Property A and uniform embedding for locally compact groups

Abstract

For locally compact groups, we define an analogue to Yu’s property A that he defined for discrete metric spaces.We show that our property A for locally compact groups agrees with Roe’s notion of property A for proper metric spaces, defined in [11]. We prove that many of the results that are known to hold in the discrete setting, hold also in the locally compact setting. In particular, we show that property A is equivalent to amenability at infinity (see [9] for the discrete case), and that a locally compact group with property A embeds uniformly into a Hilbert space (see [17] for the discrete case). We also prove that the Baum–Connes assembly map with coeffcients is split-injective, for every locally compact group that embeds uniformly into a Hilbert space. This extends results by Skandalis, Tu and Yu [13], and by Chabert, Echterhoff and Oyono-Oyono [4].