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.