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Groups with the Haagerup property. Gromov’s a-T-menability. (English) Zbl 1030.43002
Progress in Mathematics (Boston, Mass.). 197. Basel: Birkhäuser. viii, 126 p. (2001).
This book constitutes a collective volume due to five authors, featuring important breakthroughs in an intensively studied subject. The topic is presented by A. Valette. The second countable locally compact group $$G$$ is said to satisfy the Haagerup property or to be a $$T$$-menable group (in the sense of Gromov) if one of the following equivalent properties holds: 1) There exists a sequence $$(\varphi _n)$$ of normalized positive definite functions on $$G$$, converging uniformly to 1 on compact subsets. 2) There exists a continuous isometric action $$\alpha$$ of $$G$$ on an affine Hilbert space $${\mathcal H}$$ such that for any bounded subset $$B$$ of $${\mathcal H}$$, the subset $$\{g \in G : \alpha (g)B \cap B \neq \emptyset \}$$ is relatively compact in $$G$$. The famous Kazhdan property says that whenever a representation of $$G$$ on a Hilbert space weakly contains the trivial representation, then it contains it strongly, i.e., it admits a fixed vector. These properties are shared simultaneously only by compact groups. So the Haagerup property is a strong negation of the Kazhdan property. As amenable groups are Haagerup groups, $$T$$-menability is a weak amenability. In the context of $$K$$-homology, the Kasparov ring gives rise to the definition of $$K$$-amenability considered with respect to the important Baum-Connes conjecture.
P. Jolissaint characterizes the Haagerup property in terms of von Neumann algebras. P.-A. Cherix, M. Cowling, A. Valette perform a classification of connected Lie groups with the Haagerup property. P. Jolissaint, P. Julg, A. Valette deal with discrete Haagerup groups. P. Julg provides geometric proofs for the Haagerup property on SO($$n$$,1), SU($$n$$,1); he also establishes a new proof of property $$(T)$$ for Sp($$n$$,1), $$n \geq 2$$. P. Valette lists several open problems and briefly indicates new directions for further recent developments.

##### MSC:
 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 43A07 Means on groups, semigroups, etc.; amenable groups
##### Keywords:
Haagerup property; t-menability