The compactum of a semi-simple commutative Banach algebra

Abstract

Let A be a commutative semi-simple Banach algebra such that the set consisting of finite sums of elements from minimal left ideals coincides with that of finite sums of elements from minimal right ideals. Let S(A) (the socle of A) denote this set. Let C(A) denote the set of elements x in A such that the map a→xax is compact. It is shown that C(A) is the norm closure of S(A).