Characterizations of algebras arising from locally compact groups

Abstract:

Two Banach $^{\ast }$-algebras are naturally associated with a locally compact group, $G$: the group algebra, ${L^1}(G)$, and the measure algebra, $M(G)$. Either of these Banach algebras is a complete set of invariants for $G$. In any Banach $^{\ast }$-algebra, $A$, the norm one unitary elements form a group, $S$. Using $S$ we characterize those Banach $^{\ast }$-algebras, $A$, which are isometrically $^{\ast }$-isomorphic to $M(G)$. Our characterization assumes that $A$ is the dual of some Banach space and that its operations are continuous in the resulting weak $^{\ast }$ topology. The other most important condition is that the convex hull of $S$ must be weak$^{\ast }$ dense in the unit ball of $A$. We characterize Banach $^{\ast }$-algebras which are isomerically isomorphic to ${L^1}(G)$ for some $G$ as those algebras, $A$, whose double centralizer algebra, $D(A)$, satisfies our characterization for $M(G)$. In addition we require $A$ to consist of those elements of $D(A)$ on which $S$ (defined relative to $D(A)$) acts continuously with its weak$^{\ast }$ topology. Using another characterization of ${L^1}(G)$ we explicitly calculate the above isomorphism between $A$ and ${L^1}(G)$.