Algebras associated with Blaschke products of type {\it G}

Abstract

Let $\Omega$ and $Ω_{\fin}$ be the sets of all interpolating Blaschke products of type $G$ and of finite type $G$, respectively. Let $E$ and $E_{\fin}$ be the Douglas algebras generated by $H^{\infty }$ together with the complex conjugates of elements of $\Omega$ and $Ω_{\fin}$, respectively. We show that the set of all invertible inner functions in $E$ is the set of all finite products of elements of $\Omega$ , which is also the closure of $\Omega$ among the Blaschke products. Consequently, finite convex combinations of finite products of elements of $\Omega$ are dense in the closed unit ball of the subalgebra of $H^{\infty }$ generated by $\Omega$. The same results hold when we replace $\Omega$ by $Ω_{\fin}$ and $E$ by $E_{\fin}$.