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

Abstract

Let ${\mathcal{H}}^{\infty }$ be the set of all Dirichlet series $f={\sum }_{n=1}^{\infty }{a}_n{n}^{-s}$ (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 $\Vert f{\Vert }_{\infty }={\sup }_{s\in {\mathbb{C}}_0}|f(s)|<\infty$. Then ${\mathcal{H}}^{\infty }$ is a Banach algebra with pointwise operations and the supremum norm $\Vert \cdot {\Vert }_{\infty }$, and has been studied in earlier works. The article introduces a new family of Banach subalgebras ${\mathcal{H}}_S^{\infty }$ of ${\mathcal{H}}^{\infty }$. For $S\subset \mathbb{N}$, let ${\mathcal{H}}_S^{\infty }$ be the set of all elements ${\sum }_{n=1}^{\infty }{a}_n{n}^{-s}\in {\mathcal{H}}^{\infty }$ such that for all $n\in \mathbb{N}\setminus S$, we have $a_n=0$. Then ${\mathcal{H}}_S^{\infty }$ is a unital Banach subalgebra of ${\mathcal{H}}^{\infty }$ with the supremum norm if and only if $S$ is a multiplicative subsemigroup of $\mathbb{N}$ containing $1$. It is shown that for such $S$, ${\mathcal{H}}_S^{\infty }$ is the multiplier algebra of ${\mathcal{H}}_S^2$, where ${\mathcal{H}}_S^2$ is the Hilbert space of all Dirichlet series $f={\sum }_{n\in S}{a}_n{n}^{-s}$ such that $\Vert f{\Vert }_2:=({\sum }_{n\in S}|a_n{|}^2{)}^{\frac{1}{2}}<\infty$. A characterisation of the group of units in ${\mathcal{H}}_S^{\infty }$ is also given, by showing an analogue of the Wiener $1/f$ theorem for ${\mathcal{H}}_S^{\infty }$. 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 ${\mathcal{H}}_S^{\infty }$ is infinite.

19 pages. An extra result (Theorem 6.2) has been added