We consider associative algebras with involution over a field of
characteristic zero. We proved that any algebra with involution satisfies the
same identities with involution as the Grassmann envelope of some finite
dimensional Z4​-graded algebra with graded involution. As a consequence we
obtain the positive solution of the Specht problem for identities with
involution: any associative algebra with involution over a field of
characteristic zero has a finite basis of identities with involution. These
results are analogs of theorems of A.R.Kemer for ordinary identities