The compactum and finite dimensionality in Banach algebras

Abstract

ABSTRACT. Given a Banach algebra A, the compactum of A is defined to be the set of elements x g A such that the operator a xax is compact. General properties of the compactum and its relation to the socle of A are discussed. Characterizations of finite dimensionality of a seml-slmple Banach algebra are given in terms of the compactum and the socle of A.