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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 $$G= GL(n; F)$$.

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 19K56 Index theory
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