A Hilbert Module Approach to the Haagerup Property

Autor: Zhe Dong, Zhong Jin Ruan
Rok vydání: 2012
Předmět:
Zdroj: Integral Equations and Operator Theory. 73:431-454
ISSN: 1420-8989
0378-620X
DOI: 10.1007/s00020-012-1979-3
Popis: We develop a Hilbert module version of the Haagerup property for general C*-algebras \({{\mathcal{A} \subseteq \mathcal{B}}}\) . We show that if \({\alpha : \Gamma \curvearrowright \mathcal{A}}\) is an action of a countable discrete group Γ on a unital C*-algebra \({\mathcal{A}}\) , then the reduced C*-algebra crossed product \({\Gamma \ltimes _{\alpha, r} \mathcal{A}}\) has the Hilbert \({\mathcal{A}}\) -module Haagerup property if and only if the action α has the Haagerup property. We are particularly interested in the case when \({\mathcal{A} = C(X)}\) is a unital commutative C*-algebra. We compare the Haagerup property of such an action \({\alpha: \Gamma \curvearrowright C(X)}\) 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.
Databáze: OpenAIRE
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