Quasi-central bounded approximate identities in group algebras of locally compact groups

Abstract

A net in the group algebra of a locally compact group which commutes asymptotically with elements from the measure algebra is called quasi-central. In this paper we provide new characterizations of locally compact groups whose group algebras possess quasi-central bounded approximate identities. Reiter-type and structural conditions for such groups are obtained which indicate that these groups behave much like the tractable [SIN]-groups. A general notion of an amenable action on the predual of a von Neumann algebra is developed to prove these theorems. Applications to the cohomology of group and Fourier algebras are discussed.