Determining elements in Banach algebras through spectral properties

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 $C^{\ast }$-algebra, and (2) implies $a\in \mathbb{C}b$ if $A$ is a prime $C^{\ast }$-algebra. As an application of the results concerning the conditions (1) and (2) we obtain some spectral characterizations of multiplicative maps.

10 pages, accepted for publication in J. Math. Anal. Appl