Haagerup property

Summary

A locally compact group has the Haagerup property when it has a unitary representation with almost invariant vectors whose matrix coefficients vanish at infinity. Equivalently, it acts on a Hilbert space by affine isometries in a proper way; for this reason Gromov called such groups a-T-menable. The property is a weak form of amenability and is opposite to property (T): a group with both properties is compact. Haagerup showed that free groups have it, and it is used in the Baum–Connes conjecture and in the theory of von Neumann algebras.