A Hilbert Module Approach to the Haagerup Property

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.