Algebras with additional structures and multiplicities bounded by a constant

G be a finite group and A a G-graded algebra over a field F of characteristic zero. We characterize the varieties of G-graded algebras such that the multiplicities mλ appering in the n-cocharacters of A are bounded by a constant, in terms of G-identities. If A is endowed with a graded involution , i.e. if A is a (G,)-algebra, we characterize the varieties of $(G,*)$-algebras whose multiplicities in the sequence of n-cocharacters of A are bounded by $1$ by showing a list of (G,)-polynomial identities satisfied by such varieties.