标题： 某些Dirichlet级数的巴拿赫代数的最大理想空间 显示英文标题

标题： Maximal ideal space of some Banach algebras of Dirichlet series

摘要： 设$\mathscr{H}^\infty$为所有狄利克雷级数$f=\sum\limits_{n=1}^\infty \frac{a_n}{n^s}$（其中$a_n\in \mathbb{C}$对每个$n$）在每个$s\in {\mathbb{C}}_+$处收敛的集合，使得$\|f\|_{\infty}:=\sup_{s\in {\mathbb{C}}_+}|f(s)|<\infty$。 设$\mathscr{B}\subset \mathscr{H}^\infty$是一个包含狄利克雷多项式（具有有限多个非零项的狄利克雷级数）的巴拿赫代数，其范数为$\|\cdot\|_{\mathscr{B}}$，使得包含映射$\mathscr{B} \subset \mathscr{H}^\infty$是连续的。 对于$m\in \mathbb{N}=\{1,2,3,\cdots\}$，令$\partial^{-m}\mathscr{B}$表示由所有$f\in \mathscr{B}$组成的巴拿赫代数，使得$f',\cdots, f^{(m)}\in \mathscr{B}$，具有逐点运算和范数$\|f\|_{\partial^{-m}\mathscr{B}}=\sum_{\ell=0}^m \frac{1}{\ell!}\|f^{(\ell)}\|_{\mathscr{B}}$。 假设维纳$1/f$属性对$\mathscr{B}$成立（即，$\inf_{s\in {\mathbb{C}}_+} |f(s)|>0$推出$\frac{1}{f}\in \mathscr{B}$），则证明对于所有$m\in \mathbb{N}$，极大理想空间$M(\partial^{-m}\mathscr{B})$的$\partial^{-m}\mathscr{B}$与$\overline{\mathbb{D}}^{\mathbb{N}}$同胚，其中$\overline{\mathbb{D}}=\{z\in \mathbb{C}:|z|\le 1\}$。 这样的巴拿赫代数的例子包括$\mathscr{H}^\infty$，$\mathscr{A}_u$是$\mathscr{H}^\infty$的子代数，由${\mathbb{C}}_+$中的一致连续函数组成，以及狄利克雷级数的维纳代数$\mathscr{W}$，其具有$\|f\|_{\mathscr{W}}:=\sum_{n=1}^\infty |a_n|<\infty$。一些结果（对数的存在性、投射自由性、无限巴斯稳定秩）作为应用给出。 显示英文摘要

摘要： Let $\mathscr{H}^\infty$ be the set of all Dirichlet series $f=\sum\limits_{n=1}^\infty \frac{a_n}{n^s}$ (where $a_n\in \mathbb{C}$ for each $n$) that converge at each $s\in {\mathbb{C}}_+$, such that $\|f\|_{\infty}:=\sup_{s\in {\mathbb{C}}_+}|f(s)|<\infty$. Let $\mathscr{B}\subset \mathscr{H}^\infty$ be a Banach algebra containing the Dirichlet polynomials (Dirichlet series with finitely many nonzero terms) with a norm $\|\cdot\|_{\mathscr{B}}$ such that the inclusion $\mathscr{B} \subset \mathscr{H}^\infty$ is continuous. For $m\in \mathbb{N}=\{1,2,3,\cdots\}$, let $\partial^{-m}\mathscr{B}$ denote the Banach algebra consisting of all $f\in \mathscr{B}$ such that $f',\cdots, f^{(m)}\in \mathscr{B}$, with pointwise operations and the norm $\|f\|_{\partial^{-m}\mathscr{B}}=\sum_{\ell=0}^m \frac{1}{\ell!}\|f^{(\ell)}\|_{\mathscr{B}}$. Assuming that the Wiener $1/f$ property holds for $\mathscr{B}$ (that is, $\inf_{s\in {\mathbb{C}}_+} |f(s)|>0$ implies $\frac{1}{f}\in \mathscr{B}$), it is shown that for all $m\in \mathbb{N}$, the maximal ideal space $M(\partial^{-m}\mathscr{B})$ of $\partial^{-m}\mathscr{B}$ is homeomorphic to $\overline{\mathbb{D}}^{\mathbb{N}}$, where $\overline{\mathbb{D}}=\{z\in \mathbb{C}:|z|\le 1\}$. Examples of such Banach algebras are $\mathscr{H}^\infty$, the subalgebra $\mathscr{A}_u$ of $\mathscr{H}^\infty$ consisting of uniformly continuous functions in ${\mathbb{C}}_+$, and the Wiener algebra $\mathscr{W}$ of Dirichlet series with $\|f\|_{\mathscr{W}}:=\sum_{n=1}^\infty |a_n|<\infty$. Some consequences (existence of logarithms, projective freeness, infinite Bass stable rank) are given as applications.