On 2-local nonlinear surjective isometries on normed spaces and C$^*$-algebras

Michiya Mori

arxiv: 1907.02172 · v1 · pith:2PDJIQA6new · submitted 2019-07-04 · 🧮 math.FA · math.MG· math.OA

On 2-local nonlinear surjective isometries on normed spaces and C^*-algebras

Michiya Mori This is my paper

Pith reviewed 2026-05-25 09:22 UTC · model grok-4.3

classification 🧮 math.FA math.MGmath.OA

keywords 2-local isometriessurjective isometriesnormed spacesextreme pointsaffine mapsC*-algebrasnonlinear maps

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The pith

If the unit ball of a normed space has sufficiently many extreme points, then every 2-local surjective isometry on the space is affine.

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

The paper establishes that in normed spaces where the closed unit ball contains enough extreme points, any map that agrees with some surjective isometry on every pair of points must be an affine map. This property generalizes classical results on isometries by showing that the local condition implies global linearity plus translation. The result applies to a broad class of spaces and includes observations on surjectivity for mappings on C*-algebras.

Core claim

If the closed unit ball of a normed space X has sufficiently many extreme points, then every mapping Φ from X into itself with the property that for any pair of points there exists a surjective isometry on X coinciding with Φ at those two points is affine. The paper also considers surjectivity of such mappings in some special cases including C*-algebras.

What carries the argument

The 2-local property, where for any two points a surjective isometry of the whole space agrees with the given map at those points, combined with the abundance of extreme points to force affinity.

If this is right

Such maps must preserve the affine structure of the space.
In C*-algebras the maps satisfy additional surjectivity properties under the stated conditions.
The 2-local condition on pairs extends known global isometry results to nonlinear maps.

Where Pith is reading between the lines

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

The result may apply to other classes of Banach spaces if extreme points can be replaced by a different structural assumption.
Similar local-to-global arguments could be tested on maps that preserve distances only on triples or other finite sets.

Load-bearing premise

The closed unit ball of the normed space has sufficiently many extreme points.

What would settle it

A non-affine map on a normed space whose closed unit ball lacks sufficiently many extreme points, yet for every pair of points some surjective isometry of the space agrees with the map at those two points.

read the original abstract

We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $\Phi$ from $X$ into itself with the following property is affine: For any pair of points in $X$, there exists a (not necessarily linear) surjective isometry on $X$ that coincides with $\Phi$ at the two points. We also consider surjectivity of such a mapping in some special cases including C$^*$-algebras.

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.

Desk Editor's Note private letter to a colleague

The paper claims 2-local surjective isometries are affine when the unit ball has sufficiently many extreme points, but that hypothesis stays imprecise.

read the letter

The paper shows that if the closed unit ball has sufficiently many extreme points then any 2-local surjective isometry is affine. This is the central new claim, moving from linear to nonlinear maps with that extra geometric assumption. It does a fair job of extending the known results in this direction and includes some remarks on surjectivity for C star algebras. That part is useful for the special cases. The soft spot is exactly what the stress test flags. The condition on extreme points is not given a clear meaning, so it is difficult to see what spaces satisfy it or whether the proof really works when the usual results fail. The abstract states the theorem but does not include any steps or definitions that would let us verify it. If the full paper supplies a concrete characterization like the extreme points being dense in some sense or satisfying a Choquet type condition, that would make the result much more usable. The rest of the argument looks direct and there is no circularity. The result is presented as new and it seems to be. This kind of work is for experts in functional analysis who care about isometries on normed spaces. A reader following that literature would find it worth a look. The paper engages honestly with the existing results on the topic. I would bring this to a reading group as a maybe, to see the details. I would not cite it in my own papers in the next year. It deserves to go to peer review because the claim is specific enough to be checked by referees in the area. My recommendation is to send it out for review.

Referee Report

1 major / 0 minor

Summary. The paper claims to prove that if the closed unit ball of a normed space X has sufficiently many extreme points, then every mapping Φ: X → X such that for any pair of points there exists a surjective isometry on X agreeing with Φ at those two points must be affine. It also examines surjectivity of such mappings in special cases, including C*-algebras.

Significance. If the result holds with a precise hypothesis, it would extend Mazur-Ulam-type theorems on affine isometries to the 2-local nonlinear setting, providing a geometric condition on extreme points that forces affinity. The C*-algebra applications could connect to operator-algebraic preservers, but the current vagueness in the hypothesis limits immediate applicability or verification against known counterexamples to global affinity.

major comments (1)

[Abstract / Main Theorem] Abstract and main theorem statement: the hypothesis that the closed unit ball 'has sufficiently many extreme points' is invoked to conclude affinity but receives no explicit definition, characterization (e.g., density in the unit sphere, Choquet boundary condition, or cardinality requirement), or reference to a prior section. This renders the central claim untestable for concrete spaces and prevents checking whether the proof technique covers all cases where Mazur-Ulam fails.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for greater precision in the statement of the main hypothesis. We address this point below and will incorporate the necessary clarifications in the revised version.

read point-by-point responses

Referee: [Abstract / Main Theorem] Abstract and main theorem statement: the hypothesis that the closed unit ball 'has sufficiently many extreme points' is invoked to conclude affinity but receives no explicit definition, characterization (e.g., density in the unit sphere, Choquet boundary condition, or cardinality requirement), or reference to a prior section. This renders the central claim untestable for concrete spaces and prevents checking whether the proof technique covers all cases where Mazur-Ulam fails.

Authors: We agree that the phrase 'sufficiently many extreme points' is insufficiently precise as currently stated in the abstract and main theorem. The manuscript employs this condition in the proof to ensure the existence of enough extreme points to construct suitable supporting functionals that force the 2-local mapping to be affine, but the precise meaning is not spelled out or referenced at the outset. In the revision we will add an explicit definition (in terms of the geometric property used in the argument) both in the introduction and in the statement of the main theorem, together with a forward reference to the relevant section of the proof. This will make the result testable on concrete spaces and allow direct comparison with known counterexamples to the classical Mazur-Ulam theorem. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained from geometric hypothesis

full rationale

The paper states a direct implication: under the hypothesis that the closed unit ball has sufficiently many extreme points, any 2-local surjective isometry mapping is affine. No equations, definitions, or steps reduce the conclusion to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The result is presented as a theorem derived from the stated assumption on extreme points, with no evidence of renaming, smuggling via citation, or uniqueness imported from prior author work. This matches the default case of a non-circular mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard definitions from functional analysis together with the geometric hypothesis on extreme points; no free parameters or invented entities appear in the abstract.

axioms (2)

standard math Standard properties of normed spaces, isometries, and extreme points of the unit ball
Invoked to state the hypothesis and conclude affinity.
domain assumption Existence of surjective isometries for any pair of points
The 2-local property presupposes such isometries exist.

pith-pipeline@v0.9.0 · 5608 in / 1186 out tokens · 30342 ms · 2026-05-25T09:22:38.402554+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.5. Let X be a normed space with the property that the real linear span of extreme points in the closed unit ball is dense in X. Then every mapping in TL(X) is affine.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that, if the closed unit ball of a normed space X has sufficiently many extreme points, then every mapping Φ ... is affine.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 2 internal anchors

[1]

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J.B. Conway, “A course in operator theory”, Graduate Stu dies in Mathematics, 21. American Mathematical Society, Providence, RI, (2000)

work page 2000
[2]

Dang, Real isometries between JB ∗-triples, Proc

T. Dang, Real isometries between JB ∗-triples, Proc. Amer. Math. Soc. 114 (1992), no. 4, 971–980

work page 1992
[3]

2-local isometries on function spaces

O. Hatori and S. Oi, 2-local isometries on function space s, preprint, arXiv:1812.10342

work page internal anchor Pith review Pith/arXiv arXiv
[4]

Kadison, Local derivations, J

R.V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509

work page 1990
[5]

Fundamentals of the the ory of operator algebras. Vol. II

R.V. Kadison and J.R. Ringrose, “Fundamentals of the the ory of operator algebras. Vol. II”, Academic Press, Inc., Orlando, FL (1986)

work page 1986
[6]

Larson and A.R

D.R. Larson and A.R. Sourour, Local derivations and loca l automorphisms of B(X), Proc. Sympos. Pure Math. 51, Providence, Rhode Island 1990, Part 2 , pp. 187–194

work page 1990
[7]

Selected preserver problems on algebraic s tructures of linear operators and on function spaces

L. Moln´ ar, “Selected preserver problems on algebraic s tructures of linear operators and on function spaces”, Lecture Notes in Mathematics, 1895. Spri nger-Verlag, Berlin (2007)

work page 2007
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Moln´ ar, On 2-local ∗-automorphisms and 2-local isometries of B(H), to appear in J

L. Moln´ ar, On 2-local ∗-automorphisms and 2-local isometries of B(H), to appear in J. Math. Anal. Appl

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Moln´ ar and B

L. Moln´ ar and B. Zalar, On local automorphisms of group a lgebras of compact groups, Proc. Amer. Math. Soc. 128 (2000), no. 1, 93–99

work page 2000
[10]

A generalization of the Kowalski -S\{l} odkowski theorem and its application to 2-local maps on function spaces

S. Oi, A generalization of the Kowalski–S/suppress lodkowski theorem and its application to 2-local maps on function spaces, preprint, arXiv:1903.11424

work page internal anchor Pith review Pith/arXiv arXiv 1903
[11]

Pedersen, The linear span of projections in simple C∗-algebras, J

G.K. Pedersen, The linear span of projections in simple C∗-algebras, J. Operator Theory 4 (1980), no. 2, 289–296

work page 1980
[12]

Russo and H.A

B. Russo and H.A. Dye, A note on unitary operators in C ∗-algebras, Duke Math. J. 33 (1966), 413–416

work page 1966
[13]

ˇSemrl, Local automorphisms and derivations on B(H), Proc

P. ˇSemrl, Local automorphisms and derivations on B(H), Proc. Amer. Math. Soc. 125 (1997), no. 9, 2677–2680. Graduate School of Mathematical Sciences, the University of Tokyo, Komaba, Tokyo, 153-8914, Japan. E-mail address : mmori@ms.u-tokyo.ac.jp

work page 1997

[1] [1]

A course in operator theory

J.B. Conway, “A course in operator theory”, Graduate Stu dies in Mathematics, 21. American Mathematical Society, Providence, RI, (2000)

work page 2000

[2] [2]

Dang, Real isometries between JB ∗-triples, Proc

T. Dang, Real isometries between JB ∗-triples, Proc. Amer. Math. Soc. 114 (1992), no. 4, 971–980

work page 1992

[3] [3]

2-local isometries on function spaces

O. Hatori and S. Oi, 2-local isometries on function space s, preprint, arXiv:1812.10342

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

Kadison, Local derivations, J

R.V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509

work page 1990

[5] [5]

Fundamentals of the the ory of operator algebras. Vol. II

R.V. Kadison and J.R. Ringrose, “Fundamentals of the the ory of operator algebras. Vol. II”, Academic Press, Inc., Orlando, FL (1986)

work page 1986

[6] [6]

Larson and A.R

D.R. Larson and A.R. Sourour, Local derivations and loca l automorphisms of B(X), Proc. Sympos. Pure Math. 51, Providence, Rhode Island 1990, Part 2 , pp. 187–194

work page 1990

[7] [7]

Selected preserver problems on algebraic s tructures of linear operators and on function spaces

L. Moln´ ar, “Selected preserver problems on algebraic s tructures of linear operators and on function spaces”, Lecture Notes in Mathematics, 1895. Spri nger-Verlag, Berlin (2007)

work page 2007

[8] [8]

Moln´ ar, On 2-local ∗-automorphisms and 2-local isometries of B(H), to appear in J

L. Moln´ ar, On 2-local ∗-automorphisms and 2-local isometries of B(H), to appear in J. Math. Anal. Appl

work page

[9] [9]

Moln´ ar and B

L. Moln´ ar and B. Zalar, On local automorphisms of group a lgebras of compact groups, Proc. Amer. Math. Soc. 128 (2000), no. 1, 93–99

work page 2000

[10] [10]

A generalization of the Kowalski -S\{l} odkowski theorem and its application to 2-local maps on function spaces

S. Oi, A generalization of the Kowalski–S/suppress lodkowski theorem and its application to 2-local maps on function spaces, preprint, arXiv:1903.11424

work page internal anchor Pith review Pith/arXiv arXiv 1903

[11] [11]

Pedersen, The linear span of projections in simple C∗-algebras, J

G.K. Pedersen, The linear span of projections in simple C∗-algebras, J. Operator Theory 4 (1980), no. 2, 289–296

work page 1980

[12] [12]

Russo and H.A

B. Russo and H.A. Dye, A note on unitary operators in C ∗-algebras, Duke Math. J. 33 (1966), 413–416

work page 1966

[13] [13]

ˇSemrl, Local automorphisms and derivations on B(H), Proc

P. ˇSemrl, Local automorphisms and derivations on B(H), Proc. Amer. Math. Soc. 125 (1997), no. 9, 2677–2680. Graduate School of Mathematical Sciences, the University of Tokyo, Komaba, Tokyo, 153-8914, Japan. E-mail address : mmori@ms.u-tokyo.ac.jp

work page 1997