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

TL;DR

This paper explores new characterizations of the Haagerup property for groups via actions on non-commutative Lp spaces, introducing variants and establishing connections with mixing actions and affine isometries.

Contribution

It introduces a new variant (H_Lp) of the Haagerup property for orthogonal representations on non-commutative Lp spaces and relates it to existing characterizations.

Findings

Established a characterization of (H) via strongly mixing actions on non-commutative Lp spaces.

Constructed proper affine isometric actions of groups with (H) on non-commutative Lp spaces.

Linked the new variant (H_Lp) to the classical Haagerup property.

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.