DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

Abstract

AbstractLet$H^\infty ({\mathbb {D}}\times {\mathbb {N}})$be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk${\mathbb {D}}\subset {{\mathbb C}}$. We show that the dense stable rank of$H^\infty ({\mathbb {D}}\times {\mathbb {N}})$is$1$and, using this fact, prove some nonlinear Runge-type approximation theorems for$H^\infty ({\mathbb {D}}\times {\mathbb {N}})$maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra$H^\infty ({\mathbb {D}})$.