Linear and Multiplicative Maps under Spectral Conditions

IF 0.6 4区数学 Q3 MATHEMATICS

Functional Analysis and Its Applications Pub Date : 2024-03-12 DOI:10.1134/s0016266323030012

Bhumi Amin, Ramesh Golla

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引用次数: 0

Abstract

The multiplicative version of the Gleason–Kahane–Żelazko theorem for \(C^*\)-algebras given by Brits et al. in [4] is extended to maps from \(C^*\)-algebras to commutative semisimple Banach algebras. In particular, it is proved that if a multiplicative map \(\phi\) from a \(C^*\)-algebra \(\mathcal{U}\) to a commutative semisimple Banach algebra \(\mathcal{V}\) is continuous on the set of all noninvertible elements of \(\mathcal{U}\) and \(\sigma(\phi(a)) \subseteq \sigma(a)\) for any \(a \in \mathcal{U}\), then \(\phi\) is a linear map. The multiplicative variation of the Kowalski–Słodkowski theorem given by Touré et al. in [14] is also generalized. Specifically, if \(\phi\) is a continuous map from a \(C^*\)-algebra \(\mathcal{U}\) to a commutative semisimple Banach algebra \(\mathcal{V}\) satisfying the conditions \(\phi(1_\mathcal{U})=1_\mathcal{V}\) and \(\sigma(\phi(x)\phi(y)) \subseteq \sigma(xy)\) for all \(x,y \in \mathcal{U}\), then \(\phi\) generates a linear multiplicative map \(\gamma_\phi\) on \(\mathcal{U}\) which coincides with \(\phi\) on the principal component of the invertible group of \(\mathcal{U}\). If \(\mathcal{U}\) is a Banach algebra such that each element of \(\mathcal{U}\) has totally disconnected spectrum, then the map \(\phi\) itself is linear and multiplicative on \(\mathcal{U}\). It is shown that a similar statement is valid for a map with semisimple domain under a stricter spectral condition. Examples which demonstrate that some hypothesis in the results cannot be discarded.

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谱条件下的线性和乘法映射

Abstract Brits 等人在[4]中给出的 Gleason-Kahane-Żelazko 定理的乘法版本被扩展到从\(C^*\)-数到交换半简单巴拿赫数的映射。特别是证明了如果一个乘法映射（phi）来自于一个（C^*）-代数代数到交换半简单巴拿赫代数的乘法映射在 \(\mathcal{U}\) 的所有不可逆元素集合上是连续的，并且对于任何 \(a \ in \mathcal{U}\) 来说，\(\sigma(\phi(a)) \subseteq \sigma(a)\)都是连续的、那么 \(\phi\) 是一个线性映射。图雷等人在[14]中给出的科瓦尔斯基-斯沃德科夫斯基（Kowalski-Słodkowski）定理的乘法变化也得到了推广。具体来说如果 \(\phi\) 是一个来自 \(C^*\)- 代数的连续映射代数到交换半简单巴拿赫代数的连续映射，满足条件 \(\phi(1_\mathcal{U})=1_\mathcal{V}\) and \(\sigma(\phi(x)\phi(y)) \subseteq \sigma(xy)\) for all \(x、y在\mathcal{U}\)中，那么\(\phi\)在\(\mathcal{U}\)上生成一个线性乘法映射\(\gamma_\phi\)，它与\(\mathcal{U}\)可逆群的主成分上的\(\phi\)重合。如果\(\mathcal{U}\)是一个巴拿赫代数，使得\(\mathcal{U}\)的每个元素都有完全断开的谱，那么映射\(\phi\)本身在\(\mathcal{U}\)上是线性和乘法的。研究表明，在更严格的谱条件下，类似的陈述对于具有半简单域的映射也是有效的。这些例子证明结果中的某些假设是不能丢弃的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Functional Analysis and Its Applications 数学-数学

CiteScore

0.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.