A new proof on the characterization of vanishing subalgebras of group algebras

Abstract:

We give a new proof to a result due to T. Shimizu stating that for a locally compact group $G$ and the associated group algebra $L^{1}(G)$, if $S$ is a measurable subset of $G$, then a necessary and sufficient condition for the subspace of all functions in $L^{1}(G)$ that vanish almost everywhere off $S$ to be an algebra is that $S$ is equal to a subsemigroup of $G$, locally almost everywhere. Our proof bypasses a deep result of A. Ionescu Tulcea and C. Ionescu Tulcea, used in Shimizu’s proof, and is for the most part functional analytic.