On some noncommutative topological spaces associated with groups and dynamical systems

This dissertation consists of three parts. ^ In the first, we consider residually finite groups acting on a profinite completion by left translation. We study the corresponding crossed product C*-algebra for discrete countable groups that are central extensions of finitely generated abelian groups by finitely generated abelian groups. We prove that all such crossed products are classifiable by K-theoretic invariants using recent techniques from the classification theory for nuclear C*-algebras. ^ In the second part, a generalization of the Exel-Loring formula for quasi-representations of a surface group taking values in U(n) was given by the author\u27s advisor. Here we further extend this formula for quasi-representations of a surface group taking values in the unitary group of a tracial unital C*-algebra. ^ In the third part, we examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg\u27s theorem via paradoxical decompositions and a characterization of quasidiagonality for group C*-algebras in terms of embeddability of the groups. We consider several notable examples of groups, such as topological full groups associated with Cantor minimal systems and Abels\u27 celebrated example of a finitely presented solvable group that is not residually finite, and show that they have quasidiagonal C*-algebras. Finally, we study strong quasidiagonality for group C*-algebras, exhibiting classes of amenable groups with and without strongly quasidiagonal C*-algebras. ^ The second part is based on joint work with our advisor, Marius Dadarlat, and the third is based on joint work with Marius Dadarlat and Caleb Eckhardt.