On Banach subalgebras of $\mathscr{H}^\infty$ consisting of lacunary Dirichlet series
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2025
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| _version_ | 1866915465295560704 |
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| author | Sasane, Amol |
| author_facet | Sasane, Amol |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_13127 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Banach subalgebras of $\mathscr{H}^\infty$ consisting of lacunary Dirichlet series Sasane, Amol Functional Analysis Complex Variables Rings and Algebras Primary 30B50, Secondary 46J15 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. |
| title | On Banach subalgebras of $\mathscr{H}^\infty$ consisting of lacunary Dirichlet series |
| topic | Functional Analysis Complex Variables Rings and Algebras Primary 30B50, Secondary 46J15 |
| url | https://arxiv.org/abs/2508.13127 |