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Local-global principle for the Baum-Connes conjecture with coefficients. (English) Zbl 1034.46073
In the paper under review, the authors establish some version of the local-global Hasse principle for the Baum-Connes conjecture (BCC). If \(G\) is some second countable locally compact Hausdorff topological group, if \(G\) is the ascending union of open subgroups \(G_n\), and if \(A\) is a \(G-C^*\)-algebra such that each \(G_n\) satisfies the Baum-Connes conjecture with coefficients in \(A\), then \(G\) satisfies BCC with coefficients in \(A\) (Theorem 1.1). From this, the authors deduce a version of the local-global principle for the BCC (Theorem 1.2): Let \(F\) be a global field, \(\mathbb A\) its ring of adeles, \(G\) a linear algebraic group defined over \(F\). Let \(F_v\) denote a place of \(F\). If BCC is true for each local group \(G(F_v)\), then BCC is true for the adelic group \(G(\mathbb A)\). For the example of \(GL(2,F)\), BCC is established.

MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K99 \(K\)-theory and operator algebras
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