Bourgain algebras on $M(H^\infty)$

Abstract

In this paper we consider closed subalgebras of $C(M)$ and study the structure of algebras $\mathscr{A}$ satisfying $\mathscr{A}_{b}=C(M)$. We show that the Bourgain algebra of $A$ is contained in $H^{\infty}(D)+C(\overline{D})$ if $A$ is between the disk algebra $A(D)$ and $H^{\infty}(D)$ or between $H^{\infty}(D)$ and $H^{\infty}(D)+C(\overline{D})$, and the Bourgain algebra of $H^{\infty}\circ L_{m}$ is contained in $H^{\infty}(D)+C(\overline{D})$ if $m$ is a nontrivial point.