Functoriality of group algebras acting on $L^p$-spaces

TL;DR

This paper investigates the functorial properties of group algebras acting on $L^p$-spaces, revealing conditions under which $p$-pseudofunctions and their universal completions behave functorially, with applications to algebras on $bZ$.

Contribution

It establishes new functoriality results for $p$-pseudofunctions and their universal completions under specific group homomorphisms, advancing understanding of their algebraic structure.

Findings

$p$-pseudofunctions are functorial under injective homomorphisms

Universal completions are functorial with respect to quotient maps

Algebras on $bZ$ are isometrically isomorphic iff $p$ and $q$ are equal or conjugate

Abstract

We continue our study of group algebras acting on $L^p$-spaces, particularly of algebras of $p$-pseudofunctions of locally compact groups. We focus on the functoriality properties of these objects. We show that $p$-pseudofunctions are functorial with respect to homomorphisms that are either injective, or whose kernel is amenable and has finite index. We also show that the universal completion of the group algebra with respect to representations on $L^p$-spaces, is functorial with respect to quotient maps. As an application, we show that the algebras of $p$- and $q$-pseudofunctions on $\mathbb{Z}$ are (abstractly) isometrically isomorphic as Banach algebras if and only if $p$ and $q$ are either equal or conjugate.