#### Document Type

Dissertation

#### Date of Degree

Summer 2016

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Paul Muhly

#### Abstract

My research studies algebras of holomorphic functions from $d$-tuples of $n\times n$- matrices, $M_n(\bC)^d$, to $M_n(\bC)$. In particular, I study the holomorphic functions that can be approximated by \emph{polynomial matrix concomitants}, that is polynomial maps from $M_n(\bC)^d$ to $M_n(\bC)$ that satisfy the relationship

\[

f(g^{-1}\fz g) = g^{-1}f(\fz)g

\]

for every $\fz \in M_n(\bC)^d$ and $g\in GL_n(\bC)$. In a sense, these are the polynomial maps that “remember” the structure of the $d$-tuple $\fz$.

My first result is that these holomorphic matrix concomitants can be identified with holomorphic cross sections of certain matrix bundles. A holomorphic matrix bundle is a fibred space in which every fibre is $M_n(\bC)$ and the fibres are glued together in such a way that the total space has a holomorphic structure.

Once the identification between holomorphic cross sections and holomorphic concomitants is established, the structure of the matrix bundle is used to endow the algebra of continuous cross sections with a $C^*$-algebra structure. Then we study the subalgebra of cross sections that can be approximated by polynomial concomitants. By identifying the matrix concomitants with cross sections, we are able to prove interesting results about these algebras.

#### Keywords

Functional Analysis, Noncommutative Function Theory

#### Pages

vii, 86 pages

#### Bibliography

Includes bibliographical references (pages 83-86).

#### Copyright

Copyright © 2016 Erin Griesenauer