Frechet algebras in abstract Harmonic analysis

Abstract

We provide a survey of the similarities and differences between Banach and Fréchet algebras including some known results and examples. We also collect some important generalizations in abstract harmonic analysis; for example, the generalization of the concepts of vector-valued Lipschitz algebras, abstract Segal algebras, Arens regularity, amenability, weak amenability, ideal amenability, etc.

The Bochner-Schoenberg-Eberlein property for commutative Frechet algebras

“…Note that every Banach algebra (space) is a Fréchet algebra (space), but in general the converse is not true. Some differences between Fréchet and Banach algebras surveyed in [3]. In this paper, we generalize the BSE property for the commutative semisimple Fréchet algebra (A, p ℓ ) ℓ∈N .…”

Bochner-Schoenberg-Eberlein-type inequality of the direct sum, ideals and quotient of Frechet algebras

“…Note that every Fréchet (algebra) space is not necessarily a Banach (algebra) space. Some differences between Banach and Fréchet (algebras) spaces introduced in the survey paper [1].…”