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.