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 $\mu$ in $Soc(L^1(G))$ such that $\mu (G)\ne 0$. 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.

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