Geometry of Banach algebra \(\mA\) and the bidual of \(L^1(G,\mA)\)

This article is intended towards the study of the bidual of generalized group algebra \(L^1(G,\mA)\) equipped with two Arens product, where \(G\) is any locally compact group and \(\mA\) is a Banach algebra. We show that the left topological center of \((L^1(G)\hat\otimes\mA)^{**}\) is a Banach \(L^1(G)\)-module if \(G\) is abelian. Further it also holds permanance property with respect to the unitization of \(\mA\). We then use this fact to extend the remarkable result of A.M Lau and V. Losert\cite{Lau-losert}, about the topological center of \(L^1(G)^{**}\) being just \(L^1(G)\), to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of \(L^1(G,\mA)\) for non-reflexive Banach algebras \(\mA\) 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 \(\mA\) holds the Radon-Nikodym property and weak sequential completeness.