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Group representations with empty residual spectrum

Research output: Contribution to journalJournal articlepeer-review

Published
<mark>Journal publication date</mark>05/2010
<mark>Journal</mark>Integral Equations and Operator Theory
Issue number1
Volume67
Number of pages13
Pages (from-to)95-107
Publication StatusPublished
<mark>Original language</mark>English

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.

Bibliographic note

Erratum: Int. Eq. Op. Th. 69 (2011), no. 1, 149--150. DOI: 10.1007/s00020-010-1847-y