The Approximation Properties In The Uniform Roe Algebras

The coarse Roe algebras and the uniform Roe algebras are two kinds of important C*-algebras and they play an important role in both index theory and exactness problem in C*-algebra theory. In this paper, we study some approximation properties in the uniform Roe algebras. The thesis is arranged as the following:The first chapter is the introduction. We introduce the background of our problems and give our main results.In chapter 2, we introduce some preliminaries which are used in the thesis. For example, the elementary knowledge of coarse geometry, the reduced C*-algebras, the rapid decay property for groups and the metric spaces and so on.Chapter 3 contains two parts. In the first part, we studied the group invariant ap-proximation property of the uniform Roe algebra of the metric space with a group action. Let X be a metric space with a group G acting on X isometrically. Denote by Cu*(X) the uniform Roe algebra of X. We can define two C*-subalgebras of Cu*(X). One can be defined by taking the finite propagation operators in Cu*(X) which are in-variant under the group action first, then taking the hull of these operators. The other is defined by taking the group invariant operators in Cu*(X) directly. We will study when these two C*-algebras are identity. In the second part, we find the non-commutative Fejer theorem. Let T be the unit circle, C(T) be the set of all continuous functions on T. In the classical Fourier analysis, we know there are some elements in C(T) whose Fourier series are not converge to themselves uniformly. By Fourier transformation, C(T) is isomorphic to the reduced C*-algebras of the integer group Z. In other words, there are some operators in Cr*(Z) Which are not be approximated by the truncations along the diagonal of themselves. We extend this result to the finite generated groups and the discrete metric spaces. For the reduced C*-algebras of finite generated groups and the unform Roe algebras of the discrete metric spaces, there are operators which are not approximated by the truncations along the diagonal of themselves. Also, we give some sufficient condition for the operators in these two C*-algebras which can be approximated by the truncation along the diagonal of themselves.In chapter 4, we give the definition of the equi-nuclearity of the uniform Roe alge-bras and prove some permanence of nuclearity. For example, let G be a group which is hyperbolic relative to the subgroups G1, G2,…, Gn, then Cu*(G) is nuclear if and only if Cu*(G1),Gu*(G2),…,Cu*{Gn) are nuclear. we also prove that the coarse Roe algebras are not nuclear.