Haagerup property for C⁎-algebras

We define the Haagerup property for C⁎-algebras A and extend this to a notion of relative Haagerup property for the inclusion B⊆A, where B is a C⁎-subalgebra of A. Let Γ be a discrete group and Λ a normal subgroup of Γ, we show that the inclusion A⋊α,rΛ⊆A⋊α,rΓ has the relative Haagerup property if and only if the quotient group Γ/Λ has the Haagerup property. In particular, the inclusion Cr⁎(Λ)⊆Cr⁎(Γ) has the relative Haagerup property if and only if Γ/Λ has the Haagerup property; Cr⁎(Γ) has the Haagerup property if and only if Γ has the Haagerup property. We also characterize the Haagerup property for Γ in terms of its Fourier algebra A(Γ).