The compactum of a semi-simple commutative Banach algebra (Q1080123)

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 scole of A) denote this set. Let C(A) denote the set of elements x in A such that the map \(a\to xax\) is compact. It is shown that C(A) is the norm closure of S(A).