Group representations with empty residual spectrum

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.