The Corona Theorem and stable rank for the algebra C plus BH infinity

Let B be a Blaschke product. We prove in several different ways the Corona theorem for the algebra H∞B:=C+BH∞. That is, we show the equivalence of the classical Corona Condition Σ |fj | > δ> 0 on data f1, …, fn in H∞B and the solvability of the Bezout equation Σ gjfj =1 for g1, …, gn. Estimates on solutions to the Bezout equation are also obtained. We also show that the Bass stable rank of H∞B is 1. Analogous results are obtained also for A(D)B.