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

TL;DR

This paper investigates conditions under which the direct sum, ideals, and quotients of commutative semisimple Frechet algebras are BSE-algebras, providing characterizations and multiplier algebra descriptions.

Contribution

It offers a new characterization of the multiplier algebra of the direct sum of two Frechet algebras and establishes when these structures are BSE-algebras.

Findings

A  is a BSE-algebra iff A and B are BSE-algebras.

Multiplier algebra of the direct sum is characterized.

Ideals and quotients of BSE-algebras are BSE-algebras under certain conditions.

Abstract

Let A and B be two commutative semisimple Frechet algebras. We first give a characterization of the multiplier algebra of the direct sum of A and B. We then prove that A \oplus B is a BSE-algebra if and only if A and B are BSE-algebras. Furthermore, for a closed ideal I of A, we study multipliers of ideals and quotient algebras of A and show that I and A/I are BSE-algebras, under certain conditions.