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

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

2008-03-06

arxiv.org/abs/0803.0980v2

Houston J. Math. 36 (2010), no. 1, 289-302

Mathematics

Complex Variables

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

Scientific paper

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 $f_1, ..., f_n \in H^\infty_B$: \[ \forall z \in \mD, \sum_{k=1}^{n} |f_k(z)| \geq \delta >0, \] and the {\em solvability of the Bezout equation} for $g_1, ..., g_n \in H^\infty_B$: \[ \forall z\in \mD, \sum_{k=1}^n g_k (z)f_k(z)=1. \] Estimates on solutions to the Bezout equation are also obtained. We also show that the Bass stable rank of $H^\infty_B$ is 1. Let $A(\mD)_B$ be the subalgebra of all elements from $H^\infty_B$ having a continuous extension to the closed unit disk $\bar{\mD}$. Analogous results are obtained also for $A(\mD)_B$.

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