Maximal ideal space of some Banach algebras of Dirichlet series
Amol Sasane
TL;DR
The paper determines the maximal ideal space of the family $\\partial^{-m}\\mathscr{B}$ of Banach algebras contained in the Hardy algebra $\\mathscr{H}^\\infty$ of Dirichlet series. The main method constructs a family of characters $\\varphi_{\\bm{\\lambda}}$ parameterized by the infinite polydisc $\\overline{\\mathbb{D}}^{\\mathbb{N}}$ and shows every character arises in this way, yielding a homeomorphism $M(\\partial^{-m}\\mathscr{B}) \\cong \\overline{\\mathbb{D}}^{\\mathbb{N}}$. This result applies to natural examples such as $\\mathscr{H}^\\infty$, $\\mathscr{A}_u$, $\\mathscr{W}$, and $\\mathscr{H}^\\infty_S$ under Wiener property assumptions, with the corresponding $S$-extension. As a consequence, logarithms exist via the Arens–Royden framework, and the rings are projective free with infinite Bass stable rank, infinite topological stable rank, and infinite Krull dimension, illuminating the spectral and algebraic structure of these Dirichlet-series algebras.
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.