The Socle and finite dimensionality of some Banach algebras

Details The Socle and finite dimensionality of some Banach algebras The Socle and finite dimensionality of some Banach algebras

2005-08-25

arxiv.org/abs/math/0508496v1

Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 3, August 2005, pp. 327-330

Mathematics

Functional Analysis

4 pages, no figures, no tables

Scientific paper

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 $\hbox{Soc}(L^{1}(G))$ such that $\mu(G) \neq 0$. 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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