International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 2, Pages 275-280
doi:10.1155/S0161171282000246
The compactum and finite dimensionality in Banach algebras
Gulf Organization for Industrial Consulting, P.O. Box 5114, Doha, Qatar
Received 2 March 1981; Revised 23 September 1981
Copyright © 1982 Abdullah H. Al-Moajil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.