Thin interpolating sequences and three algebras of bounded functions

Abstract:

We consider the closed subalgebra ${\mathbf {A}}$ of ${H^\infty }$ generated by the thin interpolating Blaschke products, the smallest ${C^*}$ subalgebra ${\mathbf {B}}$ of ${L^\infty }$ containing ${\mathbf {A}}$, and the Douglas algebra ${\mathbf {E}}$ generated by the complex conjugates of thin interpolating Blaschke products. Our main result is that every ${\mathbf {E}}$-invertible inner function is a finite product of thin interpolating Blaschke products, making ${\mathbf {B}} = {C_{\mathbf {E}}}$. We apply results of Chang and Marshall to prove that ${\mathbf {A}} = {\mathbf {B}} \cap {H^\infty }$, that finite convex combinations of finite products of thin interpolating Blaschke products are dense in the closed unit ball of ${\mathbf {A}}$, and that the corona theorem holds for ${\mathbf {A}}$.