Dense Stable Rank and Runge Type Approximation Theorems for ${\mathbf{H}}^{\infty }$ Maps

Abstract

Let $H^{\infty }(\mathbb{D}\times \mathbb{N})$ 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 one 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 algebra $H^{\infty }(\mathbb{D})$.

26 pages

References in corpus (1)

• Oka properties of complements of holomorphically convex sets

Cited by in corpus (2)

• On Stable Rank of ${\mathbf{H}}^{\infty }$ on Coverings of Finite Bordered Riemann Surfaces
• On nonlinear Rudin-Carleson type theorem