Stable rank boundary principle

Let K denote a compact subset of the complex plane . 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.

Let G denote a plane domain whose boundary consists of finitely many closed, nonintersecting Jordan curves. We show that for a fixed function of gεC( ), g≠0, the following assertions are equivalent:

Every unimodular element (f, g) is reducible to the principal component exp(C( )).

The zero set Z_g is polynomially convex, i.e., its complement Z_g is connected.

Author Keywords: Bass' stable rank; reducible; unimodular; 1-stable; boundary principle