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 , r Λ ⊆ A ⋊ α , r Γ has the relative Haagerup property if and only if the quotient group Γ / Λ has the Haagerup property. In particular, the inclusion C r ⁎ ( Λ ) ⊆ C r ⁎ ( Γ ) has the relative Haagerup property if and only if Γ / Λ has the Haagerup property; C r ⁎ ( 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 ) .