The BSE Property for Weighted Banach Algebras of Dirichlet Series

Abstract

Let $\omega :\mathbb{N}\to [1,\infty )$ 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 $A_{\omega }={\ell }^1(\mathbb{N},\omega )$ under Dirichlet convolution. By employing Bohr's correspondence, we establish a weak-$\ast$ approximation framework for $\omega$-bounded semicharacters using scaled truncations that lie within the predual space $c_0(\mathbb{N},1/\omega )$. Utilizing the canonical $L$-projection decomposition of the bidual space $A_{\omega }^{\ast \ast }=A_{\omega }\oplus c_0(\mathbb{N},1/\omega {)}^{\perp }$ alongside the weak-$\ast$ compactness of the unit ball via the Banach-Alaoglu theorem, we prove that $A_{\omega }$ 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 $\Vert \cdot {\Vert }_{\omega }$.

6 pages