#### Date of Award

Spring 2010

#### Project Type

Dissertation

#### Program or Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### First Advisor

Don Hadwin

#### Abstract

This dissertation consists of two parts. In the first part, we obtain many new results about MF algebras. First, we continue the work on D. Voiculescu's topological free entropy dimension deltatop (x1, ..., xn) for an n-tuple x&ar; = (x1, ..., xn) of elements in a unital C*-algebra. We also introduce a new invariant that is a C*-algebra analog of the invariant K3 introduced for von Neumann algebras. Second, we discuss a full amalgamated free product of unital MF (and residually finite-dimensional) algebras with amalgamation over a finite-dimensional C*-subalgebra. Necessary and sufficient conditions are given in this situation. In the last chapter, we study the reduced amalgamated free products of C*-algebras, and we show that a reduced free product of two full matrix algebras amalgamated over a finite-dimensional C*-algebra is an MF algebra.

In the second part of the dissertation, we give an elementary proof of the Bishop-Stone-Weierstrass theorem for all unital subalgebras of M2&parl0;C&parr0; n with respect to its pure states. We show that the pure-state Bishop hull of a unital subalgebra (not necessarily selfadjoint) of M2&parl0;C&parr0; n is equal to itself.

This dissertation is partially supported by a University of New Hampshire dissertation fellowship.

#### Recommended Citation

Li, Qihui, "MF algebras and a Bishop -Stone -Weierstrass theorem result" (2010). *Doctoral Dissertations*. 600.

https://scholars.unh.edu/dissertation/600