Ideal and weak Amenability of Fréchet locally $C^*$-algebra

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.