Annihilators in the Bidual of the Generalized Group Algebra of a Discrete Group

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