arxiv: 2604.26540 · v1 · submitted 2026-04-29 · 🧮 math.FA

Recognition: unknown

Norm additive mappings between the positive cones of continuous function algebras

Natsumi Shibata, Takeshi Miura

Pith reviewed 2026-05-07 12:36 UTC · model grok-4.3

classification 🧮 math.FA

keywords norm additive mappingspositive conesC0 spacescontinuous function algebrasweighted composition operatorshomeomorphismsnonlinear functional equations

0 comments

The pith

Every bijection on positive cones of C0 spaces that preserves norm additivity must be a weighted composition map via a homeomorphism.

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

The paper studies bijections T between the sets of nonnegative continuous functions vanishing at infinity on two locally compact Hausdorff spaces X and Y. It requires that the norm of T applied to the sum of any two such functions equals the norm of the sum of the images. Unlike the compact unital case, the non-unital C0 setting lacks a unit, so the proof avoids linear extension methods and directly constructs the form of T. If this holds, the bijections are completely described by a homeomorphism of the underlying spaces together with a positive continuous scaling function.

Core claim

Every bijection T from C0+(X) to C0+(Y) satisfying ||T(f+g)|| = ||Tf + Tg|| for all positive f and g admits the representation Tf(y) = h(y) f(τ(y)), where τ is a homeomorphism from Y to X and h is a bounded continuous function from Y to the positive reals.

What carries the argument

The weighted composition operator Tf(y) = h(y) f(τ(y)), with τ a homeomorphism and h positive continuous and bounded; this form encodes the bijection while enforcing the norm-additivity condition on the cones.

If this is right

The underlying spaces X and Y must be homeomorphic.
The scaling function h must be continuous, positive, and bounded.
The map T automatically preserves the zero function and the supports in a compatible way.
This gives a full classification of all such norm-additive bijections without extra assumptions.

Where Pith is reading between the lines

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

The same norm-additivity condition may force similar multiplicative forms on positive cones of other Banach function spaces.
One could check whether the bijection hypothesis can be weakened to surjectivity while retaining the weighted composition conclusion.
The construction of τ and h might extend to maps that preserve additivity only for disjointly supported functions.

Load-bearing premise

The map T is a bijection from the positive cone of C0(X) onto the positive cone of C0(Y) and satisfies norm additivity for every pair of positive elements.

What would settle it

An explicit bijection T on the positive cones of two C0 spaces that satisfies the norm-additivity equality for all pairs but cannot be written as Tf(y) = h(y) f(τ(y)) for any homeomorphism τ and positive continuous bounded h.

read the original abstract

We study bijections between the positive cones of spaces of continuous functions vanishing at infinity that satisfy a norm additive condition. Such maps arise naturally in the study of nonlinear functional equations and norm-preserving structures on function spaces. While in the compact (unital) case these maps can often be analyzed via linear extension techniques, the non-unital setting $C_0(X)$ requires a different approach due to the absence of a distinguished unit element. In this paper, we show that every bijection $T:C_0^+(X)\to C_0^+(Y)$ between the positive cones of $C_0(X)$ and $C_0(Y)$ satisfying \[ \|T(f+g)\|=\|Tf+Tg\| \] for all $f,g\in C_0^+(X)$ admits a representation of the form \[ Tf(y)=h(y)f(\tau(y)), \] where $\tau:Y\to X$ is a homeomorphism and $h$ is a bounded continuous function from $Y$ to $(0,\infty)$. This yields a complete characterization of norm additive bijections on positive cones of $C_0^+(X)$.

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

This paper gives a direct representation for norm-additive bijections on positive cones of non-unital C0 spaces, built from the norm condition and bijectivity without linear extensions.

read the letter

The main thing to know is that this paper gives a complete characterization of norm-additive bijections on the positive cones of C0 spaces in the non-unital setting. Any such bijection T must satisfy Tf(y) = h(y) f(τ(y)) for a homeomorphism τ and a positive continuous bounded h. They do this without the linear extension tricks that work when there is a unit. Instead, they use the norm equality on sums to characterize when positive functions attain their maxima together and to match the zero sets of functions under T. This lets them define τ(y) as the unique point in X corresponding to the vanishing behavior at y, and then recover h from the values. The construction appears direct and avoids circular definitions. The stress-test note confirms there are no hidden assumptions or failures in non-sigma-compact cases, which is good because C0 spaces can be tricky there. On the downside, the boundedness of h is stated but one might want to see the explicit estimate from the norm condition. Also, proving that τ is indeed a homeomorphism requires checking continuity in both directions, which probably uses the local compactness. These are standard steps but worth confirming they go through without extra conditions. Overall, this is for specialists in preserver problems within functional analysis. If you work on maps that preserve some norm or order properties on function algebras, this gives a clean answer for the positive cone case in C0. The paper shows clear thinking by contrasting the unital and non-unital approaches and delivering the result from the given assumptions. It deserves a serious referee because it provides a new representation that was missing from the literature. I would recommend sending it to peer review rather than desk rejecting it.

Referee Report

0 major / 2 minor

Summary. The paper claims that every bijection T: C_0^+(X) → C_0^+(Y) between the positive cones of C_0 spaces on locally compact Hausdorff spaces X and Y, satisfying the norm-additivity condition ||T(f+g)|| = ||Tf + Tg|| for all f,g in C_0^+(X), admits a representation Tf(y) = h(y) f(τ(y)), where τ: Y → X is a homeomorphism and h: Y → (0,∞) is bounded and continuous. The proof constructs τ and h directly from bijectivity and the norm condition by analyzing sup-attaining behavior and vanishing sets.

Significance. If the derivation holds, this yields a complete characterization of norm-additive bijections on positive cones in the non-unital C_0 setting, extending compact-case results without relying on linear extensions. The direct construction from the given axioms, valid for general locally compact Hausdorff spaces including non-σ-compact cases, strengthens the contribution to nonlinear preserver problems in functional analysis.

minor comments (2)

[§1] §1, paragraph following the main theorem: the statement that h is 'bounded' is used in the representation but the proof sketch does not explicitly verify boundedness from the norm-additivity alone; a short remark clarifying the extraction of the bound would improve readability.
[Abstract] The abstract and introduction use C_0^+(X) without defining the positive cone explicitly; while standard, adding a one-sentence definition would aid readers outside the immediate subfield.

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 referee's summary accurately describes the main result and the approach taken in the paper.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from bijectivity and norm-additivity

full rationale

The manuscript constructs the homeomorphism τ and multiplier h directly from the given bijectivity of T on positive cones and the norm-additivity equality ||T(f+g)||=||Tf+Tg||, by recovering pointwise values via joint sup-attaining functions and corresponding vanishing-at-infinity sets. No equation or step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the argument uses only the standard locally compact Hausdorff topology on X and Y together with the cone structure, without linear extensions or imported uniqueness theorems. The result is therefore independent of its inputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition and norm of C0 spaces together with basic topological properties of locally compact Hausdorff spaces; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)

standard math C0(X) denotes the Banach space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff space X, equipped with the supremum norm.
This is the standard setting for the positive cone C0+(X) studied in the paper.
standard math A homeomorphism between locally compact Hausdorff spaces preserves the vanishing-at-infinity property under composition.
Invoked implicitly when constructing τ from the bijection T.

pith-pipeline@v0.9.0 · 5503 in / 1421 out tokens · 42714 ms · 2026-05-07T12:36:52.819762+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references

[1]

Acz ´el and J

J. Acz ´el and J. Dhombres, Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications, vol. 31, Cambridge University Press, Cambridge, 1989

1989
[2]

Fischer and Gy

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

1967
[3]

J. J. Font and S. Hernandez,On separating maps between locally compact spaces, Arch. Math.63(1994), 158–165

1994
[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]

Jarosz,Automatic continuity of separating linear isomorphisms, Canad

K. Jarosz,Automatic continuity of separating linear isomorphisms, Canad. Math. Bull.33(1990), 139–144

1990
[6]

Moln ´ar,Maps on positive cones in operator algebras preserving power means, Aequationes Math.94(2020), 703–722

L. Moln ´ar,Maps on positive cones in operator algebras preserving power means, Aequationes Math.94(2020), 703–722. Graduate School of Science and Technology , Niigata University , Niigata 950-2181, Japan Email address:f25a056h@mail.cc.niigata-u.ac.jp Department of Mathematics, Faculty of Science, Niigata University , Niigata 950-2181, Japan Email address...

2020