TY - JOUR

T1 - Group representations with empty residual spectrum

AU - Choi, Yemon

N1 - Erratum: Int. Eq. Op. Th. 69 (2011), no. 1, 149--150. DOI: 10.1007/s00020-010-1847-y

PY - 2010/5

Y1 - 2010/5

N2 - 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.

AB - 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.

KW - Residual spectrum

KW - surjunctive

KW - group von Neumann algebra

KW - directly finite

KW - amenable group

U2 - 10.1007/s00020-010-1772-0

DO - 10.1007/s00020-010-1772-0

M3 - Journal article

VL - 67

SP - 95

EP - 107

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 1420-8989

IS - 1

ER -