Geometric Properties of some Banach Algebras related to the Fourier algebra on Locally Compact Groups

TL;DR

This paper explores the geometric properties of certain Banach algebras associated with Fourier algebras on locally compact groups, revealing new results even for simple groups like the real line or integers.

Contribution

It characterizes when the Banach algebras $A_p^r(G)$ possess properties like RNP, SP, or DPP, for various groups and parameters, advancing understanding of their geometric structure.

Findings

Identifies conditions for $A_p^r(G)$ to have RNP, SP, or DPP.

Provides new results for groups $G=\mathbb{R}$ and $\mathbb{Z}$.

Enhances understanding of Banach algebra geometry related to Fourier analysis.

Abstract

Let $A_p(G)$ denote the Figa-Talamanca-Herz Banach Algebra of the locally compact group}} $G$, thus $A_2(G)$ {\it{is the Fourier Algebra of $G$. If $G$ is commutative then $A_2(G)=L^1(\hat{G})^\wedge$. Let $A_p^r(G)=A_p\cap L^r(G)$ with norm $\|u\|_{A_p^r}=\| u\|_{A_p}+\| u\|_{L^r}$.We investigate for which $p$, $r$, and $G$ do the Banach algebras $A_p^r(G)$ {\it{have the Banach space geometric properties: The Radon-Nikodym Property (RNP), the Schur Property (SP) or the Dunford-Pettis Property (DPP). The results are new even if $G=R$ (the real line) or $G=Z$ (the additive integers).