The compactum and finite dimensionality in Banach algebras

Abstract

Given a Banach algebraA, the compactum ofAis defined to be the set of elementsx∈Asuch that the operatora→xaxis compact. General properties of the compactum and its relation to the socle ofAare discussed. Characterizations of finite dimensionality of a semi-simple Banach algebra are given in terms of the compactum and the socle ofA.