Weakly Compact #?-algebras

1. A complex Banach algebra A is a compact (weakly compact) algebra if its left and right regular representations consist of compact (weakly compact) operators. Let E be any subset of A and denote by Ei and Er the left and right annihilators of E. A is an annihilator algebra if A¡= (0) —Ar, Ir^{fS) for each proper closed left ideal / and Ji t¿ (0) for each proper closed right ideal /. In [6, Theorem l], it was shown that a semi-simple compact algebra is an annihilator algebra. The first main result of the present paper (Theorem 2.1) is that a semi-simple annihilator algebra is a weakly compact algebra. Thus if 6, 6,, VP denote respectively the class of all semi-simple compact algebras, all semi-simple annihilator algebras and all weakly compact algebras, we have CCttCW. §3 is devoted to the structure theory of weakly compact S*-algebras begun in [7]. A Banach algebra A is a S'-algebra if, given aEA, there exists a^O in A such that