Annihilators in the bidual of generalized group algebra of a discrete group

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.