arXiv subjects

Izuho Matsuzaki

Publications and source records attributed to Izuho Matsuzaki.

6 recordsLinked to original sources

arXiv2026

Involution-preserving ring isomorphisms in norm between unital $C^*$-algebras

Let $A$ and $B$ be nonzero unital $C^*$-algebras with units $1_A$ and $1_B$, respectively, and let $T\colon A\to B$ be a bijection satisfying \[ \|T(a+b)\|=\|T(a)+T(b)\|, \qquad \|T(ab)\|=\|T(a)T(b)\|, \qquad T(a^*)=T(a)^* \] for all $a,b\in A$. We prove that there exist a central symmetry $u$ in $B$ and a real $*$-isomorphism $Φ\colon A\to B$ such that \[ T(a)=uΦ(a) \qquad(a\in A). \] Moreover, $u$ and $Φ$ are uniquely determined by $T$. Conversely, if $u\in B$ is a central symmetry and $Φ\colon A\to B$ is a real $ * $-isomorphism, then $T=uΦ$ is a bijection satisfying the three identities above. In particular, the two norm identities, together with involution preservation, force the normalized map $uT$ to preserve the full product, not merely the Jordan product.

math.OA↗

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

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

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.

math.FA↗

arXiv2026

On phase-isometries between the unit spheres of the Banach space of continuous real-valued functions

For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$ the Banach space of all continuous real-valued functions on $L$ vanishing at infinity, endowed with the supremum norm. In this paper, we prove that every surjective phase-isometry $T\colon S(C_0(X,\mathbb{R}))\to S(C_0(Y,\mathbb{R}))$ between the unit spheres of $C_0(X,\mathbb{R})$ and $C_0(Y,\mathbb{R})$ is a variant of a weighted composition operator in the following sense: there exist a function $\varepsilon\colon S(C_0(X,\mathbb{R}))\to\{-1,1\}$,a continuous function $α\colon Y\to \{-1,1\}$ and a homeomorphism $σ\colon Y\to X$ such that $T(f)(q)=\varepsilon(f)α(q)f(σ(q))$ for every $f\in S(C_0(X,\mathbb{R}))$ and $q\in Y$.

math.FA↗

arXiv2024

Phase-isometries between the positive cones of the Banach space of continuous real-valued functions

For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$ the Banach space of all continuous real-valued functions on $L$ vanishing at infinity equipped with the supremum norm. We prove that every surjective phase-isometry $T\colon C_0^+(X,\mathbb{R}) \to C_0^+(Y,\mathbb{R})$ between the positive cones of $C_0(X,\mathbb{R})$ and $C_0(Y,\mathbb{R})$ is a composition operator induced by a homeomorphism between $X$ and $Y$. Furthermore, we show that any surjective phase-isometry $T\colon C_0^+(X,\mathbb{R}) \to C_0^+(Y,\mathbb{R})$ extends to a surjective linear isometry from $C_0(X,\mathbb{R})$ onto $C_0(Y,\mathbb{R})$.

math.FA↗