$H^\infty(Ω)$对圆形域$Ω$的拓扑稳定秩
Topological Stable Rank of $H^\infty(Ω)$ for Circular Domains $Ω$
摘要：
设$Ω$是一个圆形区域，即一个开圆盘，去掉了1/2个不相交的闭圆盘。用$H^\infty(Ω)$表示$Ω$上所有有界全纯函数的Banach代数，它具有点态运算和上确界范数。我们证明了$H^\infty(Ω)$的拓扑稳定秩等于2。证明是基于Suarez定理，即$H^\infty(\D)$的拓扑稳定秩等于2，其中$\D$是单位圆盘.我们还证明了对于实轴对称的域，实对称代数$H^\infty_\R(Ω)$的Bass秩和拓扑稳定秩都是2。
英文摘要：
Let $Ω$ be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by $H^\infty(Ω)$ the Banach algebra of all bounded holomorphic functions on $Ω$, with pointwise operations and the supremum norm. We show that the topological stable rank of $H^\infty(Ω)$ 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 $H^\infty_\R(Ω)$ are 2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
