Subalgebras of Douglas algebras

Abstract:

A closed subalgebra $\mathcal {A}$ of ${L^\infty }$ is called a Douglas algebra in case $\mathcal {A}$ is an algebra generated by ${H^\infty }$ and a set of inverses of inner functions. It is shown that if the Douglas algebra $\mathcal {A}$ contains properly ${H^\infty } + C$, then there is another Douglas algebra $\mathcal {A}’$ such that ${H^\infty } + C \subsetneq \mathcal {A}’ \subsetneq \mathcal {A}$. Some results on subalgebras are also given for algebras generated by ${H^\infty }$ and a function of the form $f\overline B$, where $f$ is in ${H^\infty }$ and $B$ is an infinite Blaschke product.