The Corona theorem and stable rank for the algebra $\mC +BH^\infty$

Abstract

Let $B$ be a Blaschke product. We prove in several different ways the corona theorem for the algebra $H^\infty_B:=\mC+BH^\infty$. That is, we show the equivalence of the classical {\em corona condition} on data $f_1,...,f_n\in H_B^{\infty }$: \[ \forall z \in \mD, \sum_{k=1}^{n} |f_k(z)| \geq δ>0, \] and the {\em solvability of the Bezout equation} for $g_1,...,g_n\in H_B^{\infty }$: \[ \forall z\in \mD, \sum_{k=1}^n g_k (z)f_k(z)=1. \] Estimates on solutions to the Bezout equation are also obtained. We also show that the Bass stable rank of $H_B^{\infty }$ is 1. Let $A(\mD)_B$ be the subalgebra of all elements from $H_B^{\infty }$ having a continuous extension to the closed unit disk $\bar{\mD}$. Analogous results are obtained also for $A(\mD)_B$.

v1. 13 pages, v2. 13 pages, minor typos corrected, to appear in Houston J. Math