REVIEW 1 major objections 5 minor 84 references
Complemented subspaces of Banach lattices
T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper argues that the Complemented Subspace Problem for Banach lattices has a negative answer: the 1-complemented subspace PS2 of a C(K)-space is not linearly isomorphic to any Banach lattice, and the proof turns on a no-modulus…
desk verdict A well-written survey with sound new free-Banach-lattice propositions; the headline negative solution to the CSP is an external result from [38], so treat it as conditional. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the 'no modulus with respect to a norming sequence' obstruction. Given a norming sequence $(e_n^*)_{n=1}^\infty \subset B_{PS_2^*}$, a modulus of $f \in PS_2$ would be a $g \in PS_2$ with $|e_n^*(f)| = e_n^*(g)$ for every $n$; any sublattice of $\ell_\infty$ supplies such a $g$, namely $|f|$. The theorem from [38] that carries the paper's headline is that for every norming sequence there is some $\xi < \mathfrak{c}$ for which $f = 1_{B_\xi^0} - 1_{B_\xi^1}$ has no modulus. The paper's supporting machinery is the free Banach lattice $\mathrm{FBL}[E]$ with its universal property: Proposition 3.3 turns 'E is isomorphic to a Banach lattice' into the existence of an ideal $I$ with $\mathrm{FBL}[E] = \delta_E(E) \oplus I$, and Section 3 uses this to give criteria in terms of lattice homomorphisms and sets of zeros in $\mathrm{FBL}^{(\infty)}[E] = C_{ph}(B_{E^*})$.
What would settle it
Find one norming sequence $(e_n^*)_{n=1}^\infty$ in $B_{PS_2^*}$ such that for every $\xi < \mathfrak{c}$ there exists $g_\xi \in PS_2$ with $|e_n^*(1_{B_\xi^0} - 1_{B_\xi^1})| = e_n^*(g_\xi)$ for all $n$; under the paper's own reduction this would give PS2 a sublattice-of-$\ell_\infty$ structure and disprove the load-bearing non-isomorphism statement.
Extended reading notes
Core claim
The paper's central claim is that the space $PS_2$, the kernel of a norm-one averaging projection $P$ on the space $JL(\mathcal{B})$, is a 1-complemented subspace of a $C(K)$-space and is not linearly isomorphic to any Banach lattice. In the paper's telling, the argument runs through three reductions: $PS_2$ is an isomorphic predual of $\ell_1(\Gamma)$ and hence an $L_\infty$-space; any Banach lattice structure on it would therefore have to be an AM-space structure, and because $PS_2$ has a countable norming set in its dual ball, it would have to embed as a sublattice of $\ell_\infty$; but the built-in indicator differences $1_{B_\xi^0} - 1_{B_\xi^1}$ have no modulus with respect to any norming sequence in $B_{PS_2^*}$, so no such embedding exists. The manuscript states that this final no-modulus step is proved in [38, Theorem 4.4] and only outlines the route here.
Load-bearing premise
The headline result rests on the theorem cited as [38, Theorem 4.4], namely that every norming sequence in the dual ball of $PS_2$ is defeated by some vector $1_{B_\xi^0} - 1_{B_\xi^1}$ that has no modulus; this survey describes that theorem but does not prove it, so the main conclusion falls if the cited preprint's proof does not hold.
Editorial extensions
If this is right
- If the cited theorem is correct, the Complemented Subspace Problem for Banach lattices has a negative answer in the non-separable setting, while its separable version remains open.
- PS2 is a single example that answers two classical problems at once: it is a 1-complemented subspace of a $C(K)$-space and is neither isomorphic to a $C(K)$-space nor to any Banach lattice.
- A slight modification of PS2 also gives a 1-complemented subspace of a complex Banach lattice that is not linearly isomorphic to any complex Banach lattice, so the complex version of the problem fails as well.
- The free-Banach-lattice criterion reduces the question of whether a space $E$ is a Banach lattice to a splitting condition: $\mathrm{FBL}[E] = \delta_E(E) \oplus I$ for some closed ideal $I$, giving a new way to certify non-lattice spaces.
- PS2 cannot be a hyperplane of any Banach lattice: if $PS_2 \oplus_\infty \mathbb{R}$ were a Banach lattice, the hyperplanes of that AM-space would themselves be Banach lattices, forcing PS2 to be one.
Reading between the lines
- The same modulus obstruction could be used as a detection test on other $L_\infty$-spaces with countable norming sets: search for a norming sequence for which every candidate indicator difference has a modulus, which would indicate that the space is, after all, a sublattice of $\ell_\infty$.
- Because the construction is inherently non-separable, using an almost disjoint family of cardinality continuum, it gives no direct recipe for the still-open separable case; a separable counterexample, if one exists, would need a different source of no-modulus vectors.
- The ideal-splitting criterion suggests operationalizing the Hyperplane Problem for Banach lattices: a hyperplane of a Banach lattice is a lattice exactly when the corresponding evaluation kernel on $\mathrm{FBL}$ can be complemented by an ideal, and Proposition 6.5 already gives four equivalent forms of this condition.
- If the external theorem from [38] fails, most of the survey's headline consequences would collapse, but the free-Banach-lattice criteria in Section 3 would remain intact as structural results independent of PS2.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper is a research survey on complemented subspaces of Banach lattices. Its announced centrepiece is the recent negative solution of the Complemented Subspace Problem: the space PS2, a 1-complemented subspace of a C(K)-space due to Plebanek and Salguero-Alarcón, is not linearly isomorphic to any Banach lattice. The paper sketches PS2 and the reduction used in the companion paper [38], then develops a free-Banach-lattice toolbox: complementability in FBL[E] (Prop. 3.1), an ideal-decomposition criterion for being isomorphic to a Banach lattice (Prop. 3.3), descriptions of ideals in Cph(BE*) and AM-space characterizations (Prop. 3.6, Cor. 3.7), and transfer principles for lattice homomorphisms (Props. 3.8–3.11). Later sections treat projection constants, complementation in Banach lattices with extra properties, hyperplanes in Banach lattices, and several open problems. The paper proves a number of original propositions, including Props. 4.1, 4.2, 5.9, 6.4, and 6.5.
Significance. If the theorem it relies on is correct, the paper announces a major development: a 1-complemented subspace of a C(K)-space that is not isomorphic to any Banach lattice, giving a negative solution to the Complemented Subspace Problem for Banach lattices. The free-Banach-lattice criteria (Prop. 3.3, Cor. 3.7, Prop. 3.10) are elegant and likely to be useful, and the survey is careful in attributing the counterexample to [73] and [38]. The original propositions are proved in detail with standard machinery, and the open problems are well motivated. The evident weakness is that the headline counterexample is not proved here: Section 2.2 explicitly defers the decisive modulus argument to the external preprint [38], so the main advertised result is conditional on that source.
major comments (1)
- [Section 2.2] The central claim that PS2 is not linearly isomorphic to any Banach lattice is not established in this text. The paper reduces the problem to the assertion that for every norming sequence (e_n*) in the dual ball of PS2 there is xi < c such that w_xi = 1_{Bxi^0} - 1_{Bxi^1} has no modulus with respect to that sequence, but the proof of this assertion is contained only in [38, Theorem 4.4] and is described as 'a careful analysis'. Since the equivalence between being isomorphic to a sublattice of l_infinity and satisfying the modulus condition is also cited from [38, Proposition 4.1], the reader cannot verify the main theorem from the present manuscript. Please include a complete proof of the modulus assertion, or explicitly frame the negative solution as an external surveyed result and make the manuscript's own contributions the free-Banach-lattice results.
minor comments (5)
- [Section 2.1] The symbol JL(A) is reused for the modified subspace of l_infinity(bN) after having been defined as a subspace of l_infinity; although the paper alerts the reader, a distinct symbol would avoid confusion.
- [Section 2.2] The identification of PS2 with the space X = ker P from Section 2.1 is never made explicit; please state directly that PS2 is this space before discussing the vectors 1_{Bxi^0} - 1_{Bxi^1}.
- [Section 4, Proposition 4.1] The assertion that James space J and J** cannot be isomorphic to any complemented subspace of a Banach lattice is used in the contradiction but is not accompanied by a proof or reference; please supply a citation (for instance to [52] or [30]).
- [Section 5, proof of Proposition 5.9] The symbol 'FVL[E]' appears where FBL[E] is intended, and the notation for the projection P and its lattice-homomorphic extension is conflated in step (2); please clarify.
- [Section 3, Lemma 3.5] In the display after equation (3.1), the two norms in the middle of the chain are identical; likely one of them should be the quotient norm, so the chain should be rewritten for clarity.
Circularity Check
The survey's headline negative solution to the CSP is imported from the authors' own companion preprint [38, Theorem 4.4]; the rest of the paper's free-Banach-lattice results are independently derived.
-
self citation load bearing
[Section 2.2, paragraph 'PS2 is not isomorphic to a Banach lattice']
"A careful analysis of the construction of PS2 (explained in detail in [38, Theorem 4.4]) reveals that for every norming sequence in B_PS2^* there exists ξ < c such that 1_{B_ξ^0} − 1_{B_ξ^1} ∈ PS2 has no modulus with respect to that sequence, and hence PS2 cannot be isomorphic to a Banach lattice."
The decisive step of the claimed negative solution to the Complemented Subspace Problem is not proved here; it is exactly the content of [38, Theorem 4.4], a companion preprint by two of the present authors. The survey's 'sketch' stops at the modulus-without-norming-sequence assertion and then draws the headline conclusion from it. Thus the central claim's only support in this text is a self-citation to an unproved-in-this-paper theorem, rather than a derivation from the definitions of PS2 or from independently established facts.
full rationale
No definitional circularity or fitted-input-called-prediction occurs: the free Banach lattice propositions (3.1, 3.3, 3.6, 3.7, 3.10, 3.11, 4.1-4.2, 5.9, 6.4, 6.5) are proved from the universal property of FBL and standard Banach lattice facts. The flagged step is the one load-bearing self-citation: the survey's headline result that PS2 is not isomorphic to any Banach lattice depends on [38, Theorem 4.4], whose key assertion about moduli with respect to norming sequences is referenced but not demonstrated. This is not a logical loop inside the paper, but it is a central claim whose only textual support is the authors' own companion preprint, so it warrants a moderate circularity score rather than zero.
Assumptions & free parameters
assumptions (4)
- domain assumption Universal property and functional representation of free Banach lattices FBL[E] from [9,41,68].
- standard math Quotient of a Banach lattice over a norm-closed ideal is a Banach lattice.
- standard math Kakutani representation theorems identifying AM-spaces with sublattices of C(K) and AL-spaces with L1-spaces.
- domain assumption PS2 is an L_infinity-space with a countable norming set, and [38, Corollary 2.2] that an L_infinity-space isomorphic to a Banach lattice is isomorphic to an AM-space.
Cite this review
Pith. "Pith review of Complemented subspaces of Banach lattices." pith.science (2026). https://pith.science/paper/UDUGA73P
@misc{pith2026250524084,
author = {Pith},
title = {Pith review of: Complemented subspaces of Banach lattices},
year = {2026},
howpublished = {\url{https://pith.science/paper/UDUGA73P}},
note = {Machine review of arXiv:2505.24084}
}
abstract
We survey recent developments on the structure of complemented subspaces of Banach lattices, including in particular the construction of a complemented subspace of a $C(K)$-space which is not linearly isomorphic to any Banach lattice. Motivated by this, several natural questions and directions of future research are presented. We provide an approach to some of these problems using tools from the theory of free Banach lattices.
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