n\|≤ 2, we have ‖(e,h)‖2≤5‖e+h‖Gn,‖e+h‖Gn≤2‖(e,h)‖2.\|(e,h)\|_2≤ 5\,\|e+h\|_G_n, \|e+h\|_G_n≤ 2\,\|(e,h)\|_2. Consequently, when the chosen basis of GnG_n is regarded as a basis of En⊕2HnE_n _2H_n, its basis constant is at most 10K 10K. Set H=(∑n=1∞⊕Hn)2andV=Z⊕2H.H= ( _n=1^∞ H_n )_2 V=Z _2H. Then V is reflexive. Moreover, the spaces En⊕2HnE_n _2H_n form a finite-dimensional decomposition of V, whose natural projections have norms bounded by maxsupN∈ℕ‖PN‖,1. \ _N \|P_N\|,1 \. Since the chosen block bases have uniformly bounded basis constants, their concatenation is a Schauder basis of V. Finally, the embedding ι:Z→V Z→ V defined by ιz=(z,0) z=(z,0) is complemented by the projection (z,h)↦(z,0)(z,h) (z,0). ∎ 4. Finite-rank approximants We shall use the bounded approximation property in the following sequential form. This is well-known, and we include a short proof for completeness. Lemma 8.5. Let Y be separable and have the bounded approximation property. Then there are finite-rank operators (Pn)n=1∞(P_n)_n=1^∞, Pn:Y→YP_n Y→ Y, such that supn∈ℕ‖Pn‖<∞andPny→y(y∈Y). _n \|P_n\|<∞ P_ny→ y (y∈ Y). Proof. Choose a dense sequence (yj)j=1∞(y_j)_j=1^∞ in Y. Since Y has the bounded approximation property, there is a constant λ<∞λ<∞ such that, for every n∈ℕn , there is a finite-rank operator Pn:Y→YP_n Y→ Y satisfying ‖Pn‖≤λand‖Pnyj−yj‖<1n(1≤j≤n).\|P_n\|≤λ \|P_ny_j-y_j\|< 1n (1≤ j≤ n). It follows that Pnyj→yjP_ny_j→ y_j for every fixed j∈ℕj . Since the operators (Pn)n=1∞(P_n)_n=1^∞ are uniformly bounded and the set yj:j∈ℕ\y_j:j \ is dense in Y, we get Pny→yP_ny→ y for every y∈Yy∈ Y. ∎ We shall also use the following elementary metrizability fact for weakly compact sets in separable spaces. Lemma 8.6. Let Y be separable and let W⊂YW⊂ Y be weakly compact. Then W, endowed with the weak topology, is compact metrizable. Proof. Let Y0=span¯WY_0= spanW. Then Y0Y_0 is separable. If Y0=0Y_0=\0\, there is nothing to prove, so assume that Y0≠0Y_0≠\0\. Choose a norm-dense sequence (sn)n=1∞(s_n)_n=1^∞ in the unit sphere of Y0Y_0. By Hahn–Banach, choose functionals (yn∗)n=1∞⊂BY∗(y_n^*)_n=1^∞⊂ B_Y^* such that |yn∗(sn)|>12(n∈ℕ).|y_n^*(s_n)|> 12 (n ). We claim that (yn∗)n=1∞(y_n^*)_n=1^∞ separates the points of Y0Y_0. Indeed, if 0≠y∈Y00≠ y∈ Y_0, choose n∈ℕn such that ‖sn−y‖y‖<14. \|s_n- y\|y\| \|< 14. Then |yn∗(y‖y‖)|≥|yn∗(sn)|−‖sn−y‖y‖>14. |y_n^* ( y\|y\| ) |≥|y_n^*(s_n)|- \|s_n- y\|y\| \|> 14. Since W is weakly compact, it is norm-bounded, and let M=supw∈W‖w‖<∞.M= _w∈ W\|w\|<∞. Thus the map Φ:W→∏n=1∞z∈:|z|≤M,Φ(w)=(yn∗(w))n=1∞, W→ _n=1^∞\z :|z|≤ M\, (w)=(y_n^*(w))_n=1^∞, is a weakly continuous injection into a compact metric space. Since W is weakly compact, Φ is a homeomorphism of W onto its image. Hence, W is compact metrizable in its weak topology. ∎ We record a simple regularity consequence of weak compactness that will be used below. Lemma 8.7. Let Y be separable and let T:X→YT X→ Y be weakly compact. Let K be the closed unit ball of Y∗Y^*, equipped with the weak-star topology, and define F:K→X∗,F(y∗)=T∗y∗.F K→ X^*, F(y^*)=T^*y^*. Then F, regarded as an X∗X^*-valued map with the norm topology on X∗X^*, is Baire-one. Proof. Let W=T(BX)¯w⊂Y.W= T(B_X)^\,w⊂ Y. By weak compactness of T, the set W is weakly compact. By Section˜8, it is compact metrizable in its weak topology. Choose a weakly dense sequence (wj)j=1∞(w_j)_j=1^∞ in W. For y∗,z∗∈Ky^*,z^*∈ K, ‖T∗y∗−T∗z∗‖ \|T^*y^*-T^*z^*\| =supx∈BX|(y∗−z∗)(Tx)| = _x∈ B_X|(y^*-z^*)(Tx)| =supw∈W|(y∗−z∗)(w)| = _w∈ W|(y^*-z^*)(w)| =supj∈ℕ|(y∗−z∗)(wj)|. = _j |(y^*-z^*)(w_j)|. The second equality uses the weak density of T(BX)T(B_X) in W, and the third uses the weak density of (wj)j=1∞(w_j)_j=1^∞ in W, together with weak continuity on W. It follows that F(K)F(K) is norm separable, because it embeds isometrically into the separable space C(W)C(W) by restriction to W. Also, for fixed z∗∈Kz^*∈ K and r≥0r≥ 0, F−1(B¯F(K)(F(z∗),r)) F^-1 ( B_F(K)(F(z^*),r) ) =y∗∈K:‖F(y∗)−F(z∗)‖≤r =\y^*∈ K \|F(y^*)-F(z^*)\|≤ r\ =⋂j∈ℕy∗∈K:|(y∗−z∗)(wj)|≤r. = _j \y^*∈ K |(y^*-z^*)(w_j)|≤ r\. The last expression is weak-star closed in K. Hence the inverse image under F of every closed norm ball in F(K)F(K) is closed in K. Since F(K)F(K) is norm separable, the Banach space Z=span¯F(K)⊂X∗Z= span\,F(K)⊂ X^* is separable. Since Y is separable, K=(BY∗,w∗)K=(B_Y^*,w^*) is compact metrizable. Let U be a norm-open subset of F(K)F(K). Since F(K)F(K) is separable metric, U is a countable union of closed norm balls in F(K)F(K); since the inverse image under F of each such closed ball is closed in K, it follows that F−1(U)F^-1(U) is an FσF_σ subset of K. The same conclusion holds for norm-open subsets of Z, because F takes its values in F(K)F(K). Equivalently, inverse images of norm-closed subsets of Z are GδG_δ subsets of K. Hence, by [69, Theorem 4], applied to F:K→ZF K→ Z, the map F is Baire-one. ∎ We shall use the following parametrized tail form of Mazur’s theorem, which converts pointwise weak convergence into pointwise norm convergence by convexifying sufficiently far out in the sequence. Lemma 8.8 (Tail convexification). Let K be compact metrizable, let E be a Banach space, and let fn:K→Ef_n K→ E be uniformly bounded norm-continuous maps. Suppose that fn(t)→f(t)f_n(t) wf(t) for every t∈Kt∈ K, and that f:K→Ef K→ E is norm-Baire-one. Then there are gk∈convfn:n≥kg_k \f_n n≥ k\ such that ‖gk(t)−f(t)‖→0\|g_k(t)-f(t)\|→ 0 for every t∈Kt∈ K. Proof. Since f is bounded and Baire-one, choose uniformly bounded norm-continuous maps un:K→Eu_n K→ E such that un(t)→f(t)u_n(t)→ f(t) in norm for every t∈Kt∈ K. This can be done by starting with any norm-continuous approximating sequence and composing with the radial retraction onto a closed ball containing f(K)f(K); the retraction fixes f(K)f(K), so pointwise convergence to f is preserved. Set hn=fn−unh_n=f_n-u_n and L=K×(BE∗,w∗).L=K×(B_E^*,w^*). For (t,e∗)∈L(t,e^*)∈ L, define h^n(t,e∗)=e∗(hn(t)). h_n(t,e^*)=e^*(h_n(t)). Then h^n∈C(L) h_n∈ C(L), the sequence (h^n)n=1∞( h_n)_n=1^∞ is uniformly bounded, and h^n→0 h_n→ 0 pointwise on L. By dominated convergence against regular Borel measures on L, we have h^n→0 h_n→ 0 weakly in C(L)C(L). Fix k. Since 0 lies in the weak closure of convh^n:n≥kconv\ h_n n≥ k\, Mazur’s theorem gives a finite convex combination v^k=∑n≥kαk,nh^n v_k= _n≥ k _k,n h_n such that ‖v^k‖C(L)<1/k\| v_k\|_C(L)<1/k. Define gk=∑n≥kαk,nfn,vk=∑n≥kαk,nun.g_k= _n≥ k _k,nf_n, v_k= _n≥ k _k,nu_n. Then supt∈K‖gk(t)−vk(t)‖<1/k _t∈ K\|g_k(t)-v_k(t)\|<1/k. For each fixed t∈Kt∈ K, the vectors vk(t)v_k(t) are convex combinations of tails of the norm-convergent sequence (un(t))n=1∞(u_n(t))_n=1^∞, so vk(t)→f(t)v_k(t)→ f(t). Hence gk(t)→f(t)g_k(t)→ f(t) in norm. This finishes the proof. ∎ We now combine the preceding Baire-one regularity with the bounded approximation property to obtain finite-rank approximants whose adjoints converge on both sides. Proposition 8.9. Let Y be separable with the bounded approximation property, and let T:X→YT X→ Y be weakly compact. Then there are finite-rank operators Ak:X→YA_k X→ Y such that Ak∗y∗→T∗y∗(y∗∈Y∗)A_k^*y^*→ T^*y^* (y^*∈ Y^*) and Ak∗x∗→T∗x∗(x∗∈X∗)A_k^**x^**→ T^**x^** (x^**∈ X^**) in norm. Proof. Choose finite-rank operators Pn:Y→YP_n Y→ Y as in Section˜8. Let K be the closed unit ball of Y∗Y^*, equipped with the weak-star topology. Define Fn:K→X∗,Fn(y∗)=T∗Pn∗y∗,F_n K→ X^*, F_n(y^*)=T^*P_n^*y^*, and F:K→X∗,F(y∗)=T∗y∗.F K→ X^*, F(y^*)=T^*y^*. Each Fn:K→X∗F_n K→ X^* is continuous from the weak-star topology on K to the norm topology on X∗X^*, because PnP_n has finite rank. Moreover, the sequence (Fn)n=1∞(F_n)_n=1^∞ is uniformly bounded. For x∗∈X∗x^**∈ X^** and y∗∈Ky^*∈ K, ⟨Fn(y∗)−F(y∗),x∗⟩ F_n(y^*)-F(y^*),x^** =⟨Pn∗y∗−y∗,T∗x∗⟩ = P_n^*y^*-y^*,T^**x^** =⟨y∗,(Pn∗−IY∗)T∗x∗⟩. = y^*,(P_n^**-I_Y^**)T^**x^** . Since T is weakly compact, T∗x∗T^**x^** belongs to the canonical copy of Y in Y∗Y^**. Therefore PnT∗x∗→T∗x∗P_nT^**x^**→ T^**x^** in norm, and so Fn(y∗)→F(y∗)(y∗∈K).F_n(y^*) wF(y^*) (y^*∈ K). By Section˜8, F is norm-Baire-one. Applying Section˜8, choose Sk∈convPn:n≥kS_k \P_n n≥ k\ such that ‖T∗Sk∗y∗−T∗y∗‖→0(y∗∈BY∗).\|T^*S_k^*y^*-T^*y^*\|→ 0 (y^*∈ B_Y^*). By homogeneity this convergence holds for every y∗∈Y∗y^*∈ Y^*. Set Ak=SkT.A_k=S_kT. Then each AkA_k is finite-rank and Ak∗y∗=T∗Sk∗y∗→T∗y∗A_k^*y^*=T^*S_k^*y^*→ T^*y^*. Finally, Sky→yS_ky→ y for every y∈Yy∈ Y, because SkS_k is a convex combination of the tail Pn:n≥k\P_n n≥ k\. For x∗∈X∗x^**∈ X^**, weak compactness of T gives T∗x∗∈YT^**x^**∈ Y. Since Sk∗S_k^** agrees with SkS_k on the canonical copy of Y in Y∗Y^**, and since Sk→IYS_k→ I_Y strongly, we have Ak∗x∗=Sk∗T∗x∗=SkT∗x∗→T∗x∗.A_k^**x^**=S_k^**T^**x^**=S_kT^**x^**→ T^**x^**. This finishes the proof. ∎ 5. The bridge theorem We now isolate the main factorization step. The previous results produce finite-rank approximants whose adjoints converge pointwise in norm, both on Y∗Y^* and after passing to second adjoints on X∗X^**. The bridge theorem shows that this two-sided approximation is already enough to force a genuine factorization of T through a reflexive Banach space with a Schauder basis. In this way the analytic approximation information is converted into the structural conclusion needed below. Proposition 8.10 (Bridge theorem). Let T:X→YT X→ Y be a bounded operator. Suppose that there are finite-rank operators An:X→YA_n X→ Y such that An∗y∗→T∗y∗(y∗∈Y∗)A_n^*y^*→ T^*y^* (y^*∈ Y^*) (36) and An∗x∗→T∗x∗(x∗∈X∗)A_n^**x^**→ T^**x^** (x^**∈ X^**) (37) in norm. Then T factors through a reflexive Banach space with a Schauder basis. Proof. By (37) and uniform boundedness, we have M=max‖T‖,supn∈ℕ‖An‖=max‖T‖,supn∈ℕ‖An∗‖<∞.M= \\|T\|, _n \|A_n\|\= \\|T\|, _n \|A_n^**\|\<∞. Applying (37) to the canonical image of X in X∗X^** shows that Anx→TxA_nx→ Tx for every x∈Xx∈ X. It also shows that T is weakly compact, since T∗x∗T^**x^** is a norm limit in Y∗Y^** of vectors in the canonical copy of Y. Let E=c(Y)=(yn)n=1∞:(yn)n=1∞ converges in norm in YE=c(Y)=\(y_n)_n=1^∞ (y_n)_n=1^∞ converges in norm in Y\ with the supremum norm. Define R:X→E,Rx=(Anx)n=1∞,R X→ E, Rx=(A_nx)_n=1^∞, and L:E→Y,L((yn)n=1∞)=limn→∞yn.L E→ Y, L((y_n)_n=1^∞)= _n→∞y_n. Then T=LRT=LR. We first prove that R is weakly compact. Every functional Φ∈E∗ ∈ E^* has a representation Φ((zn)n=1∞)=η∗(limn→∞zn)+∑n=1∞yn∗(zn), ((z_n)_n=1^∞)=η^* ( _n→∞z_n )+ _n=1^∞y_n^*(z_n), where η∗∈Y∗η^*∈ Y^* and (yn∗)n=1∞∈ℓ1(Y∗)(y_n^*)_n=1^∞∈ _1(Y^*). This follows from the isomorphism c(Y)≅c0(Y)⊕∞Y,(zn)n=1∞↦((zn−limm→∞zm)n=1∞,limm→∞zm).c(Y) c_0(Y) _∞Y, (z_n)_n=1^∞ ((z_n- _m→∞z_m)_n=1^∞, _m→∞z_m ). For x∗∈X∗x^**∈ X^**, (37) shows that the sequence (An∗x∗)n=1∞(A_n^**x^**)_n=1^∞ belongs to E and has limit T∗x∗T^**x^**. Thus we may define R^:X∗→E,R^x∗=(An∗x∗)n=1∞. R X^**→ E, Rx^**=(A_n^**x^**)_n=1^∞. We claim that R R represents R∗R^** inside the canonical copy of E in E∗E^**. Indeed, for Φ∈E∗ ∈ E^* and x∗∈X∗x^**∈ X^**, Φ(R^x∗)=η∗(T∗x∗)+∑n=1∞yn∗(An∗x∗)=x∗(R∗Φ). ( Rx^**)=η^*(T^**x^**)+ _n=1^∞y_n^*(A_n^**x^**)=x^**(R^* ). The right-hand side is precisely (R∗x∗)(Φ)(R^**x^**)( ), while the left-hand side is (JER^x∗)(Φ)(J_E Rx^**)( ), where JE:E→E∗J_E E→ E^** is the canonical embedding. Hence R∗x∗=JER^x∗(x∗∈X∗).R^**x^**=J_E Rx^** (x^**∈ X^**). Thus R∗(X∗)⊂JE(E)R^**(X^**)⊂ J_E(E), and therefore R is weakly compact. For m∈ℕm , define Qm:E→E,Qm(y1,y2,…)=(y1,…,ym,ym,ym,…).Q_m E→ E, Q_m(y_1,y_2,…)=(y_1,…,y_m,y_m,y_m,…). Then ‖Qm‖≤1,andQmQk=Qmin(m,k).\|Q_m\|≤ 1, Q_mQ_k=Q_ (m,k). Moreover, if z=(yn)n=1∞∈Ez=(y_n)_n=1^∞∈ E, then ‖Qmz−z‖E=supn>m‖ym−yn‖→0\|Q_mz-z\|_E= _n>m\|y_m-y_n\|→ 0 as m→∞m→∞, because (yn)n=1∞(y_n)_n=1^∞ is norm-convergent in Y. Let W=R(BX)¯w⊂E.W= R(B_X)^\,w⊂ E. Since R is weakly compact, W is weakly compact. We record the key shrinking estimate. For x∈Xx∈ X, (I−Qm)Rx=(0,…,0,(Am+1−Am)x,(Am+2−Am)x,…),(I-Q_m)Rx=(0,…,0,(A_m+1-A_m)x,(A_m+2-A_m)x,…), and this sequence has limit (T−Am)x(T-A_m)x. Thus, if Φ∈E∗ ∈ E^* is represented as above then R∗(I−Qm∗)Φ=(T∗−Am∗)η∗+∑n=m+1∞(An∗−Am∗)yn∗.R^*(I-Q_m^*) =(T^*-A_m^*)η^*+ _n=m+1^∞(A_n^*-A_m^*)y_n^*. Equivalently, supv∈W|Φ(Qmv−v)|→0(Φ∈E∗). _v∈ W| (Q_mv-v)|→ 0 ( ∈ E^*). Define K to be the norm-closed absolutely convex hull of the sets QmWQ_mW, that is K=aco(⋃m=1∞QmW)¯∥⋅∥E.K= aco ( _m=1^∞Q_mW )^\|·\|_E. We claim that K is weakly compact. By the Krein–Smulian theorem, it suffices to show that ⋃m=1∞QmW _m=1^∞Q_mW is relatively weakly compact. By the Eberlein–Smulian theorem, it is enough to prove that this union is relatively weakly sequentially compact. Take a sequence (Qmjwj)j=1∞⊆⋃m=1∞QmW(Q_m_jw_j)_j=1^∞ _m=1^∞Q_mW, with wj∈Ww_j∈ W. Passing to a subsequence, we may suppose that either (mj)j=1∞(m_j)_j=1^∞ is constant or mj→∞m_j→∞. Suppose first that mj=m_j=m for all j. Then (Qmwj)j=1∞(Q_mw_j)_j=1^∞ lies in the weakly compact set QmWQ_mW, and hence has a weakly convergent subsequence. It remains to consider the case mj→∞m_j→∞. Since W is weakly compact, by passing to a further subsequence we may suppose that wj→w_j→ w weakly in W. For every Φ∈E∗ ∈ E^*, |Φ(Qmjwj−wj)|≤supv∈W|Φ(Qmjv−v)|→0,| (Q_m_jw_j-w_j)|≤ _v∈ W| (Q_m_jv-v)|→ 0, and therefore Qmjwj→wQ_m_jw_j→ w weakly. This proves weak compactness of K. We claim that W⊂KW⊂ K and that each QmQ_m leaves K invariant. For the first part, note that if w∈Ww∈ W, then Qmw∈⋃k=1∞QkWQ_mw∈ _k=1^∞Q_kW for every m∈ℕm , and Qmw→wQ_mw→ w in norm. Hence w∈Kw∈ K. For the invariance assertion, note that for w∈Ww∈ W and k,m∈ℕk,m , Qm(Qkw)=Qmin(m,k)w∈⋃ℓ=1∞QℓW.Q_m(Q_kw)=Q_ (m,k)w∈ _ =1^∞Q_ W. Thus QmQ_m maps the generating set ⋃k=1∞QkW _k=1^∞Q_kW into itself. By linearity, it maps its absolutely convex hull into itself, and by norm-continuity it maps the norm closure of that hull, namely K, into K. Apply Theorem˜8.3 to K. For j∈ℤj , set Uj=2jK+2−jBE,U_j=2^jK+2^-jB_E, let |⋅|j|·|_j be the Minkowski functional of UjU_j, and define Z=z∈E:‖z‖Z2=∑j∈ℤ|z|j2<∞.Z= \z∈ E \|z\|_Z^2= _j |z|_j^2<∞ \. By Theorem˜8.3, Z is reflexive, the inclusion J:Z→EJ Z→ E is bounded, and K is bounded as a subset of Z. Since R(BX)⊂W⊂KR(B_X)⊂ W⊂ K, the map R:X→ER X→ E lifts to a bounded operator R~:X→Z,JR~=R. R X→ Z, J R=R. Therefore T=LJR~T=LJ R, so T already factors through the reflexive space Z. It remains to show that Z has the bounded approximation property. Since QmK⊂KQ_mK⊂ K and ‖Qm‖≤1\|Q_m\|≤ 1 on E, we have QmUj⊂Uj(j∈ℤ).Q_mU_j⊂ U_j (j ). Therefore QmQ_m is contractive for each gauge |⋅|j|·|_j, and hence restricts to a contraction on Z. We now show that Qmz→zQ_mz→ z in the Z-norm as m→∞m→∞. For fixed j∈ℤj and z∈Zz∈ Z, we have |(I−Qm)z|j≤2j‖(I−Qm)z‖E→0|(I-Q_m)z|_j≤ 2^j\|(I-Q_m)z\|_E→ 0 as m→∞m→∞. On the other hand, since QmQ_m is contractive for each gauge |⋅|j|·|_j, the triangle inequality gives |(I−Qm)z|j≤|z|j+|Qmz|j≤2|z|j.|(I-Q_m)z|_j≤|z|_j+|Q_mz|_j≤ 2|z|_j. Since z∈Zz∈ Z, the sequence (|z|j)j∈ℤ(|z|_j)_j belongs to ℓ2(ℤ) _2(Z). Thus dominated convergence in the ℓ2(ℤ) _2(Z)-sum gives ‖Qmz−z‖Z2=∑j∈ℤ|(I−Qm)z|j2→0\|Q_mz-z\|_Z^2= _j |(I-Q_m)z|_j^2→ 0 as m→∞m→∞. Hence Qmz→zQ_mz→ z in Z for every z∈Zz∈ Z. We now show that Qm|ZQ_m|_Z has finite rank. Let H be the norm closure in E of the linear span of K, that is H=spanK¯∥⋅∥E.H= spanK^\|·\|_E. First, we claim that Z⊂HZ⊂ H. Indeed, let z∈Zz∈ Z and let q:E→E/Hq E→ E/H be the quotient map. Since q(K)=0q(K)=0, we have q(Uj)⊂2−jBE/Hq(U_j)⊂ 2^-jB_E/H for every j∈ℤj . Hence |z|j≥2j‖qz‖(j∈ℤ).|z|_j≥ 2^j\|qz\| (j ). If qz≠0qz≠ 0, then ∑j∈ℤ|z|j2≥∑j≥0|z|j2≥‖qz‖2∑j≥04j=∞, _j |z|_j^2≥ _j≥ 0|z|_j^2≥\|qz\|^2 _j≥ 04^j=∞, contradicting z∈Zz∈ Z. Therefore qz=0qz=0, and hence z∈Hz∈ H. Fix m∈ℕm and put Fm=∑i=1mQiR(X).F_m= _i=1^mQ_iR(X). Each QiR(X)Q_iR(X) is finite-dimensional, because QiRx=(A1x,…,Aix,Aix,Aix,…)Q_iRx=(A_1x,…,A_ix,A_ix,A_ix,…) and A1,…,AiA_1,…,A_i have finite-dimensional ranges. Hence FmF_m is finite-dimensional. Moreover, since QiR(X)Q_iR(X) is finite-dimensional, it is weakly closed in E. Since QiQ_i is weak-to-weak continuous and W=R(BX)¯wW= R(B_X)^\,w, we have QiW⊂QiR(BX)¯w⊂QiR(X)¯w=QiR(X)(1≤i≤m).Q_iW⊂ Q_iR(B_X)^\,w⊂ Q_iR(X)^\,w=Q_iR(X) (1≤ i≤ m). Hence, for w∈Ww∈ W and k≥1k≥ 1, we have QmQkw=Qmin(m,k)w∈Fm,Q_mQ_kw=Q_ (m,k)w∈ F_m, because min(m,k)≤m (m,k)≤ m and Qmin(m,k)w∈Qmin(m,k)R(X)⊂FmQ_ (m,k)w∈ Q_ (m,k)R(X)⊂ F_m. Since QmQkw∈FmQ_mQ_kw∈ F_m for every w∈Ww∈ W and k≥1k≥ 1, and since FmF_m is a linear subspace, QmQ_m maps the absolutely convex hull of ⋃k=1∞QkW _k=1^∞Q_kW into FmF_m. Because FmF_m is finite-dimensional, it is norm closed in E; hence, by the definition of K as the norm-closed absolutely convex hull, QmK⊂Fm.Q_mK⊂ F_m. Now Z⊂H=spanK¯∥⋅∥EZ⊂ H= spanK^\|·\|_E. Therefore, using the norm-continuity of QmQ_m on E, QmZ⊆QmH⊆span(QmK)¯∥⋅∥E⊆Fm.Q_mZ Q_mH span(Q_mK)^\|·\|_E F_m. Thus Qm|ZQ_m|_Z has range contained in the finite-dimensional space FmF_m, so Qm|ZQ_m|_Z is finite-rank. The finite-rank contractions Qm|ZQ_m|_Z converge strongly to the identity on Z. Thus Z has the metric approximation property. It is also separable, because ⋃m=1∞QmZ _m=1^∞Q_mZ is dense in Z and each QmZQ_mZ is finite-dimensional. Put Q0=0Q_0=0 and define Em=(Qm−Qm−1)Z(m∈ℕ).E_m=(Q_m-Q_m-1)Z (m ). Since QmQk=Qmin(m,k)Q_mQ_k=Q_ (m,k) and Qmz→zQ_mz→ z for every z∈Zz∈ Z, the sequence (Em)m=1∞(E_m)_m=1^∞ is a finite-dimensional decomposition of Z. By Theorem˜8.4, there is a reflexive Banach space V with a Schauder basis such that Z is isomorphic to a complemented subspace of V. Let i:Z→Vi Z→ V be the embedding and let P:V→iZP V→ iZ be a bounded projection. Define B:X→V,B=iR~,B X→ V, B=i R, and C:V→Y,C=(LJ)i−1P.C V→ Y, C=(LJ)i^-1P. Then CB=TCB=T. Hence T factors through the reflexive Banach space V with a Schauder basis. ∎ 6. Proof of the main theorem We now put the preceding ingredients together. The approximation result supplies the two-sided finite-rank approximants required by the bridge theorem, and the bridge theorem converts them into the desired factorization. Proof of Theorem˜8.1. By Section˜8, there are finite-rank operators An:X→YA_n X→ Y such that An∗y∗→T∗y∗(y∗∈Y∗)A_n^*y^*→ T^*y^* (y^*∈ Y^*) and An∗x∗→T∗x∗(x∗∈X∗)A_n^**x^**→ T^**x^** (x^**∈ X^**) in norm. Thus the hypotheses of Section˜8 are satisfied, and thus we obtain a factorization of T through a reflexive Banach space with a Schauder basis, finishing the proof. ∎ 9. Problem 5. Primariness of Lp(L1)L_p(L_1) Abstract. We prove that Lp(L1)L_p(L_1) has the uniform primary factorization property, and thus it is primary for every 1<p<∞1<p<∞. We also prove the corresponding scalar-compression result for the doubly cancellative product-Haar part of Lp(L1)L_p(L_1). Contents of this problem paper Run LaTeX again to generate this local table of contents. 1. Main theorem, proof structure, and notation Recall that a Banach space Y has the uniform primary factorization property if there is a constant K≥1K≥ 1 such that, for every T∈ℬ(Y)T (Y), the identity on Y factors through either T or IdY−TId_Y-T with factorization constant at most K. That is, for every T∈ℬ(Y)T (Y), there are A,B∈ℬ(Y)A,B (Y) with ‖A‖‖B‖≤K\|A\|\|B\|≤ K such that ATB=IdYorA(IdY−T)B=IdY.ATB=Id_Y A(Id_Y-T)B=Id_Y. Our main result is the following. Theorem 9.1 (Main theorem). For every 1<p<∞1<p<∞, the space X=Lp(L1)X=L_p(L_1) has the uniform primary factorization property. In particular, X is primary. In fact, Theorem˜9.1 is obtained as a consequence of a stronger factorisation result on the doubly cancellative product-Haar part X00X_00. We prove that for every operator T∈ℬ(X00)T (X_00) and every ε>0 >0, there are operators A,B∈ℬ(X00)A,B (X_00) and a scalar c∈ℝc such that AB=IdX00,‖ATB−cIdX00‖<ε.AB=Id_X_00, \|ATB-cId_X_00\|< . This scalar-compression statement is then applied to the cancellative compression of an arbitrary projection on Lp(L1)L_p(L_1), and the usual Pełczyński decomposition argument gives primarity. 1.1. Organization of the proof The main step is to show that every operator T∈ℬ(X00)T (X_00) can be reduced, after passing to suitable faithful product Haar systems, to a product Haar multiplier. This is the key reduction identified by Lechner, Motakis, Müller, and Schlumprecht [52]: once arbitrary operators can be compressed to multipliers, the scalar compression theorem for multipliers can be applied. Thus the central task of the proof is to construct faithful Haar systems for which all off diagonal coefficients of the compressed operator become small. We introduce the doubly cancellative subspace X00X_00 in Section˜9, and record the one and two parameter multiplier facts needed later in Sections˜9, 9 and 9. The first new ingredient is the local L1L_1 trace theorem, proved in Section˜9. It assigns to every operator on L10L_1^0 a limiting averaged Haar diagonal on each finite equal-dyadic set, and represents these limits by an L∞L_∞ density. We then introduce faithful outer and inner Haar systems in Section˜9. The associated exterior maps allow us to compress an operator to a faithful product Haar copy while keeping a left inverse. The construction of the faithful systems is carried out in Section˜9. Before that, the finite one step tools needed for the construction are established in Section˜9, with the outer and inner versions separated in Sections˜9 and 9. The construction itself is an alternating outer and inner tree construction. At each half stage, signs and depths are chosen so as to control three types of off diagonal coefficients: same outer strips, same inner strips, and mixed coefficients. The finite choices required at the two half stages are isolated as the technical claims in Section˜9. Once the limiting faithful systems are built, the off diagonal part of the compressed operator is the sum of three controlled pieces, while the remaining diagonal part is a bounded product Haar multiplier. This proves the reduction theorem, Theorem˜9.31. Finally, the LMMS scalar compression theorem is applied to the resulting product Haar multiplier on X00X_00. This gives scalar compression on the doubly cancellative part. The passage from X00X_00 back to the full mixed norm space is then obtained using a complemented copy of Lp(L1)L_p(L_1) inside X00X_00, and Pełczyński’s decomposition method yields primarity. 2. Notation and preliminary results We adhere to standard notation and record below the conventions needed throughout the proof. For notational convenience, all Banach spaces are over the real scalar field, and all operators are assumed to be bounded and linear unless explicitly stated otherwise. The complex case is obtained in the usual way, by inserting conjugates in coefficient pairings and using the complex versions of the multiplier results quoted below. 2.1. Notation and basic definitions Here and below, Lp(L1)L_p(L_1) denotes the Bochner space Lp([0,1];L1[0,1])L_p([0,1];L_1[0,1]), and we write X=Lp(L1)X=L_p(L_1). For nonnegative quantities A and B, we write A≲BA B if A≤CBA≤ CB for an absolute constant C, and A≈BA≈ B if A≲BA B and B≲AB A. Dependence of the implicit constant on additional parameters is indicated by subscripts; for example, A≲pBA _pB allows the constant to depend on p. Let D be the collection of dyadic intervals in [0,1)[0,1). For m≥0m≥ 0, we write m=I∈:|I|=2−mD_m=\I |I|=2^-m\ for the dyadic intervals at level m. If B∈B and n≥0n≥ 0, we write n(B)=J∈:J⊂B,|J|=2−n|B|D_n(B)=\J J⊂ B,\ |J|=2^-n|B|\ for the dyadic descendants of B at relative depth n. A measurable set B⊂[0,1)B⊂[0,1), together with a specified integer mB≥0m_B≥ 0, is called a finite equal-dyadic set if B is a finite disjoint union of intervals in mBD_m_B. Thus B=⨆ℓ=1rBℓ,Bℓ∈mB.B= _ =1^rB_ , B_ _m_B. The integer mBm_B is called the decomposition level of B. We usually suppress mBm_B from the notation and write simply B, but the decomposition level is always recorded implicitly. We extend the descendant notation relative to this specified level by n(B)=⋃ℓ=1rn(Bℓ)=J∈mB+n:J⊂B(n≥0).D_n(B)= _ =1^rD_n(B_ )=\J _m_B+n:J⊂ B\ (n≥ 0). For I∈I , let I+I^+ and I−I^- denote the two dyadic children of I, chosen so that the usual sign-valued Haar function supported on I satisfies hI=I+−I−.h_I=1_I^+-1_I^-. For 1≤q<∞1≤ q<∞, we write Lq0[0,1]=spanhI:I∈¯∥⋅∥Lq=f∈Lq[0,1]:∫01f=0.L_q^0[0,1]= span\h_I I \^\|·\|_L_q= \f∈ L_q[0,1] _0^1f=0 \. When no confusion is possible, we write Lq0L_q^0. If f and g are functions on [0,1][0,1], then (f⊗g)(s,t)=f(s)g(t)(s,t∈[0,1]).(f g)(s,t)=f(s)g(t) (s,t∈[0,1]). In particular, hI⊗hJh_I h_J denotes the function (s,t)⟼hI(s)hJ(t)(s,t) h_I(s)h_J(t). We write X00=span¯hI⊗hJ:I,J∈⊂X.X_00= span\h_I h_J I,J \⊂ X. This is the doubly cancellative product-Haar part of X. We use the normalized vectors eI=|I|−1/phI,eI∗=|I|−1/p′hIe_I=|I|^-1/ph_I, e_I^*=|I|^-1/p h_I in the outer coordinate, and fJ=|J|−1hJ,fJ∗=hJf_J=|J|^-1h_J, f_J^*=h_J in the inner coordinate. Thus uI,J=eI⊗fJ,uI,J∗=eI∗⊗fJ∗u_I,J=e_I f_J, u_I,J^*=e_I^* f_J^* is a biorthogonal product-Haar system on the algebraic product Haar span. Lemma 9.2 (The doubly cancellative projection). For f∈X=Lp(L1)f∈ X=L_p(L_1), define (2f)(s,t)=∫01f(s,u)u(E_2f)(s,t)= _0^1f(s,u)\,du and (1f)(s,t)=∫01f(r,t)r,(E_1f)(s,t)= _0^1f(r,t)\,dr, where the second integral is a Bochner integral in L1[0,1]L_1[0,1]. Then 1E_1 and 2E_2 are commuting contractive projections on X, and P00=(Id−1)(Id−2)P_00=(Id-E_1)(Id-E_2) is a projection with ‖P00‖≤4\|P_00\|≤ 4. Moreover, X00=P00X=f∈X:1f=0 and 2f=0.X_00=P_00X=\f∈ X:E_1f=0 and E_2f=0\. In particular, if x∈Lp0x∈ L_p^0 and g∈L10g∈ L_1^0, then x⊗g∈X00x g∈ X_00. Proof. For almost every s, ‖2f(s,⋅)‖1=|∫01f(s,u)u|≤‖f(s,⋅)‖1,\|E_2f(s,·)\|_1= | _0^1f(s,u)\,du |≤\|f(s,·)\|_1, so 2E_2 is contractive on X. Also, ‖1f‖Lp(L1)=‖∫01f(r,⋅)r‖1≤∫01‖f(r,⋅)‖1r≤‖f‖Lp(L1),\|E_1f\|_L_p(L_1)= \| _0^1f(r,·)\,dr \|_1≤ _0^1\|f(r,·)\|_1\,dr≤\|f\|_L_p(L_1), because the outer measure is one. Thus 1E_1 is contractive. Clearly 12=1E_1^2=E_1 and 22=2E_2^2=E_2, and the two projections commute by Fubini’s theorem. Hence P00P_00 is a bounded projection and ‖P00‖≤4\|P_00\|≤ 4. The range of P00P_00 is exactly the intersection of the kernels of 1E_1 and 2E_2. Every product Haar function hI⊗hJh_I h_J belongs to this intersection, so X00⊂P00X_00⊂ P_00X. Conversely, let f∈P00Xf∈ P_00X. Choose dyadic rectangular simple functions gng_n with gn→fg_n→ f in Lp(L1)L_p(L_1). Since P00P_00 is bounded, P00gn→fP_00g_n→ f. It is enough to check that P00gnP_00g_n belongs to the closed span of the product Haar functions. By linearity it suffices to consider a rectangle A⊗B1_A 1_B, where A and B are dyadic intervals. Then P00(A⊗B)=(A−|A|[0,1))⊗(B−|B|[0,1)).P_00(1_A 1_B)=(1_A-|A|1_[0,1)) (1_B-|B|1_[0,1)). Each factor is a dyadic step function of integral zero and hence lies in the finite linear span of the one-parameter Haar functions. Therefore the tensor above lies in the algebraic span of hI⊗hJ:I,J∈\h_I h_J:I,J \. This proves P00X⊂X00P_00X⊂ X_00. Finally, if x∈Lp0x∈ L_p^0 and g∈L10g∈ L_1^0, then both one-variable means vanish, so x⊗g∈P00X=X00x g∈ P_00X=X_00. ∎ 2.2. Multiplier terminology We shall need the following definition. Definition 9.3. Let a=(aI)I∈a=(a_I)_I be a scalar family. The scalar Haar multiplier with symbol a is the linear map Ma:spanhI:I∈⟶spanhI:I∈M_a \h_I I \ \h_I I \ defined by MahI=aIhI(I∈).M_ah_I=a_Ih_I (I ). We say that MaM_a is bounded on Lq0L_q^0 if it extends to a bounded operator Lq0→Lq0L_q^0→ L_q^0; in that case we use the same symbol for the extension, 1≤q<∞1≤ q<∞. Similarly, for a scalar array d=(dI,J)I,J∈d=(d_I,J)_I,J , the product Haar multiplier with symbol d is the linear map Md:spanhI⊗hJ:I,J∈⟶spanhI⊗hJ:I,J∈M_d \h_I h_J I,J \ \h_I h_J I,J \ defined by Md(hI⊗hJ)=dI,JhI⊗hJ(I,J∈).M_d(h_I h_J)=d_I,Jh_I h_J (I,J ). We say that MdM_d is bounded on X00X_00 if it extends to a bounded operator on X00X_00. Equivalently, the normalized vectors uI,Ju_I,J are eigenvectors with eigenvalues dI,Jd_I,J. We have the following elementary observation. Lemma 9.4 (Bounded scalar multipliers on Lp0L_p^0). Let 1<p<∞1<p<∞. There is a constant UpU_p such that, for every bounded scalar family a=(aI)I∈a=(a_I)_I , the scalar Haar multiplier MaM_a extends to a bounded operator on Lp0L_p^0 and ‖Ma‖ℬ(Lp0)≤UpsupI∈|aI|.\|M_a\|_B(L_p^0)≤ U_p _I |a_I|. Proof. This is the classical unconditionality of the Haar system in Lp[0,1]L_p[0,1] for 1<p<∞1<p<∞, restricted to the closed codimension-one subspace Lp0L_p^0; see, for instance, [7, Chapter 6]. ∎ We also use the factorization terminology of Lechner–Motakis–Müller–Schlumprecht [52, Definition 2.2]. Definition 9.5. Let E be a Banach space, let S,T∈ℬ(E)S,T (E), let C≥1C≥ 1, and let η≥0η≥ 0. We say that S projectionally C-factors through T with error η if there are operators A,B∈ℬ(E)A,B (E) such that AB=IdE,‖S−ATB‖≤η,‖A‖‖B‖≤C.AB=Id_E, \|S-ATB\|≤η, \|A\|\,\|B\|≤ C. We will recall the two main multiplier results used in the proof: the one-parameter theorem of Semenov–Uksusov [68, Theorem 3] and the bi-parameter scalar-compression theorem of Lechner–Motakis–Müller–Schlumprecht [52, Theorem 2.3]. We abbreviate the latter authors as LMMS. 2.3. The one-parameter L1L_1 Haar multiplier theorem Let MaM_a be a scalar Haar multiplier on L10[0,1]L_1^0[0,1], so that MahI=aIhI(I∈).M_ah_I=a_Ih_I (I ). For the coefficient family a=(aI)I∈a=(a_I)_I , write ‖a‖∞=supI∈|aI|,\|a\|_∞= _I |a_I|, and define the finite-chain variation norm ∥a∥W=sup∑r=0m−1|aIr+1−aIr|:I0⊃I1⊃…⊃Im,m≥1.\|a\|_W= \ _r=0^m-1|a_I_r+1-a_I_r|:I_0⊃ I_1⊃…⊃ I_m,\ m≥ 1 \. We start by recalling the one-parameter multiplier theorem of Semenov and Uksusov [68, Theorem 3], restricted to the cancellative Haar subspace L10L_1^0. This only removes the one-dimensional constant coordinate and does not change the boundedness criterion, up to absolute constants. Theorem 9.6 (Semenov–Uksusov). Let MaM_a be a scalar Haar multiplier on L10L_1^0. Then MaM_a is bounded on L10L_1^0 if and only if ‖a‖W+‖a‖∞<∞.\|a\|_W+\|a\|_∞<∞. Moreover, ‖Ma‖≈‖a‖W+‖a‖∞,\|M_a\|≈\|a\|_W+\|a\|_∞, where ‖Ma‖\|M_a\| denotes the operator norm on L10L_1^0. The norm ∥⋅∥W\|·\|_W is the Semenov–Uksusov variation norm, rewritten with dyadic intervals as indices. We shall use the following equivalent branch formulation. Corollary 9.7 (Branch variation form). For a scalar Haar multiplier MaM_a on L10L_1^0, boundedness of MaM_a is equivalent to finiteness of BVbr(a)=supI0⊃I1⊃…(|aI0|+∑k=0∞|aIk+1−aIk|),BV_br(a)= _I_0⊃ I_1⊃… (|a_I_0|+ _k=0^∞|a_I_k+1-a_I_k| ), where the supremum is over infinite dyadic branches starting at an arbitrary dyadic interval. Moreover, ‖Ma‖≈BVbr(a),\|M_a\| _br(a), where ‖Ma‖\|M_a\| denotes the operator norm on L10L_1^0. Proof. If BVbr(a)<∞BV_br(a)<∞, then ‖a‖∞≤BVbr(a)\|a\|_∞ _br(a) because the branch may start at any dyadic interval. Every finite chain can be extended to an infinite branch, so ‖a‖W≤BVbr(a)\|a\|_W _br(a). Conversely, assume ‖a‖W+‖a‖∞<∞\|a\|_W+\|a\|_∞<∞. Fix an infinite branch I0⊃I1⊃…I_0⊃ I_1⊃…. For every N, |aI0|+∑k=0N−1|aIk+1−aIk|≤‖a‖∞+‖a‖W.|a_I_0|+ _k=0^N-1|a_I_k+1-a_I_k|≤\|a\|_∞+\|a\|_W. Passing to the supremum over N gives |aI0|+∑k=0∞|aIk+1−aIk|≤‖a‖∞+‖a‖W.|a_I_0|+ _k=0^∞|a_I_k+1-a_I_k|≤\|a\|_∞+\|a\|_W. Taking the supremum over all branches yields BVbr(a)≤‖a‖∞+‖a‖W.BV_br(a)≤\|a\|_∞+\|a\|_W. The norm equivalence follows from Theorem˜9.6. ∎ 2.4. The bi-parameter multiplier reduction We also need the following scalar compression result of Lechner–Motakis–Müller–Schlumprecht [52, Theorem 2.3]. It says that, after passing through uniformly controlled complemented copies of X00X_00, every bounded product Haar multiplier is arbitrarily close to a scalar multiple of the identity. Theorem 9.8 (Lechner–Motakis–Müller–Schlumprecht). Let MdM_d be a bounded product Haar multiplier on X00X_00. Given η>0η>0, there are Ad,Bd∈ℬ(X00),λ∈ℝA_d,B_d (X_00), λ such that AdBd=IdX00,‖AdMdBd−λIdX00‖<η,‖Ad‖‖Bd‖≤1+η. gatheredA_dB_d=Id_X_00, \|A_dM_dB_d- _X_00\|<η,\\ \|A_d\|\|B_d\|≤ 1+η. gathered In the notation of [52], (δ2)V(δ^2) denotes the algebraic span of the bi-parameter Haar system [52, Notation 2.1(e)], and its completion in the Lp(L1)L^p(L^1) mixed norm is X00X_00 in our notation. By [52, Theorem 2.10], Capon’s projection [22] is unbounded on Lp(L1)L^p(L^1). Consequently, the Lp(L1)L^p(L^1) case of [52, Definition 2.2(b),(c) and Theorem 2.3] gives precisely the scalar approximate projectional factorization stated above. Finally, passing from hI⊗hJh_I h_J to the normalized vectors eI⊗fJe_I f_J does not change the coefficient array of a diagonal multiplier. 3. The local L1L_1 trace The L1L_1 coordinate is the delicate one. We begin with a technical device that will be used repeatedly below, extracting from an arbitrary operator on L10L_1^0 a limiting averaged diagonal trace on every finite equal-dyadic set. We shall need the following definition. Definition 9.9 (Finite dyadic symmetry group). Let B be a dyadic interval and let N≥0N≥ 0. We denote by GN(B)G_N(B) the group of all bijections γ of [0,1)[0,1) satisfying the following conditions. (i) γ fixes [0,1)∖B[0,1) B pointwise. (i) For each 0≤k≤N+10≤ k≤ N+1, γ maps every interval in k(B)D_k(B) onto an interval in k(B)D_k(B). (i) If L=[a,a+ℓ)∈N+1(B)L=[a,a+ ) _N+1(B) and γL=L′=[b,b+ℓ)γ L=L =[b,b+ ), then γ(t)=b+(t−a)(t∈L).γ(t)=b+(t-a) (t∈ L). Equivalently, GN(B)G_N(B) is generated by independent swaps of the two children of each dyadic interval K∈k(B)K _k(B), 0≤k≤N0≤ k≤ N. For example, if B=[0,1)B=[0,1) and N=1N=1, then G1(B)G_1(B) is generated by swapping the two halves of [0,1)[0,1) and, independently, swapping the two quarters inside each half. Hence, its elements permute the four intervals in 2([0,1))D_2([0,1)) in a way compatible with the dyadic tree structure. We will also need the following elementary observation, where we write ≤N(B)=⋃k=0Nk(B)D_≤ N(B)= _k=0^ND_k(B) for the dyadic intervals in the finite tree below B up to level N. For γ∈GN(B)γ∈ G_N(B) and J∈≤N(B)J _≤ N(B), let εγ(J)∈−1,1 _γ(J)∈\-1,1\ be defined by UγfJ=εγ(J)fγ(J),U_γf_J= _γ(J)f_γ(J), where Uγx=x∘γ−1U_γx=x γ^-1. Lemma 9.10 (Sign flip in the finite dyadic tree). Let B be a dyadic interval, let N≥0N≥ 0, and let J,L∈≤N(B)J,L _≤ N(B) with J≠LJ≠ L. Then there is an involution ρ∈GN(B)ρ∈ G_N(B) such that ρ(J)=Jρ(J)=J, ρ(L)=Lρ(L)=L, and ερ(J)ερ(L)=−1. _ρ(J) _ρ(L)=-1. Proof. Since J and L are dyadic intervals, they are either disjoint or one is properly contained in the other. If J and L are disjoint, let ρ be the symmetry which flips the two children of J and is trivial at all other nodes. Then ρ fixes both J and L as sets, changes the sign of fJf_J, and leaves fLf_L unchanged. Hence ερ(J)ερ(L)=−1 _ρ(J) _ρ(L)=-1. If, say, L⊊JL J, let ρ flip the two children of L and be trivial at all other nodes. Then ρ fixes both J and L as sets, changes the sign of fLf_L, and leaves fJf_J unchanged, because fJf_J is constant on L. Again ερ(J)ερ(L)=−1 _ρ(J) _ρ(L)=-1. The case J⊊LJ L is symmetric. ∎ We are now ready for the proof of our next result. Proposition 9.11 (Local L1L_1 trace theorem). Let R∈ℬ(L10)R (L_1^0) and let B be a finite equal-dyadic set. Then βnB(R)=1#n(B)∑J∈n(B)⟨fJ∗,RfJ⟩ _n^B(R)= 1\#D_n(B) _J _n(B) f_J^*,Rf_J converges as n→∞n→∞. Denote its limit by λB(R) _B(R). Then there is a function φR∈L∞[0,1] _R∈ L_∞[0,1] with ‖φR‖∞≲‖R‖\| _R\|_∞ \|R\| such that λB(R)=1|B|∫BφR _B(R)= 1|B| _B _R for every finite equal-dyadic set B. Proof. First assume that B is a dyadic interval. For N≥0N≥ 0, let PNBP_N^B denote the projection onto the finite-dimensional subspace VN(B)=spanfJ:J∈k(B), 0≤k≤N.V_N(B)=span\f_J J _k(B),\ 0≤ k≤ N\. Equivalently, PNBP_N^B is the local martingale-difference projection onto the first N+1N+1 Haar levels below B. Let EN+1BE_N+1^B denote conditional expectation onto the functions supported on B which are constant on each interval in N+1(B)D_N+1(B). For x∈L1[0,1]x∈ L_1[0,1], this projection is given by PNBx=EN+1B(Bx)−1|B|(∫Bx)B,P_N^Bx=E_N+1^B(1_Bx)- 1|B| ( _Bx )1_B, thus ‖PNB‖≤2\|P_N^B\|≤ 2 as an operator on L1L_1, uniformly in B and N, and PNBP_N^B maps L10L_1^0 into L10L_1^0. Let GN(B)G_N(B) be the finite dyadic symmetry group from Section˜9. Each γ∈GN(B)γ∈ G_N(B) induces an isometry UγU_γ on L10L_1^0 by Uγx=x∘γ−1U_γx=x γ^-1. Moreover, by (i) and (i), for every J∈k(B)J _k(B) with 0≤k≤N0≤ k≤ N, we have UγfJ=εγ(J)fγ(J),for someεγ(J)∈−1,1.U_γf_J= _γ(J)f_γ(J), some _γ(J)∈\-1,1\. Define the Reynolds average RN,B=1|GN(B)|∑γ∈GN(B)Uγ−1PNBRPNBUγ.R_N,B= 1|G_N(B)| _γ∈ G_N(B)U_γ^-1P_N^BRP_N^BU_γ. Since each UγU_γ is an isometry and ‖PNB‖≤2\|P_N^B\|≤ 2, we have ‖RN,B‖≤4‖R‖.\|R_N,B\|≤ 4\|R\|. Moreover, RN,BR_N,B maps VN(B)V_N(B) into itself and annihilates the complementary Haar levels. Fix J,L∈≤N(B)J,L _≤ N(B). Then ⟨fL∗,RN,BfJ⟩=1|GN(B)|∑γ∈GN(B)εγ(J)εγ(L)⟨fγ(L)∗,Rfγ(J)⟩. f_L^*,R_N,Bf_J = 1|G_N(B)| _γ∈ G_N(B) _γ(J) _γ(L) f_γ(L)^*,Rf_γ(J) . Suppose first that J≠LJ≠ L. By Section˜9, choose an involution ρ∈GN(B)ρ∈ G_N(B) such that ρ(J)=Jρ(J)=J, ρ(L)=Lρ(L)=L, and ερ(J)ερ(L)=−1. _ρ(J) _ρ(L)=-1. We pair the term indexed by γ with the term indexed by γργρ. Since ρ fixes J and L as sets, we have (γρ)(J)=γ(J)(γρ)(J)=γ(J) and (γρ)(L)=γ(L)(γρ)(L)=γ(L), while εγρ(J)εγρ(L)=−εγ(J)εγ(L). _γρ(J) _γρ(L)=- _γ(J) _γ(L). Thus the two terms cancel. Therefore ⟨fL∗,RN,BfJ⟩=0(J,L∈≤N(B),J≠L). f_L^*,R_N,Bf_J =0 (J,L _≤ N(B),\ J≠ L). It remains to identify the diagonal coefficients. Let J∈k(B)J _k(B) for some 0≤k≤N0≤ k≤ N. The orbit of J under GN(B)G_N(B) is precisely k(B)D_k(B), with each interval occurring equally often. Hence ⟨fJ∗,RN,BfJ⟩=1#k(B)∑M∈k(B)⟨fM∗,RfM⟩=βkB(R). f_J^*,R_N,Bf_J = 1\#D_k(B) _M _k(B) f_M^*,Rf_M = _k^B(R). Thus RN,BR_N,B is the finite Haar multiplier with coefficient βkB(R) _k^B(R) on every node in k(B)D_k(B), 0≤k≤N0≤ k≤ N, and coefficient zero on Haar functions outside ≤N(B)D_≤ N(B). Regard RN,BR_N,B as a Haar multiplier on the whole space L10L_1^0, with symbol equal to βkB(R) _k^B(R) on every node in k(B)D_k(B) for 0≤k≤N0≤ k≤ N, and equal to zero on all other Haar nodes. Choose any infinite dyadic branch beginning at B. Along this branch, the symbol has one entry jump from 0 to β0B(R) _0^B(R), then the successive jumps |β1B(R)−β0B(R)|,…,|βNB(R)−βN−1B(R)|,| _1^B(R)- _0^B(R)|,…,| _N^B(R)- _N-1^B(R)|, and finally the jump from βNB(R) _N^B(R) to 0 below the truncated tree. Hence this branch has contribution |β0B(R)|+∑k=0N−1|βk+1B(R)−βkB(R)|+|βNB(R)|.| _0^B(R)|+ _k=0^N-1| _k+1^B(R)- _k^B(R)|+| _N^B(R)|. By Section˜9, |β0B(R)|+∑k=0N−1|βk+1B(R)−βkB(R)|+|βNB(R)|≲‖RN,B‖.| _0^B(R)|+ _k=0^N-1| _k+1^B(R)- _k^B(R)|+| _N^B(R)| \|R_N,B\|. Since ‖RN,B‖≲‖R‖\|R_N,B\| \|R\|, with an absolute constant independent of B and N, we obtain |β0B(R)|+∑k=0N−1|βk+1B(R)−βkB(R)|+|βNB(R)|≲‖R‖.| _0^B(R)|+ _k=0^N-1| _k+1^B(R)- _k^B(R)|+| _N^B(R)| \|R\|. Hence the finite initial variations of the sequence (βnB(R))n≥0( _n^B(R))_n≥ 0 are uniformly bounded, and therefore ∑k=0∞|βk+1B(R)−βkB(R)|<∞. _k=0^∞| _k+1^B(R)- _k^B(R)|<∞. It follows that (βnB(R))n≥0( _n^B(R))_n≥ 0 is Cauchy. We write λB(R)=limn→∞βnB(R). _B(R)= _n→∞ _n^B(R). If B0,B1B_0,B_1 are the dyadic children of B, then for every n, βn+1B(R)=12βnB0(R)+12βnB1(R). _n+1^B(R)= 12 _n^B_0(R)+ 12 _n^B_1(R). Passing to the limit gives λB(R)=12λB0(R)+12λB1(R). _B(R)= 12 _B_0(R)+ 12 _B_1(R). (38) Equivalently, |B|λB(R)=|B0|λB0(R)+|B1|λB1(R).|B| _B(R)=|B_0| _B_0(R)+|B_1| _B_1(R). For m≥0m≥ 0, define φm=∑B∈mλB(R)B. _m= _B _m _B(R)1_B. By (38), (φm)m≥0( _m)_m≥ 0 is a dyadic martingale. Moreover, supm≥0‖φm‖∞≲‖R‖. _m≥ 0\| _m\|_∞ \|R\|. By Banach–Alaoglu, choose a weak-star cluster point φR∈L∞[0,1]=(L1[0,1])∗ _R∈ L_∞[0,1]=(L_1[0,1])^* of (φm)m≥0( _m)_m≥ 0. Fix a dyadic interval B. If m is at least the level of B, then, by iterating (38), we have λB(R)=1#m−ℓ(B)(B)∑A∈m−ℓ(B)(B)λA(R), _B(R)= 1\#D_m- (B)(B) _A _m- (B)(B) _A(R), where ℓ(B) (B) denotes the dyadic level of B. Since φm _m is equal to λA(R) _A(R) on each such A, we get ∫Bφm=∑A∈m−ℓ(B)(B)|A|λA(R)=|B|λB(R). _B _m= _A _m- (B)(B)|A| _A(R)=|B| _B(R). Since weak-star convergence preserves the pairing with B∈L1[0,1]1_B∈ L_1[0,1], passing to the cluster point gives ∫BφR=|B|λB(R). _B _R=|B| _B(R). Thus λB(R)=1|B|∫BφR. _B(R)= 1|B| _B _R. Finally, if B is a finite equal-dyadic set, write it as a disjoint union of dyadic intervals B1,…,BrB_1,…,B_r of the same length. The descendants of B are the union of the descendants of the BℓB_ , and therefore λB(R)=1r∑ℓ=1rλBℓ(R)=1|B|∫BφR. _B(R)= 1r _ =1^r _B_ (R)= 1|B| _B _R. ∎ 4. Faithful Haar systems We shall replace the standard Haar system by copies supported on dyadic sets which may be geometrically scattered, but which preserve the dyadic tree structure exactly. The following definitions separate the tree structure from the normalizations used in the outer LpL_p and inner L1L_1 coordinates. Definition 9.12 (Faithful trees). A rooted dyadic subtree is a set ℱ⊂ F such that [0,1)∈ℱ[0,1)∈ F, every predecessor of every I∈ℱI∈ F also belongs to ℱ F, and, for every I∈ℱI∈ F, either both children I+I^+ and I−I^- belong to ℱ F, or neither child belongs to ℱ F. If ℱ F is finite, we call it a finite rooted dyadic subtree. We write ∂ℱ=I∈ℱ:I+∉ℱ and I−∉ℱ∂ F=\I∈ F I^+∉ F and I^-∉ F\ for the terminal nodes, or leaves, of ℱ F. A faithful tree indexed by ℱ F is a family (ΓI)I∈ℱ( _I)_I∈ F of finite equal-dyadic sets such that Γ[0,1)=[0,1) _[0,1)=[0,1), |ΓI|=|I|| _I|=|I|, and, whenever I,I+,I−∈ℱI,I^+,I^-∈ F, the sets ΓI+ _I^+ and ΓI− _I^- partition ΓI _I into two equal-measure subsets. If ℱ F is finite, we call (ΓI)I∈ℱ( _I)_I∈ F a finite faithful tree. The terminal sets of (ΓI)I∈ℱ( _I)_I∈ F are the sets ΓI _I with I∈∂ℱI∈∂ F. Example 9.13. A faithful tree need not be the standard dyadic tree. For instance, at the first level one may take Γ[0,1)=[0,1), _[0,1)=[0,1), and Γ[0,1/2)=[0,1/4)∪[1/2,3/4),Γ[1/2,1)=[1/4,1/2)∪[3/4,1). _[0,1/2)=[0,1/4)∪[1/2,3/4), _[1/2,1)=[1/4,1/2)∪[3/4,1). These two sets are finite equal-dyadic sets, they partition [0,1)[0,1), and both have measure 1/21/2, but Γ[0,1/2)≠[0,1/2) _[0,1/2)≠[0,1/2). Continuing one more level, we may split Γ[0,1/2)=Γ[0,1/4)⊔Γ[1/4,1/2) _[0,1/2)= _[0,1/4) _[1/4,1/2) by setting Γ[0,1/4)=[0,1/8)∪[1/2,5/8),Γ[1/4,1/2)=[1/8,1/4)∪[5/8,3/4). _[0,1/4)=[0,1/8)∪[1/2,5/8), _[1/4,1/2)=[1/8,1/4)∪[5/8,3/4). Again, both children are finite equal-dyadic sets, and both have measure 1/41/4. One can split Γ[1/2,1) _[1/2,1) in the same way. Thus the indexing tree is the usual dyadic tree, but the sets ΓI _I may be scattered finite unions of equal-length dyadic intervals. We shall use faithful trees in both coordinates. The underlying tree structure is the same in the outer and inner variables, but the Haar blocks are normalized according to the ambient space: the outer coordinate uses the LpL_p normalization, while the inner coordinate uses the L1L_1 normalization. Definition 9.14 (Outer faithful Haar blocks). Let (ΓI)I∈( _I)_I be a faithful tree in the outer coordinate. The associated outer Haar blocks are HI=|ΓI|−1/p(ΓI+−ΓI−),HI∗=|ΓI|−1/p′(ΓI+−ΓI−).H_I=| _I|^-1/p(1_ _I^+-1_ _I^-), H_I^*=| _I|^-1/p (1_ _I^+-1_ _I^-). Definition 9.15 (Inner faithful Haar blocks). Let (ΔJ)J∈( _J)_J be a faithful tree in the inner coordinate. The associated inner Haar blocks are KJ=|ΔJ|−1(ΔJ+−ΔJ−),KJ∗=ΔJ+−ΔJ−.K_J=| _J|^-1(1_ _J^+-1_ _J^-), K_J^*=1_ _J^+-1_ _J^-. The faithful trees give copies of the product Haar basis inside X00X_00. We shall use the following exterior maps to pass between the original product-Haar coordinates and these copies. The map B embeds the original basis into the Haar blocks, while A is the corresponding coordinate projection back onto the original basis. Lemma 9.16 (Exterior maps). Let (ΓI)I∈( _I)_I be a faithful outer tree, with associated blocks (HI,HI∗)I∈(H_I,H_I^*)_I , and let (ΔJ)J∈( _J)_J be a faithful inner tree, with associated blocks (KJ,KJ∗)J∈(K_J,K_J^*)_J . Then the map BuI,J=HI⊗KJBu_I,J=H_I K_J extends to an isometric embedding B:X00→X00B X_00→ X_00. Moreover, there is a contraction A:X00→X00A X_00→ X_00 such that A restricts to the inverse isometry on B(X00)B(X_00); in particular, AB=IdX00.AB=Id_X_00. For every bounded operator T on X00X_00, ⟨uI,J∗,ATBuI′,J′⟩=⟨HI∗⊗KJ∗,T(HI′⊗KJ′)⟩. u_I,J^*,ATBu_I ,J = H_I^* K_J^*,T(H_I K_J ) . Proof. Let ΣΓ _ and ΣΔ _ be the σ-algebras generated by the faithful trees (ΓI)I∈( _I)_I and (ΔJ)J∈( _J)_J , respectively. Let UΓ(r):Lr[0,1]⟶Lr(ΣΓ)U_ ^(r) L_r[0,1] L_r( _ ) be the isometric lattice embedding determined on dyadic simple functions by UΓ(r)I=ΓI(I∈),U_ ^(r)1_I=1_ _I (I ), and define UΔ(r)U_ ^(r) similarly. Let VΓ(r)V_ ^(r) and VΔ(r)V_ ^(r) denote the inverse isometries on the corresponding ranges. Define B=UΓ(p)⊗UΔ(1)B=U_ ^(p) U_ ^(1) on the algebraic product Haar span. Since UΔ(1)U_ ^(1) preserves the L1L_1-norm on each inner fibre and UΓ(p)U_ ^(p) is induced by a measure-preserving lattice embedding in the outer variable, for every f∈X00f∈ X_00 we have ‖(UΓ(p)⊗UΔ(1))f‖Lp(L1)=‖f‖Lp(L1). \|(U_ ^(p) U_ ^(1))f \|_L_p(L_1)=\|f\|_L_p(L_1). Hence B=UΓ(p)⊗UΔ(1)B=U_ ^(p) U_ ^(1) extends uniquely to an isometric embedding of X00X_00 into X00X_00. Moreover, BuI,J=B(eI⊗fJ)=HI⊗KJ.Bu_I,J=B(e_I f_J)=H_I K_J. Let EΔ(t)E_ ^(t) be conditional expectation in the inner variable onto ΣΔ _ , and let EΓ(s)E_ ^(s) be Bochner conditional expectation in the outer variable onto ΣΓ _ . Put EΓ,Δ=EΓ(s)EΔ(t)=EΔ(t)EΓ(s).E_ , =E_ ^(s)E_ ^(t)=E_ ^(t)E_ ^(s). For almost every s, ‖EΔ(t)f(s,⋅)‖1≤‖f(s,⋅)‖1,\|E_ ^(t)f(s,·)\|_1≤\|f(s,·)\|_1, and therefore EΔ(t)E_ ^(t) is contractive on Lp(L1)L_p(L_1). Also EΓ(s)E_ ^(s) is the usual Bochner conditional expectation on Lp(L1)L_p(L_1) and is contractive. Hence EΓ,ΔE_ , is contractive on Lp(L1)L_p(L_1). Define A=(VΓ(p)⊗VΔ(1))EΓ,Δ.A=(V_ ^(p) V_ ^(1))E_ , . The map VΓ(p)⊗VΔ(1)V_ ^(p) V_ ^(1) is an isometry on the range of EΓ,ΔE_ , , so A is contractive on Lp(L1)L_p(L_1), hence on X00X_00. Since EΓ,ΔE_ , is the identity on B(X00)B(X_00), the definition gives AB=IdX00AB=Id_X_00. Finally, for every x∈X00x∈ X_00, ⟨uI,J∗,Ax⟩=⟨HI∗⊗KJ∗,x⟩. u_I,J^*,Ax = H_I^* K_J^*,x . Applying this with x=TBuI′,J′=T(HI′⊗KJ′)x=TBu_I ,J =T(H_I K_J ) gives ⟨uI,J∗,ATBuI′,J′⟩=⟨HI∗⊗KJ∗,T(HI′⊗KJ′)⟩.∎ u_I,J^*,ATBu_I ,J = H_I^* K_J^*,T(H_I K_J ) . 5. Potential shadows and trace operators The shadow averages used below must be defined before the future faithful tree has been chosen. We therefore define them from the ordinary dyadic descendants of a current terminal set, viewed as a finite equal-dyadic set. Let A⊂[0,1)A⊂[0,1) be a finite equal-dyadic set and m≥0m≥ 0. For L∈m(A)L _m(A), we use the standard normalized Haar vectors eL=|L|−1/phL,eL∗=|L|−1/p′hL.e_L=|L|^-1/ph_L, e_L^*=|L|^-1/p h_L. For x∈X00x∈ X_00 define the outer coefficient πLx=∫01eL∗(s)x(s,⋅)s∈L10. _Lx= _0^1e_L^*(s)x(s,·)\,ds∈ L_1^0. Thus πLx _Lx is the eLe_L-coefficient of x in the outer variable, regarded as an element of the inner space. Definition 9.17 (Potential traces and shadows). Let A be a finite equal-dyadic set. For Q∈ℬ(Lp0)Q (L_p^0) and m≥0m≥ 0, define αmA(Q)=1#m(A)∑L∈m(A)⟨eL∗,QeL⟩. _m^A(Q)= 1\#D_m(A) _L _m(A) e_L^*,Qe_L . For T∈ℬ(X00)T (X_00) and m≥0m≥ 0, define ΩA,mT∈ℬ(L10) _A,m^T (L_1^0) by ΩA,mTg=1#m(A)∑L∈m(A)πL(T(eL⊗g))(g∈L10). _A,m^Tg= 1\#D_m(A) _L _m(A) _L (T(e_L g) ) (g∈ L_1^0). Indeed, for each L∈m(A)L _m(A), the map g↦πL(T(eL⊗g))g _L(T(e_L g)) is a bounded operator L10→L10L_1^0→ L_1^0 of norm at most ‖T‖\|T\|, since ‖eL‖Lp=‖eL∗‖Lp′=1\|e_L\|_L_p=\|e_L^*\|_L_p =1. Hence the finite average defining ΩA,mT _A,m^T belongs to ℬ(L10)B(L_1^0) and satisfies ‖ΩA,mT‖≤‖T‖\| _A,m^T\|≤\|T\|. These quantities depend only on the current terminal set A, viewed as a finite equal-dyadic set, and on the standard dyadic structure below it. They do not depend on any future choices in the faithful tree. For a dyadic interval L and σ∈−1,1σ∈\-1,1\, write Lσ=L+L^σ=L^+ if σ=1σ=1 and Lσ=L−L^σ=L^- if σ=−1σ=-1. If A is a finite equal-dyadic set and n≥0n≥ 0, let ε=(εL)L∈n(A)∈−1,1n(A) =( _L)_L _n(A)∈\-1,1\^D_n(A) be a sign selection on the depth-n descendants of A. Define A+(ε)=⋃L∈n(A)LεL,A−(ε)=⋃L∈n(A)L−εL.A_+( )= _L _n(A)L _L, A_-( )= _L _n(A)L^- _L. (39) Both A+(ε)A_+( ) and A−(ε)A_-( ) are regarded as finite equal-dyadic sets with decomposition level mA+(ε)=mA−(ε)=mA+n+1m_A_+( )=m_A_-( )=m_A+n+1. The associated outer random Haar block is HA(ε)=|A|−1/p(A+(ε)−A−(ε)),HA∗(ε)=|A|−1/p′(A+(ε)−A−(ε)).H_A( )=|A|^-1/p (1_A_+( )-1_A_-( ) ), 3.0ptH_A^*( )=|A|^-1/p (1_A_+( )-1_A_-( ) ). (40) The associated inner random Haar block is KA(ε)=|A|−1(A+(ε)−A−(ε)),KA∗(ε)=A+(ε)−A−(ε).K_A( )=|A|^-1 (1_A_+( )-1_A_-( ) ), K_A^*( )=1_A_+( )-1_A_-( ). (41) If N=#n(A)N=\#D_n(A), then HA(ε)=N−1/p∑L∈n(A)εLeL,HA∗(ε)=N−1/p′∑L∈n(A)εLeL∗.H_A( )=N^-1/p _L _n(A) _Le_L, H_A^*( )=N^-1/p _L _n(A) _Le_L^*. (42) Similarly, KA(ε)=N−1∑L∈n(A)εLfL,KA∗(ε)=∑L∈n(A)εLfL∗.K_A( )=N^-1 _L _n(A) _Lf_L, K_A^*( )= _L _n(A) _Lf_L^*. (43) Consequently, for every Q∈ℬ(Lp0)Q (L_p^0), expanding the random block, we have ⟨HA∗(ε),QHA(ε)⟩=1N∑L,M∈n(A)εLεM⟨eL∗,QeM⟩. H_A^*( ),QH_A( ) = 1N _L,M _n(A) _L _M e_L^*,Qe_M . Since the signs (εL)L∈n(A)( _L)_L _n(A) are independent and symmetric, ε[εLεM]=1,L=M,0,L≠M.E_ [ _L _M]= cases1,&L=M,\\ 0,&L≠ M. cases Thus, all off-diagonal terms vanish after taking expectation, and only the diagonal terms remain, which gives ε[⟨HA∗(ε),QHA(ε)⟩]=1N∑L∈n(A)⟨eL∗,QeL⟩=αnA(Q).E_ [ H_A^*( ),QH_A( ) ]= 1N _L _n(A) e_L^*,Qe_L = _n^A(Q). (44) Moreover, for r≥0r≥ 0, ε[αrA+(ε)(Q)]=ε[αrA−(ε)(Q)]=αn+1+rA(Q).E_ [ _r^A_+( )(Q) ]=E_ [ _r^A_-( )(Q) ]= _n+1+r^A(Q). (45) Indeed, αrA+(ε)(Q) _r^A_+( )(Q) is the average, over L∈n(A)L _n(A), of the average of the diagonal coefficients below the randomly selected half LεL _L. Averaging in the signs therefore averages over both halves of every L, which is precisely the average over n+1+r(A)D_n+1+r(A). The next elementary estimate says that, after a sufficiently fine random split, the child traces concentrate around their expected values. This is the probabilistic result used later to preserve infinite tails through the construction. Lemma 9.18 (Variance of child traces). Let A be a finite equal-dyadic set, let Q∈ℬ(Lp0)Q (L_p^0), and let n,r≥0n,r≥ 0. Let ε=(εL)L∈n(A) =( _L)_L _n(A) be uniformly distributed on −1,1n(A)\-1,1\^D_n(A). For each sign θ∈+,−θ∈\+,-\, we have Varε(αrAθ(ε)(Q))≤‖Q‖2#n(A).Var_ ( _r^A_θ( )(Q) )≤ \|Q\|^2\#D_n(A). (46) Proof. We prove the estimate for A+A_+. Write αrA+(ε)(Q)=1#n(A)∑L∈n(A)XL(εL), _r^A_+( )(Q)= 1\#D_n(A) _L _n(A)X_L( _L), where XL(1)=2−r∑M∈r(L+)⟨eM∗,QeM⟩,XL(−1)=2−r∑M∈r(L−)⟨eM∗,QeM⟩.X_L(1)=2^-r _M _r(L^+) e_M^*,Qe_M , X_L(-1)=2^-r _M _r(L^-) e_M^*,Qe_M . The random variables XL(εL)X_L( _L) are independent as L varies. Moreover, since ‖eM‖Lp=‖eM∗‖Lp′=1\|e_M\|_L_p=\|e_M^*\|_L_p =1, we have |⟨eM∗,QeM⟩|≤‖Q‖| e_M^*,Qe_M |≤\|Q\|, and hence |XL(εL)|≤‖Q‖|X_L( _L)|≤\|Q\|. Therefore Varε(αrA+(ε)(Q))=1#n(A)2∑L∈n(A)Varε(XL(εL))≤‖Q‖2#n(A).Var_ ( _r^A_+( )(Q) )= 1\#D_n(A)^2 _L _n(A)Var_ (X_L( _L) )≤ \|Q\|^2\#D_n(A). The proof for A−A_- is identical. ∎ We will need the following well-known elementary result; we include a proof for completeness. Lemma 9.19 (L1L_1 disjointification criterion). Let A be a bounded subset of L1L_1. If A is not uniformly integrable, then there are a sequence (fk)k≥1(f_k)_k≥ 1 in A, pairwise disjoint measurable sets Ek⊂[0,1]E_k⊂[0,1], and a number δ>0δ>0 such that ∫Ek|fk|>δ(k≥1). _E_k|f_k|>δ (k≥ 1). Proof. Since A is not uniformly integrable, there are η>0η>0, functions fn∈f_n , and measurable sets An⊂[0,1]A_n⊂[0,1] such that μ(An)<2−n,∫An|fn|>η(n≥1).μ(A_n)<2^-n, _A_n|f_n|>η (n≥ 1). We pass to a subsequence, still denoted by (fn,An)(f_n,A_n), with the following extra property. After f1,A1,…,fk,Akf_1,A_1,…,f_k,A_k have been chosen, choose δi>0 _i>0 for 1≤i≤k1≤ i≤ k so that μ(E)<δi⟹∫E|fi|<η/2μ(E)< _i _E|f_i|<η/2 for every measurable E. Since the original witnessing sets have measures tending to zero, the remaining subsequence may be chosen so that, for every j>ij>i, μ(Aj)<2−jδi.μ(A_j)<2^-j _i. Consequently, for every fixed i, μ(⋃j>iAj)≤∑j>iμ(Aj)<δi.μ ( _j>iA_j )≤ _j>iμ(A_j)< _i. Now define Ei=Ai∖⋃j>iAj.E_i=A_i _j>iA_j. The sets EiE_i are pairwise disjoint. Moreover, ∫Ei|fi|≥∫Ai|fi|−∫⋃j>iAj|fi|>η−η/2=η/2. _E_i|f_i|≥ _A_i|f_i|- _ _j>iA_j|f_i|>η-η/2=η/2. Thus the conclusion holds with δ=η/2δ=η/2. ∎ The next proposition is the compactness step behind the shadow construction. Although ΩA,mT _A,m^T is defined by averaging more and more outer dyadic descendants of A, its orbit on each fixed g∈L10g∈ L_1^0 remains uniformly integrable. Thus, along any free ultrafilter supported on an infinite set of depths, the weak limit is still represented by an element of L10L_1^0, rather than only by an element of L1∗L_1^**. Proposition 9.20 (Potential outer shadows are L1L_1-valued). Let A be a finite equal-dyadic set, let T∈ℬ(X00)T (X_00), and let S⊂ℕS be infinite. For every g∈L10g∈ L_1^0, the set ΩA,mTg:m∈S\ _A,m^Tg:m∈ S\ is uniformly integrable in L1L_1. Consequently, if U is a free ultrafilter containing S, then ΩAT,g=weaklimm→ΩA,mTg _A^T,Ug=weak\! _m _A,m^Tg exists in L10L_1^0, and ΩAT,:L10→L10 _A^T,U L_1^0→ L_1^0 is bounded with ‖ΩAT,‖≤‖T‖.\| _A^T,U\|≤\|T\|. Proof. Fix g∈L10g∈ L_1^0 and m∈Sm∈ S, and put Nm=#m(A)N_m=\#D_m(A). We record the randomization identity used below. For a sign family σ=(σL)L∈m(A)σ=( _L)_L _m(A), set xmσ=Nm−1/p∑L∈m(A)σLeL,(xmσ)∗=Nm−1/p′∑L∈m(A)σLeL∗.x_m^σ=N_m^-1/p _L _m(A) _Le_L, (x_m^σ)^*=N_m^-1/p _L _m(A) _Le_L^*. The supports are pairwise disjoint, so ‖xmσ‖Lp=1\|x_m^σ\|_L_p=1 and ‖(xmσ)∗‖Lp′=1\|(x_m^σ)^*\|_L_p =1. For every ϕ∈L∞φ∈ L_∞, ⟨ϕ,ΩA,mTg⟩=σ[⟨(xmσ)∗⊗ϕ,T(xmσ⊗g)⟩]. φ, _A,m^Tg =E_σ [ (x_m^σ)^* φ,T(x_m^σ g) ]. (47) Indeed, after expanding the right hand side, the terms with L≠L′L≠ L have mean zero, while the terms with L=L′L=L give exactly the defining average of ΩA,mTg _A,m^Tg. In particular, ‖ΩA,mTg‖1≤‖T‖‖g‖1.\| _A,m^Tg\|_1≤\|T\|\,\|g\|_1. (48) Suppose that ΩA,mTg:m∈S\ _A,m^Tg:m∈ S\ is not uniformly integrable for some ‖g‖1=1\|g\|_1=1. By Section˜9, after passing to a sequence m1<m2<…m_1<m_2<… in S, there are pairwise disjoint measurable sets Ek⊂[0,1]E_k⊂[0,1] and a number δ>0δ>0 such that ∫Ek|ΩA,mkTg|>δ(k≥1). _E_k| _A,m_k^Tg|>δ (k≥ 1). Choose ψk∈L∞ _k∈ L_∞ supported on EkE_k, with ‖ψk‖∞≤1\| _k\|_∞≤ 1, such that ⟨ψk,ΩA,mkTg⟩>δ. _k, _A,m_k^Tg >δ. Fix M≥1M≥ 1. For each 1≤k≤M1≤ k≤ M, let σkσ^k be an independent sign family on mk(A)D_m_k(A), and let η1,…,ηM _1,…, _M be independent signs, independent also of all the families σkσ^k. Put ak(σ)=(xmkσk)∗⊗ψk,bk(σ)=xmkσk⊗g.a_k(σ)=(x_m_k^σ^k)^* _k, b_k(σ)=x_m_k^σ^k g. By (47), for each k we have δ<σk[⟨ak(σ),Tbk(σ)⟩].δ<E_σ^k [ a_k(σ),Tb_k(σ) ]. Summing over k gives δM<σ[∑k=1M⟨ak(σ),Tbk(σ)⟩].δ M<E_σ [ _k=1^M a_k(σ),Tb_k(σ) ]. For fixed σ, averaging in the signs η1,…,ηM _1,…, _M gives η[⟨∑k=1Mηkak(σ),T(∑ℓ=1Mηℓbℓ(σ))⟩]=∑k,ℓ=1Mη[ηkηℓ]⟨ak(σ),Tbℓ(σ)⟩=∑k=1M⟨ak(σ),Tbk(σ)⟩.E_η [ _k=1^M _ka_k(σ),T ( _ =1^M _ b_ (σ) ) ]= _k, =1^ME_η[ _k _ ] a_k(σ),Tb_ (σ) = _k=1^M a_k(σ),Tb_k(σ) . Therefore δM<σ,η[⟨∑k=1Mηk(xmkσk)∗⊗ψk,T(∑k=1Mηkxmkσk⊗g)⟩].δ M<E_σ,η [ _k=1^M _k(x_m_k^σ^k)^* _k,T ( _k=1^M _kx_m_k^σ^k g ) ]. For fixed signs, the first vector has norm at most one in Lp′(L∞)L_p (L_∞). Indeed, the functions ψk _k have disjoint supports in the inner variable and |(xmkσk)∗|=|A|−1/p′A.|(x_m_k^σ^k)^*|=|A|^-1/p 1_A. Hence, using the normalization of g, |⟨∑k=1Mηk(xmkσk)∗⊗ψk,T(∑k=1Mηkxmkσk⊗g)⟩|≤‖T‖‖∑k=1Mηkxmkσk‖Lp. | _k=1^M _k(x_m_k^σ^k)^* _k,T ( _k=1^M _kx_m_k^σ^k g ) |≤\|T\| \| _k=1^M _kx_m_k^σ^k \|_L_p. The Khintchine inequality, applied pointwise in the outer variable, gives σ,η‖∑k=1Mηkxmkσk‖Lp≤CpM1/2.E_σ,η \| _k=1^M _kx_m_k^σ^k \|_L_p≤ C_pM^1/2. Consequently, δM≤Cp‖T‖M1/2(M≥1),δ M≤ C_p\|T\|M^1/2 (M≥ 1), which is impossible for large M. This proves uniform integrability. By the Dunford–Pettis criterion for L1L_1, see for example [7, Theorem 5.2.9], uniform integrability implies relative weak compactness in L1L_1. Hence, the ultrafilter weak limit belongs to L1L_1. Since L10L_1^0 is weakly closed and each ΩA,mTg _A,m^Tg lies in L10L_1^0, the limit lies in L10L_1^0. Linearity is inherited from the scalar ultralimits, and (48) gives ‖ΩAT,g‖1≤‖T‖‖g‖1.\| _A^T,Ug\|_1≤\|T\|\,\|g\|_1. ∎ The next lemma establishes the compatibility between outer-averaged diagonals and shadow operators. If the inner coordinate is fixed as v and then tested against v∗v^*, the averaged outer diagonal of the resulting one-parameter operator is exactly the corresponding coefficient of the shadow. Lemma 9.21 (The same-outer identity). Let A be a finite equal-dyadic set in the outer variable, let T∈ℬ(X00)T (X_00), v∈L10v∈ L_1^0, and v∗∈L∞v^*∈ L_∞. Define Qv∗,vx=(Id⊗v∗)T(x⊗v)(x∈Lp0).Q_v^*,vx=(Id v^*)T(x v) (x∈ L_p^0). Then Qv∗,v∈ℬ(Lp0)Q_v^*,v (L_p^0) and, for every m≥0m≥ 0, αmA(Qv∗,v)=⟨v∗,ΩA,mTv⟩. _m^A(Q_v^*,v)= v^*, _A,m^Tv . (49) Consequently, if U is a free ultrafilter for which ΩAT,v _A^T,Uv is defined, then limm→αmA(Qv∗,v)=⟨v∗,ΩAT,v⟩. _m _m^A(Q_v^*,v)= v^*, _A^T,Uv . Proof. The boundedness follows from ‖Qv∗,vx‖p≤‖v∗‖∞‖T‖‖x‖p‖v‖1.\|Q_v^*,vx\|_p≤\|v^*\|_∞\|T\|\|x\|_p\|v\|_1. Since T(x⊗v)∈X00T(x v)∈ X_00, taking the inner v∗v^*-coefficient leaves an element of Lp0L_p^0. For L∈m(A)L _m(A), ⟨eL∗,Qv∗,veL⟩=⟨eL∗⊗v∗,T(eL⊗v)⟩=⟨v∗,πL(T(eL⊗v))⟩. e_L^*,Q_v^*,ve_L = e_L^* v^*,T(e_L v) = v^*, _L(T(e_L v)) . Averaging over L∈m(A)L _m(A) gives (49). The ultrafilter statement follows by applying v∗v^* to the weak ultralimit. ∎ We shall also need the complementary construction, where the local trace is taken in the inner L1L_1 coordinate and the outer variable is left alive. Fixing an inner finite equal-dyadic set B, the local L1L_1 trace theorem turns each outer matrix coefficient of T into a scalar. The next lemma packages these scalars as a bounded operator on Lp0L_p^0. Lemma 9.22 (Outer trace operator associated with an inner set). Let B be a finite equal-dyadic set in the inner variable and let T∈ℬ(X00)T (X_00). There is a unique operator ℐBT∈ℬ(Lp0)I_B^T (L_p^0) such that, for all x∈Lp0x∈ L_p^0 and all x∗∈Lp′x^*∈ L_p , ⟨x∗,ℐBTx⟩=λB(Rx∗,xT), x^*,I_B^Tx = _B (R_x^*,x^T ), (50) where Rx∗,xTg=∫01x∗(s)T(x⊗g)(s,⋅)s(g∈L10).R_x^*,x^Tg= _0^1x^*(s)T(x g)(s,·)\,ds (g∈ L_1^0). Moreover, ‖ℐBT‖≲p‖T‖.\|I_B^T\| _p\|T\|. Proof. For g∈L10g∈ L_1^0, ‖Rx∗,xTg‖1≤‖x∗‖p′‖T(x⊗g)‖Lp(L1)≤‖x∗‖p′‖T‖‖x‖p‖g‖1.\|R_x^*,x^Tg\|_1≤\|x^*\|_p \|T(x g)\|_L_p(L_1)≤\|x^*\|_p \|T\|\|x\|_p\|g\|_1. Since T(x⊗g)∈X00T(x g)∈ X_00 is cancellative in the inner variable, Rx∗,xTgR_x^*,x^Tg has inner integral zero. Hence Rx∗,xT∈ℬ(L10)R_x^*,x^T (L_1^0). By Section˜9, |λB(Rx∗,xT)|≲‖Rx∗,xT‖≤‖x∗‖p′‖T‖‖x‖p.| _B(R_x^*,x^T)| \|R_x^*,x^T\|≤\|x^*\|_p \|T\|\|x\|_p. Thus, for fixed x, the map x∗↦λB(Rx∗,xT)x^* _B(R_x^*,x^T) is a bounded functional on Lp′L_p . Since 1<p<∞1<p<∞, it is represented by a unique element y∈Lpy∈ L_p. It remains to check that y belongs to Lp0L_p^0. This uses the outer cancellation. If x∗=[0,1)x^*=1_[0,1), then R[0,1),xTg=∫01T(x⊗g)(s,⋅)s=0,R_1_[0,1),x^Tg= _0^1T(x g)(s,·)\,ds=0, because T(x⊗g)∈X00T(x g)∈ X_00 has zero outer integral. Therefore ⟨[0,1),y⟩=0 1_[0,1),y =0, so y∈Lp0y∈ L_p^0. Define ℐBTx=yI_B^Tx=y. Linearity follows from the linearity of x↦λB(Rx∗,xT)x _B(R_x^*,x^T) and uniqueness of the representing vector. The displayed estimate gives ‖ℐBT‖≲p‖T‖\|I_B^T\| _p\|T\|, and uniqueness follows because an element of Lp0L_p^0 annihilated by all x∗∈Lp′x^*∈ L_p is zero. ∎ 6. Admissible constraints and outer tail preservation We now isolate the side constraints that will be imposed during the random splittings introduced above. These are the coefficients that can be made small by randomness. The self-diagonal coefficients are excluded, because their averages are the trace quantities controlled separately. In the finite splitting estimates below, where E is either Lp0L_p^0 or L10L_1^0, we adopt a generic notation in order to treat the two normalizations simultaneously. A random normalized Haar block means the block constructed from the split (39), with the normalization appropriate to E. More precisely, when E=Lp0E=L_p^0, we use the outer normalization (40) and write uA(ε)=HA(ε),uA∗(ε)=HA∗(ε).u_A( )=H_A( ), u_A^*( )=H_A^*( ). When E=L10E=L_1^0, we use the inner normalization (41) and write uA(ε)=KA(ε),uA∗(ε)=KA∗(ε).u_A( )=K_A( ), u_A^*( )=K_A^*( ). Definition 9.23 (Admissible finite constraints). Let E=Lp0E=L_p^0 or E=L10E=L_1^0. Let A1,…,AdA_1,…,A_d be pairwise disjoint finite equal-dyadic sets. For each a=1,…,da=1,…,d, let ua,ua∗u_a,u_a^* be a random normalized Haar block supported on AaA_a, defined by an independent sign selection on sufficiently fine dyadic descendants of AaA_a. A finite collection C of inequalities involving the random blocks ua,ua∗u_a,u_a^* is called admissible if each element C∈C∈ C is an inequality of the form |XC|<ηC,|X_C|< _C, where ηC>0 _C>0 is called the tolerance of C, and where the scalar-valued random variable XCX_C has one of the following forms: XC=⟨ua∗,SCzC⟩,XC=⟨zC∗,SCua⟩,X_C= u_a^*,S_Cz_C , X_C= z_C^*,S_Cu_a , or XC=⟨ua∗,SCub⟩(a≠b).X_C= u_a^*,S_Cu_b (a≠ b). Here SC∈ℬ(E)S_C (E), zC∈Ez_C∈ E, zC∗∈E∗z_C^*∈ E^*, the indices a,ba,b, and the number ηC>0 _C>0 are fixed before the random signs defining the blocks uau_a are chosen. The self-diagonal inequality |⟨ua∗,Sua⟩|<η| u_a^*,Su_a |<η is deliberately not included. Such terms are not side constraints: their expectations are averaged diagonal traces, and they are controlled separately by the trace estimates. 7. Finite splitting estimates At each finite stage of the construction, only finitely many current finite equal-dyadic sets are split. The estimates below are finite-dimensional probability estimates for the Haar blocks produced by such splits. In particular, we have the following elementary observation. Lemma 9.24 (Coordinate projections on dyadic descendants). Let E=Lp0E=L_p^0, 1<p<∞1<p<∞, or E=L10E=L_1^0. Let L1,…,LNL_1,…,L_N be pairwise disjoint dyadic intervals of the same length. In the case E=Lp0E=L_p^0, put ui=eLi,ui∗=eLi∗(1≤i≤N),u_i=e_L_i, u_i^*=e_L_i^* (1≤ i≤ N), and in the case E=L10E=L_1^0, put ui=fLi,ui∗=fLi∗(1≤i≤N).u_i=f_L_i, u_i^*=f_L_i^* (1≤ i≤ N). Define Px=∑i=1N⟨ui∗,x⟩ui.Px= _i=1^N u_i^*,x u_i. Then P:E→EP E→ E is a contraction. Moreover, the map (ai)i=1N⟼∑i=1Naiui(a_i)_i=1^N _i=1^Na_iu_i identifies the span of u1,…,uNu_1,…,u_N isometrically with ℓpN _p^N in the LpL_p case and with ℓ1N _1^N in the L1L_1 case. Proof. The supports of the uiu_i are disjoint, and the chosen normalizations give the displayed ℓp _p and ℓ1 _1 norms by direct integration. The projection is the sum of the local Haar-coordinate projections on the intervals LiL_i. On each LiL_i it has the form x⟼|Li|−1hLi∫LihLix.x |L_i|^-1h_L_i _L_ih_L_ix. For E=Lp0E=L_p^0, Hölder’s inequality gives ‖|Li|−1hLi∫LihLix‖p≤‖Lix‖p. \||L_i|^-1h_L_i _L_ih_L_ix \|_p≤\|1_L_ix\|_p. For E=L10E=L_1^0, the same formula gives ‖|Li|−1hLi∫LihLix‖1≤∫Li|x|. \||L_i|^-1h_L_i _L_ih_L_ix \|_1≤ _L_i|x|. Since the intervals LiL_i are disjoint, summing over i gives ‖Px‖E≤‖x‖E\|Px\|_E≤\|x\|_E. ∎ The next estimate says that the random self-diagonal coefficient of a new Haar block is concentrated around the average of the diagonal coefficients on the dyadic descendants from which the block is built. Lemma 9.25 (Centered quadratic estimate). Let E=Lp0E=L_p^0 or E=L10E=L_1^0, let Q∈ℬ(E)Q (E), and let A be a finite equal-dyadic set. Fix n≥0n≥ 0, write n(A)=L1,…,LND_n(A)=\L_1,…,L_N\, and let ε=(εi)i=1N =( _i)_i=1^N be uniformly distributed on −1,1N\-1,1\^N. Let u(ε),u∗(ε)u( ),u^*( ) be the random normalized Haar block and its biorthogonal functional associated with A, n, and ε . For each i, let ui,ui∗u_i,u_i^* be the normalized Haar vector and biorthogonal functional associated with LiL_i, using the normalization of E. Then ε[|⟨u∗(ε),Qu(ε)⟩−1N∑i=1N⟨ui∗,Qui⟩|2]≲p‖Q‖2N.E_ [ | u^*( ),Qu( ) - 1N _i=1^N u_i^*,Qu_i |^2 ] _p \|Q\|^2N. In the case E=L10E=L_1^0, the subscript p is omitted. Proof. Write qij=⟨ui∗,Quj⟩q_ij= u_i^*,Qu_j . In the LpL_p case, u(ε)=N−1/p∑i=1Nεiui,u∗(ε)=N−1/p′∑i=1Nεiui∗.u( )=N^-1/p _i=1^N _iu_i, u^*( )=N^-1/p _i=1^N _iu_i^*. In the L1L_1 case, u(ε)=N−1∑i=1Nεiui,u∗(ε)=∑i=1Nεiui∗.u( )=N^-1 _i=1^N _iu_i, u^*( )= _i=1^N _iu_i^*. Thus, in both cases, ⟨u∗(ε),Qu(ε)⟩−1N∑i=1Nqii=1N∑i≠jεiεjqij. u^*( ),Qu( ) - 1N _i=1^Nq_i= 1N _i≠ j _i _jq_ij. By independence of the signs, ε[|1N∑i≠jεiεjqij|2]≤2N−2∑i≠j|qij|2.E_ [ | 1N _i≠ j _i _jq_ij |^2 ]≤ 2N^-2 _i≠ j|q_ij|^2. It remains to bound the Hilbert–Schmidt sum. For fixed j, the column (qij)i=1N(q_ij)_i=1^N is the coordinate vector of PQujPQu_j, where P is the coordinate projection from Section˜9. In the L1L_1 case, its ℓ1 _1 norm is at most ‖Q‖\|Q\|, hence its ℓ2 _2 norm is at most ‖Q‖\|Q\|. In the LpL_p case with 1<p≤21<p≤ 2, its ℓp _p norm is at most ‖Q‖\|Q\|, hence its ℓ2 _2 norm is at most ‖Q‖\|Q\|. Summing over j gives ∑i,j=1N|qij|2≲pN‖Q‖2. _i,j=1^N|q_ij|^2 _pN\|Q\|^2. For p≥2p≥ 2, the same argument is applied to the rows, equivalently to Q∗Q^* on Lp′L_p with p′≤2p ≤ 2. Combining this estimate with the preceding second-moment bound proves the lemma. ∎ The next estimates handle the genuinely one-sided coefficients. If one side of the pairing is fixed before the signs are chosen, then a sufficiently fine random Haar block is almost orthogonal to it on average. Lemma 9.26 (One-sided estimates). Let E=Lp0E=L_p^0 or E=L10E=L_1^0, and let A be a finite equal-dyadic set. Then, for every η>0η>0, every pair of integers M1,M2≥0M_1,M_2≥ 0, and every pair of finite families ℱ1=(Sb,zb):1≤b≤M1⊂ℬ(E)×EF_1=\(S_b,z_b):1≤ b≤ M_1\ (E)× E and ℱ2=(Ra,za∗):1≤a≤M2⊂ℬ(E)×E∗,F_2=\(R_a,z_a^*):1≤ a≤ M_2\ (E)× E^*, there is n0≥0n_0≥ 0 such that, for every n≥n0n≥ n_0, the following holds. If n(A)=L1,…,LND_n(A)=\L_1,…,L_N\ and u(ε),u∗(ε)u( ),u^*( ) is the random normalized Haar block and its biorthogonal functional associated with A, n, and ε∈−1,1N ∈\-1,1\^N, then ε[|⟨u∗(ε),Sbzb⟩|2]<η2(1≤b≤M1),E_ [ | u^*( ),S_bz_b |^2 ]<η^2 (1≤ b≤ M_1), and ε[|⟨Ra∗za∗,u(ε)⟩|2]<η2(1≤a≤M2).E_ [ | R_a^*z_a^*,u( ) |^2 ]<η^2 (1≤ a≤ M_2). Proof. For each depth n, write n(A)=L1,…,LND_n(A)=\L_1,…,L_N\ and denote the corresponding random block by un(ε),un∗(ε)u_n( ),u_n^*( ). It suffices to prove that, for every fixed y∈Ey∈ E and every fixed y∗∈E∗y^*∈ E^*, limn→∞ε[|⟨un∗(ε),y⟩|2]=0,limn→∞ε[|⟨y∗,un(ε)⟩|2]=0. _n→∞E_ [ | u_n^*( ),y |^2 ]=0, _n→∞E_ [ | y^*,u_n( ) |^2 ]=0. Indeed, after applying these two limits to the finitely many fixed vectors yb=Sbzby_b=S_bz_b and fixed functionals ya∗=Ra∗za∗y_a^*=R_a^*z_a^*, one chooses a single depth n large enough for all of them. First consider E=L10E=L_1^0. If z∈L10z∈ L_1^0, then ε[|⟨u∗(ε),z⟩|2]=∑i=1N|∫LifLi∗z|2≤(max1≤i≤N∫Li|z|)‖z‖1.E_ [ | u^*( ),z |^2 ]= _i=1^N | _L_if_L_i^*z |^2≤ ( _1≤ i≤ N _L_i|z| )\|z\|_1. The last expression tends to zero as n→∞n→∞ by absolute continuity of the integral. If z∗∈(L10)∗z^*∈(L_1^0)^*, choose an L∞L_∞ representative, still denoted by z∗z^*; then ε[|⟨z∗,u(ε)⟩|2]=N−2∑i=1N|⟨z∗,fLi⟩|2≤N−1‖z∗‖∞2,E_ [ | z^*,u( ) |^2 ]=N^-2 _i=1^N| z^*,f_L_i |^2≤ N^-1\|z^*\|_∞^2, which tends to zero. Now consider E=Lp0E=L_p^0. In this case u(ε)=N−1/p∑i=1Nεiui,u∗(ε)=N−1/p′∑i=1Nεiui∗.u( )=N^-1/p _i=1^N _iu_i, u^*( )=N^-1/p _i=1^N _iu_i^*. If z∈Lp0z∈ L_p^0, then ε[|⟨u∗(ε),z⟩|2]=N−2/p′∑i=1N|⟨ui∗,z⟩|2,|⟨ui∗,z⟩|≤‖Liz‖p.E_ [ | u^*( ),z |^2 ]=N^-2/p _i=1^N| u_i^*,z |^2, | u_i^*,z |≤\|1_L_iz\|_p. For 1<p≤21<p≤ 2, this tends to zero because ∑i=1N‖Liz‖p2≤(max1≤i≤N‖Liz‖p)2−p‖z‖p. _i=1^N\|1_L_iz\|_p^2≤ ( _1≤ i≤ N\|1_L_iz\|_p )^2-p\|z\|_p^p. For p>2p>2, Hölder’s inequality gives ∑i=1N‖Liz‖p2≤N1−2/p‖z‖p2, _i=1^N\|1_L_iz\|_p^2≤ N^1-2/p\|z\|_p^2, and hence ε[|⟨u∗(ε),z⟩|2]≤N−1‖z‖p2.E_ [ | u^*( ),z |^2 ]≤ N^-1\|z\|_p^2. Thus, the vector estimate tends to zero in all cases. The estimate for fixed functionals is the same argument with p replaced by p′p , after representing the functional by an element of Lp′L_p modulo constants. Applying these estimates to the fixed vectors SbzbS_bz_b and the fixed functionals Ra∗za∗R_a^*z_a^*, and then choosing n0n_0 large enough so that all estimates hold for every n≥n0n≥ n_0, proves the lemma. ∎ The next estimate is the two-sided random analogue of the preceding one-sided estimates. When two new Haar blocks are built on disjoint finite equal-dyadic sets, and their signs are chosen independently, the mixed coefficient has second moment of order the reciprocal of the number of descendants used in the split. Lemma 9.27 (Bilinear estimate for two new blocks). Let E=Lp0E=L_p^0 or E=L10E=L_1^0, and let S∈ℬ(E)S (E). Let AaA_a and AbA_b be disjoint finite equal-dyadic sets. Fix depths na,nb≥0n_a,n_b≥ 0, write na(Aa)=L1a,…,LNaa,nb(Ab)=L1b,…,LNbb,D_n_a(A_a)=\L_1^a,…,L_N_a^a\, _n_b(A_b)=\L_1^b,…,L_N_b^b\, and let N0=minNa,NbN_0= \N_a,N_b\. Let ua(εa),ua∗(εa)u_a( ^a),u_a^*( ^a) and ub(εb),ub∗(εb)u_b( ^b),u_b^*( ^b) be the random normalized Haar blocks and their biorthogonal functionals associated with these two splittings, where the sign selections εa∈−1,1Na ^a∈\-1,1\^N_a and εb∈−1,1Nb ^b∈\-1,1\^N_b are independent. Then εa,εb[|⟨ua∗(εa),Sub(εb)⟩|2]≲p‖S‖2N0.E_ ^a, ^b [ | u_a^*( ^a),Su_b( ^b) |^2 ] _p \|S\|^2N_0. In the case E=L10E=L_1^0 the subscript p is omitted. Proof. Let ua,i,ua,i∗u_a,i,u_a,i^* be the normalized Haar vector and biorthogonal functional associated with LiaL_i^a, and let ub,j,ub,j∗u_b,j,u_b,j^* be the corresponding objects associated with LjbL_j^b. Write sij=⟨ua,i∗,Sub,j⟩(1≤i≤Na, 1≤j≤Nb).s_ij= u_a,i^*,Su_b,j (1≤ i≤ N_a,\ 1≤ j≤ N_b). The products εiaεjb _i^a _j^b are orthogonal in L2L_2 of the product sign space, so the mixed terms vanish after averaging. First consider E=L10E=L_1^0. Then ua∗(εa)=∑i=1Naεiaua,i∗,ub(εb)=Nb−1∑j=1Nbεjbub,j.u_a^*( ^a)= _i=1^N_a _i^au_a,i^*, u_b( ^b)=N_b^-1 _j=1^N_b _j^bu_b,j. Hence εa,εb[|⟨ua∗(εa),Sub(εb)⟩|2]=Nb−2∑i=1Na∑j=1Nb|sij|2.E_ ^a, ^b [ | u_a^*( ^a),Su_b( ^b) |^2 ]=N_b^-2 _i=1^N_a _j=1^N_b|s_ij|^2. For each fixed j, the column (sij)i=1Na(s_ij)_i=1^N_a is the coordinate vector of the projection of Sub,jSu_b,j onto the span of ua,1,…,ua,Nau_a,1,…,u_a,N_a. By Section˜9, its ℓ1 _1 norm is at most ‖S‖\|S\|, hence its ℓ2 _2 norm is at most ‖S‖\|S\|. Therefore ∑i=1Na∑j=1Nb|sij|2≤Nb‖S‖2, _i=1^N_a _j=1^N_b|s_ij|^2≤ N_b\|S\|^2, and so εa,εb[|⟨ua∗(εa),Sub(εb)⟩|2]≤Nb−1‖S‖2≤N0−1‖S‖2.E_ ^a, ^b [ | u_a^*( ^a),Su_b( ^b) |^2 ]≤ N_b^-1\|S\|^2≤ N_0^-1\|S\|^2. Now consider E=Lp0E=L_p^0. Then ua∗(εa)=Na−1/p′∑i=1Naεiaua,i∗,ub(εb)=Nb−1/p∑j=1Nbεjbub,j.u_a^*( ^a)=N_a^-1/p _i=1^N_a _i^au_a,i^*, u_b( ^b)=N_b^-1/p _j=1^N_b _j^bu_b,j. Thus εa,εb[|⟨ua∗(εa),Sub(εb)⟩|2]=Na−2/p′Nb−2/p∑i=1Na∑j=1Nb|sij|2.E_ ^a, ^b [ | u_a^*( ^a),Su_b( ^b) |^2 ]=N_a^-2/p N_b^-2/p _i=1^N_a _j=1^N_b|s_ij|^2. If 1<p≤21<p≤ 2, then for each fixed j, the column (sij)i=1Na(s_ij)_i=1^N_a has ℓp _p norm at most ‖S‖\|S\|, hence ℓ2 _2 norm at most ‖S‖\|S\|. Therefore ∑i=1Na∑j=1Nb|sij|2≤Nb‖S‖2, _i=1^N_a _j=1^N_b|s_ij|^2≤ N_b\|S\|^2, and consequently εa,εb[|⟨ua∗(εa),Sub(εb)⟩|2]≤Na−2/p′Nb1−2/p‖S‖2≤N0−1‖S‖2.E_ ^a, ^b [ | u_a^*( ^a),Su_b( ^b) |^2 ]≤ N_a^-2/p N_b^1-2/p\|S\|^2≤ N_0^-1\|S\|^2. If p≥2p≥ 2, we use rows instead. For each fixed i, the row (sij)j=1Nb(s_ij)_j=1^N_b has ℓp′ _p norm at most ‖S‖\|S\|, hence ℓ2 _2 norm at most ‖S‖\|S\|. Thus ∑i=1Na∑j=1Nb|sij|2≤Na‖S‖2, _i=1^N_a _j=1^N_b|s_ij|^2≤ N_a\|S\|^2, and therefore εa,εb[|⟨ua∗(εa),Sub(εb)⟩|2]≤Na1−2/p′Nb−2/p‖S‖2≤N0−1‖S‖2.E_ ^a, ^b [ | u_a^*( ^a),Su_b( ^b) |^2 ]≤ N_a^1-2/p N_b^-2/p\|S\|^2≤ N_0^-1\|S\|^2. This proves the lemma. ∎ 8. Finite one-step constructions The preceding estimates are probabilistic results. We now use them to prove two finite splitting results that are used in the recursive construction. Each result starts from finitely many current terminal sets and finitely many constraints fixed before the random signs are chosen. 8.1. Outer one-step construction The outer coordinate is constructed by repeatedly splitting finite equal-dyadic sets. At a single outer step, one has finitely many current outer sets, one-parameter operators on Lp0L_p^0, and admissible side constraints. The step has two tasks. First, the newly born outer Haar blocks must satisfy the prescribed side constraints. Second, each child must inherit an infinite tail of depths on which the relevant averaged diagonal estimates remain valid. The next lemma is the precise one-step form of this operation. Lemma 9.28 (Additive preservation of outer tails). Let A1,…,AdA_1,…,A_d be pairwise disjoint finite equal-dyadic sets in the outer coordinate. Assume that the following is given. (a) For each 1≤a≤d1≤ a≤ d, a finite index set FaF_a and a family (Qa,s)s∈Fa(Q_a,s)_s∈ F_a in ℬ(Lp0)B(L_p^0). (b) For each 1≤a≤d1≤ a≤ d, an infinite set Sa⊂ℕS_a . (c) Positive numbers θa,s _a,s for 1≤a≤d1≤ a≤ d and s∈Fas∈ F_a such that |αmAa(Qa,s)|<θa,s(1≤a≤d,m∈Sa,s∈Fa).| _m^A_a(Q_a,s)|< _a,s (1≤ a≤ d,\ m∈ S_a,\ s∈ F_a). (51) (d) Positive numbers ηa,s _a,s and δa,s _a,s for 1≤a≤d1≤ a≤ d and s∈Fas∈ F_a. (e) A finite collection C of admissible side constraints, in the sense of Section˜9, involving the new outer blocks to be constructed on A1,…,AdA_1,…,A_d. Then there exist depths na∈San_a∈ S_a and sign selections εa∈−1,1na(Aa)(1≤a≤d) ^a∈\-1,1\^D_n_a(A_a) (1≤ a≤ d) with the following properties. Put Aa,+=Aa,+(εa),Aa,−=Aa,−(εa)A_a,+=A_a,+( ^a), A_a,-=A_a,-( ^a) and let HAa(εa),HAa∗(εa)H_A_a( ^a),H_A_a^*( ^a) be the associated new outer block and its biorthogonal functional. (i) The chosen signs make every admissible side constraint in C true for the blocks HAa(εa),HAa∗(εa)H_A_a( ^a),H_A_a^*( ^a). (i) For every 1≤a≤d1≤ a≤ d and every s∈Fas∈ F_a, |⟨HAa∗(εa),Qa,sHAa(εa)⟩−αnaAa(Qa,s)|<ηa,s. | H_A_a^*( ^a),Q_a,sH_A_a( ^a) - _n_a^A_a(Q_a,s) |< _a,s. (52) (i) For every 1≤a≤d1≤ a≤ d and every ϑ∈+,− ∈\+,-\, there is an infinite set Sa,ϑ⊂ℕS_a, such that, for every r∈Sa,ϑr∈ S_a, and every s∈Fas∈ F_a, na+1+r∈Saand|αrAa,ϑ(Qa,s)|<θa,s+δa,s.n_a+1+r∈ S_a | _r^A_a, (Q_a,s)|< _a,s+ _a,s. (53) Proof. Throughout the proof, ℙP denotes probability with respect to the random sign choices. For every finite equal-dyadic set A, write Nn(A)=#n(A)N_n(A)=\#D_n(A) for the number of depth n descendants of A. Choose numbers 0<γ<1/40<γ<1/4 and 0<κ<1/(8d)0<κ<1/(8d). The proof has four steps: first we choose the depths so that the diagonal and side-constraint failures are rare; then we estimate the probability that a child tail fails at a fixed future depth; then we use Markov’s inequality to force many future depths to be good; finally we use finiteness of the sign space to pass from arbitrarily many good depths to infinitely many good depths. Step 1: Choosing the splitting depths: We first choose depths na∈San_a∈ S_a, 1≤a≤d1≤ a≤ d. Since each SaS_a is infinite and Nn(Aa)→∞N_n(A_a)→∞ as n→∞n→∞ along SaS_a, we may choose the depths so large that the following three requirements hold. Let ℰdiagE_diag be the event that conclusion (i) holds for all 1≤a≤d1≤ a≤ d and s∈Fas∈ F_a. For 1≤a≤d1≤ a≤ d and s∈Fas∈ F_a, let ℬa,sB_a,s be the event |⟨HAa∗(εa),Qa,sHAa(εa)⟩−αnaAa(Qa,s)|≥ηa,s. | H_A_a^*( ^a),Q_a,sH_A_a( ^a) - _n_a^A_a(Q_a,s) |≥ _a,s. By Section˜9, [|⟨HAa∗(εa),Qa,sHAa(εa)⟩−αnaAa(Qa,s)|2]≲p‖Qa,s‖2Nna(Aa).E [ | H_A_a^*( ^a),Q_a,sH_A_a( ^a) - _n_a^A_a(Q_a,s) |^2 ] _p \|Q_a,s\|^2N_n_a(A_a). Hence Chebyshev’s inequality gives ℙ(ℬa,s)≲p‖Qa,s‖2Nna(Aa)ηa,s2.P(B_a,s) _p \|Q_a,s\|^2N_n_a(A_a) _a,s^2. Since ℰdiagc⊆⋃a=1d⋃s∈Faℬa,sE_diag^c _a=1^d _s∈ F_aB_a,s, the finite union bound gives ℙ(ℰdiagc)≲p∑a=1d∑s∈Fa‖Qa,s‖2Nna(Aa)ηa,s2.P(E_diag^c) _p _a=1^d _s∈ F_a \|Q_a,s\|^2N_n_a(A_a) _a,s^2. Increasing the depths na∈San_a∈ S_a if necessary, we may arrange that ℙ(ℰdiagc)<γ/2.P(E_diag^c)<γ/2. Let ℰsideE_side be the event that conclusion (i) holds for the finite admissible collection C. For each constraint C∈C∈ C, let XCX_C denote the random scalar appearing in that constraint, and let τC>0 _C>0 denote its tolerance. By admissibility, each XCX_C is of one of the following forms: ⟨HAa∗(εa),SCzC⟩,⟨RC∗zC∗,HAa(εa)⟩,⟨HAa∗(εa),SCHAb(εb)⟩(a≠b). H_A_a^*( ^a),S_Cz_C , R_C^*z_C^*,H_A_a( ^a) , H_A_a^*( ^a),S_CH_A_b( ^b) (a≠ b). Choose numbers ρC>0 _C>0, C∈C∈ C, such that ∑C∈ρC<γ/2. _C∈ C _C<γ/2. By Sections˜9 and 9, after increasing the depths na∈San_a∈ S_a if necessary, we may arrange that [|XC|2]<ρCτC2(C∈).E[|X_C|^2]< _C _C^2 (C∈ C). Let ℬC=|XC|≥τCB_C=\|X_C|≥ _C\ be the event that the constraint C fails. By Markov’s inequality, ℙ(ℬC)≤[|XC|2]τC2<ρC.P(B_C)≤ E[|X_C|^2] _C^2< _C. Since ℰsidec⊆⋃C∈ℬCE_side^c _C∈ CB_C, the finite union bound gives ℙ(ℰsidec)≤∑C∈ℙ(ℬC)<∑C∈ρC<γ/2.P(E_side^c)≤ _C∈ CP(B_C)< _C∈ C _C<γ/2. Finally, we choose the same depths so large that, for every 1≤a≤d1≤ a≤ d, ∑s∈Fa2‖Qa,s‖2Nna(Aa)δa,s2<κ. _s∈ F_a 2\|Q_a,s\|^2N_n_a(A_a) _a,s^2<κ. (54) Set ℰ=ℰdiag∩ℰsideE=E_diag _side. Then ℙ(ℰ)>1−γ.P(E)>1-γ. (55) Step 2: Estimating child tail failures: Fix 1≤a≤d1≤ a≤ d. Since SaS_a is infinite, the set of integers r≥0r≥ 0 such that na+1+r∈San_a+1+r∈ S_a is infinite. Enumerate it increasingly as ra,1<ra,2<…,na+1+ra,ℓ∈Sa(ℓ≥1).r_a,1<r_a,2<…, n_a+1+r_a, ∈ S_a ( ≥ 1). For each ℓ≥1 ≥ 1, let ℋa,ℓH_a, be the event that, for both children and every s∈Fas∈ F_a, |αra,ℓAa,+(Qa,s)|<θa,s+δa,s,|αra,ℓAa,−(Qa,s)|<θa,s+δa,s.| _r_a, ^A_a,+(Q_a,s)|< _a,s+ _a,s, | _r_a, ^A_a,-(Q_a,s)|< _a,s+ _a,s. Fix ℓ≥1 ≥ 1 and put r=ra,ℓr=r_a, . For s∈Fas∈ F_a, set Xs+=αrAa,+(Qa,s),Xs−=αrAa,−(Qa,s).X_s^+= _r^A_a,+(Q_a,s), X_s^-= _r^A_a,-(Q_a,s). By (45), both random variables have the same expectation εa[Xs+]=εa[Xs−]=αna+1+rAa(Qa,s).E_ ^a[X_s^+]=E_ ^a[X_s^-]= _n_a+1+r^A_a(Q_a,s). Denote this common scalar by μs=αna+1+rAa(Qa,s). _s= _n_a+1+r^A_a(Q_a,s). Since na+1+r∈San_a+1+r∈ S_a, the hypothesis (51) gives |μs|<θa,s.| _s|< _a,s. Hence, if either child fails the desired estimate for this s, then the corresponding random variable must deviate from its expectation by more than δa,s _a,s. Indeed, |Xsϑ|≥θa,s+δa,s⟹|Xsϑ−μs|≥|Xsϑ|−|μs|>δa,s(ϑ∈+,−).|X_s |≥ _a,s+ _a,s |X_s - _s|≥|X_s |-| _s|> _a,s ( ∈\+,-\). Therefore ℋa,ℓc⊆⋃s∈Fa⋃ϑ∈+,−|Xsϑ−μs|>δa,s.H_a, ^c _s∈ F_a _ ∈\+,-\\|X_s - _s|> _a,s\. By (46) and Chebyshev’s inequality, for each s∈Fas∈ F_a and ϑ∈+,− ∈\+,-\, ℙ(|Xsϑ−μs|>δa,s)≤‖Qa,s‖2Nna(Aa)δa,s2.P(|X_s - _s|> _a,s)≤ \|Q_a,s\|^2N_n_a(A_a) _a,s^2. Taking the union bound over both children and over s∈Fas∈ F_a, and using (54), we obtain ℙ(ℋa,ℓc)≤∑s∈Fa2‖Qa,s‖2Nna(Aa)δa,s2<κ.P(H_a, ^c)≤ _s∈ F_a 2\|Q_a,s\|^2N_n_a(A_a) _a,s^2<κ. Step 3: Forcing many good future depths: Fix M≥1M≥ 1. For each 1≤a≤d1≤ a≤ d, let a,MG_a,M be the event that at least M of the events ℋa,1,…,ℋa,2MH_a,1,…,H_a,2M occur. Let Za,M=∑ℓ=12Mℋa,ℓcZ_a,M= _ =1^2M1_H_a, ^c be the number of failures among these 2M2M events. By Step 2, [Za,M]=∑ℓ=12Mℙ(ℋa,ℓc)<2Mκ.E[Z_a,M]= _ =1^2MP(H_a, ^c)<2Mκ. If a,MG_a,M fails, then fewer than M of the 2M2M events occur, and hence at least M+1M+1 of them fail. Therefore a,Mc=Za,M≥M+1⊆Za,M≥M.G_a,M^c=\Z_a,M≥ M+1\ \Z_a,M≥ M\. By Markov’s inequality, ℙ(a,Mc)≤ℙ(Za,M≥M)≤[Za,M]M<2κ.P(G_a,M^c) (Z_a,M≥ M)≤ E[Z_a,M]M<2κ. Using (55) and a union bound over 1≤a≤d1≤ a≤ d, we get ℙ(ℰ∩⋂a=1da,M)≥1−ℙ(ℰc)−∑a=1dℙ(a,Mc)>1−γ−2dκ>0.P (E∩ _a=1^dG_a,M )≥ 1-P(E^c)- _a=1^dP(G_a,M^c)>1-γ-2dκ>0. Thus, for every M≥1M≥ 1, there is a choice of signs for which ℰE and all the events a,MG_a,M hold. Step 4: Extracting one sign choice with infinitely many good depths: The depths nan_a are now fixed, so the product sign space ∏a=1d−1,1na(Aa) _a=1^d\-1,1\^D_n_a(A_a) is finite. For each M≥1M≥ 1, choose one sign choice for which ℰ∩⋂a=1da,ME∩ _a=1^dG_a,M holds. Since the product sign space is finite, one of these sign choices occurs for infinitely many values of M. Fix such a sign choice, and let ℳ⊂ℕM be an infinite set such that this fixed choice works for every M∈ℳM . Since ℰE holds for this choice, conclusions (i) and (i) hold. Moreover, for every 1≤a≤d1≤ a≤ d and every M∈ℳM , the event a,MG_a,M holds. Since ℳM is infinite, it is unbounded. Hence, for each fixed a, infinitely many of the events ℋa,ℓH_a, hold. Indeed, if only finitely many of them held, say RaR_a of them, then choosing M∈ℳM with M>RaM>R_a would contradict the fact that a,MG_a,M holds. For each 1≤a≤d1≤ a≤ d, define Sa,+=Sa,−=ra,ℓ:ℋa,ℓ holds.S_a,+=S_a,-=\r_a, :H_a, holds\. These sets are infinite by the preceding paragraph. The same infinite set works for both children because ℋa,ℓH_a, requires the estimates for Aa,+A_a,+ and Aa,−A_a,- simultaneously. If r∈Sa,ϑr∈ S_a, , then r=ra,ℓr=r_a, for some ℓ with ℋa,ℓH_a, true, so na+1+r∈San_a+1+r∈ S_a and, for every s∈Fas∈ F_a, |αrAa,ϑ(Qa,s)|<θa,s+δa,s.| _r^A_a, (Q_a,s)|< _a,s+ _a,s. This proves conclusion (i), and hence the lemma. ∎ 8.2. Inner one-step construction The inner coordinate is controlled differently. The relevant one-parameter multipliers live on L10L_1^0, and the local trace from Section˜9 replaces the outer tail averages. At one inner step, the new block is chosen so that its diagonal coefficient is close to the trace over the parent set, and the two children are chosen so that their traces remain close to the same value. Lemma 9.29 (One-step L1L_1 split with trace control). Suppose that the following objects are fixed. (a) Operators R1,…,Rm∈ℬ(L10)R_1,…,R_m (L_1^0). (b) A finite equal-dyadic set B in the inner coordinate. (c) A number δ>0δ>0. (d) A finite admissible collection C of one-sided side constraints, to be imposed on the new inner Haar block produced by the split of B. For a split at depth n, choose independent signs ε=(εJ)J∈n(B) =( _J)_J _n(B) and let B±(ε)B_±( ) and KB(ε),KB∗(ε)K_B( ),K_B^*( ) be the associated random split and inner Haar block. The constraints in C are interpreted with this block KB(ε)K_B( ) and this functional KB∗(ε)K_B^*( ). Then, for all sufficiently large n, there is a choice of signs ε such that the following hold. (i) For every 1≤r≤m1≤ r≤ m, |⟨KB∗(ε),RrKB(ε)⟩−λB(Rr)|<δ. | K_B^*( ),R_rK_B( ) - _B(R_r) |<δ. (56) (i) For every 1≤r≤m1≤ r≤ m, |λB+(ε)(Rr)−λB(Rr)|<δ,|λB−(ε)(Rr)−λB(Rr)|<δ.| _B_+( )(R_r)- _B(R_r)|<δ, | _B_-( )(R_r)- _B(R_r)|<δ. (57) (i) All one-sided side constraints in C hold. Proof. We show that, for all sufficiently large depths n, the desired set of sign choices has positive probability. Throughout the proof, probability and expectation are taken with respect to the random signs defining the split of B. By Section˜9, for each 1≤r≤m1≤ r≤ m we have βnB(Rr)⟶λB(Rr)(n→∞). _n^B(R_r) _B(R_r) (n→∞). Since there are only finitely many operators in (a), there is n0n_0 such that, for every n≥n0n≥ n_0, |βnB(Rr)−λB(Rr)|<δ/2(1≤r≤m).| _n^B(R_r)- _B(R_r)|<δ/2 (1≤ r≤ m). (58) Fix n≥n0n≥ n_0 and put N=#n(B)N=\#D_n(B). By the expansion of the random inner block, KB(ε)=N−1∑J∈n(B)εJfJ,KB∗(ε)=∑J∈n(B)εJfJ∗.K_B( )=N^-1 _J _n(B) _Jf_J, K_B^*( )= _J _n(B) _Jf_J^*. Therefore, for each 1≤r≤m1≤ r≤ m, ε[⟨KB∗(ε),RrKB(ε)⟩]=βnB(Rr).E_ [ K_B^*( ),R_rK_B( ) ]= _n^B(R_r). Moreover, applying Section˜9 in the case E=L10E=L_1^0 gives ε[|⟨KB∗(ε),RrKB(ε)⟩−βnB(Rr)|2]≲‖Rr‖2N.E_ [ | K_B^*( ),R_rK_B( ) - _n^B(R_r) |^2 ] \|R_r\|^2N. Thus, by Chebyshev’s inequality and a finite union bound over 1≤r≤m1≤ r≤ m, ℙ(∃ 1≤r≤m:|⟨KB∗(ε),RrKB(ε)⟩−βnB(Rr)|≥δ/2)⟶0P (∃\,1≤ r≤ m: | K_B^*( ),R_rK_B( ) - _n^B(R_r) |≥δ/2 ) 0 as n→∞n→∞. Increasing n0n_0 if necessary, this failure probability is smaller than 1/41/4 for all n≥n0n≥ n_0. On the complementary event, (58) implies (i). Next we control the traces of the two children. For each 1≤r≤m1≤ r≤ m, let φr∈L∞[0,1] _r∈ L_∞[0,1] be supplied by Section˜9, so that λC(Rr)=1|C|∫Cφr _C(R_r)= 1|C| _C _r for every finite equal-dyadic set C, and ‖φr‖∞≲‖Rr‖.\| _r\|_∞ \|R_r\|. For the plus child, λB+(ε)(Rr)=2|B|∑J∈n(B)∫JεJφr. _B_+( )(R_r)= 2|B| _J _n(B) _J _J _r. The summands are independent as J varies, and ε[λB+(ε)(Rr)]=λB(Rr).E_ [ _B_+( )(R_r) ]= _B(R_r). Since every J∈n(B)J _n(B) has measure |B|/N|B|/N, we also have Varε(λB+(ε)(Rr))≤∑J∈n(B)(‖φr‖∞N)2≲‖Rr‖2N.Var_ ( _B_+( )(R_r) )≤ _J _n(B) ( \| _r\|_∞N )^2 \|R_r\|^2N. The same argument gives ε[λB−(ε)(Rr)]=λB(Rr),Varε(λB−(ε)(Rr))≲‖Rr‖2N.E_ [ _B_-( )(R_r) ]= _B(R_r), _ ( _B_-( )(R_r) ) \|R_r\|^2N. Chebyshev’s inequality and a finite union bound over 1≤r≤m1≤ r≤ m and the two signs ± show that the failure probability of (i) tends to zero as n→∞n→∞. Increasing n0n_0 again, this failure probability is strictly smaller than 1/41/4 for all n≥n0n≥ n_0. It remains to impose the side constraints from (d). Since C is finite and admissible, every constraint is one of the one-sided forms handled by Section˜9. For each constraint, Markov’s inequality applied to the corresponding second-moment estimate shows that its failure probability can be made arbitrarily small by taking the splitting depth sufficiently large. Hence, after increasing n0n_0 once more and using a finite union bound over all constraints in C, the probability that some side constraint fails is strictly smaller than 1/41/4 for all n≥n0n≥ n_0. For every n≥n0n≥ n_0, the probability that any of the three required conclusions fails is therefore strictly smaller than 3/43/4. Hence the probability that (i), (i), and (i) all hold is positive. Thus there is a choice of signs satisfying all three conclusions. ∎ The previous lemma handles one inner set and one new inner block. We now record the finite simultaneous version needed at an inner splitting stage. The only extra point is that bilinear constraints between two distinct new blocks are allowed; these are handled by the bilinear estimate, using independence of the sign choices on distinct sets. Lemma 9.30 (Simultaneous finite L1L_1 splitting). Let B1,…,BdB_1,…,B_d be pairwise disjoint finite equal-dyadic sets in the inner coordinate. Suppose that the following objects are fixed. (a) For each 1≤a≤d1≤ a≤ d, a finite family ℛa⊂ℬ(L10)R_a (L_1^0). (b) For each 1≤a≤d1≤ a≤ d and each R∈ℛaR _a, a positive tolerance δa,R>0 _a,R>0. (c) A finite admissible collection C of side constraints involving the new inner blocks to be constructed on B1,…,BdB_1,…,B_d. Then there are depths n1,…,ndn_1,…,n_d and independent sign choices εa=(εJa)J∈na(Ba)(1≤a≤d) ^a=( _J^a)_J _n_a(B_a) (1≤ a≤ d) such that, writing Ba,±=Ba,±(εa)B_a,±=B_a,±( ^a) and denoting the associated inner Haar block by KBa(εa)K_B_a( ^a), with functional KBa∗(εa)K_B_a^*( ^a), the following hold. (i) For every 1≤a≤d1≤ a≤ d and every R∈ℛaR _a, |⟨KBa∗(εa),RKBa(εa)⟩−λBa(R)|<δa,R. | K_B_a^*( ^a),RK_B_a( ^a) - _B_a(R) |< _a,R. (i) For every 1≤a≤d1≤ a≤ d, every R∈ℛaR _a, and every ϑ∈+,− ∈\+,-\, |λBa,ϑ(R)−λBa(R)|<δa,R.| _B_a, (R)- _B_a(R)|< _a,R. (i) All side constraints in C hold. Proof. We choose the depths probabilistically. Throughout the proof, probability and expectation are taken with respect to the independent sign choices εa=(εJa)J∈na(Ba)(1≤a≤d). ^a=( _J^a)_J _n_a(B_a) (1≤ a≤ d). Since only finitely many operators and side constraints occur, it is enough to choose the depths so that each of the relevant failure probabilities is small. First consider the trace estimates. Fix 1≤a≤d1≤ a≤ d and R∈ℛaR _a. By the proof of Section˜9, applied to the set BaB_a and the operator R, the failure probability of |⟨KBa∗(εa),RKBa(εa)⟩−λBa(R)|<δa,R | K_B_a^*( ^a),RK_B_a( ^a) - _B_a(R) |< _a,R and |λBa,+(εa)(R)−λBa(R)|<δa,R,|λBa,−(εa)(R)−λBa(R)|<δa,R| _B_a,+( ^a)(R)- _B_a(R)|< _a,R, | _B_a,-( ^a)(R)- _B_a(R)|< _a,R tends to zero as na→∞n_a→∞. Hence, by increasing the finitely many depths nan_a, we may assume that the probability that any trace estimate in (i) or (i) fails is less than 1/31/3. It remains to impose the side constraints in C. Since C is admissible, each constraint is either one-sided or bilinear between two distinct new blocks. For every one-sided constraint, Section˜9 and Markov’s inequality show that its failure probability can be made arbitrarily small by increasing the depth of the corresponding split. For every bilinear constraint between the blocks over BaB_a and BbB_b, with a≠ba≠ b, Section˜9 and Markov’s inequality show that its failure probability can be made arbitrarily small by increasing both depths nan_a and nbn_b. Because there are only finitely many constraints, we may increase the depths again so that the probability that any constraint in C fails is less than 1/31/3. For these choices of depths, the probability that one of the trace estimates or one of the side constraints fails is less than 2/32/3. Therefore the probability that all conclusions (i), (i), and (i) hold is positive. Thus, there is a choice of signs satisfying all the required estimates simultaneously. ∎ 9. The multiplier reduction construction We now prove the reduction from an arbitrary operator on X00X_00 to a product Haar multiplier. Fix T∈ℬ(X00).T (X_00). We construct an outer faithful Haar system (HI,HI∗)I∈(H_I,H_I^*)_I and an inner faithful Haar system (KJ,KJ∗)J∈(K_J,K_J^*)_J so that, after passing to the associated faithful product-Haar copy, the matrix of T is small away from the product diagonal. The product matrix coefficients are aJ,J′I,I′=⟨HI∗⊗KJ∗,T(HI′⊗KJ′)⟩.a_J,J ^I,I = H_I^* K_J^*,T(H_I K_J ) . (59) The diagonal coefficients are aJ,JI,Ia_J,J^I,I, and they define the multiplier which remains after the construction. The off-diagonal coefficients have three types: I=I′,J≠J′,I≠I′,J=J′,I≠I′,J≠J′.I=I ,\ J≠ J , I≠ I ,\ J=J , I≠ I ,\ J≠ J . The first type is controlled by outer tail preservation, the second by inner trace variation, and the third by summable coefficient estimates. We shall use the following terminology throughout the construction. We say that an outer block HIH_I is born at the outer half stage where the set ΓI _I is split into its two children. We say that an inner block KJK_J is born at the inner half stage where the set ΔJ _J is split into its two children. A product coefficient is available once all Haar blocks appearing in it have been born. A frontier is the finite family of indices whose sets have already been chosen but have not yet been split. Thus, at a full state n0 S_n^0, both frontiers are indexed by nD_n, while at a half state n1/2 S_n^1/2 the outer frontier is indexed by n+1D_n+1 and the inner frontier is indexed by nD_n. An element of one of these finite generations is called a frontier index. A reservoir tail for an outer frontier index A is an infinite set SA⊂ℕS_A along which the outer tail estimates over ΓA _A are imposed. For J≠J′J≠ J , the family (aJ,J′I,I)I∈(a_J,J ^I,I)_I is called the same outer strip indexed by (J,J′)(J,J ). This strip is registered at the inner half stage where the later of the two blocks KJK_J and KJ′K_J is born. From that half stage onward, its outer coefficient estimates and reservoir-tail estimates are included in every later legal state. For I≠I′I≠ I , the family (aJ,JI,I′)J∈(a_J,J^I,I )_J is called the same inner strip indexed by (I,I′)(I,I ). This strip is registered at the outer half stage where the later of the two blocks HIH_I and HI′H_I is born. From that half stage onward, its inner trace and variation estimates are included in every later legal state. The coefficients aJ,J′I,I′a_J,J ^I,I with I≠I′I≠ I and J≠J′J≠ J are called mixed coefficients. A mixed coefficient is registered at the half stage where the last of the four blocks HI,HI′,KJ,KJ′H_I, H_I , K_J, K_J is born. At that half stage its assigned summable estimate is imposed. Thorough the proof, αmA(Q) _m^A(Q) denotes the potential outer trace average from Section˜9, and λB(R) _B(R) denotes the local L1L_1 trace from Section˜9. The proof below invokes construction claims exactly at the points where choices have to be made: registrations are locally finite, the outer half stage can be performed, and the inner half stage can be performed. The final norm estimate also uses a few elementary auxiliary lemmas. The formal statements and proofs are given in Section˜9. Theorem 9.31 (Arbitrary operators reduce to product Haar multipliers). Let 1<p<∞1<p<∞. For every T∈ℬ(X00)T (X_00) and every ε>0 >0, there are faithful outer and inner Haar systems, exterior maps A,B∈ℬ(X00)A,B (X_00), and a bounded product Haar multiplier M on X00X_00 such that AB=IdX00,‖ATB−M‖ℬ(X00)<ε.AB=Id_X_00, \|ATB-M\|_B(X_00)< . Moreover, ‖M‖ℬ(X00)≤‖T‖ℬ(X00)+ε.\|M\|_B(X_00)≤\|T\|_B(X_00)+ . If the faithful systems are denoted by (HI,HI∗)I∈(H_I,H_I^*)_I and (KJ,KJ∗)J∈(K_J,K_J^*)_J , then MuI,J=⟨HI∗⊗KJ∗,T(HI⊗KJ)⟩uI,J(I,J∈).Mu_I,J= H_I^* K_J^*,T(H_I K_J) u_I,J (I,J ). Proof. Fix T∈ℬ(X00)T (X_00) and ε>0 >0. Let UpU_p be the constant from Section˜9, and let CSUC_ SU be such that the estimate ‖Ma‖ℬ(L10)≤CSUBVbr(a)\|M_a\|_B(L_1^0)≤ C_ SUBV_br(a) holds in Section˜9. Put C0=8,Cout=Up,Cin=CSU.C_0=8, C_ out=U_p, C_ in=C_ SU. Choose positive summable families (σJ,J′)J≠J′,(τI,I′)I≠I′,(μI,I′,J,J′)I≠I′,J≠J′( _J,J )_J≠ J , ( _I,I )_I≠ I , ( _I,I ,J,J )_I≠ I ,\,J≠ J such that Cout∑J≠J′σJ,J′<ε3,Cin∑I≠I′τI,I′<ε3,∑I≠I′,J≠J′μI,I′,J,J′<ε3.C_ out _J≠ J _J,J < 3, C_ in _I≠ I _I,I < 3, _I≠ I ,\,J≠ J _I,I ,J,J < 3. (60) The number σJ,J′ _J,J is reserved for the same outer strip indexed by (J,J′)(J,J ), the number τI,I′ _I,I is reserved for the same inner strip indexed by (I,I′)(I,I ), and the number μI,I′,J,J′ _I,I ,J,J is reserved for the mixed coefficient indexed by (I,I′,J,J′)(I,I ,J,J ). Put ≤n=⋃m=0nm,<n=⋃m=0n−1m.D_≤ n= _m=0^nD_m, _<n= _m=0^n-1D_m. The recursion alternates outer and inner half stages: 00→outer01/2→inner10→outer11/2→inner20→outer…. S_0^0 outer S_0^1/2 inner S_1^0 outer S_1^1/2 inner S_2^0 outer…. At the beginning of stage n, denoted by n0 S_n^0, the outer sets ΓI _I and inner sets ΔJ _J have been chosen for I,J∈≤nI,J _≤ n, while the outer and inner Haar blocks have been born for I,J∈<nI,J _<n. Thus both current frontiers are indexed by nD_n. The construction is by induction on n. In the induction hypothesis we first list the objects in place, and then the conditions which the construction has achieved for the corresponding coefficients and strip operators. A full state n0 S_n^0 is called legal if the objects in (Q1)–(Q5) have been specified and the estimates in (E1)–(E5) hold. A half state n1/2 S_n^1/2 is legal when the analogous assertion holds with outer frontier n+1D_n+1 and inner frontier nD_n. In the base case these estimates are empty because no strip or mixed coefficient has yet been registered. Base case. Choose Γ[0,1)=[0,1),Δ[0,1)=[0,1),S[0,1)=ℕ. _[0,1)=[0,1), _[0,1)=[0,1), S_[0,1)=N. No Haar block has yet been born, no same outer or same inner strip has been registered, and no mixed coefficient has been registered. The estimates in the induction hypothesis below are therefore vacuous, except for the reservoir tail S[0,1)S_[0,1). Hence 00 S_0^0 is a legal initial state. Induction hypothesis at stage n. Suppose that n0 S_n^0 has been constructed and is legal. The following objects are in place. (Q1) The outer and inner sets ΓI _I and ΔJ _J for I,J∈≤nI,J _≤ n, and the born blocks HI,HI∗,KJ,KJ∗H_I,H_I^*,K_J,K_J^* for I,J∈<nI,J _<n. (Q2) For every current outer frontier index A∈nA _n, an infinite reservoir tail SA⊂ℕS_A . (Q3) For every same outer strip indexed by (J,J′)(J,J ) which has already been registered and every current outer frontier index A∈nA _n, positive numbers θAJ,J′ _A^J,J and ωAJ,J′ _A^J,J . (Q4) For every same inner strip indexed by (I,I′)(I,I ) which has already been registered, the finite inner tree I,I′oldT_I,I old and the finite inner frontier ℛI,I′R_I,I present when (I,I′)(I,I ) was registered, the future forest ℱI,I′=C∈:C⊆B for some B∈ℛI,I′,F_I,I =\C :C B for some B _I,I \, and positive future tolerances ηCI,I′ _C^I,I and ρC,C′I,I′ _C,C ^I,I for every pair C,C′∈ℱI,I′C,C _I,I with C a dyadic parent of C′C , which we will denote by C→C′C→ C . (Q5) The finite collection of mixed coefficients whose registration half stage has already occurred. For a same outer strip indexed by (J,J′)(J,J ) which has already been registered, we write QJ,J′x=(Id⊗KJ∗)T(x⊗KJ′)(x∈Lp0).Q_J,J x=(Id K_J^*)T(x K_J ) (x∈ L_p^0). (61) For a same inner strip indexed by (I,I′)(I,I ) which has already been registered, we write RI,I′v=(HI∗⊗Id)T(HI′⊗v)(v∈L10).R_I,I v=(H_I^* )T(H_I v) (v∈ L_1^0). (62) The construction has been carried out so that the following conditions hold for these operators and coefficients. (E1) (Existing same outer-strip are small) For every same outer strip indexed by (J,J′)(J,J ) which has already been registered and every already born outer index I, |aJ,J′I,I|<σJ,J′.|a_J,J ^I,I|< _J,J . (63) (E2) (Room to make future same outer-strip choices small) For every same outer strip indexed by (J,J′)(J,J ) which has already been registered and every current outer frontier index A, |αmΓA(QJ,J′)|<θAJ,J′(m∈SA)| _m _A(Q_J,J )|< _A^J,J (m∈ S_A) (64) and θAJ,J′+ωAJ,J′<σJ,J′. _A^J,J + _A^J,J < _J,J . (65) (E3) (Existing same inner-strip is small) For every same inner-strip indexed by (I,I′)(I,I ) which has already been registered, set GI,I′= G_I,I = [0,1)∈I,I′old|a[0,1),[0,1)I,I′|+∑P→QP,Q∈I,I′old|aP,PI,I′−aQ,QI,I′| 1_\[0,1) _I,I old\|a_[0,1),[0,1)^I,I |+ _ subarraycP→ Q\\ P,Q _I,I old subarray|a_P,P^I,I -a_Q,Q^I,I | (66) +∑B∈ℛI,I′p(B)∈I,I′old|ap(B),p(B)I,I′|, + _ subarraycB _I,I \\ p(B) _I,I old subarray|a_p(B),p(B)^I,I |, here p(B)p(B) denotes the dyadic parent of B, when this parent belongs to the old inner tree I,I′oldT_I,I old. The part of the strip which was already present at the registration half stage is controlled by GI,I′+∑B∈ℛI,I′|λΔB(RI,I′)|<τI,I′C0.G_I,I + _B _I,I | _ _B(R_I,I )|< _I,I C_0. (67) Here GI,I′G_I,I measures the variation on the inner tree which had already been built, while the trace terms λΔB(RI,I′) _ _B(R_I,I ), B∈ℛI,I′B _I,I , measure the size at the roots where the future tree will be attached. (E4) (Room to make future same inner-strip choices small) For every same inner strip indexed by (I,I′)(I,I ) which has already been registered, the future tolerances are chosen so that 4∑C∈ℱI,I′ηCI,I′+∑C→C′,C,C′∈ℱI,I′ρC,C′I,I′<τI,I′C0.4 _C _I,I _C^I,I + _C→ C ,\ C,C _I,I _C,C ^I,I < _I,I C_0. (68) After the registration half stage, whenever a future inner block KCK_C is born after (I,I′)(I,I ) is born, that is, C is born but C∉I,I′oldC _I,I old, we impose |⟨KC∗,RI,I′KC⟩−λΔC(RI,I′)|<ηCI,I′,| K_C^*,R_I,I K_C - _ _C(R_I,I )|< _C^I,I , (69) and whenever a future edge C→C′C→ C in ℱI,I′F_I,I has been created, we impose |λΔC′(RI,I′)−λΔC(RI,I′)|<ρC,C′I,I′.| _ _C (R_I,I )- _ _C(R_I,I )|< _C,C ^I,I . (70) Thus (E3) controls the part of the strip present at registration, while (E4) controls the coefficients born after registration. (E5) (Mixed coefficients are small) Every registered mixed coefficient satisfies |aJ,J′I,I′|<μI,I′,J,J′.|a_J,J ^I,I |< _I,I ,J,J . (71) By Claim 1, every off-diagonal coefficient is assigned to exactly one of these mechanisms, and only finitely many new strips, coefficients, and estimates occur at each half stage. We now argue how to continue the inductive process, provided we are starting from a legal state at stage n. Outer half stage at level n. We split every current outer set ΓI _I, I∈nI _n, birth HI,HI∗H_I,H_I^* for I∈nI _n, and produce a half state n1/2 S_n^1/2 whose outer frontier is indexed by n+1D_n+1 and whose inner frontier is indexed by nD_n. By Claim 2, the splits can be chosen together with the reservoir tails SA+S_A^+ and SA−S_A^- below each A∈nA _n, the numbers θAϑJ,J′ _A ^J,J and ωAϑJ,J′ _A ^J,J for every same outer strip registered by the half state, and, for every same inner strip registered at this half stage, the objects listed in (Q4). These choices can be made so that, (E1) and (E2) hold for the same outer strips registered by the half state, (E3) and (E4) hold for every same inner strip registered by the half state, and (E5) holds for every mixed coefficient registered at this half stage. Inner half stage at level n. We split every current inner set ΔJ _J, J∈nJ _n, birth KJ,KJ∗K_J,K_J^* for J∈nJ _n, and produce the next full state n+10 S_n+1^0. By Claim 3, the splits can be chosen together with the possibly shrunk reservoir tails SAS_A for A∈n+1A _n+1, the initial numbers θAJ,J′ _A^J,J and ωAJ,J′ _A^J,J for every same outer strip registered at this half stage, and the future estimates in (E4) for all same inner strips registered by that time whose relevant future blocks or future edges are created at this half stage. These choices can be made so that, (E1) and (E2) hold for all same outer strips registered by the new full state, (E3) and (E4) hold for all same inner strips registered by that time, and (E5) holds for every mixed coefficient registered at this half stage. This proves the induction step from n0 S_n^0 to n+10 S_n+1^0. Repeating the two half stages gives the sequence 00→outer01/2→inner10→outer11/2→inner…. S_0^0 outer S_0^1/2 inner S_1^0 outer S_1^1/2 inner…. Therefore every dyadic index is eventually split in both coordinates, and we obtain faithful outer and inner Haar systems (HI,HI∗)I∈,(KJ,KJ∗)J∈.(H_I,H_I^*)_I , (K_J,K_J^*)_J . Let A and B be the associated exterior maps. By Section˜9, ‖A‖≤1,‖B‖≤1,AB=IdX00,\|A\|≤ 1, \|B\|≤ 1, AB=Id_X_00, and ⟨uI,J∗,ATBuI′,J′⟩=aJ,J′I,I′(I,I′,J,J′∈). u_I,J^*,ATBu_I ,J =a_J,J ^I,I (I,I ,J,J ). (72) On the algebraic product Haar span define M0uI,J=aJ,JI,IuI,J.M_0u_I,J=a_J,J^I,Iu_I,J. The matrix coefficients of ATB−M0ATB-M_0 are the off-diagonal coefficients from (59). We estimate the three off-diagonal pieces. For J≠J′J≠ J , let OJ,J′outO_J,J out be given on product Haar vectors by OJ,J′outuI,L=aJ,J′I,IuI,J,L=J′,0,L≠J′.O_J,J outu_I,L= casesa_J,J ^I,Iu_I,J,&L=J ,\\ 0,&L≠ J . cases The construction gives |aJ,J′I,I|<σJ,J′|a_J,J ^I,I|< _J,J for all I∈I , and Section˜9 gives ‖OJ,J′out‖≤CoutσJ,J′.\|O_J,J out\|≤ C_ out _J,J . Thus ‖∑J≠J′OJ,J′out‖≤Cout∑J≠J′σJ,J′. \| _J≠ J O_J,J out \|≤ C_ out _J≠ J _J,J . (73) For I≠I′I≠ I , let OI,I′inO_I,I in be given on product Haar vectors by OI,I′inuL,J=aJ,JI,I′uI,J,L=I′,0,L≠I′.O_I,I inu_L,J= casesa_J,J^I,I u_I,J,&L=I ,\\ 0,&L≠ I . cases The estimates (67), (68), (69), and (70), together with Section˜9, imply that the branch variation of (aJ,JI,I′)J∈(a_J,J^I,I )_J is less than τI,I′ _I,I . Hence Section˜9 and Section˜9 gives ‖OI,I′in‖≤CinτI,I′.\|O_I,I in\|≤ C_ in _I,I . Therefore ‖∑I≠I′OI,I′in‖≤Cin∑I≠I′τI,I′. \| _I≠ I O_I,I in \|≤ C_ in _I≠ I _I,I . (74) Finally, define Omix=∑I≠I′,J≠J′aJ,J′I,I′uI,J⊗uI′,J′∗.O mix= _I≠ I ,\,J≠ J a_J,J ^I,I u_I,J u_I ,J ^*. Since ‖uI,J‖=‖uI′,J′∗‖=1\|u_I,J\|=\|u_I ,J ^*\|=1 and (71) holds for every mixed coefficient, the mixed series converges absolutely in operator norm and ‖Omix‖≤∑I≠I′,J≠J′μI,I′,J,J′.\|O mix\|≤ _I≠ I ,\,J≠ J _I,I ,J,J . (75) Set O=∑J≠J′OJ,J′out+∑I≠I′OI,I′in+Omix.O= _J≠ J O_J,J out+ _I≠ I O_I,I in+O mix. Combining (60), (73), (74), and (75), we obtain ‖O‖<ε.\|O\|< . Moreover, by construction and (72), the operators ATB−M0ATB-M_0 and O have the same product Haar coefficients on the algebraic product Haar span. By Section˜9, they agree there. Define M=ATB−O.M=ATB-O. Then M∈ℬ(X00)M (X_00), and M agrees with M0M_0 on the algebraic product Haar span. Hence M is the bounded product Haar multiplier with diagonal MuI,J=aJ,JI,IuI,J=⟨HI∗⊗KJ∗,T(HI⊗KJ)⟩uI,J.Mu_I,J=a_J,J^I,Iu_I,J= H_I^* K_J^*,T(H_I K_J) u_I,J. Finally, ‖ATB−M‖=‖O‖<ε,\|ATB-M\|=\|O\|< , and, because A and B are contractions, ‖M‖≤‖ATB‖+‖ATB−M‖≤‖T‖+ε.\|M\|≤\|ATB\|+\|ATB-M\|≤\|T\|+ . This completes the proof. ∎ 10. Technical claims for the multiplier reduction construction In this section we prove the auxiliary lemmas and construction claims invoked in the proof of Theorem˜9.31. The terminology is the one fixed at the beginning of the construction section. Lemma 9.32 (One-coordinate strip estimates). Let (dI)I∈(d_I)_I define a scalar Haar multiplier MdM_d on Lp0L_p^0, and let J≠J′J≠ J . The product strip operator OJ,J′outuI,L=dIuI,J,L=J′,0,L≠J′O_J,J outu_I,L= casesd_Iu_I,J,&L=J ,\\ 0,&L≠ J cases satisfies ‖OJ,J′out‖ℬ(X00)≤‖Md‖ℬ(Lp0).\|O_J,J out\|_B(X_00)≤\|M_d\|_B(L_p^0). In particular, with UpU_p as in Section˜9, ‖OJ,J′out‖ℬ(X00)≤UpsupI∈|dI|.\|O_J,J out\|_B(X_00)≤ U_p _I |d_I|. Similarly, if (eJ)J∈(e_J)_J defines a scalar Haar multiplier NeN_e on L10L_1^0 and I≠I′I≠ I , then the product strip operator OI,I′inuL,J=eJuI,J,L=I′,0,L≠I′O_I,I inu_L,J= casese_Ju_I,J,&L=I ,\\ 0,&L≠ I cases satisfies ‖OI,I′in‖ℬ(X00)≤‖Ne‖ℬ(L10).\|O_I,I in\|_B(X_00)≤\|N_e\|_B(L_1^0). Proof. For the same outer strip, define PJ′,J:L10→L10P_J ,J L_1^0→ L_1^0 by PJ′,Jv=⟨fJ′∗,v⟩fJ.P_J ,Jv= f_J ^*,v f_J. Then ‖PJ′,J‖≤1\|P_J ,J\|≤ 1. For x∈X00x∈ X_00, let g(s)=⟨fJ′∗,x(s,⋅)⟩g(s)= f_J ^*,x(s,·) . Since x has zero outer integral, g∈Lp0g∈ L_p^0. Also ‖g‖Lp≤‖x‖Lp(L1)\|g\|_L_p≤\|x\|_L_p(L_1), and the same outer strip is (Mdg)⊗fJ(M_dg) f_J. Hence ‖OJ,J′outx‖Lp(L1)≤‖Md‖‖x‖Lp(L1).\|O_J,J outx\|_L_p(L_1)≤\|M_d\|\|x\|_L_p(L_1). The estimate with UpU_p follows from Section˜9. For the same inner strip, define PI′,I:Lp0→Lp0P_I ,I L_p^0→ L_p^0 by PI′,Ix=⟨eI′∗,x⟩eI.P_I ,Ix= e_I ^*,x e_I. Then ‖PI′,I‖≤1\|P_I ,I\|≤ 1. For x∈X00x∈ X_00, let v(t)=∫01eI′∗(s)x(s,t)s.v(t)= _0^1e_I ^*(s)x(s,t)\,ds. Since x has zero inner integral, v∈L10v∈ L_1^0. By Hölder’s inequality, ‖v‖1≤‖x‖Lp(L1)\|v\|_1≤\|x\|_L_p(L_1), and the same inner strip is eI⊗Neve_I N_ev. Hence ‖OI,I′inx‖Lp(L1)≤‖Ne‖‖x‖Lp(L1).\|O_I,I inx\|_L_p(L_1)≤\|N_e\|\|x\|_L_p(L_1). ∎ Lemma 9.33 (Product Haar coefficients separate points). If x∈X00x∈ X_00 and ⟨uI,J∗,x⟩=0(I,J∈), u_I,J^*,x =0 (I,J ), then x=0x=0. In particular, if two bounded operators on X00X_00 have the same product Haar coefficients on every product Haar vector, then they agree on the algebraic product Haar span. Proof. Let PmP_m be the finite product-Haar projection onto the span of uI,Ju_I,J with I,J∈⋃k=0mkI,J∈ _k=0^mD_k. These projections are obtained from the dyadic martingale difference projections in the two variables, and they converge strongly to the identity on X00⊂Lp(L1)X_00⊂ L_p(L_1). If all product Haar coefficients of x vanish, then Pmx=0P_mx=0 for every m, hence x=limmPmx=0x= _mP_mx=0. The operator statement follows by applying this to the difference of the two images of each product Haar vector. ∎ Lemma 9.34 (Inner strip branch variation estimate). Let (I,I′)(I,I ) be a registered same inner strip, and put dJ=aJ,JI,I′d_J=a_J,J^I,I . If the old-part estimate in (E3) and the future-part estimates in (E4) hold for this strip, then BVbr((dJ)J∈)<τI,I′.BV_br((d_J)_J )< _I,I . Proof. Fix an infinite dyadic branch J0⊃J1⊃…J_0⊃ J_1⊃…. The branch is the union of an initial part contained in the old finite tree I,I′oldT_I,I old and, possibly after crossing one index B∈ℛI,I′B _I,I , a tail contained in the future forest ℱI,I′F_I,I . On the old part, the contribution is bounded by the corresponding terms in GI,I′G_I,I . If the branch crosses from an old parent p(B)p(B) into B∈ℛI,I′B _I,I , then |dp(B)−dB|≤|ap(B),p(B)I,I′|+|λΔB(RI,I′)|+ηBI,I′.|d_p(B)-d_B|≤|a_p(B),p(B)^I,I |+| _ _B(R_I,I )|+ _B^I,I . If the branch starts inside the future forest, let B∈ℛI,I′B _I,I be the frontier root containing J0J_0. The value |dJ0||d_J_0| is bounded by |λΔB(RI,I′)|| _ _B(R_I,I )|, the ρ-sum along the path from B to J0J_0, and ηJ0I,I′ _J_0^I,I . Finally, for every future edge C→C′C→ C on the branch, |dC−dC′|≤ηCI,I′+ρC,C′I,I′+ηC′I,I′.|d_C-d_C |≤ _C^I,I + _C,C ^I,I + _C ^I,I . Therefore the branch variation is bounded by GI,I′+3∑B∈ℛI,I′|λΔB(RI,I′)|+4∑C∈ℱI,I′ηCI,I′+∑C→C′,C,C′∈ℱI,I′ρC,C′I,I′.G_I,I +3 _B _I,I | _ _B(R_I,I )|+4 _C _I,I _C^I,I + _C→ C ,\ C,C _I,I _C,C ^I,I . By (67), the first two terms satisfy GI,I′+3∑B∈ℛI,I′|λΔB(RI,I′)|≤3(GI,I′+∑B∈ℛI,I′|λΔB(RI,I′)|)<3τI,I′C0.G_I,I +3 _B _I,I | _ _B(R_I,I )|≤ 3 (G_I,I + _B _I,I | _ _B(R_I,I )| )< 3 _I,I C_0. Together with (68) and C0=8C_0=8, this gives a bound smaller than τI,I′ _I,I . Taking the supremum over all branches gives the claim. ∎ Claim 1 (Registration is well defined and locally finite). Every off diagonal product coefficient is assigned to exactly one of the three mechanisms in the construction. At every half stage, only finitely many new strips and mixed coefficients are registered, only finitely many reservoir tails are updated, and the side constraint lists submitted to the finite splitting lemmas are finite. When a same inner strip is registered, the countable family of future tolerances attached to it is chosen after the finite constraints have been imposed. Proof. Let aJ,J′I,I′a_J,J ^I,I be an off-diagonal coefficient. If I=I′I=I and J≠J′J≠ J , the coefficient belongs to the same outer strip indexed by (J,J′)(J,J ), and this strip is registered when the later of KJK_J and KJ′K_J is born. If I≠I′I≠ I and J=J′J=J , the coefficient belongs to the same inner strip indexed by (I,I′)(I,I ), and this strip is registered when the later of HIH_I and HI′H_I is born. If I≠I′I≠ I and J≠J′J≠ J , the coefficient is mixed, and it is registered when the last of HI,HI′,KJ,KJ′H_I,H_I ,K_J,K_J is born. These three alternatives are disjoint and exhaust the off-diagonal cases. At a fixed half stage, only finitely many blocks have been born and only finitely many new blocks are born. Hence only finitely many strips can be registered for the first time, and only finitely many mixed coefficients can have their last block born at that half stage. The current frontiers are finite dyadic generations, so only finitely many reservoir tails are created or shrunk at that half stage. Therefore every side constraint list submitted to a finite splitting lemma is finite. The future tolerance family attached to a newly registered same inner strip is countable, but it is chosen separately after the finite splitting step and is not part of the side constraint list. ∎ We now prove the main claims in the construction, in other words, that we can make appropriate choices in each of the half-stages. Claim 2 (The outer half stage is possible). Assume that n0 S_n^0 is legal. Then one can choose the outer splits, the reservoir tails for the new outer frontier, the propagated same outer numbers θ and ω, and, for every same inner strip registered at this half stage, the objects in (Q4), so that the resulting half state n1/2 S_n^1/2 is legal. The chosen outer splits determine the born blocks HI,HI∗H_I,H_I^* for I∈nI _n. More precisely, (E1) and (E2) hold for the same outer strips registered by the half state, (E5) holds for every mixed coefficient registered at this half stage, and every same inner strip registered at this half stage satisfies the old-part condition (E3) and is assigned future tolerances so that the budget in (E4) holds. Proof. At the beginning of the outer half stage, the inner tree is fixed and the current outer frontier is indexed by nD_n. For each current outer frontier index A and each same outer strip indexed by (J,J′)(J,J ) already registered, choose positive numbers ηAJ,J′ _A^J,J and δAJ,J′ _A^J,J so small that ηAJ,J′<ωAJ,J′2,δAJ,J′<ωAJ,J′2. _A^J,J < _A^J,J 2, _A^J,J < _A^J,J 2. We shall apply Section˜9 with the current common tail SAS_A, the bound θAJ,J′ _A^J,J , and the tolerances ηAJ,J′,δAJ,J′ _A^J,J , _A^J,J , together with the admissible side constrain now defined. We form one finite admissible outer side constraint list. First, include every mixed coefficient whose registration time is the present outer half stage: |aJ,J′I,I′|<μI,I′,J,J′.|a_J,J ^I,I |< _I,I ,J,J . The inner blocks are fixed. If exactly one of HI,HI′H_I,H_I is new, the constraint is one-sided in the new outer block. If both are new, then I≠I′I≠ I , so the two new outer blocks are supported on distinct current frontier sets, and the constraint is bilinear between distinct new blocks. Thus the constraint is admissible. Next consider a same inner strip indexed by (I,I′)(I,I ) whose registration time is the present outer half stage. Let I,I′oldT_I,I old be the already born inner tree and let ℛI,I′R_I,I be the current inner frontier. For every P∈I,I′oldP _I,I old, the coefficient is aP,PI,I′=⟨HI∗⊗KP∗,T(HI′⊗KP)⟩.a_P,P^I,I = H_I^* K_P^*,T(H_I K_P) . Choose tolerances for the finitely many old coefficients so small that, if all corresponding inequalities hold, then GI,I′<τI,I′4C0.G_I,I < _I,I 4C_0. This is possible because GI,I′G_I,I is a finite sum of terms |aP,PI,I′||a_P,P^I,I |, |aP,PI,I′−aQ,QI,I′||a_P,P^I,I -a_Q,Q^I,I |, and |ap(B),p(B)I,I′||a_p(B),p(B)^I,I |, and making all finitely many coefficients |aP,PI,I′||a_P,P^I,I | sufficiently small makes that finite sum small. These inequalities are admissible outer constraints, because the inner blocks are fixed and at least one of the two outer blocks is new. For each B∈ℛI,I′B _I,I , the trace value is λΔB(RI,I′)=⟨HI∗,ℐΔBTHI′⟩, _ _B(R_I,I )= H_I^*,I_ _B^TH_I , by Section˜9. Choose positive numbers ζBI,I′ _B^I,I , B∈ℛI,I′B _I,I , such that ∑B∈ℛI,I′ζBI,I′<τI,I′4C0, _B _I,I _B^I,I < _I,I 4C_0, and add the admissible constraints |λΔB(RI,I′)|=|⟨HI∗,ℐΔBTHI′⟩|<ζBI,I′(B∈ℛI,I′).| _ _B(R_I,I )|=| H_I^*,I_ _B^TH_I |< _B^I,I (B _I,I ). The side constraint list is finite by Claim 1. Apply Section˜9 simultaneously to the current outer frontier sets, the operators QJ,J′Q_J,J for the already registered same outer strips, the common tails SAS_A, the chosen tolerances, and this finite side constraint list. All side constraints are then satisfied. Hence all mixed coefficients registered at this half stage satisfy their μ bounds, and every same inner strip indexed by (I,I′)(I,I ) registered at this half stage satisfies GI,I′+∑B∈ℛI,I′|λΔB(RI,I′)|<τI,I′C0.G_I,I + _B _I,I | _ _B(R_I,I )|< _I,I C_0. This is precisely the old-part condition (E3) for the newly registered same inner strips. For a same outer strip indexed by (J,J′)(J,J ) already registered, Section˜9 gives |⟨HA∗,QJ,J′HA⟩|<θAJ,J′+ηAJ,J′<θAJ,J′+ωAJ,J′<σJ,J′| H_A^*,Q_J,J H_A |< _A^J,J + _A^J,J < _A^J,J + _A^J,J < _J,J for every newly born outer block HAH_A. For each child AϑA , ϑ∈+,− ∈\+,-\, the same lemma gives an infinite tail SAϑS_A such that |αmΓAϑ(QJ,J′)|<θAJ,J′+δAJ,J′(m∈SAϑ).| _m _A (Q_J,J )|< _A^J,J + _A^J,J (m∈ S_A ). Define θAϑJ,J′=θAJ,J′+δAJ,J′,ωAϑJ,J′=ωAJ,J′−δAJ,J′. _A ^J,J = _A^J,J + _A^J,J , _A ^J,J = _A^J,J - _A^J,J . Then ωAϑJ,J′>0 _A ^J,J >0 and θAϑJ,J′+ωAϑJ,J′=θAJ,J′+ωAJ,J′<σJ,J′. _A ^J,J + _A ^J,J = _A^J,J + _A^J,J < _J,J . Thus the same outer tail estimate is preserved. For each same inner strip indexed by (I,I′)(I,I ) registered at this half stage, choose positive numbers ηCI,I′ _C^I,I and ρC,C′I,I′ _C,C ^I,I on the future forest ℱI,I′F_I,I so that (68) holds. Thus the future tolerance condition in (E4) is initialized for the newly registered same inner strips; the corresponding node and edge estimates will be imposed in later inner half stages when those future blocks and edges are born. Therefore the half state is legal. ∎ Claim 3 (The inner half stage is possible). Assume that n1/2 S_n^1/2 is legal. Then one can choose the inner splits, the possibly shrunk reservoir tails for the current outer frontier, and the initial same outer numbers θ and ω for every same outer strip registered at this half stage so that the resulting full state n+10 S_n+1^0 is legal. The chosen inner splits determine the born blocks KJ,KJ∗K_J,K_J^* for J∈nJ _n. More precisely, the relevant node and edge estimates from the future-part condition (E4) are imposed for the same inner strips already registered, (E5) holds for every mixed coefficient registered at this half stage, and every same outer strip registered at this inner half stage satisfies (E1) and (E2). Proof. At the beginning of the inner half stage, the outer tree is fixed. We form one finite admissible inner side constraint list. First include every mixed coefficient whose registration time is the present inner half stage: |aJ,J′I,I′|<μI,I′,J,J′.|a_J,J ^I,I |< _I,I ,J,J . The outer blocks are fixed. If exactly one of KJ,KJ′K_J,K_J is new, the constraint is one-sided in the new inner block. If both are new, then J≠J′J≠ J , so the two new inner blocks are supported on distinct current frontier sets, and the constraint is bilinear between distinct new blocks. Thus the constraint is admissible. For each current outer frontier index A in the half state n1/2 S_n^1/2, choose once and for all a free ultrafilter AU_A containing the current reservoir tail SAS_A. For each same outer strip indexed by (J,J′)(J,J ) whose registration time is the present inner half stage, add, for every already born outer index I, the constraint |⟨HI∗⊗KJ∗,T(HI⊗KJ′)⟩|<σJ,J′.| H_I^* K_J^*,T(H_I K_J ) |< _J,J . The outer block is fixed, so this is an admissible inner constraint. If exactly one of KJ,KJ′K_J,K_J is new it is one-sided; if both are new it is bilinear between distinct current inner frontier sets. For the same newly registered same outer strip, add, for every current outer frontier index A, the shadow constraint |⟨KJ∗,ΩΓAT,AKJ′⟩|<σJ,J′8.| K_J^*, _ _A^T,U_AK_J |< _J,J 8. By Section˜9, ΩΓAT,A∈ℬ(L10) _ _A^T,U_A (L_1^0), so this is an admissible inner constraint. The same ultrafilter AU_A is used for every same outer strip registered at this half stage. Now add the same inner estimates. For every same inner strip indexed by (I,I′)(I,I ) already registered and every current inner frontier index C∈ℱI,I′C _I,I , include the operator RI,I′R_I,I with tolerance δCI,I′=minηCI,I′,ρC,C+I,I′,ρC,C−I,I′. _C^I,I = \ _C^I,I , _C,C^+^I,I , _C,C^-^I,I \. The side constraint list and the list of already registered same inner strips are finite by Claim 1. Apply Section˜9 to the current inner frontier sets, the finite families of operators RI,I′R_I,I for the already registered same inner strips, the tolerances δCI,I′ _C^I,I , and the finite admissible side constraint list. The chosen signs impose all listed side constraints. Therefore every mixed coefficient registered at this half stage satisfies its μ bound. Since the old finite tree and registration frontier attached to a same inner strip do not change after that strip is registered, (E3) remains true for every previously registered same inner strip. The conclusions of Section˜9 also give |⟨KC∗,RI,I′KC⟩−λΔC(RI,I′)|<ηCI,I′| K_C^*,R_I,I K_C - _ _C(R_I,I )|< _C^I,I for every same inner strip indexed by (I,I′)(I,I ) already registered and every current inner frontier index C∈ℱI,I′C _I,I , and |λΔC′(RI,I′)−λΔC(RI,I′)|<ρC,C′I,I′| _ _C (R_I,I )- _ _C(R_I,I )|< _C,C ^I,I for the two children C′C of each current C. These are exactly the node and edge estimates required in the future-part condition (E4) at this half stage. Let (J,J′)(J,J ) index a same outer strip registered at this half stage, and write QJ,J′Q_J,J as in (61). The old coefficient constraints give |aJ,J′I,I|<σJ,J′|a_J,J ^I,I|< _J,J for every already born outer index I. For each current outer frontier index A, the shadow constraint and Section˜9 give limm→AαmΓA(QJ,J′)=⟨KJ∗,ΩΓAT,AKJ′⟩. _m _A _m _A(Q_J,J )= K_J^*, _ _A^T,U_AK_J . Hence m∈ℕ:|αmΓA(QJ,J′)|<σJ,J′/4∈A.\m :| _m _A(Q_J,J )|< _J,J /4\ _A. For fixed A, all such good sets, as (J,J′)(J,J ) ranges over the finitely many same outer strips registered at the present half stage, belong to the same ultrafilter AU_A. Hence their finite intersection with the old reservoir tail SAS_A also belongs to AU_A, and is infinite. Replace SAS_A by this finite intersection. For every newly registered same outer strip indexed by (J,J′)(J,J ), this reservoir satisfies |αmΓA(QJ,J′)|<σJ,J′4(m∈SA).| _m _A(Q_J,J )|< _J,J 4 (m∈ S_A). We record this estimate by setting θAJ,J′=σJ,J′4 _A^J,J = _J,J 4 and choose, for instance, ωAJ,J′=σJ,J′2. _A^J,J = _J,J 2. Then θAJ,J′+ωAJ,J′=3σJ,J′/4<σJ,J′ _A^J,J + _A^J,J =3 _J,J /4< _J,J . Thus (64) and (65) hold for all newly registered same outer strips, while they remain true for the previously registered same outer strips because the common tail has only been shrunk. Hence n+10 S_n+1^0 is legal. ∎ 11. Scalar compression and primarity We now combine the arbitrary-to-multiplier reduction with the LMMS scalar compression theorem for bounded product Haar multipliers. This gives the scalar compression statement on X00X_00. Theorem 9.35 (Scalar compression on X00X_00). For every S∈ℬ(X00)S (X_00) and every ε>0 >0, there are A,B∈ℬ(X00)A,B (X_00) and a scalar c∈ℝc such that AB=IdX00,‖ASB−cIdX00‖ℬ(X00)<ε.AB=Id_X_00, \|ASB-cId_X_00\|_B(X_00)< . Moreover, A and B may be chosen so that ‖A‖‖B‖≤1+ε.\|A\|\,\|B\|≤ 1+ . Proof. Choose δ>0δ>0 small enough so that (1+δ)δ+δ<ε,1+δ<1+ε.(1+δ)δ+δ< , 1+δ<1+ . Apply Theorem˜9.31 to S with accuracy δ. We obtain A0,B0∈ℬ(X00)A_0,B_0 (X_00) and a bounded product Haar multiplier M on X00X_00 such that A0B0=IdX00,‖A0SB0−M‖ℬ(X00)<δ.A_0B_0=Id_X_00, \|A_0SB_0-M\|_B(X_00)<δ. By Section˜9, the maps A0A_0 and B0B_0 are contractions. Apply Theorem˜9.8 to M with accuracy δ. There are A1,B1∈ℬ(X00)A_1,B_1 (X_00) and c∈ℝc such that A1B1=IdX00,‖A1MB1−cIdX00‖ℬ(X00)<δ,‖A1‖‖B1‖≤1+δ.A_1B_1=Id_X_00, \|A_1MB_1-cId_X_00\|_B(X_00)<δ, \|A_1\|\,\|B_1\|≤ 1+δ. Set A=A1A0,B=B0B1.A=A_1A_0, B=B_0B_1. Then AB=A1A0B0B1=A1B1=IdX00.AB=A_1A_0B_0B_1=A_1B_1=Id_X_00. Moreover, ‖ASB−cIdX00‖ℬ(X00) \|ASB-cId_X_00\|_B(X_00) ≤‖A1(A0SB0−M)B1‖ℬ(X00)+‖A1MB1−cIdX00‖ℬ(X00) ≤\|A_1(A_0SB_0-M)B_1\|_B(X_00)+\|A_1MB_1-cId_X_00\|_B(X_00) <‖A1‖‖B1‖δ+δ≤(1+δ)δ+δ<ε. <\|A_1\|\,\|B_1\|δ+δ≤(1+δ)δ+δ< . Finally, since A0A_0 and B0B_0 are contractions, ‖A‖‖B‖≤‖A1‖‖B1‖≤1+δ<1+ε.\|A\|\,\|B\|≤\|A_1\|\,\|B_1\|≤ 1+δ<1+ . ∎ It remains to pass from the doubly cancellative space back to X. The passage uses a standard complemented cancellative copy. Lemma 9.36 (A one-complemented cancellative copy). The space X00X_00 contains an isometric copy of X which is the range of a norm-one projection on X. Proof. Let r(t)=[0,1/2)(t)−[1/2,1)(t),σ(t)=2t(mod1).r(t)=1_[0,1/2)(t)-1_[1/2,1)(t), σ(t)=2t 1. Define U:X→XU X→ X by (Uf)(s,t)=r(s)r(t)f(σ(s),σ(t)).(Uf)(s,t)=r(s)r(t)f(σ(s),σ(t)). Since σ preserves Lebesgue measure and |r|=1|r|=1, we have ‖Uf‖Lp(L1)=‖f‖Lp(L1)(f∈X),\|Uf\|_L_p(L_1)=\|f\|_L_p(L_1) (f∈ X), so U is an isometry. Moreover, for almost every s, ∫01(Uf)(s,t)t=0, _0^1(Uf)(s,t)\,dt=0, and, as an L1L_1-valued integral in the outer variable, ∫01(Uf)(s,⋅)s=0. _0^1(Uf)(s,·)\,ds=0. Thus U(X)⊂X00U(X)⊂ X_00. Define V:X→XV X→ X by (Vg)(u,v)=14∑α,β∈0,1(−1)α+βg(u+α2,v+β2).(Vg)(u,v)= 14 _α,β∈\0,1\(-1)^α+βg ( u+α2, v+β2 ). For f∈Xf∈ X we have Uf(u+α2,v+β2)=(−1)α+βf(u,v),Uf ( u+α2, v+β2 )=(-1)^α+βf(u,v), and therefore VUf=f.VUf=f. It remains to check that V is contractive. For g∈Xg∈ X, put a(s)=‖g(s,⋅)‖L1.a(s)=\|g(s,·)\|_L_1. Then, for almost every u, ‖Vg(u,⋅)‖L1≤12(a(u/2)+a((u+1)/2)).\|Vg(u,·)\|_L_1≤ 12 (a(u/2)+a((u+1)/2) ). By convexity and a change of variables, ‖Vg‖Lp(L1)p≤∫0112(a(u/2)p+a((u+1)/2)p)u=‖g‖Lp(L1)p.\|Vg\|_L_p(L_1)^p≤ _0^1 12 (a(u/2)^p+a((u+1)/2)^p )\,du=\|g\|_L_p(L_1)^p. Thus ‖V‖≤1\|V\|≤ 1. Now set P=UV.P=UV. Since VU=IdXVU=Id_X, we have P2=UVUV=U(VU)V=UV=P,P^2=UVUV=U(VU)V=UV=P, and ranP=U(X)⊂X00.ranP=U(X)⊂ X_00. Also ‖P‖≤‖U‖‖V‖≤1\|P\|≤\|U\|\|V\|≤ 1, and since P is a non-zero projection, ‖P‖=1\|P\|=1. Hence U(X)U(X) is an isometric copy of X contained in X00X_00 and one-complemented in X. ∎ Finally, we are ready for the proof of our main result. Proof of Theorem˜9.1. Let P00=(Id−1)(Id−2):X→X00,P_00=(Id-E_1)(Id-E_2) X→ X_00, where iE_i denotes integration in the i-th variable. By Section˜9, P00P_00 is a bounded projection onto X00X_00 and ‖P00‖≤4\|P_00\|≤ 4. Let ι:X00→X X_00→ X denote the inclusion. Fix T∈ℬ(X)T (X), and define S=P00Tι∈ℬ(X00).S=P_00T (X_00). Apply Theorem˜9.35 to S with accuracy η=1/8η=1/8. We obtain A0,B0∈ℬ(X00)A_0,B_0 (X_00) and c∈ℝc such that A0B0=IdX00,‖A0SB0−cIdX00‖ℬ(X00)<1/8,‖A0‖‖B0‖≤9/8.A_0B_0=Id_X_00, \|A_0SB_0-cId_X_00\|_B(X_00)<1/8, \|A_0\|\,\|B_0\|≤ 9/8. Either |c|≥1/2|c|≥ 1/2 or |1−c|≥1/2|1-c|≥ 1/2. First suppose that |c|≥1/2|c|≥ 1/2. Then A0SB0A_0SB_0 is invertible on X00X_00, and ‖(A0SB0)−1‖≤1|c|−1/8≤83.\|(A_0SB_0)^-1\|≤ 1|c|-1/8≤ 83. Set R=(A0SB0)−1.R=(A_0SB_0)^-1. Since S=P00TιS=P_00T , we have RA0P00TιB0=IdX00.RA_0P_00T B_0=Id_X_00. Now let U:X→X00U X→ X_00 and V:X00→XV X_00→ X be the maps from Section˜9, so that U is an isometry, V is a contraction, and VU=IdXVU=Id_X. Composing the preceding factorization with U on the right and V on the left gives VRA0P00TιB0U=IdX.VRA_0P_00T B_0U=Id_X. Thus IdXId_X factors through T, with factorization constant at most ‖V‖‖R‖‖A0‖‖P00‖‖B0‖‖U‖≤83⋅98‖P00‖=3‖P00‖.\|V\|\,\|R\|\,\|A_0\|\,\|P_00\|\,\|B_0\|\,\|U\|≤ 83· 98\,\|P_00\|=3\|P_00\|. Now suppose that |1−c|≥1/2|1-c|≥ 1/2. Since A0(IdX00−S)B0=IdX00−A0SB0,A_0(Id_X_00-S)B_0=Id_X_00-A_0SB_0, we have ‖A0(IdX00−S)B0−(1−c)IdX00‖ℬ(X00)<1/8.\|A_0(Id_X_00-S)B_0-(1-c)Id_X_00\|_B(X_00)<1/8. Thus A0(IdX00−S)B0A_0(Id_X_00-S)B_0 is invertible and its inverse has norm at most 8/38/3. Since IdX00−S=P00(IdX−T)ι,Id_X_00-S=P_00(Id_X-T) , the same argument gives V(A0(IdX00−S)B0)−1A0P00(IdX−T)ιB0U=IdX.V (A_0(Id_X_00-S)B_0 )^-1A_0P_00(Id_X-T) B_0U=Id_X. Hence IdXId_X factors through IdX−TId_X-T, again with factorization constant at most 3‖P00‖3\|P_00\|. Therefore, X has the uniform primary factorization property, with constant K=3‖P00‖K=3\|P_00\|. Since 1E_1 and 2E_2 are contractive projections, ‖P00‖≤4\|P_00\|≤ 4, so one may take K≤12K≤ 12. Primariness follows automatically from the UPFP and Pełczyński’s decomposition method. ∎ Part I The Automated Pipeline 10. Technical and methodological considerations 10.1. An experimental search process The computational part of the project began as an experimental search procedure rather than as a fully developed autonomous system. In the initial runs, a small number of agents inspected papers, extracted questions, attempted proofs or counterexamples, and retained promising outputs for later review. After these runs produced several promising proof candidates, we formalized the procedure into the pipeline described below. The subsequent refinements were primarily organizational rather than mathematical. The agent instructions did not include problem-specific lemmas, proof strategies, or techniques. Agents were instructed to read the source paper, assess whether an extracted question was approachable, and attempt it using the local context and any references they chose to consult. Later changes concerned classification, record keeping, and coordination: distinguishing full from partial results, identifying answers already present in the literature, preserving unsuccessful attempts, preventing duplication across parallel agents, and standardizing the resulting packets. The pipeline supplied candidate targets, while the protocol provided a common structure in which attempts could be recorded and reviewed; neither prescribed a solution path for an individual problem. Human involvement remained substantial: we selected the broad mathematical area, revised the protocol in response to observed failure modes, chose packets for closer examination, and edited or rewrote the material retained for inclusion. 10.2. Pipeline overview Figure˜1 gives the operational view of the stable version of the experiment. The diagram separates two automatic stages, candidate setup and agent work. Run memory denotes shared state consulted by agents, while human review sits outside the automatic loop. Candidate setupAgent work Corpus arXiv metadata TeX/PDF source Source signals question and conjecture cues Target queue ranked targets lane assignment Run memory global index past outcomes Agent attempt read source try proof/example Make packet result or attempt note Human review Figure 1. Operational view of the automatic proof-discovery pipeline. Candidate setup builds the target queue; agent work consults run memory, makes a packet or attempt note, and passes the result to human review. The implementation combined scripts with protocol files read by the model. Scripts handled the deterministic parts: collecting source material, ranking papers, assigning lanes, and rebuilding the global indexes. The mathematical choices were left to the agent. At the start of a run the agent read a protocol file, selected targets from its lane, inspected the source, and decided whether to attempt a proof, search for a counterexample, record a literature answer, or leave an attempt note. This is not a fully scripted proof engine. It is a controlled way to run Codex sessions as mathematical agents, while preserving enough structure for their outputs to be compared across many papers and sessions. Once the protocols stabilized, most deviations concerned formatting or coordination rather than changes in the mathematical task. 10.3. Candidate setup The candidate setup corresponds to the three boxes in the upper part of Figure˜1. The corpus consisted of arXiv metadata together with TeX source whenever it was available. TeX source was important because open questions, theorem environments, definitions, and references can be located more reliably in source form than from PDF text extraction. The second box, source signals, is a deliberately simple extraction stage. We searched the source for phrases such as “Question”, “Problem”, “Conjecture”, “we ask”, and “we do not know”. These signals did not define the target by themselves. They only marked passages that an agent should read. This distinction matters because many papers ask a question and then answer it nearby, or quote an older problem only to report a known solution. The final setup box is the target queue. Ranking was a triage device, not a measure of absolute importance. For the Banach-space run, the ranker favoured functional-analytic terminology, explicit question or conjecture language, publication and affiliation signals, and references to central work in the area. The goal is practical: to give the agents a stream of papers likely to contain serious questions that were still specific enough for an autonomous attempt. 10.4. Agent work The agent workflow in Figure˜1 begins with the run memory. Before attacking a target, an agent checks whether the paper, the question, or a close variant has already appeared in the run. The same check helps avoid same paper false positives, in which a paper states a question and then answers it nearby. If the target has already been covered, the agent is instructed to record only what is useful for future avoidance and then move on. The next stage is the mathematical attempt. After reading the relevant source passage, the agent decides whether to seek a proof or counterexample, establish a special case, isolate an obstruction, or identify an applicable theorem from the literature. The instructions give priority to full proofs and counterexamples, followed by substantial partial results and conditional theorems. Literature searches are used to identify directly relevant prior results, particularly before an output is classified as a full solution; they are not intended to constitute an exhaustive review. If an existing answer is found, the agent records it and moves to another target. Otherwise, it proceeds with the mathematical attempt. The final stage in the agent workflow is make packet. If the attempt produces a substantive mathematical claim, the agent writes either a packet or a compact attempt note. The protocol instructs agents not to call a proof complete when it depends on an unproved lemma, to label conditional arguments honestly, and to record novelty checks for claimed full solutions or counterexamples. These safeguards reduce overclaiming, but do not eliminate it. Several packets later changed status after review. Thus packets should be read as structured claims for inspection, not as formal certificates. In the case of partial results, agents can be encouraged to push the argument to a full result, in which case they can produce an additional packet. The early runs used one or two agents under close human supervision. Later runs are organized to support parallel work. In the main Banach run, each paper is assigned to one of twenty possible lanes by a deterministic hash of its arXiv identifier. An agent launched on a lane asks a queue helper for suitable targets in that lane. 10.5. Run memory and packet records Run memory is the shared state behind the experiment. Each substantial result or literature identification is recorded there, together with the papers and questions already associated with full solutions, counterexamples, partial results, conditional results, literature answers, proof gaps, or attempts. Agents are instructed to consult this memory before committing to a target and to update it after each attempt. In parallel runs, the same memory also records which targets are currently being pursued, reducing collisions among agents. The output of an agent run is not necessarily a solution packet. If an attempt fails but contains useful information, the agent can leave an attempt note. If it produces a result, a serious reduction, a proof gap, or a literature identification, it updates the run memory and, when appropriate, writes a human readable packet. The packet is the persistent record. It identifies the source paper, the question or conjecture being addressed, the proposed result, and enough evidence for a human reader to reconstruct the claim. Once the packet format had been standardized, packets also included the original arXiv PDF, a brief account of the proof idea, the references consulted, and any code or exact computations used in the argument. The packet categories exist because early runs made clear that “solved” was too coarse a label. We use the following distinctions. (i) Full solution. A claimed proof of the extracted question as stated in the source paper. These packets are treated as high priority candidates for human review, not as formally certified proofs. (i) Counterexample. A construction satisfying the hypotheses of a question or conjecture while violating its proposed conclusion. As with full solutions, these require later checks of both the proof and the novelty status. (i) Partial result. A solved subcase, theorem adjacent to the original problem, meaningful reduction, sharp obstruction, or quantitative improvement that does not settle the full source question. (iv) Conditional result. An argument whose remaining dependency is isolated explicitly, for example an unproved lemma, a scalar inequality, or a computational check. (v) Literature answer. A record that the question is already answered elsewhere. Within this category we distinguish two cases. An explicit literature answer means that another paper explicitly answers the original question. An implied literature answer means that an existing theorem answers the question only after an identification or reformulation made by the agent. Both kinds of record are useful for avoiding duplication, but neither is counted as a new result of the pipeline. (vi) Proof gap. A possible gap identified incidentally in a source paper or in one of our earlier packets. This category is included because agents began pointing out such gaps while attempting to solve other problems. We do not systematically search for proof gaps, and a gap is not counted as a solution unless it subsequently leads to a corrected theorem or a genuine counterexample. These gaps have not been independently verified and may therefore reflect errors made by the model. These categories respond to concrete failure modes. Sometimes an agent finds a theorem in the literature and initially writes as if it has proved something new. Sometimes it solves a nearby subcase but not the actual question. Sometimes it finds a real obstruction, but only under an additional assumption. Sometimes, while attempting a target, it notices a possible gap in the source paper or in an earlier packet. The taxonomy is a way of making these outcomes visible rather than hiding them inside a single success/failure label. 10.6. Scale of the math.FA run The Banach-space run grew in stages. The first small-scale pass began with a ranked metadata pool of 439 papers. From this pool we selected 35 papers for source probing, and the deterministic source scan recorded 166 open-problem signals across 28 of them. This first pass was mainly a feasibility test of whether the model could find real questions and produce packets that a human would want to read. The main pipeline run, from which the principal results reported in this paper are drawn, then used a source backed queue for math.FA. Candidate papers were ranked using deterministic metadata and source text signals. Papers were admitted to the queue if source text could be extracted, at least one deterministic open problem signal was detected, and the paper passed a functional analysis relevance filter. This procedure produced a queue of 1,433 papers. All aggregate counts, rates, and quantitative results reported in this paper refer to this 1,433 paper run. After this run, we expanded the source corpus to include the full modern math.FA collection of 35,297 papers published from 2000 through 2026. Source text was successfully extracted from 34,890 of these papers. The deterministic scan identified 42,483 open problem signals across 15,666 papers. This expanded corpus will be used for live updates on the project website111Future live updates may include solutions generated by the newer GPT 5.6 model. None of the results discussed in this paper were generated by that model., but it is not included in the aggregate counts reported in this paper. 10.7. Summary of Results This section summarizes the aggregate output of the discovery run. We separate active run records from packets that entered human review, since these represent different stages of the workflow. The tables below report both the total number of records or packets and the corresponding number of unique sources. Unique sources are counted by primary arXiv identifier, so the two columns need not agree: a single source paper may contribute more than one result, and multiple attempts may target the same paper. Category Records Unique sources Full solutions 126 124 Counterexamples 102 101 Partial results 211 185 Conditional results 14 14 Literature already answered 140 137 Literature implied answers 116 114 Proof gaps 18 17 Attempt records 925 822 All active solution packets 709 653 All registry records 1127 945 Table 1. Aggregate counts for records in the active run indexes. Packets moved to human review are reported separately in Table˜2. Table˜1 reports the active state of the run. The solution categories count packets that remain in the active solution folders; packets moved to human review are excluded from these counts and reported separately. Differences between the record count and the unique source count arise when a single source paper contributes more than one record. For instance, arXiv:1901.07866 contributes two records; arXiv:2312.14711 contributes two records in the full solution category. Partial results can likewise include multiple records for the same target, since distinct partial attacks may yield different intermediate results that the model judged worth preserving. Attempt records should not be interpreted as disjoint mathematical results. Rather, they are bookkeeping entries for searches, failed attacks, same paper triage, and partial investigations. The registry is broader still: it serves as the long term memory of the run, recording active packets, reviewed packets, failures, and other outcomes used to avoid rediscovering the same target. Human review outcome Packets Unique sources Verified 31 29 Rejected 10 10 Table 2. Human reviewed packets grouped by recorded review outcome. These packets have been moved out of the active solution folders and are therefore not counted as active solution packets in Table˜1. Table˜2 reports the corresponding outcomes for packets that entered human review. These counts are kept separate from the active run counts because review changes the status of a packet. Once a packet is moved into the human review workflow, it no longer represents an unresolved active solution packet. The table therefore records the assigned review outcome while preserving the same distinction between total packets and unique source papers. 10.8. Scope, verification, novelty, and attribution The labels “full solution” and “counterexample” describe the pipeline’s current classification of an attempt. They should not be interpreted as guarantees that the argument is correct, that it addresses the source question exactly as intended, or that the result is novel. An attempt may contain a mathematical error, rely on an incorrect interpretation or extraction of the problem statement, overlook a hypothesis, or establish only a nearby result. As reflected in the verified and rejected outcomes reported in Table˜2, subsequent model and human review can reveal errors, gaps, or misunderstandings that were not apparent in the initial packet. We conducted targeted literature searches and recorded known answers where they were found, but these searches were not exhaustive. In particular, the absence of a result from the references inspected by the pipeline should not be taken as evidence that the result is new. A proof or counterexample classified as potentially new may already appear, either explicitly or implicitly, in the existing literature, possibly in different terminology or as a consequence of a more general theorem. Claims of novelty therefore remain provisional until the relevant literature has been examined more thoroughly by researchers familiar with the area. All reported attempts should be treated as candidates for mathematical inspection rather than as independently certified results. Establishing a result requires more than producing a plausible argument. It requires understanding the proof, checking that it addresses the intended problem, identifying and correcting errors where necessary, determining its relationship to existing work, and explaining its significance within the surrounding mathematical context. For any result that is ultimately established, mathematical credit belongs to the people who formulated the question and to the mathematicians who understand, verify, correct, develop, and contextualize the argument. These contributions are central to both the mathematical value of the result and its proper attribution. The model and pipeline are best understood as research tools that can support the search for connections, the generation of candidate ideas, and the organization of possible arguments. 11. Selected results from the automated pipeline This section presents a small selection of results obtained by the automated pipeline described above. The examples were chosen arbitrarily by the first-named author and are intended only to illustrate the range and character of the mathematics produced by the pipeline; they should not be regarded as exhaustive, representative, or selected according to mathematical significance. For readability, the original arguments have been rewritten and reorganised to improve their exposition. The complete collection of results and their accompanying records is available at the project website. Each example below records its origin, the question addressed, and the resulting answer and proof. 11.1. A tube construction for uniformly Lipschitz maps on decomposable Banach balls Origin. Question 2 in the paper of Barroso and Ferreira [13] asks whether the closed unit ball of every infinite-dimensional Banach space admits a fixed-point-free uniformly Lipschitz self-map with null minimal displacement. Question 11.1. Let X be an infinite-dimensional Banach space. Does there exist a map T:BX→BXT B_X→ B_X such that Fix(T)=∅,d(T,BX):=infx∈BX‖Tx−x‖=0,supn≥1Lip(Tn)<∞?Fix(T)= , d(T,B_X):= _x∈ B_X\|Tx-x\|=0, _n≥ 1Lip(T^n)<∞? Theorem 11.2. Suppose that X=Y⊕ZX=Y Z as a topological direct sum, where Y and Z are closed infinite-dimensional subspaces. Then there is a map T:BX→BXT B_X→ B_X such that Fix(T)=∅,d(T,BX)=0,supn≥1Lip(Tn)<∞.Fix(T)= , d(T,B_X)=0, _n≥ 1Lip(T^n)<∞. Proof. We begin with three auxiliary observations. For a normed space E, define the radial retraction κE:E→BE _E E→ B_E by κE(x)=x,‖x‖≤1,x‖x‖,‖x‖>1. _E(x)= casesx,&\|x\|≤ 1,\\[2.84526pt] x\|x\|,&\|x\|>1. cases This map is 22-Lipschitz. Indeed, suppose that ‖x‖≤‖y‖\|x\|≤\|y\|. The assertion is immediate if x,y∈BEx,y∈ B_E. If ‖x‖≤1<‖y‖\|x\|≤ 1<\|y\|, then ‖x−y‖y‖≤‖x−y‖+‖y−y‖y‖=‖x−y‖+‖y‖−1≤2‖x−y‖. \|x- y\|y\| \|≤\|x-y\|+ \|y- y\|y\| \|=\|x-y\|+\|y\|-1≤ 2\|x-y\|. If 1<‖x‖≤‖y‖1<\|x\|≤\|y\|, then ‖x‖x‖−y‖y‖ \| x\|x\|- y\|y\| \| ≤‖x−y‖x‖+‖y‖|1‖x‖−1‖y‖| ≤ \|x-y\|\|x\|+\|y\| | 1\|x\|- 1\|y\| | =‖x−y‖x‖+‖y‖−‖x‖x‖≤2‖x−y‖. = \|x-y\|\|x\|+ \|y\|-\|x\|\|x\|≤ 2\|x-y\|. We next record an estimate for selecting a point on a sphere whose radius varies. Let RZ:BZ→SZR_Z B_Z→ S_Z be a Lipschitz retraction, put L=Lip(RZ),L=Lip(R_Z), and, for r>0r>0 and q∈Zq∈ Z, define H(r,q)=rRZ(κZ(q/r)).H(r,q)=rR_Z ( _Z(q/r) ). Then ∥H(r,q)−H(s,p)∥≤2L∥q−p∥+(1+L)|r−s|(r,s>0,p,q∈Z).\|H(r,q)-H(s,p)\|≤ 2L\|q-p\|+(1+L)|r-s| (r,s>0,\ p,q∈ Z). For fixed r>0r>0, the estimate for κZ _Z gives ‖H(r,q)−H(r,p)‖≤2L‖q−p‖.\|H(r,q)-H(r,p)\|≤ 2L\|q-p\|. To compare the radii, suppose that 0<r≤s0<r≤ s. For fixed q∈Zq∈ Z, ‖H(r,q)−H(s,q)‖≤s−r+rL‖κZ(q/r)−κZ(q/s)‖.\|H(r,q)-H(s,q)\|≤ s-r+rL\| _Z(q/r)- _Z(q/s)\|. Writing c=‖q‖c=\|q\|, a direct consideration of the cases c≤rc≤ r, r<c<sr<c<s, and s≤cs≤ c gives r‖κZ(q/r)−κZ(q/s)‖≤s−r.r\| _Z(q/r)- _Z(q/s)\|≤ s-r. The claimed estimate follows. Finally, every infinite-dimensional Banach space Y admits a Lipschitz function η:SY→(0,1]η S_Y→(0,1] whose infimum is zero. To see this, use Riesz’ lemma to choose a sequence (un)n=1∞⊂SY(u_n)_n=1^∞⊂ S_Y and δ>0δ>0 such that ‖un−um‖≥δ(n≠m).\|u_n-u_m\|≥δ (n≠ m). Let (εn)n=1∞( _n)_n=1^∞ be a decreasing sequence of positive numbers converging to zero, and define η(u)=min1,infn∈ℕ(εn+‖u−un‖)(u∈SY).η(u)= \1, _n ( _n+\|u-u_n\| ) \ (u∈ S_Y). The function η is 11-Lipschitz and satisfies η(un)≤εn(n∈ℕ),η(u_n)≤ _n (n ), so its infimum is zero. It is nevertheless strictly positive at every point. Indeed, at most one member of the separated sequence can lie strictly within distance δ/2δ/2 of a given u∈SYu∈ S_Y; all the remaining terms in the infimum are at least δ/2δ/2, while the possible exceptional term is also positive. We now construct the required map. Let P:X→YP X→ Y and Q:X→ZQ X→ Z be the bounded projections associated with the decomposition, and choose a,b>0a,b>0 such that a+b≤1.a+b≤ 1. By the theorem of Benyamini and Sternfeld [19], there are Lipschitz retractions RY:BY→SY,RZ:BZ→SZ.R_Y B_Y→ S_Y, R_Z B_Z→ S_Z. Define A:Y→aSYA Y→ aS_Y by A(y)=aRY(κY(y/a)).A(y)=aR_Y ( _Y(y/a) ). Then A is Lipschitz and fixes every point of aSYaS_Y. Using the function η constructed above, define r(y)=bη(y/a)(y∈aSY).r(y)=bη(y/a) (y∈ aS_Y). Thus r is Lipschitz, 0<r(y)≤b(y∈aSY),0<r(y)≤ b (y∈ aS_Y), and infy∈aSYr(y)=0. _y∈ aS_Yr(y)=0. For x∈BXx∈ B_X, set y(x)=A(Px),ρ(x)=r(y(x)),y(x)=A(Px), ρ(x)=r(y(x)), and Φ(x)=H(ρ(x),Qx)=ρ(x)RZ(κZ(Qx/ρ(x))). (x)=H(ρ(x),Qx)=ρ(x)R_Z ( _Z(Qx/ρ(x)) ). Finally, define T(x)=y(x)−Φ(x).T(x)=y(x)- (x). The preceding estimates show that T is Lipschitz. Moreover, ‖y(x)‖=a,‖Φ(x)‖=ρ(x)≤b,\|y(x)\|=a, \| (x)\|=ρ(x)≤ b, and hence ‖T(x)‖≤a+b≤1.\|T(x)\|≤ a+b≤ 1. Thus T maps BXB_X into itself. Define also S(x)=y(x)+Φ(x).S(x)=y(x)+ (x). We claim that T2=S,T3=T.T^2=S, T^3=T. Fix x∈BXx∈ B_X, and write y=y(x),ρ=ρ(x),Φ(x)=ρzy=y(x), ρ=ρ(x), (x)=ρ z for some z∈SZz∈ S_Z. Since P(y−ρz)=yP(y-ρ z)=y and A(y)=yA(y)=y, the base point and the radius are unchanged after applying T. Furthermore, Q(y−ρz)ρ=−z∈SZ. Q(y-ρ z)ρ=-z∈ S_Z. Both κZ _Z and RZR_Z fix points of SZS_Z, and therefore T(Tx)=y+ρz=S(x).T(Tx)=y+ρ z=S(x). The same argument applied to S(x)=y+ρzS(x)=y+ρ z gives T(Sx)=y−ρz=T(x).T(Sx)=y-ρ z=T(x). Thus the iterates of T alternate between the two Lipschitz maps T and S, and consequently supn≥1Lip(Tn)<∞. _n≥ 1Lip(T^n)<∞. The map T has no fixed point. Indeed, if T(x)=xT(x)=x, then T2(x)=T(x)T^2(x)=T(x). On the other hand, the formulas above give T(x)=y−ρz,T2(x)=y+ρz.T(x)=y-ρ z, T^2(x)=y+ρ z. This would imply 2ρz=02ρ z=0, which is impossible because ρ>0ρ>0 and z∈SZz∈ S_Z. It remains to prove that the minimal displacement is zero. The following scaling-and-radial-retraction argument follows [13, Proposition 4.12]. Fix z0∈SZz_0∈ S_Z and use the sequence (un)n=1∞(u_n)_n=1^∞ occurring in the construction of η. Put xn=aun+bη(un)z0(n∈ℕ).x_n=au_n+bη(u_n)z_0 (n ). Since ‖xn‖≤a+b≤1\|x_n\|≤ a+b≤ 1, the sequence lies in BXB_X. The defining formula for T gives T(xn)=aun−bη(un)z0,T(x_n)=au_n-bη(u_n)z_0, and therefore ‖T(xn)−xn‖=2bη(un)≤2bεn⟶0.\|T(x_n)-x_n\|=2bη(u_n)≤ 2b _n 0. Hence d(T,BX)=0d(T,B_X)=0. ∎ 11.2. Uniform Hölder nonexpansive maps on decomposable Banach balls Origin. The final open-question list in Barroso’s paper [14] asks whether the closed unit ball BXB_X fails the fixed point property for uniformly α-Hölder nonexpansive maps with null minimal displacement in several important classes of Banach spaces. Question 11.3. Let X be an infinite-dimensional Banach space and let α∈(0,1)α∈(0,1). Does there exist a map T:BX→BXT B_X→ B_X such that Fix(T)=∅,d(T,BX):=infx∈BX‖Tx−x‖=0,Fix(T)= , d(T,B_X):= _x∈ B_X\|Tx-x\|=0, and ∥Tnx−Tny∥≤∥x−y∥α(x,y∈BX,n≥1)?\|T^nx-T^ny\|≤\|x-y\|^α (x,y∈ B_X,\ n≥ 1)? In particular, what happens when X is isomorphic to a Hilbert space or when X is reflexive and has an unconditional basis? Answer and proof. The answer is affirmative whenever X=Y⊕Z,X=Y Z, where Y and Z are closed infinite-dimensional subspaces; equivalently, whenever X is decomposable. This includes spaces isomorphic to an infinite-dimensional Hilbert space and spaces with an unconditional basis. By Theorem˜11.2, there is a map U:BX→BXU B_X→ B_X such that Fix(U)=∅,d(U,BX)=0,M:=supn≥1Lip(Un)<∞.Fix(U)= , d(U,B_X)=0, M:= _n≥ 1Lip(U^n)<∞. It remains to convert these uniformly Lipschitzian dynamics into uniformly α-Hölder nonexpansive dynamics. Put C=2MC=2M and η=min2,C−1/(1−α).η= \2,C^-1/(1-α)\. Choose r>0r>0 sufficiently small that 2r≤ηα2r≤η^α. In particular, r<1r<1. Let Rr:BX→rBXR_r B_X→ rB_X be the radial retraction Rr(x)=x,‖x‖≤r,rx‖x‖,‖x‖>r.R_r(x)= casesx,&\|x\|≤ r,\\[2.84526pt] rx\|x\|,&\|x\|>r. cases This retraction is 22-Lipschitz on every normed space. Define S:rBX→rBXS rB_X→ rB_X by S(z)=rU(z/r).S(z)=rU(z/r). Then Sn(z)=rUn(z/r),Lip(Sn)≤M(n≥1),S^n(z)=rU^n(z/r), (S^n)≤ M (n≥ 1), and S is fixed-point free with null minimal displacement on rBXrB_X. Set V=SRr:BX⟶BX.V=SR_r B_X B_X. Because S(rBX)⊆rBXS(rB_X) rB_X and RrR_r is the identity on rBXrB_X, Vn=SnRr(n≥1).V^n=S^nR_r (n≥ 1). For x,y∈BXx,y∈ B_X, write d=‖x−y‖d=\|x-y\|. Since Vnx,Vny∈rBXV^nx,V^ny∈ rB_X, ‖Vnx−Vny‖≤minM‖Rrx−Rry‖,2r≤minCd,2r.\|V^nx-V^ny\|≤ \M\|R_rx-R_ry\|,2r\≤ \Cd,2r\. If d≤ηd≤η, then the definition of η gives Cd≤dαCd≤ d^α. If d≥ηd≥η, then 2r≤ηα≤dα2r≤η^α≤ d^α. Consequently, ∥Vnx−Vny∥≤dα(x,y∈BX,n≥1).\|V^nx-V^ny\|≤ d^α (x,y∈ B_X,\ n≥ 1). If Vx=xVx=x, then x∈rBXx∈ rB_X, so Rrx=xR_rx=x and Sx=xSx=x, contradicting Fix(S)=∅Fix(S)= . Hence V has no fixed point. Finally, because d(S,rBX)=0d(S,rB_X)=0, there are zj∈rBXz_j∈ rB_X such that ‖Szj−zj‖→0\|Sz_j-z_j\|→ 0. For these points Rrzj=zjR_rz_j=z_j, and therefore ‖Vzj−zj‖=‖Szj−zj‖⟶0.\|Vz_j-z_j\|=\|Sz_j-z_j\| 0. Thus d(V,BX)=0d(V,B_X)=0. For the stated consequences, split a Hilbert space into two closed orthogonal summands of infinite dimension and transport the decomposition through an isomorphism. If X has an unconditional basis, its odd and even coordinate subspaces give the required decomposition. This argument does not settle the arbitrary reflexive case. ∎ 11.3. Nearly isometric embeddability need not imply almost Lipschitz embeddability Origin. This is Problem 4 in Section 5 of Baudier and Lancien [15]. Question 11.4. Exhibit metric spaces X and Y such that X nearly isometrically embeds into Y, but X does not almost Lipschitz embed into Y. More precisely, X almost Lipschitz embeds into Y if there are constants r>0r>0 and D≥1D≥ 1 such that, for every continuous function φ:[0,∞)→[0,1) [0,∞)→[0,1) satisfying φ(0)=0,φ(t)>0(t>0), (0)=0, (t)>0 (t>0), there is a map fφ:X→Yf_ X→ Y for which rdX(x,y)φ(dX(x,y))≤dY(fφ(x),fφ(y))≤DrdX(x,y).rd_X(x,y) (d_X(x,y))≤ d_Y(f_ (x),f_ (y))≤ Drd_X(x,y). For nearly isometric embeddability, let P consist of the continuous functions ρ satisfying ρ(t)=t(0≤t≤1),ρ(t)≤t(t≥1),ρ(t)t⟶0(t→∞),ρ(t)=t (0≤ t≤ 1), ρ(t)≤ t (t≥ 1), ρ(t)t 0 (t→∞), and let Ω consist of the functions ω satisfying ω(0) ω(0) =0, =0, t t ≤ω(t) ≤ω(t) (0≤t≤1), (0≤ t≤ 1), ω(t) ω(t) =t =t (t≥1), (t≥ 1), ω(t)t ω(t)t ⟶∞ ∞ (t↓0). (t 0). The space X nearly isometrically embeds into Y if, for every (ρ,ω)∈×Ω(ρ,ω) × , there is a map f:X→Yf X→ Y such that ρ(dX(x,y))≤dY(f(x),f(y))≤ω(dX(x,y)).ρ(d_X(x,y))≤ d_Y(f(x),f(y))≤ω(d_X(x,y)). Answer and proof. We first record a concave-majorant construction. Given ρ∈ρ , there is a continuous, nondecreasing, concave, unbounded function g:[0,∞)→[0,∞)g [0,∞)→[0,∞) satisfying g(0)=0,g(t)=t(0≤t≤1),ρ(t)≤g(t)≤t,g(t)t⟶0.g(0)=0, g(t)=t (0≤ t≤ 1), ρ(t)≤ g(t)≤ t, g(t)t 0. To see this, define h(t)=t,0≤t≤1,1+logt,t≥1,ρ0(t)=maxρ(t),h(t).h(t)= casest,&0≤ t≤ 1,\\ 1+ t,&t≥ 1, cases _0(t)= \ρ(t),h(t)\. Choose a sequence εj↓0 _j 0. Since ρ0(t)/t→0 _0(t)/t→ 0, each number Aj=supt≥0(ρ0(t)−εjt)A_j= _t≥ 0 ( _0(t)- _jt ) is finite. Set g(t)=inf(t∪Aj+εjt:j≥1).g(t)= (\t\∪\A_j+ _jt:j≥ 1\ ). Every affine function in this infimum dominates ρ0 _0. Thus ρ≤ρ0≤g≤tρ≤ _0≤ g≤ t. The function g is concave and nondecreasing. For each fixed j, lim supt→∞g(t)t≤lim supt→∞Aj+εjt=εj, _t→∞ g(t)t≤ _t→∞ A_j+ _jtt= _j, so g(t)/t→0g(t)/t→ 0. Finally, g≥hg≥ h, and hence g is unbounded. A nondecreasing concave function with g(0)=0g(0)=0 is subadditive. Consequently, dg(m,n)=g(|m−n|)d_g(m,n)=g(|m-n|) defines a metric on ℕ0N_0. Take X=ℕ0X=N_0 with its usual metric. For every ρ∈ρ , choose a function gρg_ρ as above, and let YρY_ρ be a copy of ℕ0N_0 with metric dρ(m,n)=gρ(|m−n|).d_ρ(m,n)=g_ρ(|m-n|). Let Y be the metric wedge of the family (Yρ)ρ∈(Y_ρ)_ρ , obtained by identifying all zero points to a common basepoint o. If a∈Yρa∈ Y_ρ and b∈Yσb∈ Y_σ, define dY(a,b)=gρ(|a−b|),ρ=σ,gρ(a)+gσ(b),ρ≠σ.d_Y(a,b)= casesg_ρ(|a-b|),&ρ=σ,\\[2.84526pt] g_ρ(a)+g_σ(b),&ρ≠σ. cases We first show that X nearly isometrically embeds into Y. Fix ρ∈ρ and ω∈Ωω∈ , and send n∈ℕ0n _0 to the point n in the ρ-component. For k=|m−n|≥1k=|m-n|≥ 1, ρ(k)≤gρ(k)=dY(Fρ,ω(m),Fρ,ω(n))≤k=ω(k).ρ(k)≤ g_ρ(k)=d_Y(F_ρ,ω(m),F_ρ,ω(n))≤ k=ω(k). The case k=0k=0 is immediate. We next prove that Y contains no bi-Lipschitz copy of the usual integer ray. Suppose that F:ℕ0→YF _0→ Y and constants c,C>0c,C>0 satisfy c|m−n|≤dY(F(m),F(n))≤C|m−n|(m,n∈ℕ0).c|m-n|≤ d_Y(F(m),F(n))≤ C|m-n| (m,n _0). Put Rn=dY(F(n),o)R_n=d_Y(F(n),o). Then Rn≥dY(F(n),F(0))−R0≥cn−R0,R_n≥ d_Y(F(n),F(0))-R_0≥ cn-R_0, so RnR_n grows at least linearly. On the other hand, dY(F(n+1),F(n))≤Cd_Y(F(n+1),F(n))≤ C. If F(n)F(n) and F(n+1)F(n+1) belong to different wedge components, their distance is Rn+Rn+1R_n+R_n+1, which is eventually larger than C. Thus, from some index onward, the whole sequence lies in a single component Yρ0Y_ _0. Write F(n)=an∈Yρ0F(n)=a_n∈ Y_ _0 for n≥Nn≥ N. Since gρ0(|an+1−an|)=dY(F(n+1),F(n))≤Cg_ _0(|a_n+1-a_n|)=d_Y(F(n+1),F(n))≤ C and gρ0g_ _0 is nondecreasing and unbounded, there is B<∞B<∞ such that |an+1−an|≤B(n≥N).|a_n+1-a_n|≤ B (n≥ N). Hence |am−an|≤B|m−n|(m,n≥N),|a_m-a_n|≤ B|m-n| (m,n≥ N), and therefore dY(F(m),F(n))≤gρ0(B|m−n|).d_Y(F(m),F(n))≤ g_ _0(B|m-n|). Because gρ0(t)/t→0g_ _0(t)/t→ 0, the right-hand side is o(|m−n|)o(|m-n|) as |m−n|→∞|m-n|→∞, contradicting the lower estimate dY(F(m),F(n))≥c|m−n|d_Y(F(m),F(n))≥ c|m-n|. Thus no bi-Lipschitz embedding ℕ0→YN_0→ Y exists. Finally, suppose that X almost Lipschitz embeds into Y. Choose φ(t)=t/2,0≤t≤1,1/2,t≥1. (t)= casest/2,&0≤ t≤ 1,\\ 1/2,&t≥ 1. cases Every nonzero distance in ℕ0N_0 is at least 11, so the corresponding map would satisfy r2|m−n|≤dY(fφ(m),fφ(n))≤Dr|m−n|(m≠n). r2|m-n|≤ d_Y(f_ (m),f_ (n))≤ Dr|m-n| (m≠ n). This would be a bi-Lipschitz embedding of the integer ray into Y, a contradiction. Hence X does not almost Lipschitz embed into Y. ∎ 11.4. A c0c_0-vector outside the canonical closed span in a variable-exponent space Origin. This answers negatively the open question repeated on page 8 of Talponen’s paper [70]. Question 11.5. Let p:ℕ→[1,∞]p →[1,∞]. For x=(xn)∈ℓ∞x=(x_n)∈ ^∞, define |||x|||(1)=|x1|⊞p(1)|x2|,|||x|||(k)=|||x|||(k−1)⊞p(k)|xk+1|, \! \! x \! \! _(1)=|x_1| _p(1)|x_2|, \! \! x \! \! _(k)= \! \! x \! \! _(k-1) _p(k)|x_k+1|, where a⊞rb=(ar+br)1/ra _rb=(a^r+b^r)^1/r when r<∞r<∞, and put Φ(x)=limk→∞|||x|||(k),ℓp(⋅)=x∈ℓ∞:Φ(x)<∞. (x)= _k→∞ \! \! x \! \! _(k), ^p(·)=\x∈ ^∞: (x)<∞\. Writing [(en)][(e_n)] for the closed linear span in ℓp(⋅) ^p(·) of the canonical unit vectors, must one always have ℓp(⋅)∩c0=[(en)]? ^p(·)∩ c_0=[(e_n)]? Answer and proof. No. We construct a function p:ℕ→[2,∞)p →[2,∞) taking only finite values and an element x∈ℓp(⋅)∩c0∖[(en)].x∈ ^p(·)∩ c_0 [(e_n)]. Let PnP_n denote truncation after the nnth coordinate and let Qn=I−PnQ_n=I-P_n. We first note that, for every y∈ℓp(⋅)y∈ ^p(·), y∈[(en)]⟺‖Qny‖ℓp(⋅)⟶0.y∈[(e_n)] \|Q_ny\|_ ^p(·) 0. Indeed, the reverse implication follows because PnyP_ny is finitely supported and Pny→yP_ny→ y. Conversely, if y∈[(en)]y∈[(e_n)] and ε>0 >0, choose a finitely supported z, supported in 1,…,N\1,…,N\, such that ‖y−z‖ℓp(⋅)<ε\|y-z\|_ ^p(·)< . The recursive norm is coordinatewise monotone in the absolute values, so every coordinate projection is contractive. Hence, for n≥Nn≥ N, ‖Qny‖ℓp(⋅)=‖Qn(y−z)‖ℓp(⋅)≤‖y−z‖ℓp(⋅)<ε.\|Q_ny\|_ ^p(·)=\|Q_n(y-z)\|_ ^p(·)≤\|y-z\|_ ^p(·)< . Fix 0<c<10<c<1, and for j≥1j≥ 1 put rj=j+1,mj=jrj.r_j=j+1, m_j=j^r_j. Set N1=2N_1=2 and Nj+1=Nj+mjN_j+1=N_j+m_j, and let Bj=Nj,Nj+1,…,Nj+1−1.B_j=\N_j,N_j+1,…,N_j+1-1\. Define x∈ℓ∞x∈ ^∞ by x1=1,xn=cj(n∈Bj).x_1=1, x_n= cj (n∈ B_j). Since c/j→0c/j→ 0, we have x∈c0x∈ c_0. Define p:ℕ→[2,∞)p →[2,∞) by p(k)=rjwheneverNj−1≤k≤Nj+1−2.p(k)=r_j N_j-1≤ k≤ N_j+1-2. These intervals partition ℕN, so p is well defined. Let Aj=‖PNj+1−1x‖ℓp(⋅)A_j= \|P_N_j+1-1x \|_ ^p(·) be the norm of the initial segment consisting of x1x_1 and the first j complete blocks. Then A0=1A_0=1, and all exponents used while adjoining block j are equal to rjr_j. Therefore Aj=(Aj−1rj+mj(cj)rj)1/rj=(Aj−1rj+crj)1/rj.A_j= (A_j-1^r_j+m_j ( cj )^r_j )^1/r_j= (A_j-1^r_j+c^r_j )^1/r_j. Since Aj−1≥1A_j-1≥ 1, Aj=Aj−1(1+(cAj−1)rj)1/rj≤Aj−1(1+crj)1/rj,A_j=A_j-1 (1+ ( cA_j-1 )^r_j )^1/r_j≤ A_j-1(1+c^r_j)^1/r_j, and consequently Aj≤∏i=1j(1+cri)1/ri.A_j≤ _i=1^j(1+c^r_i)^1/r_i. This infinite product converges because ∑i=1∞log(1+cri)ri≤∑i=1∞criri<∞. _i=1^∞ (1+c^r_i)r_i≤ _i=1^∞ c^r_ir_i<∞. Every finite prefix ending inside a block has norm at most the norm after the complete block. Thus supn‖Pnx‖ℓp(⋅)<∞ _n\|P_nx\|_ ^p(·)<∞, and hence x∈ℓp(⋅)x∈ ^p(·). It remains to show that x∉[(en)]x∉[(e_n)]. Consider the tail QNj−1xQ_N_j-1x. Its projection onto BjB_j has mjm_j coordinates equal to c/jc/j, with all relevant exponents equal to rjr_j, and therefore has norm (mj(cj)rj)1/rj=c. (m_j ( cj )^r_j )^1/r_j=c. By coordinatewise monotonicity, ‖QNj−1x‖ℓp(⋅)≥c(j≥1).\|Q_N_j-1x\|_ ^p(·)≥ c (j≥ 1). The tails therefore do not converge to zero in norm, so the preceding equivalence yields x∉[(en)]x∉[(e_n)]. ∎ 11.5. A radial-twist counterexample for a coarse sum Origin. This is Problem 5.7 in Braga’s paper [20]. It appears on page 20 of the arXiv version. Question 11.6. Let X,Y1,Y2X,Y_1,Y_2 be Banach spaces, and let f=(f1,f2):X⟶Y1⊕Y2f=(f_1,f_2) X Y_1 Y_2 be a coarse Lipschitz embedding. Must there exist an infinite-dimensional subspace X0⊆X_0 X such that either f1|X0:X0→Y1f_1|_X_0 X_0→ Y_1 or f2|X0:X0→Y2f_2|_X_0 X_0→ Y_2 is a coarse Lipschitz embedding? Answer and proof. No. We construct a bi-Lipschitz embedding F=(F1,F2):ℓ2⟶ℓ2⊕2ℓ2F=(F_1,F_2) _2 _2 _2 _2 such that neither coordinate restricts to a coarse embedding on any nonzero linear subspace of ℓ2 _2. Let H=ℓ2H= _2 over the real scalars. For t≥0t≥ 0, define d4(t)=dist(t,4ℤ),φ(t)=π4mind4(t),2,d_4(t)=dist(t,4Z), (t)= π4 \d_4(t),2\, and, for r≥0r≥ 0, set θ(r)=φ(log(1+r)).θ(r)= ( (1+r)). The function φ is π/4π/4-Lipschitz and takes values in [0,π/2][0,π/2]. Moreover, for every k∈ℕ0k _0, θ(e4k−1)=0,θ(e4k+2−1)=π2.θ(e^4k-1)=0, θ(e^4k+2-1)= π2. Define F(x)=(cos(θ(‖x‖))x,sin(θ(‖x‖))x)∈H⊕2H,F(x)= ( (θ(\|x\|))x, (θ(\|x\|))x )∈ H _2H, and write F=(F1,F2)F=(F_1,F_2). We first prove that F is bi-Lipschitz. For t∈[0,π/2]t∈[0,π/2], let Jt:H⟶H⊕2H,Jtz=(cos(t)z,sin(t)z).J_t H H _2H, J_tz=( (t)z, (t)z). Each JtJ_t is an isometry and F(x)=Jθ(‖x‖)xF(x)=J_θ(\|x\|)x. Let r=‖x‖r=\|x\| and s=‖y‖s=\|y\|. If x,y≠0x,y≠ 0, put u=xr,v=ys,a=⟨u,v⟩,c=cos(θ(r)−θ(s)).u= xr, v= ys, a= u,v , c= (θ(r)-θ(s)). Since θ takes values in [0,π/2][0,π/2], we have c∈[0,1]c∈[0,1], and a direct calculation gives ‖F(x)−F(y)‖2=r2+s2−2rsca,‖x−y‖2=r2+s2−2rsa.\|F(x)-F(y)\|^2=r^2+s^2-2rsca, \|x-y\|^2=r^2+s^2-2rsa. If a≥0a≥ 0, then ca≤aca≤ a, so ‖F(x)−F(y)‖≥‖x−y‖\|F(x)-F(y)\|≥\|x-y\|. If a<0a<0, then ‖F(x)−F(y)‖2≥r2+s2≥12‖x−y‖2.\|F(x)-F(y)\|^2≥ r^2+s^2≥ 12\|x-y\|^2. The same estimate is immediate if one of x,yx,y is zero. Hence ‖F(x)−F(y)‖≥12‖x−y‖.\|F(x)-F(y)\|≥ 1 2\|x-y\|. For the upper estimate, assume without loss of generality that r≥sr≥ s. Since ‖Jt−Jt′‖≤|t−t′|\|J_t-J_t \|≤|t-t |, we obtain ‖F(x)−F(y)‖ \|F(x)-F(y)\| ≤‖Jθ(r)(x−y)‖+‖(Jθ(r)−Jθ(s))y‖ ≤\|J_θ(r)(x-y)\|+\|(J_θ(r)-J_θ(s))y\| ≤‖x−y‖+s|θ(r)−θ(s)|. ≤\|x-y\|+s|θ(r)-θ(s)|. The Lipschitz estimate for φ yields |θ(r)−θ(s)|≤π4log1+r1+s.|θ(r)-θ(s)|≤ π4 1+r1+s. Using log(1+u)≤u (1+u)≤ u with u=(r−s)/(1+s)u=(r-s)/(1+s), we get slog1+r1+s≤s1+s(r−s)≤r−s≤‖x−y‖.s 1+r1+s≤ s1+s(r-s)≤ r-s≤\|x-y\|. Therefore 12‖x−y‖≤‖F(x)−F(y)‖≤(1+π4)‖x−y‖, 1 2\|x-y\|≤\|F(x)-F(y)\|≤ (1+ π4 )\|x-y\|, so F is bi-Lipschitz and hence a coarse Lipschitz embedding. It remains to show that neither coordinate works on any nonzero subspace. Let X0⊆HX_0 H be a nonzero linear subspace, and choose u∈X0u∈ X_0 with ‖u‖=1\|u\|=1. For Rk=e4k+2−1,xk=Rku,R_k=e^4k+2-1, x_k=R_ku, we have θ(Rk)=π/2θ(R_k)=π/2, and hence F1(xk)=F1(−xk)=0,‖xk−(−xk)‖=2Rk⟶∞.F_1(x_k)=F_1(-x_k)=0, \|x_k-(-x_k)\|=2R_k ∞. Thus F1|X0F_1|_X_0 is not a coarse embedding. Similarly, with Sk=e4k−1,yk=Sku,S_k=e^4k-1, y_k=S_ku, we have θ(Sk)=0θ(S_k)=0, and therefore F2(yk)=F2(−yk)=0,‖yk−(−yk)‖=2Sk⟶∞.F_2(y_k)=F_2(-y_k)=0, \|y_k-(-y_k)\|=2S_k ∞. Thus F2|X0F_2|_X_0 is not a coarse embedding either. This holds for every nonzero X0X_0, and gives a negative answer to the question. ∎ Acknowledgements. This paper forms part of the first-named author’s PhD research at Lancaster University, conducted under the supervision of Professor N. J. Laustsen. He gratefully acknowledges the support of the Engineering and Physical Sciences Research Council (EPSRC), grant number EP/W524438/1, which has supported his studies. The authors would like to thank Kevin Beanland and Tomasz Kania for proposing problems and providing comments for this project, which helped to shape some of the mathematical directions pursued here. They are also grateful to Ondřej Kalenda, Tommaso Russo, Bruno de Mendonça Braga, Anna Pelczar-Barwacz, Jesús María Fernández Castillo, and Samya Kumar Ray for their kind and careful responses to our original questions concerning the automatic pipeline, and for taking the time to look at the problems and material we sent them. The first named author would like to express his special gratitude to Kevin Beanland for his enthusiastic response to the project, as well as for his encouragement and support. He is also especially grateful to Tomasz Kania for many long conversations about mathematics and artificial intelligence, and for his generosity in answering questions, even at inconvenient times. He would also like to thank Enrique Rozas García and Felix Schwarzfischer for their comments on a preliminary version of this manuscript and for their thoughtful and stimulating conversations about artificial intelligence and its role in mathematical research. Finally, he would like to express his heartfelt thanks to his supervisor, Niels Laustsen, for giving him the freedom to explore, including when that exploration led in unconventional directions. AI usage statement. Large language models were central to the exploratory and drafting stages of this project, and we refer to the main text for a full description of AI usage. The proofs printed here have been edited, checked against the cited literature, and integrated by the authors, who take responsibility for the mathematical content. For the purpose of open access, the author has applied a Creative Commons Attribution (C BY) licence to any Author Accepted Manuscript version arising. Data availability. The source materials used in this study consist of publicly available arXiv papers and their associated source files. For papers for which the pipeline produced a packet, the relevant paper information, packet, and model output are available at the project website. No proprietary, personal, or restricted data were used. Conflict of interest. The authors declare that they have no conflict of interest. References [1] M. Abouzaid, A. J. Blumberg, M. Hairer, J. Kileel, T. G. Kolda, P. D. Nelson, D. Spielman, N. Srivastava, R. Ward, S. Weinberger, and L. Williams, First Proof, arXiv:2602.05192, 2026. [2] M. Abouzaid, N. Srivastava, R. Ward, and L. Williams, First Proof Second Batch, arXiv:2606.18119, 2026. [3] G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. 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