A Hilbert Module Approach to the Haagerup Property

Zhe Dong, Zhong Jin Ruan

Research output: Contribution to journalArticlepeer-review

Abstract

We develop a Hilbert module version of the Haagerup property for general C*-algebras A ⊆ B. We show that if α : Γ {right curved arrow} A is an action of a countable discrete group Γ on a unital C*-algebra A, then the reduced C*-algebra crossed product Γ ⋉ α,r A has the Hilbert A-module Haagerup property if and only if the action α has the Haagerup property. We are particularly interested in the case when A = C(X) is a unital commutative C*-algebra. We compare the Haagerup property of such an action α : Γ {right curved arrow} C(X) with the two special cases when (1) Γ has the Haagerup property and (2) Γ is coarsely embeddable into a Hilbert space. We also prove a contractive Schur mutiplier characterization for groups coarsely embeddable into a Hilbert space, and a uniformly bounded Schur multiplier characterization for exact groups.

Original languageEnglish (US)
Pages (from-to)431-454
Number of pages24
JournalIntegral Equations and Operator Theory
Volume73
Issue number3
DOIs
StatePublished - Jul 2012

Keywords

  • Haagerup property
  • Hilbert modules
  • coarse embedding
  • exact groups
  • multipliers

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

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