Geometry of Banach algebra [formula omitted] and the bidual of [formula omitted

Abstract

This article is intended towards the study of the bidual of generalized group algebra L1(G,A) equipped with two Arens products, where G is any locally compact group and A is a Banach algebra. We show that the left topological center of (L1(G)⊗ˆA)⁎⁎ is a Banach L1(G)-module if G is abelian. Further it also holds permanence property with respect to the unitization of A. We then use this fact to extend the remarkable result of A.M. Lau and V. Losert [7], about the topological center of L1(G)⁎⁎ being just L1(G), to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of L1(G,A)⁎⁎ for non-reflexive Banach algebras A and give a partial characterization for elements of pseudo-center using the Cohen's factorization theorem. In the running we also observe few consequences when A holds the Radon-Nikodym property and weak sequential completeness.