Topological Stable Rank of $H^\infty(Ω)$ for Circular Domains $Ω$

Details Topological Stable Rank of $H^\infty(Ω)$ for Circular Domains $Ω$ Topological Stable Rank of $H^\infty(Ω)$ for Circular Domains $Ω$

2009-09-14

arxiv.org/abs/0909.2533v1

Anal. Math., 36 (2010) no. 4, 287--297

Mathematics

Complex Variables

v1: 9 pages

Scientific paper

10.1007/s10476-010-0403-y

Let $\Omega$ be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by $H^\infty(\Omega)$ the Banach algebra of all bounded holomorphic functions on $\Omega$, with pointwise operations and the supremum norm. We show that the topological stable rank of $H^\infty(\Omega)$ is equal to 2. The proof is based on Suarez's theorem that the topological stable rank of $H^\infty(\D)$ is equal to 2, where $\D$ is the unit disk. We also show that for domains symmetric to the real axis, the Bass and topological stable ranks of the real symmetric algebra $H^\infty_\R(\Omega)$ are 2.

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