## Doctoral Dissertations

Spring 1994

Dissertation

Mathematics

#### Degree Name

Doctor of Philosophy

#### Abstract

This paper deals with universal $C\sp\*$-algebras generated by matricial relations on the generators, for example, the universal $C\sp\*$-algebra with generators $a\sb{ij}, 1 \leq i,j \leq n$, subject to the condition that the matrix ($a\sb{ij}$) be normal and have spectrum in a designated compact subset ${\cal K}$ of the complex plane.

The main thrust of the paper is to compute the K-groups of some of these $C\sp\*$-algebras and to determine when they contain non-trivial projections. In the above example, we show that the K-groups of the algebra coincide with the topological K-groups of the set ${\cal K}$. We show, in general, that if the algebra has a multiplicative linear functional, then the K-theory is independent of n, when the matricial constraints are fixed.

It is also shown that if the constraints are fixed and ${\cal A}\sb n$ is the algebra with $n\sp2$ generators, then the tensor product of ${\cal A}\sb n$ with the algebra $M\sb n$ of complex n x n matrices is isomorphic to the free product of ${\cal A}\sb1$ with $M\sb n$.

Also in the example above, the algebra contains no non-trivial projections when n is not less than the number of connected components of ${\cal K}$. These results have also been extended to include the case in which the constraints are in several variables.

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