## 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 language | English (US) |
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Pages (from-to) | 431-454 |

Number of pages | 24 |

Journal | Integral Equations and Operator Theory |

Volume | 73 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2012 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory