Index Theory and K-Theory for Operator Algebras

这个研究项目以P. Baum和a . Connes提出的一个猜想为中心。如果这个猜想是正确的，那么它就给出了理解和计算群C*-代数的k理论问题的答案。该猜想的有效性意味着Novikov猜想（高特征的同伦不变性）、稳定的Gromov-Lawson-Rosenberg猜想（闭自旋流形承认一个正标量曲率的黎曼度规的充分必要条件）和Kadison-Kaplansky猜想（无扭离散群的约化C*-代数中不存在幂等幂）的有效性。当应用于李群时，该猜想精确地将半简单李群G的缓和表示理论与麦基和维格纳与G相关的半直接积李群的表示理论进行了麦基-维格纳类比。因此，该猜想的不同之处在于，它跨越了几个不同的数学领域，并统一了许多以前看起来不相关的问题和问题。19世纪和20世纪数学的一个主题是，数学系统的某些特征乍一看似乎是解析的（即基于微分和积分等微积分方法），实际上是拓扑的（即仅依赖于初等连续几何）。本研究项目在非交换几何的新背景下发展了这一主题。本文将研究一个猜想（由P. Baum和A. Connes提出），它与局部紧群的解析不变量和拓扑不变量有关。如果这个猜想是正确的，那么它将是一个潜在的原理，揭示了许多数学问题和问题中意想不到的统一性。＊＊＊

9704001 Baum This research project centers on a conjecture formulated by P. Baum and A. Connes. If true, this conjecture gives an answer to the problem of understanding and calculating the K-theory of group C*-algebras. Validity of the conjecture implies validity for the Novikov conjecture (homotopy invariance of higher signatures), the stable Gromov-Lawson-Rosenberg conjecture (necessary and sufficient conditions for a closed Spin manifold to admit a Riemannian metric of positive scalar curvature), and the Kadison-Kaplansky conjecture (non-existence of idempotents in the reduced C*-algebra of a torsion-free discrete group). When applied to Lie groups, the conjecture makes precise the Mackey-Wigner analogy between the tempered representation theory of a semi-simple Lie group G and the representation theory of the semi-direct product Lie group that Mackey and Wigner associate to G. Thus the conjecture is unusual in that it cuts across several different areas of mathematics and unifies a number of problems and issues that previously appeared to be unrelated. A theme in nineteenth and twentieth century mathematics has been that certain features of mathematical systems that at first glance seem to be analytical (i.e., based on methods of calculus such as differentiation and integration) in fact are topological (i.e., depend only on elementary continuous geometry). This research project develops this theme within the new context of non-commutative geometry. A conjecture (formulated by P. Baum and A. Connes) will be studied that relates analytic and topological invariants of locally compact groups. If true, the conjecture will be an underlying principle revealing an unexpected unity in a number of mathematical problems and issues. ***