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Skanalis, 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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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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