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International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 4, Pages 821-822
http://dx.doi.org/10.1155/S0161171284000855

The compactum of a semi-simple commutative Banach algebra

Gulf Organization For Industrial Consulting, P. O. Box 5114, Doha, Qatar

Received 18 May 1983

Copyright © 1984 Hindawi Publishing Corporation. 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

Let A be a commutative semi-simple Banach algebra such that the set consisting of finite sums of elements from minimal left ideals coincides with that of finite sums of elements from minimal right ideals. Let S(A) (the socle of A) denote this set. Let C(A) denote the set of elements x in A such that the map axax is compact. It is shown that C(A) is the norm closure of S(A).