Determining elements in Banach algebras through spectral properties

Abstract

Let A be a Banach algebra. By σ(x) and r(x), we denote the spectrum and the spectral radius of x∈A, respectively. We consider the relationship between elements a,b∈A that satisfy one of the following two conditions: (1) σ(ax)=σ(bx) for all x∈A, (2) r(ax)≤r(bx) for all x∈A. In particular, we show that (1) implies that a=b if A is a C∗-algebra, and (2) implies that a∈Cb 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.