Maximal ideal space of some Banach algebras of Dirichlet series

Abstract

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

10 pages