  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.
