The Socle and finite dimensionality of some Banach algebras

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 $\mu(G) \neq 0$\mathrm{.}Wealsoprovethat$G$isfiniteif$\hbox{Soc}(M(G))$ is closed and every nonzero left ideal in $M(G)$ contains a minimal left ideal.