The coarse Baum-Connes conjecture and groupoids. II (Q416772)

Let \((X,d)\) be a metric space, proper (i.e., all closed balls are compact), not necessarily discrete. A subset \(E\) of \(X\times X\) is controlled if \(d_{|E}\) is bounded. In a previous paper by \textit{G. Skandalis} et al., see Part I [Topology 41, No. 4, 807--834 (2002; Zbl 1033.19003)], it was shown that \(G(X)=\bigcup_{E\text{\;controlled}}\) \(\overline{E}\subset \beta(X\times X)\) can be endowed with the structure of an étale, locally compact, \(\sigma-\)compact groupoid. Moreover, the coarse Baum-Connes conjecture for \(X\), see \textit{G. Yu} [K-Theory 9, No. 3, 199--221 (1995; Zbl 0829.19004)], which states that a certain assembly map NEWLINE\[NEWLINE\lim_d K_*(P_d(X))\to K(C^*(X))NEWLINE\]NEWLINE is an isomorphism, is equivalent to the Baum-Connes conjecture for \(G(X)\) with coefficients in \(\ell^\infty(X,{\mathcal K})\).NEWLINENEWLINEIn this paper, the main result from that paper is extended in the following two directions: first, the construction is extended to a large class of locally compact, proper metric spaces that are not necessarily discrete; secondly, a coarse Baum-Connes with coefficients is defined. The author shows that it is stable under taking closed subspaces.