arXiv subjects

Shiho Oi

Publications and source records attributed to Shiho Oi.

14 recordsLinked to original sources

arXiv2026

Linear maps preserving $C$-symmetry operators

Let $H$ be a separable complex Hilbert space. In this paper, we study a continuous bijective linear map $T$ on the algebra of all bounded linear operators on $H$. We characterize the map $T$ that maps the set of all $C$-symmetric operators onto the set of all $\psi(C)$-symmetric operators, where $\psi$ is a commutativity-preserving bijection on the set of all conjugations. Furthermore, we show that this condition is equivalent to the existence of a bijection $\varphi$ on the set of all orthonormal bases of $H$ such that $T$ maps the set of all $\{e_n\}$-diagonal operators onto the set of all $\varphi(\{e_n\})$-diagonal operators.

math.FA↗

arXiv2026

Spectral decomposition of doubly power-bounded elements in Banach algebras

We establish a characterization of doubly power-bounded elements with finite spectrum in Banach algebras. In particular, we present a spectral decomposition for such elements, extending a classical theorem of Gelfand concerning doubly power-bounded elements with singleton spectrum. Furthermore, we generalize a theorem of Koehler and Rosenthal for doubly power-bounded elements to the setting of Banach algebras. In the final section, we are initiating a study to investigate whether the properties of doubly power-bounded elements can offer insight into the commutativity of Banach algebras.

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arXiv2025

Spectrally additive maps on the positive cones of the Wiener algebra

We study surjective maps between the positive cones of the Wiener algebra that preserve the spectrum of the sum of every two elements. We show that such maps can be extended to isometric real-linear isomorphisms of the Wiener algebra.

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arXiv2025

Spectral decomposition of power-bounded operators: The finite spectrum case

In this paper, we investigate power-bounded operators, including surjective isometries, on Banach spaces. Koehler and Rosenthal asserted that an isolated point in the spectrum of a surjective isometry on a Banach space lies in the point spectrum, with the corresponding eigenspace having an invariant complement. However, they did not provide a detailed proof of this claim, at least as understood by the authors of this manuscript. Here, by applications of a theorem of Gelfand and the Riesz projections, we demonstrate that the theorem of Koehler and Rosenthal holds for any power-bounded operator on a Banach space. This not only furnishes a detailed proof of the theorem but also slightly generalizes its scope. As a result, we establish that if $T: X \to X$ is a power-bounded operator on a Banach space $X$ whose spectrum consists of finitely many points ${\lambda_1, \lambda_2, \dots, \lambda_m}$, then for every $1 \leq i, j \leq m$, there exist projections $P_j$ on $X$ such that $P_iP_j=\delta_{ij}P_i$, $\sum_{j=1}^mP_j=I$, and $T=\Sigma_{j=1}^m \lambda_j P_j$. It follows that such an operator $T$ is an algebraic operator.

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arXiv2024

Multiplicatively spectrum-preserving maps on $C^{*}$-algebras

We study surjective maps between the sets of all self-adjoint elements of unital $C^*$-algebras which satisfy the multiplicatively spectrum-preserving property. We show that such maps are characterized by Jordan isomorphisms and central symmetries. This is an answer to a problem posed by Moln\'ar.

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arXiv2024

Non-linear characterization of Jordan $$-isomorphisms via maps on positive cones of $C^$-algebras

We study maps between positive definite or positive semidefinite cones of unital $C^*$-algebras. We describe surjective maps that preserve (1) the norm of the quotient or multiplication of elements; (2) the spectrum of the quotient or multiplication of elements; (3) the spectral seminorm of the quotient or multiplication of elements. These maps relate to the Jordan $*$-isomorphisms between the specified $C^*$-algebras. While a surjection between positive definite cones that preserves the norm of the quotient of elements may not be extended to a linear map between the underlying $C^*$-algebras, the other types of surjections can be extended to a Jordan $*$-isomorphism or a Jordan $*$-isomorphism followed by the implementation by a positive invertible element. We also study conditions for the centrality of positive invertible elements. We generalize "the corollary" regarding surjections between positive semidefinite cones of unital $C^*$-algebras. Applying it, we provide positive solutions to the problem posed by Moln\'ar for general unital $C^*$-algebras.

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arXiv2023

Isometries between groups of invertible elements in Fourier-Stieltjes algebras

We prove that if open subgroups of the groups of invertible elements in two Fourier-Stieltjes algebras are isometric as metric spaces, then the underlying locally compact groups are topologically isomorphic. We describe the structure of isometric real algebra isomorphisms between Fourier-Stieltjes algebras and apply it to prove the above result.

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arXiv2023

Linear preservers on idempotents of Fourier algebras

In this article, we give a representation of bounded complex linear operators which preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is moreover positive or contractive, we show that the operator is induced by either a continuous group homomorphism or a continuous group anti-homomorphism. If the groups are totally disconnected, bounded homomorphisms on the Fourier algebra can be realised by the idempotent preserving operators.

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arXiv2022

Jordan $$-homomorphisms on the spaces of continuous maps taking values in $C^{}$-algebras

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. We consider Jordan $*$-homomorphisms on $C(X, \mathcal{A})$ and Jordan $*$-homomorphisms on $\operatorname{Lip}(X,\mathcal{A})$. More precisely, for any unital $C^{*}$-algebra $\mathcal{A}$, we prove that every Jordan $*$-homomorphism on $C(X, \mathcal{A})$ and every Jordan $*$-homomorphism on $\operatorname{Lip}(X,\mathcal{A})$ is represented as a weighted composition operator by using the irreducible representations of $\mathcal{A}$. In addition, when $\mathcal{A}_1$ and $\mathcal{A}_2$ are primitive $C^{*}$-algebras, we characterize the Jordan $*$-isomorphisms. These results unify and enrich previous works on algebra $*$-homomorphisms on $C(X, \mathcal{A})$ and $\operatorname{Lip}(X,\mathcal{A})$ for several concrete examples of $\mathcal{A}$.

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arXiv2021

Isometries and hermitian operators on spaces of vector-valued Lipschitz maps

We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm: $\|\cdot\|_{\infty}+L(\cdot)$. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on $\operatorname{Lip}(X,E)$, where $E$ is a complex Banach space, is a generalized composition operator. Secondly, we give a complete description of unital surjective complex linear isometries on $\operatorname{Lip}(X,\mathcal{A})$ where $\mathcal{A}$ is a unital factor $C^{*}$-algebra. These results improve previous results stated by the author.

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arXiv2021

Tingley's problem on uniform algebras

We prove that a surjective isometry between the unit spheres of two uniform algebras is extended to a surjective real-linear isometry between the uniform algebras. It provides the first positive solution for Tingley's problem on a Banach space of analytic functions.

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arXiv2019

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

In this paper, we extend a spherical variant of the Kowalski-S\{l}odkowski theorem due to Li, Peralta, Wang and Wang. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to the problem on 2-local isometries posed by Molnár.

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arXiv2018

2-local isometries on function spaces

We study 2-local reflexivity of the set of all surjective isometries between certain function spaces. We do not assume linearity for isometries. We prove that a 2-local isometry in the group of all surjective isometries on the algebra of all continuously differentiable functions on the closed unit interval with respect to several norms is a surjective isometry. We also prove that a 2-local isometry in the group of all surjective isometries on the Banach algebra of all Lipschitz functions on the closed unit interval with the sum-norm is a surjective isometry.

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arXiv2018

Isometries on Banach algebras of vector-valued maps

We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative $C^*$-algebras. As a consequence we confirm the statement of \cite[Example 8]{jp} on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.

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