Index Theory and K-Theory for Operator Algebras

算子代数的索引理论和 K 理论

• 批准号: 9704001
• 负责人: Paul Baum
• 金额: $ 10.5万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-05-15 至 2001-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704001&HistoricalAwards=false
• 关键词: Index; Theory; Operator; Algebras

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. ***

9704001鲍姆本研究项目以P·鲍姆和A·康尼斯提出的一个猜想为中心。如果为真，则这个猜想回答了群C*-代数的K-理论的理解和计算问题。这个猜想的有效性意味着Novikov猜想(高签名的同伦不变性)、稳定的Gromov-Lawson-Rosenberg猜想(闭自旋流形允许正标量曲率的黎曼度量的充要条件)和Kadison-Kaplansky猜想(在无挠离散群的约化C*-代数中不存在幂等元)。当应用于李群时，该猜想使半单李群G的调和表示理论与Mackey和Wigner联系到G的半直积李群的表示理论之间的Mackey-Wigner类比更加精确。因此，这个猜想是不寻常的，因为它跨越了数学的几个不同领域，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)。本研究项目在非对易几何的新背景下发展了这一主题。我们将研究一个猜想(由P.Baum和A.Connes提出)，它把局部紧群的解析不变量和拓扑不变量联系起来。如果是真的，这个猜想将是一个基本原理，揭示了许多数学问题和问题中意想不到的统一性。***