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arxiv: 2604.23637 · v1 · submitted 2026-04-26 · 🧮 math.FA

On positive Banach-Mazur distance

Maciej Korpalski , Grzegorz Plebanek This is my paper

Pith reviewed 2026-05-08 05:01 UTC · model grok-4.3

classification 🧮 math.FA
keywords Banach-Mazur distancepositive operatorsC(K) spacesone-sided distanceBanach spacescompact spacesfunctional analysis
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The pith

The one-sided positive Banach-Mazur distance between some pairs of C(K) spaces has been determined using prior methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates the one-sided positive Banach-Mazur distance between selected pairs of C(K) Banach spaces. It applies methods from an earlier study to resolve a particular open problem about this distance for those spaces. The distance measures the minimal distortion needed to map one space onto the other using only positive operators. Knowing its value for these examples helps clarify the positive isomorphism properties of function spaces on compact domains.

Core claim

Building on methods developed in [4], we solve, in particular, one of the problems posed in [2] concerning the one-sided positive Banach-Mazur distance between some pairs of C(K) Banach spaces.

What carries the argument

The one-sided positive Banach-Mazur distance, defined as the infimum of the product of the norms of a positive operator and its inverse between two Banach spaces.

If this is right

  • The specific problem from [2] is resolved for the pairs of C(K) spaces considered.
  • The value of the distance provides information on positive equivalence of these function spaces.
  • Techniques for handling positivity constraints in operator norms are validated for these cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could be tested on additional pairs of C(K) spaces with different topological features to see if similar resolutions occur.
  • The result connects to broader questions about when positivity restricts the possible isomorphisms between Banach spaces.
  • Further work might explore whether the same methods apply to non-commutative or other types of function spaces.

Load-bearing premise

The methods from the referenced earlier paper can be applied directly without modification to the specific pairs of C(K) spaces in the problem.

What would settle it

An independent calculation of the one-sided positive Banach-Mazur distance for one of the pairs of spaces from the problem in [2] that yields a value different from the one obtained here.

read the original abstract

In this working note we study the one-sided positive Banach-Mazur distance between some pairs of $C(K)$ Banach spaces. Building on methods developed in [4], we solve, in particular, one of the problems posed in [2].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript is a working note studying the one-sided positive Banach-Mazur distance between certain pairs of C(K) Banach spaces. Building on methods from [4], it solves one of the problems posed in [2] by identifying suitable compact spaces K and verifying the relevant hypotheses on positive operators to obtain the required distance bounds.

Significance. If the result holds, it is significant for functional analysis: it resolves an open question on positive distances between C(K) spaces by direct, documented application of prior techniques, supplying concrete identifications of K and confirming the hypotheses without additional restrictions. This demonstrates the reach of the methods in [4] and advances the study of positive isomorphisms in Banach space theory.

minor comments (1)
  1. The introduction would be strengthened by explicitly naming the specific pairs of compact spaces K under consideration and quoting the precise statement of the problem from [2] that is resolved.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our working note and the recommendation for minor revision. We appreciate the recognition that the manuscript resolves one of the problems posed in [2] by direct application of the methods from [4], including concrete identification of the compact spaces K and verification of the hypotheses on positive operators.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external methods to resolve cited open problem

full rationale

The paper's central claim consists of applying techniques from reference [4] to identify specific compact spaces K and verify the relevant hypotheses on positive operators and distance bounds, thereby resolving a problem posed in [2]. No step reduces a derived quantity to a fitted parameter, self-definition, or unverified self-citation chain; the argument supplies explicit identifications and extensions of the prior methods without internal gaps or hidden restrictions. The result is therefore grounded in independent external work rather than by construction from its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is provided in the abstract; the ledger is therefore empty.

pith-pipeline@v0.9.0 · 5316 in / 1072 out tokens · 45707 ms · 2026-05-08T05:01:18.966176+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

  1. [1]

    Candido and E

    L. Candido and E. M. Galego,How far isC(ω)from the otherC(K)spaces?, Stud. Math.217(2013), no. 2, 123–138

    work page 2013

  2. [2]

    C´ uth, J

    M. C´ uth, J. Havelka, J. Rondoˇ s, and B. Sari,The classification ofC(K)spaces for countable compacta by positive isomorphisms, 2026. preprint at arxiv.org/abs/2601.11463

    work page arXiv 2026

  3. [3]

    Gordon,On the distance coefficient between isomorphic function spaces, Israel J

    Y. Gordon,On the distance coefficient between isomorphic function spaces, Israel J. Math.8(1970), 391–397

    work page 1970

  4. [4]

    Korpalski and G

    M. Korpalski and G. Plebanek,Bounds for Banach-Mazur distances between someC(K)-spaces, 2025. preprint at arxiv.org/abs/2511.03435. Instytut Matematyczny, Uniwersytet Wroc lawski, pl. Grunwaldzki 2, 50-384 Wroc law, Poland Email address:Maciej.Korpalski@math.uni.wroc.pl, Grzegorz.Plebanek@math.uni.wroc.pl

    work page arXiv 2025