Quasidiagonality is a matricial approximation property which asymptotically captures the multiplicative structure of C* -algebras. Quasidiagonal C* -algebras must be stably finite. It has been conjectured by Blackadar and Kirchberg that stably finiteness implies quasidiagonality for the class of separable nuclear C* -algebras. It has also been conjectured that separable exact quasidiagonal C* -algebras are AF embeddable. In this thesis, we study the behavior of these conjectures in the context of extensions 0 → I → E → B → 0. Specifically, we show that if I is exact and connective and B is separable, nuclear, and quasidiagonal (AF embeddable), then E is quasidiagonal (AF embeddable). Additionally, we show that if I is of the form C(X) ⊗ K for a compact metrizable space X and B is separable, nuclear, quasidiagonal (AF embeddable), and satisfies the UCT, then E is quasidiagonal (AF embeddable) if and only if E is stably finite.

<p dir="ltr">On this thesis, we study the validity of the Blackadar-Kirchberg conjecture for C*-<br>algebras that arise as extensions of separable, nuclear, quasidiagonal C*-algebras that satisfy<br>the Universal Coefficient Theorem. More specifically, we show that the conjecture for the<br>C*-algebra in the middle has an affirmative answer if the ideal lies in a class of C*-algebras<br>that is closed under local approximations and contains all separable ASH-algebras, as well<br>as certain classes of simple, unital C*-algebras and crossed products of unital C*-algebras<br>with Z. We also investigate when discrete, amenable groups have C*-algebras of real rank<br>zero. While it is known that this happens when the group is locally finite, the converse in<br>an open problem. We show that if C*(G) has real rank zero, then all normal subgroups of<br>G that are elementary amenable and have finite Hirsch length must be locally finite.<br><br></p>