Stable rank boundary principle

Abstract

(1) Let K denote a compact subset of the complex plane C . We present correct proof that the stable rank of A( K) is one. Hereby, A ( K) is the algebra of all continuous functions on K which are analytic in the interior of K. (2) Let G denote a plane domain whose boundary consists of finitely many closed, nonintersecting Jordan curves. We show that for a fixed function of gϵ C( Ḡ), g≠0, the following assertions are equivalent: (i) Every unimodular element ( f, g) is reducible to the principal component exp( C( Ḡ)). (ii) The zero set Z g is polynomially convex, i.e., its complement C ⧹ Z g is connected.