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

Source abstract

Let G G be a locally compact group and let V N ( G ) VN(G) denote the von Neumann algebra generated by the left translations of G G on L 2 ( G ) {L_2}(G) . Then V N ( G ) ∗ VN{(G)^{\ast }} , when regarded as the second conjugate space of the Fourier algebra of G G , is a Banach algebra with the Arens product. We prove among other things that when G G is amenable, V N ( G ) ∗ VN{(G)^{\ast }} is neither commutative nor semisimple unless G G is finite. We study in detail the class of maximal regular left ideals in V N ( G ) ∗ VN{(G)^{\ast }} . We also show that if G 1 {G_1} and G 2 {G_2} are discrete groups, then G 1 {G_1} and G 2 {G_2} are isomorphic if and only if V N ( G 1 ) ∗ VN{({G_1})^{\ast }} and V N ( G 2 ) ∗ VN{({G_2})^{\ast }} are isometric order isomorphic.