The BSE Property for Weighted Banach Algebras of Dirichlet Series
Abstract
Let ω: ℕ → [1, ∞) be a weight on the multiplicative semigroup of natural numbers. We investigate the Bochner-Schoenberg-Eberlein (BSE) property for the weighted Banach algebra of Dirichlet series Aω= ℓ1(ℕ, ω) under Dirichlet convolution. By employing Bohr's correspondence, we establish a weak-∗ approximation framework for ω-bounded semicharacters using scaled truncations that lie within the predual space c0(ℕ, 1/ω). Utilizing the canonical L-projection decomposition of the bidual space Aω∗∗ = Aω⊕ c0(ℕ, 1/ω)⊥ alongside the weak-∗ compactness of the unit ball via the Banach-Alaoglu theorem, we prove that Aω is a BSE-algebra for any weight. Furthermore, we show that the intrinsic BSE-norm on the Gel'fand transforms is isometric to the natural weighted algebra norm |·|ω.
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BibTeX
@misc{dabhi2026property,
title = {{The BSE Property for Weighted Banach Algebras of Dirichlet Series}},
author = {Prakash A. Dabhi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15332},
eprint = {2609.15332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15332},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The BSE Property for Weighted Banach Algebras of Dirichlet Series
AU - Prakash A. Dabhi
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15332
UR - https://arxiv.org/abs/2609.15332
AB - Let ω: ℕ → [1, ∞) be a weight on the multiplicative semigroup of natural numbers. We investigate the Bochner-Schoenberg-Eberlein (BSE) property for the weighted Banach algebra of Dirichlet series A_ω= ℓ^1(ℕ, ω) under Dirichlet convolution. By employing Bohr's correspondence, we establish a weak-∗ approximation framework for ω-bounded semicharacters using scaled truncations that lie within the predual space c_0(ℕ, 1/ω). Utilizing the canonical L-projection decomposition of the bidual space A_ω^∗∗ = A_ω⊕ c_0(ℕ, 1/ω)^⊥ alongside the weak-∗ compactness of the unit ball via the Banach-Alaoglu theorem, we prove that A_ω is a BSE-algebra for any weight. Furthermore, we show that the intrinsic BSE-norm on the Gel'fand transforms is isometric to the natural weighted algebra norm |·|_ω.
ER -