On the Fischer-Musz\'ely equation for the positive cones of $C^*$-algebras

Daisuke Hirota; Jyamira Oppekepenguin

arxiv: 2606.28665 · v1 · pith:XBUWSXVCnew · submitted 2026-06-27 · 🧮 math.FA · math.OA

On the Fischer-Musz\'ely equation for the positive cones of C^*-algebras

Daisuke Hirota , Jyamira Oppekepenguin This is my paper

Pith reviewed 2026-06-30 09:04 UTC · model grok-4.3

classification 🧮 math.FA math.OA

keywords Fischer-Muszély equationpositive semidefinite conesC*-algebrasJordan *-isomorphismfunctional equationspositive definite cones

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The pith

Bijections between positive semidefinite cones of unital C-algebras that satisfy the Fischer-Muszély equation extend to Jordan -isomorphisms followed by multiplication by a positive element on both sides.

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

The paper establishes that any bijection between the positive semidefinite cones of two unital C*-algebras which satisfies the Fischer-Muszély functional equation extends to a Jordan *-isomorphism of the underlying algebras composed with left and right multiplication by a positive element. This provides a complete algebraic description of the maps that preserve the given equation on these cones. The same conclusion applies to bijections on the positive definite cones as a corollary.

Core claim

Any bijection between the positive semidefinite cones satisfying the Fischer-Muszély equality extends to a Jordan *-isomorphism followed by multiplication on both sides by a positive element.

What carries the argument

The Fischer-Muszély functional equation, which the bijection between positive semidefinite cones is required to satisfy exactly.

Load-bearing premise

The map must be an exact bijection that satisfies the Fischer-Muszély equality on the positive semidefinite cones.

What would settle it

A concrete bijection between the positive semidefinite cones of two unital C*-algebras that satisfies the Fischer-Muszély equality but fails to extend to any Jordan *-isomorphism scaled by a positive element.

read the original abstract

We study the Fischer-Musz\'ely functional equation for the positive semidefinite and the positive definite cones of unital $C^*$-algebras. We show that any bijection between the positive semidefinite cones satisfying the Fischer-Musz\'ely equality extends to a Jordan $*$-isomorphism followed by multiplication on both sides by a positive element. As a corollary, we obtain a similar result for the positive definite cones of unital $C^*$-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 shows bijections on positive semidefinite cones of unital C*-algebras that satisfy the Fischer-Muszély equation extend to Jordan *-isomorphisms composed with positive multiplications.

read the letter

The main point is that any bijection between the positive semidefinite cones of unital C*-algebras satisfying the Fischer-Muszély equation extends to a Jordan *-isomorphism followed by left and right multiplication by a positive element. The same pattern holds for the positive definite cones as a corollary.

This application of the equation to the positive cones in the C*-algebra setting looks new. The abstract states the result directly and keeps the scope to exact preservation under bijections.

The statement is clean and the corollary follows without extra machinery. The authors stick to what the equation forces on these cones.

The soft spot is that only the abstract is visible here, so the actual proof steps and any reliance on prior results for the equation cannot be checked in detail. The bijectivity assumption is stated explicitly, which avoids hidden issues, but it does mean the result does not cover approximate or non-bijective maps.

This is for people working on preserver problems or functional equations inside operator algebras. A reader already familiar with Jordan *-isomorphisms and cone maps would see the most value. The claim is precise enough and the setting is standard enough that it deserves a serious referee to examine the proofs and references.

Referee Report

0 major / 2 minor

Summary. The paper studies the Fischer-Muszély functional equation on the positive semidefinite cones (and, as a corollary, the positive definite cones) of unital C*-algebras. Its central claim is that any bijection between these cones that satisfies the equation exactly extends to a Jordan *-isomorphism followed by left and right multiplication by a fixed positive element.

Significance. If the result holds, it supplies a clean structural characterization of exact preservers of the Fischer-Muszély relation on positive cones, extending the literature on functional equations and Jordan-isomorphism preservers in operator algebras. The bijection hypothesis and exact equality are stated explicitly, and the corollary for positive definite cones follows the same pattern without additional assumptions.

minor comments (2)

The precise statement of the Fischer-Muszély equation (presumably of the form f(x+y)+f(x-y)=2f(x)+2f(y) or an operator-algebra analogue) should be recalled in the introduction or §1 for readers who may not have the reference at hand.
Notation for the positive semidefinite cone (e.g., A_+ or A_sa^+) and the precise domain of the bijection should be fixed uniformly across the abstract, introduction, and main theorem statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation to accept. The report accurately captures the main result on bijections preserving the Fischer-Muszély equation on positive semidefinite cones of unital C*-algebras.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct existence theorem: any bijection between positive semidefinite cones of unital C*-algebras that satisfies the Fischer-Muszély equality extends to a Jordan *-isomorphism composed with multiplication by a positive element. The abstract and corollary make no reference to fitted parameters, self-definitional constructions, or load-bearing self-citations. The result is framed as a consequence of the exact preservation condition on bijections, with no equations or derivations in the provided text that reduce the conclusion to its inputs by construction. This is a standard functional-equation result in operator algebras whose proof chain is independent of the target statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard properties of unital C*-algebras and the definition of the Fischer-Muszély equation; no free parameters or invented entities are introduced.

axioms (1)

domain assumption The structures under study are unital C*-algebras with their positive semidefinite and positive definite cones.
Invoked in the first sentence of the abstract as the setting for the functional equation.

pith-pipeline@v0.9.1-grok · 5608 in / 1168 out tokens · 18245 ms · 2026-06-30T09:04:44.336807+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Y. Dong, L. Li, L. Molnár and N.-C. Wong, Transformations preserving the normofmeansbetweenpositiveconesofgeneralandcommutative C ∗-algebras, J. Oper. Theory88 (2022), 365–406

2022
[2]

Fischer and Gy

P. Fischer and Gy. Muszély, On some new generalizations of the functional equation of Cauchy, Canad. Math. Bull.10 (1967), 197–205

1967
[3]

Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Będlewo, Poland, May 17–23, 2015, (2017), 71–106

R.Ger, Fischer-Muszélyadditivity: ahalfcenturystory, Developmentsinfunc- tional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Będlewo, Poland, May 17–23, 2015, (2017), 71–106

2015
[4]

Hirota, The Cauchy equation and norm additive mappings between positive cones of commutativeC ∗-algebras, J

D. Hirota, The Cauchy equation and norm additive mappings between positive cones of commutativeC ∗-algebras, J. Math. Anal. Appl.561 (2026), 130606

2026
[5]

Norm additive mappings between the positive cones of continuous function algebras

M. Shibata and T. Miura, Norm additive mappings between the positive cones of continuous function algebras, arXiv:2604.26540v1

work page internal anchor Pith review Pith/arXiv arXiv
[6]

Tabor, Stability of the Fischer–Muszély functional equation, Publ

J. Tabor, Stability of the Fischer–Muszély functional equation, Publ. Math. Debrecen 62 (2003), 205–211

2003
[7]

Zhang, M.-C

J. Zhang, M.-C. Tsai and N.-C. Wong, Norm of positive sum preservers of smooth Banach lattices and Lp(µ), J. Nonlinear Convex Anal., 20 (2019), 2613–2621. National Institute of Technology, Tsuruoka College, Yamagata 997-8511, Japan Email address: dhirota@tsuruoka-nct.ac.jp IKAKEN+HIBIKEN, Niigata, Japan Email address: oppekepenguin@gmail.com

2019

[1] [1]

Y. Dong, L. Li, L. Molnár and N.-C. Wong, Transformations preserving the normofmeansbetweenpositiveconesofgeneralandcommutative C ∗-algebras, J. Oper. Theory88 (2022), 365–406

2022

[2] [2]

Fischer and Gy

P. Fischer and Gy. Muszély, On some new generalizations of the functional equation of Cauchy, Canad. Math. Bull.10 (1967), 197–205

1967

[3] [3]

Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Będlewo, Poland, May 17–23, 2015, (2017), 71–106

R.Ger, Fischer-Muszélyadditivity: ahalfcenturystory, Developmentsinfunc- tional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Będlewo, Poland, May 17–23, 2015, (2017), 71–106

2015

[4] [4]

Hirota, The Cauchy equation and norm additive mappings between positive cones of commutativeC ∗-algebras, J

D. Hirota, The Cauchy equation and norm additive mappings between positive cones of commutativeC ∗-algebras, J. Math. Anal. Appl.561 (2026), 130606

2026

[5] [5]

Norm additive mappings between the positive cones of continuous function algebras

M. Shibata and T. Miura, Norm additive mappings between the positive cones of continuous function algebras, arXiv:2604.26540v1

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

Tabor, Stability of the Fischer–Muszély functional equation, Publ

J. Tabor, Stability of the Fischer–Muszély functional equation, Publ. Math. Debrecen 62 (2003), 205–211

2003

[7] [7]

Zhang, M.-C

J. Zhang, M.-C. Tsai and N.-C. Wong, Norm of positive sum preservers of smooth Banach lattices and Lp(µ), J. Nonlinear Convex Anal., 20 (2019), 2613–2621. National Institute of Technology, Tsuruoka College, Yamagata 997-8511, Japan Email address: dhirota@tsuruoka-nct.ac.jp IKAKEN+HIBIKEN, Niigata, Japan Email address: oppekepenguin@gmail.com

2019