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DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR ${\boldsymbol H}^{\boldsymbol{\infty}}$ MAPS

Part of: Spaces and algebras of analytic functions

Published online by Cambridge University Press:  12 May 2021

ALEXANDER BRUDNYI
ALEXANDER BRUDNYI*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, CanadaT2N 1N4
*
e-mail: albru@math.ucalgary.ca
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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 $1$ 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 $H^\infty ({\mathbb {D}})$ .

Keywords

bounded holomorphic functionBanach algebradense stable rankmaximal ideal spaceOka manifoldopen polyhedronRunge theorem

MSC classification

Secondary: 30H05: Bounded analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 3 , December 2022 , pp. 289 - 317
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Finnur Larusson

This research is supported in part by NSERC Grant No. 10010444.

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