Journal of Noncommutative Geometry

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Volume 9, Issue 3, 2015, pp. 797–819
DOI: 10.4171/JNCG/208

Published online: 2015-10-29

Property A and uniform embedding for locally compact groups

Steven Deprez[1] and Kang Li[2]

(1) University of Copenhagen, Denmark
(2) University of Copenhagen, Denmark

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 coecients 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, Echterho and Oyono-Oyono [4].

Keywords: Property A, Baum–Connes conjecture, uniform embeddability, locally compact groups

Deprez Steven, Li Kang: Property A and uniform embedding for locally compact groups. J. Noncommut. Geom. 9 (2015), 797-819. doi: 10.4171/JNCG/208