Centralizers in semisimple algebras, and descent spectrum in Banach algebras

We prove that semisimple algebras containing some algebraic element whose centralizer is semiperfect are artinian. As a consequence, semisimple complex Banach algebras containing some element whose centralizer is algebraic are finite-dimensional. This answers affirmatively a question raised in Burgos et al. (2006) [4], , and is applied to show that an element a in a semisimple complex Banach algebra A does not perturb the descent spectrum of every element commuting with a if and only if some of power of a lies in the socle of A. This becomes a Banach algebra version of a theorem in Burgos et al. (2006) [4], , Kaashoek and Lay (1972) [9] for bounded linear operators on complex Banach spaces.