  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.
