arXiv 2201.001832022-01-06math.FAmath.ACmath.GN

多圆盘上对称函数的巴拿赫代数

Banach algebras of symmetric functions on the polydisc

Amol Sasane

备注：: 19 pages

摘要：

令${\mathbb{D}}=\{z\in \mathbb{C}:|z|<1\}$，对于整数$d\geq 1$，令$S_d$表示对称群，由集合$\{1,\cdots, d\}$的所有排列组成。函数$f:{\mathbb{D}}^d\rightarrow \mathbb{C}$是对称的，如果对于所有的$σ\in S_d$和所有的$(z_1,\cdots, z_d)\in {\mathbb{D}}^d$都是$f(z_1,\cdots, z_d)=f(z_{σ(1)},\cdots, z_{σ(d)})$。多圆盘代数$A({\mathbb{D}}^d)$是多圆盘${\mathbb{D}}^d$上的所有全纯函数$f$的Banach代数，它可以连续扩展到${\mathbb{C}}^d$中的多圆盘的闭包，具有逐点运算和上确界范数（由$\|f\|_\infty:=\sup_{\mathbf{z} \in {\mathbb{D}}^d} |f(\mathbf{z})|$给出）。设$A_{\textrm{sym}}({\mathbb{D}}^d)$是多圆盘代数中所有对称函数组成的$A({\mathbb{D}}^d)$的Banach子代数.研究了$A_{\textrm{sym}}({\mathbb{D}}^d)$的代数分析性质。特别是，以下结果显示：电晕定理，极大理想空间的描述和它的可压缩性，厄米性，射影自由，和非相干。

英文摘要：

Let ${\mathbb{D}}=\{z\in \mathbb{C}:|z|<1\}$ and for an integer $d\geq 1$, let $S_d$ denote the symmetric group, consisting of of all permutations of the set $\{1,\cdots, d\}$. A function $f:{\mathbb{D}}^d\rightarrow \mathbb{C}$ is symmetric if $f(z_1,\cdots, z_d)=f(z_{σ(1)},\cdots, z_{σ(d)})$ for all $σ\in S_d$ and all $(z_1,\cdots, z_d)\in {\mathbb{D}}^d$. The polydisc algebra $A({\mathbb{D}}^d)$ is the Banach algebra of all holomorphic functions $f$ on the polydisc ${\mathbb{D}}^d$ that can be continuously extended to the closure of the polydisc in ${\mathbb{C}}^d$, with pointwise operations and the supremum norm (given by $\|f\|_\infty:=\sup_{\mathbf{z} \in {\mathbb{D}}^d} |f(\mathbf{z})|$). Let $A_{\textrm{sym}}({\mathbb{D}}^d)$ be the Banach subalgebra of $A({\mathbb{D}}^d)$ consisting of all symmetric functions in the polydisc algebra. Algebraic-analytic properties of $A_{\textrm{sym}}({\mathbb{D}}^d)$ are investigated. In particular, the following results are shown: the corona theorem, description of the maximal ideal space and its contractibility, Hermiteness, projective-freeness, and non-coherence.

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。

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