The structure of Banach algebras of bounded continuous functions on the open disk that contain $H^{∞}$, the Hoffman algebra, and nontangential limits

Abstract

Representable in the form $\mathcal{H}_B \bigcap G$, where $G = C(M(H^{\infty})) \overset{\rm def}{=} \text{alg}(H^{\infty}, \overline{H^{\infty}})$ and $\mathcal{H}_B$ is a closed subalgebra in $C(D)$ consisting of the functions that have nontangential limits almost everywhere on $\mathbb{T}$, and these limits belong to the Douglas algebra $B$. In this paper we describe the space $M(\mathcal{H}^G_B)$ of maximal ideals of the algebra $\mathcal{H}^G_B$ and prove that $M(\mathcal{H}^G_B) = M(B) \bigcup M(\mathcal{H}^G_0)$ and prove that $M(\mathcal{H}^G_0)$, where $\mathcal{H}^G_0$ is a closed ideal in $G$ consisting of functions having nontangential limits equal to zero almost everywhere on $\mathbb{T}$. Moreover, it is established that $H^{\infty \supset } [\overline Z ] \ne \mathcal{H}_{H^\infty + C}^G$ on the disk. The Chang-Marshall theorem is generalized for the Banach algebras $\mathcal{H}^G_B$. We also prove that $\mathcal{H}^G_B = {\rm alg}(\mathcal{H}^G_{H^{\infty}}, \overline{IB})$ for any Douglas algebra $B$, where $IB = \{u_{\alpha}\}_B$ are inner functions such that $\overline{u_{\alpha}} \in B$ on $\mathbb{T}$.