#### Document Type

Dissertation

#### Date of Degree

Summer 2010

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Paul Muhly

#### First Committee Member

Paul S Muhly

#### Second Committee Member

Palle Jorgensen

#### Third Committee Member

Richard Baker

#### Fourth Committee Member

Jonathan Simon

#### Fifth Committee Member

Victor Camillo

#### Abstract

The principal objects of study in this thesis are the noncommutative Hardy algebras introduced by Muhly and Solel in 2004, also called simply ``Hardy algebras,'' and their quotients by ultraweakly closed ideals. The Hardy algebras form a class of nonselfadjoint dual operator algebras that generalize the classical Hardy algebra, the noncommutative analytic Toeplitz algebras introduced by Popescu in 1991, and other classes of operator algebras studied in the literature.

It is known that a quotient of a noncommutative analytic Toeplitz algebra by a weakly closed ideal can be represented completely isometrically as the compression of the algebra to the complement of the range of the ideal, but there is no known general extension of this result to Hardy algebras. An analogous problem on representing quotients of Hardy algebras as compressions of images of induced representations is considered in Chapter 2. Using Muhly and Solel's generalization of Beurling's theorem together with factorizations of weakly continuous linear functionals on infinite multiplicity operator spaces, it is shown that compressing onto the complement of the range of an ultraweakly closed ideal in the space of an infinite multiplicity induced representation yields a completely isometric isomorphism of the quotient.

A generalization of Pick's interpolation theorem for elements of Hardy algebras evaluated on their spaces of representations was proved by Muhly and Solel. In Chapter 3, a general theory of reproducing kernel W*-correspondences and their multipliers is developed, generalizing much of the classical theory of reproducing kernel Hilbert space. As an application, it is shown using the generalization of Pick's theorem that the function space representation of a Hardy algebra is isometrically isomorphic (with its quotient norm) to the multiplier algebra of a reproducing kernel W*-correspondence constructed from a generalization of the Szegõ kernel on the unit disk. In Chapter 4, properties of polynomial approximation and analyticity of these functions are studied, with special attention given to the noncommutative analytic Toeplitz algebras.

In the final chapter, the canonical curvatures for a class of Hermitian holomorphic vector bundles associated with a C*-correspondence are computed. The Hermitian metrics are closely related to the generalized Szegõ kernels, and when specialized to the disk, the bundle is the Cowen-Douglas bundle associated with the backward shift operator.

#### Pages

vi, 64 pages

#### Bibliography

Includes bibliographical references (pages 62-64).

#### Copyright

Copyright 2010 Jonas R Meyer