Group representations with empty residual spectrum

TL;DR

This paper investigates the residual spectrum of group algebra operators on Banach spaces, establishing conditions under which the spectrum is empty, especially for spaces like (\u0393) and group von Neumann algebras, and introduces the concept of surjunctive pairs.

Contribution

It introduces the notion of surjunctive pairs and analyzes residual spectra for group actions on various Banach spaces, providing new insights and partial results for (\u0393) and related spaces.

Findings

Operators on (\u0393) and von Neumann algebras have empty residual spectrum.

Partial results for ^p(93) spaces when 93 is amenable.

Existence of necessary conditions on 93 for residual spectrum properties.

Abstract

Let $X$ be a Banach space on which a discrete group $\Gamma$ acts by isometries. For certain natural choices of $X$, every element of the group algebra, when regarded as an operator on $X$, has empty residual spectrum. We show, for instance, that this occurs if $X$ is $\ell^2(\Gm)$ or the group von Neumann algebra $VN(\Gm)$. In our approach, we introduce the notion of a {\em surjunctive pair}, and develop some of the basic properties of this construction. The cases $X=\ell^p(\Gm)$ for $1<p<2$ or $2<p<\infty$ are more difficult. If $\Gm$ is amenable we can obtain partial results, using a majorization result of Herz; an example of Willis shows that some condition on $\Gm$ is necessary.