On the Arens regularity of Frechet algebras and their biduals

TL;DR

This paper investigates the Arens regularity of Frechet algebras and their biduals, establishing conditions under which these algebras and their sequences are Arens regular, and clarifying the structure of weakly almost periodic functions.

Contribution

It proves that WAP(A) equals wap(A) for Frechet algebras and characterizes Arens regularity of A** and sequences of Frechet algebras, providing new insights into their structure.

Findings

WAP(A) = wap(A) for Frechet algebras

A** is Arens regular iff A and WAP(A)* are Arens regular

Sequence of Frechet algebras is Arens regular iff each is Arens regular

Abstract

In This paper, we study the concept of weakly almost periodic functions on Frechet algebras. For a Frechet algebra A, we show that WAP(A)=wap(A). We also show that A** is Arens regular if and only if both A and WAP(A)* are Arens regular. Finally, for a sequence of Frechet algebras (An), we prove that the Frechet algebra \ell^1-\prod An is Arens regular if and only if each An is Arens regular.