  We consider associative algebras with involution graded by a finite abelian
group G over a field of characteristic zero. Suppose that the involution is
compatible with the grading. We represent conditions permitting
PI-representability of such algebras. Particularly, it is proved that a
finitely generated (Z/qZ)-graded associative PI-algebra with involution
satisfies exactly the same graded identities with involution as some finite
dimensional (Z/qZ)-graded algebra with involution for any prime q or q = 4.
This is an analogue of the theorem of A.Kemer for ordinary identities, and an
extension of the result of the author for identities with involution. The
similar results were proved also recentely for graded identities.
