arXiv subjects

Natsumi Shibata

Publications and source records attributed to Natsumi Shibata.

4 recordsLinked to original sources

arXiv2026

Ring isomorphisms in norm between Banach algebras of continuous functions

Let $X$ and $Y$ be locally compact Hausdorff spaces and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. We say that a bijection $T\colon C_0(X,\mathbb{K})\to C_0(Y,\mathbb{K})$ is a ring isomorphism in norm if \[ \|T(f+g)\|=\|T(f)+T(g)\|,\qquad \|T(fg)\|=\|T(f)T(g)\| \] for every $f,g\in C_0(X,\mathbb{K})$. We determine the form of such maps. When $\mathbb{K}=\mathbb{C}$, under the additional assumption that $\|T(\overline f)\|=\|T(f)\|$ for every $f\in C_0(X,\mathbb{C})$, there exist a continuous function $w\colon Y\to\{λ\in\mathbb{C}:|λ|=1\}$, a homeomorphism $φ\colon Y\to X$, and a closed and open subset $Y_0\subset Y$ such that \[ T(f)(y)= \begin{cases} w(y)f(φ(y)),& y\in Y_0,\\ w(y)\overline{f(φ(y))},& y\in Y\setminus Y_0, \end{cases} \] for every $f\in C_0(X,\mathbb{C})$ and $y\in Y$. When $\mathbb{K}=\mathbb{R}$, there exist a continuous function $w\colon Y\to\{\pm1\}$ and a homeomorphism $φ\colon Y\to X$ such that \[ T(f)(y)=w(y)f(φ(y)) \] for every $f\in C_0(X,\mathbb{R})$ and $y\in Y$. In particular, the real case extends the norm version of the Gelfand--Kolmogoroff theorem to the locally compact setting.

math.FA↗

arXiv2026

Order Isomorphisms between Positive Cones of $C_0(X)$

Let $X$ and $Y$ be locally compact Hausdorff spaces. We study order isomorphisms \[ T:C_0^+(X)\to C_0^+(Y), \] where $C_0(X)$ denotes the Banach space of all real-valued continuous functions on $X$ vanishing at infinity, and \[ C_0^+(X)=\{f\in C_0(X):f\ge0\} \] is its positive cone. We assume that $T$ is positive homogeneous. That is, \[ T(rf)=rT(f) \qquad (r>0,\,f\in C_0^+(X)). \] Under this assumption, we prove that $T$ is represented as a weighted composition operator induced by a homeomorphism from $Y$ onto $X$ and a bounded continuous weight function. Moreover, we show that $T$ extends uniquely to a linear order isomorphism between $C_0(X)$ and $C_0(Y)$.

math.FA↗

arXiv2026

Norm additive mappings between the positive cones of continuous function algebras

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 \[ Tf(y)=h(y)f(τ(y)), \] where $τ:Y\to X$ 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 $C_0^+(X)$.

math.FA↗

arXiv2026

Additive and multiplicative maps in norm on the positive cone of continuous function algebras

Let $X$ and $Y$ be locally compact Hausdorff spaces. We denote by $C_0^+(X)$ the positive cone of all real-valued continuous functions on $X$ vanishing at infinity. In this paper, we consider a bijection $T\colon C_0^+(X) \to C_0^+(Y)$ satisfying the following two norm conditions for all $f, g \in C_0^+(X)$: \[ \|T(f+g)\| = \|T(f)+T(g)\|,\qquad \|T(f \cdot g)\| = \|T(f) \cdot T(g)\|. \] The main result of this paper is that such a map $T$ is a composition operator of the form $T(f) = f \circ τ$, induced by a homeomorphism $τ\colon Y \to X$.

math.FA↗