Date of Award
Doctor of Philosophy
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
Ekstrand, Jason, "Positivity in function algebras" (2015). Graduate Theses and Dissertations. 14341.