Determining elements in Banach algebras through spectral properties

TL;DR

This paper investigates how spectral properties of elements in Banach algebras determine their relationships, establishing conditions under which elements are equal or scalar multiples, especially in $C^*$-algebras, and applies these to characterize multiplicative maps.

Contribution

It provides new spectral criteria for identifying when elements in Banach algebras are equal or scalar multiples, especially within $C^*$-algebras, and characterizes multiplicative maps using spectral properties.

Findings

Condition (1) implies a=b in $C^*$-algebras.

Condition (2) implies a is a scalar multiple of b in prime $C^*$-algebras.

Spectral properties can characterize multiplicative maps.

Abstract

Let $A$ be a Banach algebra. By $\sigma(x)$ and $r(x)$ we denote the spectrum and the spectral radius of $x\in A$, respectively. We consider the relationship between elements $a,b\in A$ that satisfy one of the following two conditions: (1) $\sigma(ax) = \sigma(bx)$ for all $x\in A$, (2) $r(ax) \le r(bx)$ for all $x\in A$. In particular we show that (1) implies $a=b$ if $A$ is a $C^*$-algebra, and (2) implies $a\in \mathbb C b$ if $A$ is a prime $C^*$-algebra. As an application of the results concerning the conditions (1) and (2) we obtain some spectral characterizations of multiplicative maps.