The compactum and finite dimensionality in Banach algebras - Abdullah H. Al-Moajil (International Journal of Mathematics and Mathematical Sciences)
Given a Banach algebra A, the compactum of A is defined to be the set of elements x∈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 semi-simple Banach algebra are given in terms of the compactum and the socle of A.