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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Baum-Connes conjecture ： ウィキペディア英語版 Baum–Connes conjecture In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the C *-algebra of a group and the K-homology of the corresponding classifying space of proper actions of that group. The conjecture sets up a correspondence between different areas of mathematics, with the K-homology being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the reduced $C^$ *-algebra is a purely analytical object. The conjecture, if true, would have some older famous conjectures as consequences. For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for a discrete torsion-free group, and the injectivity is closely related to the Novikov conjecture. The conjecture is also closely related to index theory, as the assembly map $\backslash mu$ is a sort of index, and it plays a major role in Alain Connes' noncommutative geometry program. The origins of the conjecture go back to Fredholm theory, the Atiyah–Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects. ==Formulation== Let Γ be a second countable locally compact group (for instance a countable discrete group). One can define a morphism :$\backslash mu\backslash colon\; RK^\backslash Gamma\_$ *(\underline) \to K_ *(C^ *_\lambda(\Gamma)), called the assembly map, from the equivariant K-homology with $\backslash Gamma$-compact supports of the classifying space of proper actions $\backslash underline$ to the K-theory of the reduced C *-algebra of Γ. The index * can be 0 or 1. Paul Baum and Alain Connes introduced the following conjecture (1982) about this morphism: :The assembly map μ is an isomorphism. As the left hand side tends to be more easily accessible than the right hand side, because there are hardly any general structure theorems of the $C^$ *-algebra, one usually views the conjecture as an "explanation" of the right hand side. The original formulation of the conjecture was somewhat different, as the notion of equivariant K-homology was not yet common in 1982. In case $\backslash Gamma$ is discrete and torsion-free, the left hand side reduces to the non-equivariant K-homology with compact supports of the ordinary classifying space $B\backslash Gamma$ of $\backslash Gamma$. There is also more general form of the conjecture, known as Baum–Connes conjecture with coefficients, where both sides have coefficients in the form of a $C^$ *-algebra $A$ on which $\backslash Gamma$ acts by $C^$ *-automorphisms. It says in KK-language that the assembly map :$\backslash mu\_\backslash colon\; RKK^\backslash Gamma\_$ *(\underline,A) \to K_ *(A\rtimes_\lambda \Gamma), is an isomorphism, containing the case without coefficients as the case $A=\backslash mathbb$. However, counterexamples to the conjecture with coefficients were found in 2002 by Nigel Higson, Vincent Lafforgue and Georges Skandalis, basing on not universally accepted, as of 2008, results of Gromov on expanders in Cayley graphs. Even provided validity of Higson, Lafforgue & Skandalis, conjecture with coefficients remains an active area of research, since it is, not unlike the classical conjecture, often seen as a statement concerning particular groups or class of groups. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Baum–Connes conjecture」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.