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The socle and finite dimensionality of some Banach algebras. (English) Zbl 1098.46034
Summary: 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 $$\text{Soc}(L^1(G))$$ such that $$\mu(G)\neq 0$$. We also prove that $$G$$ is finite if $$\text{Soc} (M(G))$$ is closed and every nonzero left ideal in $$M(G)$$ contains a minimal left ideal.

##### MSC:
 46H20 Structure, classification of topological algebras 46H10 Ideals and subalgebras
##### Keywords:
locally compact group
Full Text:
##### References:
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