Degree Type

Dissertation

Date of Award

2015

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Justin Peters

Abstract

In this dissertation, we consider algebras of holomorphic functions on the unit disc and

other domains. We begin with the disc algebra A(D) which is a well-studied example in the

field of Banach and operator algebras. To this algebra can be applied an involution f → f*

given by f*(z) = f (z). With this involution, A(D) becomes a Banach ∗-algebra that is not

a C*-algebra. We study the positive elements of this Banach ∗-algebra and compare them to

the classical C*-algebra case. In particular, we use the classical BSF factorization on H^p(D),

to show that f = g*g for some g ∈ A(D) if and only if f([−1, 1]) ⊆ R_+. A similar result is

proved for H^p(D); 1 ≤ p ≤ ∞. These results are then extended, first to holomorphic functions

on an annulus, and then to holomorphic functions on any domain G that is symmetric with

respect to the real line and where ∂G is the union of finitely many disjoint Jordan curves.

Connections are also made between these results and the representation theory of holomorphic

function algebras.

DOI

https://doi.org/10.31274/etd-180810-3893

Copyright Owner

Jason Ekstrand

Language

en

File Format

application/pdf

File Size

102 pages

Included in

Mathematics Commons

Share

COinS