Group representations with empty residual spectrum

Choi, Yemon (2010) Group representations with empty residual spectrum. Integral Equations and Operator Theory, 67 (1). pp. 95-107. ISSN 1420-8989

Abstract

Let X be a Banach space on which a discrete group Γ 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 ℓ 2(Γ) or the group von Neumann algebra VN(Γ). In our approach, we introduce the notion of a surjunctive pair, and develop some of the basic properties of this construction. The cases X = ℓ p (Γ) for 1 ≤ p < 2 or 2 < p < ∞ are more difficult. If Γ is amenable we can obtain partial results, using a majorization result of Herz; an example of Willis shows that some condition on Γ is necessary.