The Second Conjugate Algebra of the Fourier Algebra of a Locally Compact Group

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