Positive Elements in Function Algebras

Abstract

Given a domain $G \subseteq \mathbb {C}$, we let $H^\infty (G)$ be the Banach algebra of all bounded holomorphic functions on $G$ under the supremum norm and $\mathcal {A}(G)$ be the subalgebra of $H^\infty (G)$ of those functions which have continuous extension to the closure $\overline{G}$. If the domain $G$ is symmetric with respect to the real axis, we may give these algebras the involution $f \mapsto f^*$; $f^*(z) = \overline{f(\bar{z})}$. We study the positive elements of the resulting Banach $*$-algebras. Under certain restrictions on the domain $G$, we are able to show that, for any $f \in \mathcal {A}(G)$, $f|_{G \cap \mathbb {R}} \ge 0$ if and only $f = g^*g$ for some $g \in \mathcal {A}(G)$. Similar results are proved in $H^\infty (G)$ and $H^p(G)$ where appropriate.