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

TL;DR

This paper investigates the structure of the second dual of a generalized group algebra, revealing explicit annihilators and showing that it is not strongly Arens irregular, thus advancing understanding of dual algebra properties.

Contribution

It explicitly constructs annihilators in the second dual of generalized group algebras and demonstrates their properties, providing new insights into the algebra's Arens regularity.

Findings

Constructs explicit annihilators using non-principal ultrafilters.

Shows the algebra is not strongly Arens irregular.

Provides insights into the structure of the second dual of generalized group algebras.

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