  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.
