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Skandalis, Tu and Yu in "The coarse Baum-Connes conjecture and groupoids" proved that:

Let $\Gamma$ be a countable group with a proper left-invariant metric $d$. If $\Gamma$ admits a uniform embedding into Hilbert space, then Baum-Connes assembly map with coefficients is split injective.

My question is that is it still true if we replace $\Gamma$ by a locally compact second countable Hausdorff topological group $G$? known or unknown?

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Here is the link for the Skandalis-Tu-Yu paper: – Alain Valette Jan 3 '13 at 20:18
up vote 7 down vote accepted

I believe that, in full generality, this is open. However, the point of the Skandalis-Tu-Yu paper is to construct a locally compact groupoid of the form $X\rtimes\Gamma$ (with $X$ a compact $\Gamma$-space), which admits a proper isometric action on a continuous field of Hilbert spaces (then previous results by J.-L. Tu do apply). A sufficient condition for that is to find an $X$ on which $\Gamma$ acts amenably. For a general locally compact group $G$, the proof most probably goes through provided there is a compact $G$-space on which $G$ acts amenably, which happens in many cases (e.g. connected groups).

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Dear Alain For a general locally compact group $G$, why amenability at infinity condition is sufficient for existence of proper affine isometric action of $X\rtimes \gamma$ on a continuous field of affine Euclidean spaces? I only can see this for discrete group $\Gamma$. Indeed, amenable at infinity implies that uniform embeddable into a Hilbert space and then applying proposition 6.5 in "The coarse Baum-Connes conjecture and groupoids" – m07kl Feb 27 '13 at 15:51

Property A and uniform embedding for locally compact groups by Steven Deprez and Kang Li arXiv:1309.7290

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