On the extreme non-Arens regularity of Banach algebras

TL;DR

This paper investigates the conditions under which Banach algebras exhibit extreme non-Arens regularity, providing new criteria applicable to key algebras in harmonic analysis, and demonstrating their effectiveness through specific examples.

Contribution

It introduces new criteria based on $ ext{ell}^1$-bases for establishing extreme non-Arens regularity in Banach algebras, extending understanding of their structure.

Findings

Criteria apply to group, measure, semigroup, and Fourier algebras.

Bounded approximate identities and TI-nets are key tools.

Criteria successfully used on specific harmonic analysis algebras.

Abstract

As is well-know, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper we show that $\ell^1$-bases which approximate upper and lower triangles of products of elements in the algebra produce large sets of functionals that are not weakly almost periodic. This leads to criteria for extreme non-Arens regularity of Banach algebras in the sense of Granirer. We find in particular that bounded approximate identities (bai's) and bounded nets converging to invariance (TI-nets) both fall into this approach, suggesting that this is indeed the main tool behind most known constructions of non-Arens regular algebras. These criteria can be applied to the main algebras in harmonic analysis such as the group algebra, the measure algebra, the semigroup algebra (with certain weights) and the Fourier algebra. In this paper, we apply our criteria to the Lebesgue-Fourier algebra, the 1-Segal Fourier algebra and the Fig\`a-Talamanca Herz algebra.