The weak Haagerup property II: Examples

TL;DR

This paper characterizes when connected simple Lie groups have the weak Haagerup property, showing it coincides with weak amenability for these groups and exploring related constants and examples.

Contribution

It establishes a precise criterion for the weak Haagerup property in connected simple Lie groups and compares it with weak amenability and related constants.

Findings

Connected simple Lie groups have the weak Haagerup property iff their real rank is zero or one.

For these groups, the weak Haagerup constant equals the weak amenability constant.

The semidirect product of R^2 by SL(2,R) does not have the weak Haagerup property.

Abstract

The weak Haagerup property for locally compact groups and the weak Haagerup constant was recently introduced by the second author. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author and the Haagerup property introduced by Connes and Choda. In this paper it is shown that a connected simple Lie group G has the weak Haagerup property if and only if the real rank of G is zero or one. Hence for connected simple Lie groups the weak Haagerup property coincides with weak amenability. Moreover, it turns out that for connected simple Lie groups the weak Haagerup constant coincides with the weak amenability constant, although this is not true for locally compact groups in general. It is also shown that the semidirect product of R^2 by SL(2,R) does not have the weak Haagerup property.