A remark on Bourgain algebras on the disk

Abstract:

It is shown that the Bourgain algebra ${X_b}$ of the space $X = {H^\infty }$ considered as a subalgebra of $\mathcal {U} = \operatorname {alg} \{ {H^\infty },{\overline H ^\infty }\}$ is ${H^\infty }(\mathbb {D}) + UC(\mathbb {D})$ where $UC(\mathbb {D})$ is the algebra of uniformly continuous functions on the open unit disk $\mathbb {D}$. This uses and extends a recent result of Cima-Janson-Yale on the Bourgain algebra of ${H^\infty }$ on $\partial \mathbb {D}$. Further, ${({X_b})_b} = {X_b}$.