Introduction

Abstract

For a second countable, locally compact group G,consider the following four properties:

• (1) there exists a continuous function which is conditionally negative definite and proper, that is, lim

• (2) G has the Haagerup approximation property in the sense of C.A. Ake-mann and M. Walter [AW81] and M. Choda [Cho83], or property C_o in the sense of V. Bergelson and J. Rosenblatt [BR88]: there exists a sequence(φ _n ) _n∈ℕ of continuous, normalized (i.e., ø_n(1) = 1) positive definite functions on G, vanishing at infinity on G, and converging to 1 uniformly on compact subsets of G;

• (3) G is a-T-menable as M. Gromov meant it in 1986 ([Gro88, 4.5.C]): there exists a (strongly continuous, unitary) representation of G, weakly containing the trivial representation, whose matrix coefficients vanish at infinity on G (a representation with matrix coefficients vanishing at infinity will be called a C _o -representation);

• (4) G is a-T-menable as Gromov meant it in 1992 ([Gro93, 7.A and 7.E]): there exists a continuous, isometric action a of G on some affine Hilbert space H, which is metrically proper (that is, for all bounded subsets B of H the set is relatively compact in G).