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

Abstract

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 non-commutative Lp spaces. We introduce a variant (H_Lp) for orthogonal representations with vanishing coefficients on Lp, and study its relationships with property (H). We also give a characterization of (H) by the means of strongly mixing actions on a non-commutative Lp space. We construct proper actions of groups with (H) by affine isometries on some non-commutative Lp space, such as the Lp space associated to the hyperfinite II infinite factor.