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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 [31], and an extension of the result of the author for identities with involution [42]. The similar results were proved also for graded identities [1], [41]. MSC: Primary 16R50; Secondary 16W20, 16W50, 16W10The first results on PI-representability of associative algebras belong to A.Kemer. He proves that any finitely generated associative (Z/2Z)-graded PI-algebra over a field of characteristic zero satisfies the same (Z/2Z)-graded identities as some finite dimensional (Z/2Z)-graded algebra over the same field [31], [34]. Later, he proves also that a finitely generated associative PI-algebra over an infinite field satisfies the same ordinary (non-graded) polynomial identities as some finite dimensional algebra [32]. PI-representability of finitely generated associative algebras over a commutative associative Noetherian ring with respect to ordinary polynomial identities was studied in [14]- [22].The series of results were obtained also for graded identities and identities with involution of associative algebras over a field of characteristic zero. If G is a finite abelian group then a finitely generated G-graded associative PI-algebra over an algebraically closed field of characteristic zero satisfies the same graded identities as some finite dimensional G-graded algebra over the same field [41]. For more general case of a finite (not necessarily abelian) group G it was proved that a finitely generated G-graded associative PI-algebra over a field of characteristic zero satisfies the same graded identities as some finite dimensional G-graded algebra over some extension of the base field [1]. As the direct consequences of [1], [41] we have also the similar results for G-identities if G is a finite abelian group of automorphisms of an associative algebra.Recently, PI-representability was proved also for identities with involution [42]. A finitely generated associative PI-algebra with involution over a field of characteristic zero satisfies the same identities with involution as some finite dimensional algebra with involution over the same field.The special interest to graded identities in the case of characteristic zero is explained by the super-trick and relation with the Specht problem [31]. Therefore the problem of PI-representability of graded algebras with involution is of current interest also.We consider associative algebras over a field F of characteristic zero. Further they will be called algebras.An F -algebra A is graded by a group G (G...

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