On the stable rank of Q p ∩ H ∞

Abstract

Let 0 < p < 1. The stable rank of the Banach algebra Q p ∩ H ∞ is 1 if given f 1, f 2 in Q p ∩ H ∞ such that inf zED( l f1(z)I + l f2(z)I) > 0, there exists g in Q p ∩ H ∞ such that f 1 + gf 2 is invertible in Q p ∩ H ∞. As a partial answer to this problem, we prove the result when f 1 is an inner function in Q p .