Determining elements in Banach algebras through spectral properties

Let AA be a Banach algebra. By σ(x)σ(x) and r(x)r(x), we denote the spectrum and the spectral radius of x∈Ax∈A, respectively. We consider the relationship between elements a,b∈Aa,b∈A that satisfy one of the following two conditions: (1) σ(ax)=σ(bx)σ(ax)=σ(bx) for all x∈Ax∈A, (2) r(ax)≤r(bx)r(ax)≤r(bx) for all x∈Ax∈A. In particular, we show that (1) implies that a=ba=b if AA is a C∗C∗-algebra, and (2) implies that a∈Cba∈Cb if AA is a prime C∗C∗-algebra. As an application of the results concerning the conditions (1) and (2), we obtain some spectral characterizations of multiplicative maps.