Azumaya algebras with involution, polarizations, and linear generalized identities

An algebra R with anti-isomorphism (*) is shown to be Azumaya if (*) is given by an element of R ⊗ R^op; in particular, this is the case if the canonical map R ⊗ _CR^op → End_C(R) is onto. Consequently, the existence of a strict polarization often implies that an algebra is Azumaya. On the other hand, all simple rings have polarizations, and algebras with involution of the second kind have polarizations. These results are obtained via the theory of generalized polynomial identities.