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

This article is intended towards the study of the bidual of generalized group algebra $L^1(G,\text{mA})$ equipped with two Arens product, where $G$ is any locally compact group and $\text{mA}$ is a Banach algebra. We show that the left topological center of $(L^1(G)\widehat{\otimes }\text{mA}{)}^{\ast \ast }$ is a Banach $L^1(G)$-module if $G$ is abelian. Further it also holds permanance property with respect to the unitization of $\text{mA}$. 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 $L^1(G{)}^{\ast \ast }$ being just $L^1(G)$, to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of $L^1(G,\text{mA})$ for non-reflexive Banach algebras $\text{mA}$ 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 $\text{mA}$ holds the Radon-Nikodym property and weak sequential completeness.