Symmetrized pseudofunction algebras from $L^p$-representations and amenability of locally compact groups

TL;DR

This paper explores the structure of $L^p$-pseudofunction algebras for locally compact groups, linking them with group amenability and providing new characterizations and duality results in the symmetrized setting.

Contribution

It introduces a unified framework connecting $L^p$-pseudofunction algebras with group amenability and extends existing characterizations to symmetrized algebras.

Findings

$L^p$-pseudofunction algebras sit between $L^1(G)$ and $C^*(G)$

Characterizations of group amenability are extended to symmetrized algebras

Dual space descriptions for these algebras are provided

Abstract

We show via an application of techniques from complex interpolation theory how the $L^p$-pseudofunction algebras of a locally compact group $G$ can be understood as sitting between $L^1(G)$ and $C^*(G)$. Motivated by this, we collect and review various characterizations of group amenability connected to the $p$-pseudofunction algebra of Herz and generalize these to the symmetrized setting. Along the way, we describe the Banach space dual of the symmetrized pseudofuntion algebras on $G$ associated with representations on reflexive Banach spaces.