Spectral Invariant Subalgebras Of Reduced Crossed Product C～*-algebras And Quasidiagonality Of Roe Algebras

Roe algebras'arose from the index theory on noncompact complete Rieman-nian manifolds, and is a class of concrete C-algebras associated to the coarse structure of metric spaces. This paper mainly study the relations between some algebraic properties of Roe algebras and the coarse geometry of metric spaces, and its applications to the reduced crossed product C-algebras and the coarse Baum-Connes conjecture. The paper consists of two chapters. The first chapter deals with spectral invariant subalgebras of reduced crossed product C-algebras. Let G be a discrete group with a proper length function l. We prove that the Schwartz space Sl2(G, A) is a spectral invariant dense subalgebra of the reduced crossed product Cr(G,A) for all commutative C-algebra A with a G-action if and only if (7 has polynomial growth with respect to l. The second chapter study how quasidiagonality of Roe algebras was related to topological invariants. We obtain some sufficient and necessary conditions for Roe algebras to be quasidiagonal, which involve both the indices of Fredholm operators and coarse connectedness of metric spaces. An application to the coarse Baum-Connes conjecture is given.