Date of Award

Fall 2005

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Donald Hadwin

Abstract

In the first part of this paper we will consider a generalization of D. Hadwin and E. Nordgren's work on multiplier pairs. Here we will not assume the existence of an identity, but rather just ask for the existence of a bounded approximate identity. Without the assumption of the identity, we find a new result concerning the relationship between the norm closure of the left multiplication operators and the approximate double commutant of the left multiplication operators.

In the second part we will suppose f, g : T→T are continuous functions on the unit circle T and let B (f, g) denote the universal C*-algebra generated by U and V subject to the conditions that U and V are a unitary, and Uf( V)U-1 = g( V). We then will prove that this C*-algebra may be represented as a crossed product. Next we will show that under certain conditions on f or g, B (f, g) will be nuclear, weakly quasidiagonal and we will be able to compute its Ext group. In the last two sections we will give a partial description of the K1-group of B (f, g) and then using the results from [DH] calculate the free entropy dimension of B (f, g).

In the third and last part of this paper we show that the standard family of independent unitary n x n random matrices remains an asymptotically free Haar unitary with respect to any state 4:Mn C →C . The result was originally stated by Voiculescu for the normalized trace. Our work here will follow the modified version of Voiculescu's theorem given by D. Hadwin and M. Dostal in [DH].

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