The Weak Haagerup Property II: Examples

The weak Haagerup property for locally compact groups and the weak Haagerup constant were recently introduced by the second author [27]. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author [9] and the Haagerup property introduced by Connes [6] and Choda [5]. 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 R-2 x SL(2, R) does not have the weak Haagerup property.