| Abstract | This dissertation is split into two parts. In the first part we expand upon work by G\'abor Elek on C*-algebras of Uniformly Recurrent Subgroups. We construct a dynamical system from the set of subgroups of a finitely-generated discrete group. This has a nice correspondence with a Cayley-like graph of a subgroup's cosets. From these structures we construct a C*-algebra. We then apply techniques from other constructions to reveal properties of the new C*-algebra and relate them to properties of the graph, the dynamical system, and the subgroup itself. In the second we expand upon work from many hands on the decomposition of nuclear maps. Such maps can be characterized by their ability to be approximately written as the composition of maps to and from matrices. Under certain conditions (such as quasidiagonality), we can find a decomposition whose maps behave nicely, such as preserving multiplication up to an arbitrary degree of accuracy. We investigate these conditions and relate them to a W*-analog. |
