Baum, Paul; Millington, Stephen; Plymen, Roger Local-global principle for the Baum-Connes conjecture with coefficients. (English) Zbl 1034.46073 \(K\)-Theory 28, No. 1, 1-18 (2003). 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. Reviewer: Do Ngoc Diep (Hanoi) Cited in 4 Documents MSC: 46L80 \(K\)-theory and operator algebras (including cyclic theory) 19K99 \(K\)-theory and operator algebras Keywords:I-UTEN: adelic groups; C*-algebras; Baum-Connes conjecture PDFBibTeX XMLCite \textit{P. Baum} et al., \(K\)-Theory 28, No. 1, 1--18 (2003; Zbl 1034.46073) Full Text: DOI arXiv