Maximal ideal space of some Banach algebras of Dirichlet series

Abstract

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.