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This article is intended towards the study of the bidual of generalized groupalgebra $L^1(G,\mA)$ equipped with two Arens product, where $G$ is any locallycompact group and $\mA$ is a Banach algebra. We show that the left topologicalcenter of $(L^1(G)\hat\otimes\mA)^{**}$ is a Banach $L^1(G)$-module if $G$ isabelian. Further it also holds permanance property with respect to theunitization of $\mA$. We then use this fact to extend the remarkable result ofA.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 caseusing the theory of vector measures. We further explore pseudo-center of$L^1(G,\mA)$ for non-reflexive Banach algebras $\mA$ and give a partialcharacterization for elements of pseudo-center using the Cohen's factorizationtheorem. In the running we also observe few consequences when $\mA$ holds theRadon-Nikodym property and weak sequential completeness.

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