Positivity in function algebras
Date
Authors
Major Professor
Advisor
Committee Member
Journal Title
Journal ISSN
Volume Title
Publisher
Altmetrics
Authors
Research Projects
Organizational Units
Journal Issue
Is Version Of
Versions
Series
Department
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.