Variants of a-T-menability for actions on non-commutative Lp-spaces

We investigate various characterizations of the Haagerup property (H) for a second countable locally compact group G, in terms of orthogonal representations of G on Lp(M), the non-commutative Lp-space associated with the von Neumann algebra M. For a semi-finite von Neumann algebra M, we introduce a variant of property (H), namely property HLp(M), defined in terms of orthogonal representations on Lp(M) which have vanishing coefficients. We study the relationships between properties (H) and (HLp(M)) for various von Neumann algebras M. We also characterize property (H) in terms of strongly mixing actions on Lp(M) for some finite von Neumann algebras M. We finally give constructions of proper actions of groups with the Haagerup property by affine isometries on Lp(M) for some algebras M, such as the hyperfinite II∞ factor B(l2)⊗R.