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arxiv: 2606.22320 · v1 · pith:EIEWK65Ynew · submitted 2026-06-21 · 🧮 math.FA

Globalization of local sign structures for phase-isometries on uniform algebras

Yuta Enami , Daisuke Hirota , Izuho Matsuzaki , Takeshi Miura This is my paper

Pith reviewed 2026-06-26 10:10 UTC · model grok-4.3

classification 🧮 math.FA
keywords uniform algebrasphase-isometriesChoquet boundarysign structuresBanach-Stone theoremmaximal convex setshomeomorphismclopen decomposition
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The pith

Every surjective phase-isometry between unit spheres of uniform algebras admits a global boundary representation via sign function, weight, homeomorphism, and clopen linear split.

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

The paper shows that local sign ambiguities in phase-isometries on uniform algebras can be resolved globally on the Choquet boundary. A refined additive Bishop-type construction propagates sign information consistently across maximal convex sets tied to boundary points. Once globalized, the isometry yields an explicit boundary representation that includes a single sign function, a unimodular weight, a homeomorphism of Choquet boundaries, and a clopen decomposition into complex-linear and conjugate-linear pieces. This representation then extends to the maximal ideal spaces, producing a real-algebraic form of the Banach-Stone theorem for phase-isometries. Readers care because the result removes the main obstruction that had blocked classical isometry arguments from applying directly to phase-preserving maps.

Core claim

Surjective phase-isometries preserve maximal convex sets up to signs, yet local sign ambiguity blocks direct Banach-Stone arguments. By refining an additive Bishop-type construction, the local sign structures on maximal convex sets associated with boundary points propagate consistently on the Choquet boundary. Every such map therefore admits a boundary representation consisting of a global sign function, a unimodular weight, a homeomorphism between the Choquet boundaries, and a clopen decomposition into complex-linear and conjugate-linear parts. The same data extend to the maximal ideal spaces and furnish the corresponding real-algebraic Banach-Stone type representation.

What carries the argument

Refined additive Bishop-type construction that propagates sign information consistently among maximal convex sets on the Choquet boundary

If this is right

  • Every surjective phase-isometry admits an explicit boundary representation with one global sign function.
  • The representation splits the domain into clopen sets on which the map is either complex-linear or conjugate-linear.
  • A unimodular weight and a homeomorphism of Choquet boundaries complete the boundary data.
  • The boundary representation extends to the full maximal ideal spaces, giving a real-algebraic Banach-Stone theorem.

Where Pith is reading between the lines

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

  • The propagation method may apply to phase-isometries on other classes of function algebras beyond uniform algebras.
  • Similar globalization of local data could be attempted for non-surjective phase-isometries when additional continuity or density conditions hold.
  • The clopen decomposition suggests possible links to automatic continuity results or to representation questions in real Banach algebras.

Load-bearing premise

The local sign structures on maximal convex sets can be propagated consistently to the whole Choquet boundary by the refined additive Bishop-type construction.

What would settle it

A concrete surjective phase-isometry on some uniform algebra for which no consistent global sign function exists on the Choquet boundary, shown by exhibiting a point where the Bishop-type propagation fails to match the local signs.

read the original abstract

We study surjective phase-isometries between the unit spheres of uniform algebras. Although such maps preserve maximal convex sets up to signs, the resulting local sign ambiguity prevents a direct application of the usual Banach--Stone type arguments for isometries. The main point of the paper is to prove that these local sign structures can be globalized on the Choquet boundary. To this end, we refine an additive Bishop-type construction and use it to propagate the sign information among the maximal convex sets associated with boundary points. As a consequence, every surjective phase-isometry admits a boundary representation by means of a global sign function, a unimodular weight, a homeomorphism between the Choquet boundaries, and a clopen decomposition into complex-linear and conjugate-linear parts. We then extend this representation to the maximal ideal spaces and obtain the corresponding real-algebraic Banach--Stone type representation.

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 / 0 minor

Summary. The paper claims to show that local sign structures for phase-isometries on uniform algebras can be globalized on the Choquet boundary using a refined additive Bishop-type construction. This globalization yields a representation of surjective phase-isometries by a global sign function, a unimodular weight, a homeomorphism between the Choquet boundaries, and a clopen decomposition into complex-linear and conjugate-linear parts. The representation is extended to the maximal ideal spaces to obtain a real-algebraic Banach-Stone type representation.

Significance. If the result holds, it is a significant contribution to the theory of isometries on uniform algebras, as it resolves the local sign ambiguity that blocks standard Banach-Stone arguments. The refined Bishop-type construction for propagating sign information is a technical achievement. The paper does not provide machine-checked proofs or reproducible code, but the result is framed as a direct consequence of the construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of our manuscript and for acknowledging the potential significance of the result in resolving local sign ambiguity for surjective phase-isometries on uniform algebras. The referee notes an uncertain recommendation. No specific major comments were enumerated in the report, so we provide no point-by-point responses below. We remain available to supply further details or clarifications should the referee identify particular points of concern.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's central result is obtained by refining an additive Bishop-type construction to propagate local sign information consistently across maximal convex sets on the Choquet boundary, yielding the global sign function, unimodular weight, homeomorphism, and clopen decomposition. This is presented as a direct mathematical consequence of the construction rather than a reduction to fitted parameters, self-definitions, or load-bearing self-citations. No quoted equations or steps equate the claimed representation to its inputs by construction; the argument remains self-contained against external benchmarks such as standard Banach-Stone techniques for isometries.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard properties of uniform algebras, Choquet boundaries, and Bishop-type constructions from prior literature; no free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption Uniform algebras possess Choquet boundaries and maximal convex sets with the usual properties used in Banach-Stone arguments
    Invoked implicitly when discussing preservation of maximal convex sets and propagation on the boundary.

pith-pipeline@v0.9.1-grok · 5688 in / 1266 out tokens · 28048 ms · 2026-06-26T10:10:53.820839+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

19 extracted references

  1. [1]

    Banach, Th´ eorie des op´ erations lin´ eaires, Monografie Matematyczne, Vol

    S. Banach, Th´ eorie des op´ erations lin´ eaires, Monografie Matematyczne, Vol. 1, PWN, Warszawa, 1932

    1932

  2. [2]

    Browder, Introduction to function algebras, W

    A. Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam 1969 xii+273 pp

    1969

  3. [3]

    Fleming and J

    R. Fleming and J. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 129, Chapman & Hall/CRC, Boca Raton, FL, 2003

    2003

  4. [4]

    Hatori,The Mazur–Ulam property for a Banach space which satisfies a separation condition, RIMS Kˆ okyˆ uroku Bessatsu,B93(2023), 29–82

    O. Hatori,The Mazur–Ulam property for a Banach space which satisfies a separation condition, RIMS Kˆ okyˆ uroku Bessatsu,B93(2023), 29–82

    2023

  5. [5]

    Hatori and T

    O. Hatori and T. Miura,Real linear isometries between function algebras. II, Cent. Eur. J. Math.11(2013), no. 10, 1838–1842

    2013

  6. [6]

    Hatori, T

    O. Hatori, T. Miura and H. Takagi,Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties, Proc. Amer. Math. Soc.134(2006), no. 10, 2923–2930

    2006

  7. [7]

    Hatori, S

    O. Hatori, S. Oi and R. Shindo-Togashi,Tingley’s problems on uniform algebras, J. Math. Anal. Appl.503 (2021), no. 2, Paper No. 125346, 14 pp

    2021

  8. [8]

    Hirota, I

    D. Hirota, I. Matsuzaki and T. Miura,Phase-isometries between positive cones of Banach space of continuous real-valued functions, Ann. Funct. Anal.15(2024), no. 4, Paper No. 77, 11 pp

    2024

  9. [9]

    Huang, J

    X. Huang, J. Liu and S. Wang,The Wigner property of smooth normed spaces, Bull. Aust. Math. Soc.110 (2024), no. 3, 545–553

    2024

  10. [10]

    Iliˇ sevi´ c, M

    D. Iliˇ sevi´ c, M. Omladiˇ c and A. Turnˇ sek,Phase-isometries between normed spaces, Linear Algebra Appl.612 (2021), 99–111

    2021

  11. [11]

    Li and D

    Y. Li and D. Tan,Wigner’s theorem on the Tsirelson spaceT, Ann. Funct. Anal.10(2019), no. 4, 515–524

    2019

  12. [12]

    J. Liu, X. Huang and S. Wang,On Wigner’s theorem in complex smooth normed spaces, J. Math. Anal. Appl. 538(2024), no. 2, Paper No. 128404, 14 pp

    2024

  13. [13]

    Maksa and Z

    G. Maksa and Z. P´ ales,Wigner’s theorem revisited, Publ. Math. Debrecen81(2012), no. 1-2, 243–249

    2012

  14. [14]

    Tan and Y

    D. Tan and Y. Gao,Phase-isometries on the unit sphere ofC(K), Ann. Funct. Anal.12(2021), no. 1, Paper No. 15, 14 pp

    2021

  15. [15]

    Tan and X

    D. Tan and X. Huang,Phase-isometries on real normed spaces, J. Math. Anal. Appl.488(2020), no. 1, 124058, 17 pp

    2020

  16. [16]

    Tan and X

    D. Tan and X. Huang,The Wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces, Proc. Edinb. Math. Soc. (2)64(2021), no. 2, 183–199

    2021

  17. [17]

    D. Tan, F. Zhang and X. Huang,Phase-isometries on the unit sphere of CL-spaces, J. Math. Anal. Appl.527 (2023), no. 2, Paper No. 127568, 9 pp

    2023

  18. [18]

    Tingley,Isometries of the unit sphere, Geom

    D. Tingley,Isometries of the unit sphere, Geom. Dedicata22(1987), no. 3, 371–378

    1987

  19. [19]

    Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, J

    E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, J. W. Edwards, Ann Arbor, MI, 1944. (Y. Enami)Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan Email address:f21j008j@mail.cc.niigata-u.ac.jp (D. Hirota)National Institute of Technology, Tsuruoka College, Yamagata 997-8511, Japan Em...

    1944