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

The notion of Fr'{e}chet locally $C^{\ast }$-algebra generalizes the notion of $C^{\ast }$-algebra. In this paper, we first present some definitions and basic facts about locally $C^{\ast }$-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 $C^{\ast }$-algebra is ideally amenable.