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The Socle and finite dimensionality of some Banach algebras

Ali Ghaffari·Ali Reza Medghalchi
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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 $\hbox{Soc}(L^{1}(G))$ such that $μ(G) \neq 0$. We also prove that $G$ is finite if $\hbox{Soc}(M(G))$ is closed and every nonzero left ideal in $M(G)$ contains a minimal left ideal.

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Ali Ghaffari, Ali Reza Medghalchi. 2005-08-25. The Socle and finite dimensionality of some Banach algebras. https://arxiv.org/abs/math/0508496

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