Operator algebras, matrix bundles, and Riemann surfaces

Let $\overline{R}$ be a finitely bordered Riemann surface, and let $\mathfrak{E}_\rho(\overline{R})$ be a flat matrix $PU_n(\mathbb{C})$-bundle over $\overline{R}$. Let $\Gamma_c(\overline{R}, \mathfrak{E}(\overline{R}))$ denote the $C^*$-algebra of continuous cross-sections of $\mathfrak{E}(\overline{R})$, and let $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ denote the subalgebra consisting of the continuous holomorphic sections, i.e.~the continuous cross-sections that are holomorphic on the interior of $\overline{R}$. The algebra $\Gamma_c(\overline{R}, \mathfrak{E}(\overline{R}))$ is an example of an $n$-homogeneous $C^*$-algebra, and the subalgebra $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ is the principal object of study of this thesis. The algebras $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ appeared in the earlier works \cite{Abrahamse1976} and \cite{Blecher2000}. Operators that can be viewed as elements in $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ are the subject of \cite{Abrahamse1976}. The Morita theory of $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$, under the guise of a fixed-point algebra and in the special case of an annulus $R$, is studied in \cite[Ex.~8.3]{Blecher2000}. This thesis studies these algebras and their topological data $\mathfrak{E}_\rho(\overline{R})$ motivated by several problems in the theory of nonselfadjoint operator algebras.

Boundary representations are an invariant of operator algebras that were introduced by Arveson in 1969. However, it took nearly 50 years to show that boundary representations existed in sufficient abundance in all cases. I show that every boundary representation of $\Gamma_c(\overline{R}, \mathfrak{E}(\overline{R}))$ for $\Gamma_h(\overline{R}, \mathfrak{E}(\overline{R}))$ is given by evaluation at some point $r \in \partial R$. As a corollary, the $C^*$-envelope of $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ is $\Gamma_c(\partial R, \mathfrak{E}(\partial R))$. Using the $C^*$-envelope, I show that for certain choices of fibre and base space, $\Gamma_h(\overline{R}, \mathfrak{E}_\rho(\overline{R}))$ is not completely isometrically isomorphic to $A(\overline{R})\otimes M_n(\mathbb{C})$ unless the representation $\rho$ is the trivial representation.

I also show that $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ is an Azumaya over its center. Azumaya algebras are the ``pure-algebra'' analogues to $n$-homogeneous $C^*$-algebras \cite{Artin1969}. Thus the structure of the nonselfadjoint subalgebra $\Gamma_h(\overline{R},\mathfrak{E}(\overline{R}))$ reflects some of the structure of its $C^*$-envelope (which is $n$-homogeneous). Finally, I answer a question raised in \cite[Ex.~8.3]{Blecher2000} on the $cb$ and strong Morita theory of $\Gamma_h(\overline{R},\mathfrak{E}_\rho(\overline{R}))$, showing in particular that $\Gamma_h(\overline{R},\mathfrak{E}_\rho(\overline{R}))$ is $cb$ Morita equivalent to its center $A(\overline{R})$. As suggested in \cite[Ex.~8.3]{Blecher2000}, I provide additional evidence that $\Gamma_h(\overline{R},\mathfrak{E}_\rho(\overline{R}))$ may not be strongly Morita equivalent to its center. This evidence, in turn, suggests that there may be a Brauer group -like analysis for these algebras.