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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
##### Keywords:
I-UTEN: adelic groups; C*-algebras; Baum-Connes conjecture
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