K-Theory Of The C*-Algebra Of Reductive Lie Groups And The Functoriality With That Of Their Uniform Lattices

An important problem in C*-algebras is to tell whether two C*-algebras are iso-morphic or not.K-theory is a central invariant of C*-algebras,and is effective in the classification of C*-algebras.However,there is no universal method for calculating the K-theory of C*-algebras.The Baum-Connes conjecture proposed an algorithm of cal-culating the K-theory of reduced group C*-algebras.The Baum-Connes conjecture for connected Lie groups has been verified.The conjecture for general discrete groups is still an open problem.The calculation of the K-theory of group C*-algebras of discrete groups is helpful for determining the structure and properties of groups,which provides examples which may satisfy the Baum-Connes conjecture.Comparing the C*-algebra of discrete groups and connected Lie groups is helpful to provide new ideas for the subsequent research on the Baum-Connes conjecture.This motivates the master thesis.In this paper,we first introduce the reduced group C*-algebra and clarify some re-lated concepts,with a detailed description.Based on the theory of tempered representa-tions of reductive groups,we preliminarily describe some properties of the C*-algebras of reductive groups via parabolic induction.With the above discussion,the structure of C*-algebras is clarified according to the properties of Hilbert modules.This lays a theoretical foundation for the subsequent calculation of the K-theory.Then we use the above results to calculate precisely the K-theory of the reduced groups C*-algebras of some connected Lie groups.Finally,based on the theory of crossed product,this paper constructs an imprim-itivity module for a connected Lie groups and the quotient of the groups modulo their uniform lattices.By using the properties of the module,we construct a monomorphism from the group C*-algebras of connected Lie groups to the tensors product of the group C*-algebras of discrete cocompact closed subgroup and a compact operator algebras.By taking an appropriate representative of the equivalence class of K-theory,a homo-morphism between the K-theory of the C*-algebra of a connected Lie group and that of its cocompact closed subgroups is established.Finally,using the Baum-Connes iso-morphism for connected Lie groups,some partial structure of the K-theory of the group C*-algebra of the uniform lattice can be obtained.