On Banach subalgebras of $\mathscr{H}^\infty$ consisting of lacunary Dirichlet series

TL;DR

This paper introduces a new family of Banach subalgebras of the algebra of bounded Dirichlet series, characterizes their structure, units, and stable rank, extending classical results to lacunary series with multiplicative index sets.

Contribution

It defines and analyzes Banach subalgebras of Dirichlet series based on multiplicative subsets of natural numbers, establishing their properties and connections to multiplier algebras and stable ranks.

Findings

Subalgebras are Banach algebras iff the index set is multiplicative and contains 1.

Identifies these subalgebras as multiplier algebras of associated Hilbert spaces.

Shows the stable rank is infinite under certain generator conditions.

Abstract

Let $\mathscr{H}^\infty$ be the set of all Dirichlet series $f=\sum_{n=1}^\infty a_nn^{-s}$ (where $a_n\in\mathbb{C}$ for all $n\in\mathbb{N}=\{1,2,3,\cdots\}$) that converge at each $s$ in $\mathbb{C}_0=\{s\in \mathbb{C}:\text{Re}(s)>0\}$, such that $\|f\|_{\infty}=\sup_{s\in\mathbb{C}_0}|f(s)|<\infty$. Then $\mathscr{H}^\infty$ is a Banach algebra with pointwise operations and the supremum norm $\|\cdot\|_\infty$, and has been studied in earlier works. The article introduces a new family of Banach subalgebras $\mathscr{H}^\infty_{S}$ of $\mathscr{H}^\infty$. For $S\subset\mathbb{N}$, let $\mathscr{H}^\infty_{S}$ be the set of all elements $\sum_{n=1}^\infty a_nn^{-s}\in \mathscr{H}^\infty$ such that for all $n\in \mathbb{N}\setminus S$, we have $a_n=0$. Then $\mathscr{H}^\infty_{S}$ is a unital Banach subalgebra of $\mathscr{H}^\infty$ with the supremum norm if and only if $S$ is a multiplicative subsemigroup of $\mathbb{N}$ containing $1$. It is shown that for such $S$, $\mathscr{H}^\infty_{S}$ is the multiplier algebra of $\mathscr{H}^2_S$, where $\mathscr{H}^2_S$ is the Hilbert space of all Dirichlet series $f=\sum_{n\in S} a_nn^{-s}$ such that $\|f\|_2:=(\sum_{n\in S} |a_n|^2)^{\frac{1}{2}}<\infty$. A characterisation of the group of units in $\mathscr{H}^\infty_{S}$ is also given, by showing an analogue of the Wiener $1/f$ theorem for $\mathscr{H}^\infty_{S}$. If $S$ has a set of generators allowing a unique representation of each element of $S$, then it is shown that the Bass stable rank of $\mathscr{H}^\infty_S$ is infinite.