Scattered elements of Banach algebras

A scattered element of a Banach algebra A is an element with at most countable spectrum. The set of all scattered elements is denoted by S(A): The scattered radical R_sc(A) is the largest ideal consisting of scattered elements. We characterize in several ways central elements of A modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) S(A)+S(A) ⊂ S(A); (ii) S(A)S(A) ⊂ S(A); (iii) [S(A);A] ⊂ R_sc(A).