In this short note, the second dual of generalized group algebra \((\ell ^1(G,\mathcal {A}),*)\) equipped with both Arens product 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 toplogical center. As a consequence, we also deduce the fact that \(\ell ^1(G,\mathcal {A})\) is not Strongly Arens irregular.

在这篇简短的笔记中，研究了配备两个 Arens乘积的广义群代数\((\ell ^1(G,\mathcal {A}),*)\) 的第二个对偶，其中G是任何离散群，而\( \mathcal {A}\)是一个 Banach 代数，包含\((\ell ^1(\mathbb N),\bullet )\)的补代数副本。我们在代数\(\ell ^1(G,\mathcal {A})^{**}\)中给出了一个显式的歼灭子族（均是 Arens 乘积），由\({ \mathbb {N}}\)并且不在拓扑中心。因此，我们还推断出\(\ell ^1(G,\mathcal {A})\)不是强阿伦斯不规则。