The Socle and finite dimensionality of some Banach algebras

2005/08/25 by Ali Ghaffari, Ali Reza Medghalchi
Mathematics · #math.FA #msc:22A25 #msc:46H15

paper · pdf

published as Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 3, August 2005, pp. 327-330 · 4 pages, no figures, no tables

arxiv created 2005/08/25 · arxiv updated 2009/12/01

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 \hboxSoc(L¹(G)) such that μ(G) ≠ 0. We also prove that G is finite if \hboxSoc(M(G)) is closed and every nonzero left ideal in M(G) contains a minimal left ideal.

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