  In this short note, the second dual of generalized group algebra
$(\ell^1(G,\mathcal A),\ast)$ equipped with both Arens products is
investigated, where $G$ is any discrete group and $\mathcal A$ is a Banach
algebra containing a complemented algebraic copy of $(\ell^1(\mathbb
N),\bullet)$. We give an explicit family of annihilators(w.r.t both the Arens
product) in the algebra $\ell^1(G,\mathcal A)^{**}$, arising from non-principal
ultrafilters on $\mathbb N$ and which are not in the topological center. As a
consequence, we also deduce the fact that $\ell^1(G,\mathcal A)$ is not
Strongly Arens irregular.
