Mathematical Sciences: Noncommutative Geometric Methods in Operator Algebras

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9500886 Ji This project involves the study of operator algebras associated with discrete groups and C*-dynamical systems by the method of noncommutative differential geometry. Central to this study is the construction of smooth and dense subalgebras of the corresponding crossed product C*-algebras and the related analysis to the smooth subalgebras. The main techniques employed are Connes' cyclic theory for associative algebras and the Schwartz cohomology for discrete groups introduced by the investigator. Completion of the projects described in this proposal help elucidate deep properties of operator algebras, algebraic topology and differential geometry, such as, a further understanding of noncommutative Toeplitz index theory; a solution to the rational injectivity of the Baum-Connes map, the Novikov conjecture and the Kadison-Kaplansky conjecture for the class of rapidly decaying discrete groups. This research lies at the interface between geometry, analysis and algebra. A modern approach of Connes studies geometrical invariants using non-commutative algebraic structures. A good example of a noncommutative structure is the multiplicative structure on the collection of square matrices of fixed size. This project will employ Connes machinery to elucidate other important non-commutative algebras called operator algebras which can be viewed as collections of infinite matrices. The results will impact algebra, geometry and analysis. ***