The second conjugate algebra of the Fourier algebra of a locally compact group

Abstract:

Let $G$ be a locally compact group and let $VN(G)$ denote the von Neumann algebra generated by the left translations of $G$ on ${L_2}(G)$. Then $VN{(G)^{\ast }}$, when regarded as the second conjugate space of the Fourier algebra of $G$, is a Banach algebra with the Arens product. We prove among other things that when $G$ is amenable, $VN{(G)^{\ast }}$ is neither commutative nor semisimple unless $G$ is finite. We study in detail the class of maximal regular left ideals in $VN{(G)^{\ast }}$. We also show that if ${G_1}$ and ${G_2}$ are discrete groups, then ${G_1}$ and ${G_2}$ are isomorphic if and only if $VN{({G_1})^{\ast }}$ and $VN{({G_2})^{\ast }}$ are isometric order isomorphic.