Ideal and weak Amenability of Fr\'{e}chet locally $C^*$-algebra

TL;DR

This paper explores the concepts of ideal and weak amenability in Fréchet locally $C^*$-algebras, establishing that all such algebras are ideally amenable, thus extending the understanding of their algebraic properties.

Contribution

It introduces and studies the notions of ideal and weak amenability specifically for Fréchet locally $C^*$-algebras, proving their ideal amenability.

Findings

Every Fréchet locally $C^*$-algebra is ideally amenable.

Provides foundational definitions and properties of locally $C^*$-algebras.

Extends amenability concepts from $C^*$-algebras to the Fréchet locally $C^*$-algebra setting.

Abstract

The notion of Fr\'{e}chet locally $C^*$-algebra generalizes the notion of $C^*$-algebra. In this paper, we first present some definitions and basic facts about locally $C^*$-algebra, and then we introduce and study the notion of ideal and weak amenability for these algebras. Also, we show that every Fr\'{e}chet locally $C^*$-algebra is ideally amenable.