The Socle and finite dimensionality of some Banach algebras

Abstract

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact group G, G is compact if there exists a measure µ in Soc(L 1 (G)) such that µ(G) = 0. We also prove that G is finite if Soc(M(G)) is closed and every nonzero left ideal in M(G) contains a minimal left ideal.

On ideals in some Banach algebras associated with a foundation semigroup

“…Suppose first that I is a left ideal and μ ∈ I. Let x ∈ S, and let (e α ) α∈D be a bounded approximate identity in M a (S) [9]. Since M a (S) is an ideal in M(S) [5],…”

“…The proof is now complete. In recent years there has been shown considerable interest by harmonic analysts in the ideal problem of L 1 (G), LUC(S) * and L 1 (G) * * of a locally compact group G (see [2], [6], [7] and [9]). It seems to the author that the ideal problem of corresponding algebras of topological semigroups has not been touched so far.…”