BSE PROPERTY OF FRÉCHET ALGEBRA

Abstract

A class of commutative Banach algebras which satisfy a Bochner–Schoenberg–Eberlein-type inequality was introduced by Takahasi and Hatori. We generalize this property for the commutative Fréchet algebra ${(\mathcal{𝒜},p_{\ell })}_{\ell \in \mathbb{N}}$. Moreover, we verify and generalize some of the main results in the class of Banach algebras, for the Fréchet case. We prove that all Fréchet ${\mathrm{C}}^{*}$-algebras and also uniform Fréchet algebras are BSE algebras. Also, we show that $C^{\infty }[0,1]$ is not a Fréchet BSE algebra.