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

Published online by Cambridge University Press:  12 May 2021

ALEXANDER BRUDNYI*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

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 $H^\infty ({\mathbb {D}})$.

MSC classification

Type
Research Article
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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DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR $H^\infty$ MAPS
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