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

Abstract

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 ${\ell }^1(G,\mathcal{A}{)}^{\ast \ast }$, 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.

Major changes done in accordance with the referee report

References in corpus (1)

• Geometry of Banach algebra $\mA$ and the bidual of $L^1(G,\mA)$