论文标题
在polydisc上的对称函数的Banach代数
Banach algebras of symmetric functions on the polydisc
论文作者
论文摘要
令$ {\ mathbb {d}} = \ {z \ in \ mathbb {c}：| z | | <1 \} $，对于整数$ d \ geq 1 $，让$ s_d $表示对称组，由$ \ {1，\ cdots，d \ cdots的所有排列组成的所有排列组成。 a函数$ f：{\ mathbb {d}}^d \ rightarrow \ mathbb {c} $是对称的，如果$ f（z_1，\ cdots，z_d）= f（z__ {σ（1）}，\ cdots，z_______________________________________________________ z_d）\ in {\ mathbb {d}}^d $。 polydisc代数$ a（{\ mathbb {d}}^d）$是polydisc $ {\ mathbb {d}}^d $上所有Holomorthic函数的Banach代数$ f $，可以连续地扩展到$ {c} $ {C} copteries the Polydisc^d. turemum norm（由$ \ | f \ | _ \ infty：= \ sup _ {\ mathbf {z} \ in {\ mathbb {d}}}^d} | f（\ mathbf {z}）| $）| $）。令$ a _ {\ textrm {sym}}（{\ mathbb {d}}^d）$为$ a（{\ m athbb {d}}^d）的banach subalgebra of $ a（{\ mathbb {d}}^d）$，由polydisc algebra中的所有对称功能组成。研究了$ a _ {\ textrm {sym}}（{\ Mathbb {d}}^d）$的代数分析属性。特别是显示了以下结果：电晕定理，最大理想空间的描述及其合同，庇护所，投射性柔性和不相关性。
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.
