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A proof of the Baum-Connes conjecture for \(p\)-adic \(GL(n)\). (English. Abridged French version) Zbl 0918.46061
Let \(F\) be a non-Archimedean local field of characteristic 0. Since \(F\) is a locally compact field, \(G= GL(n;F)\) is a locally compact topological group. The group \(C^*\)-algebra, denoted \(C^*(G)\), is the enveloping \(C^*\)-algebra of the convolution algebra \(L_1(G)\). The reduced \(C^*\)-algebra of \(G\), denoted \(C^*_{red}(G)\), is the completion of \(L_1(G)\) in its regular representation as bounded operators on \(L_2(G)\). The action of \(G\) on the affine building of \(G\), denoted \(\beta G\), is proper, and in fact \(\beta G\) is universal proper \(G\)-space. The Baum-Connes conjecture asserts that the index map \[ \mu: K^G_j(\beta G)\to K_j(C^*_{red}(G)),\quad j= 0,1, \] is an isomorphism of Abelian groups, where \(K^G_j(\beta G)\) is the equivariant \(K\)-homology of \(\beta G\), in the sense of Kasparov, and \(K_j(C^*_{red}(G))\) is the \(K\)-theory of \(C^*\)-algebra \(C^*_{red}(G)\).
The purpose of this paper is to prove the Baum-Connes conjecture for \(G= GL(n; F)\).

46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K56 Index theory
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