  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.
