Star-polynomial identities: Computing the exponential growth of the codimensions

Can one compute the exponential rate of growth of the ⁎-codimensions of a PI-algebra with involution ⁎ over a field of characteristic zero? It was shown in [2] that any such algebra A has the same ⁎-identities as the Grassmann envelope of a finite dimensional superalgebra with superinvolution B  . Here, by exploiting this result we are able to provide an exact estimate of the exponential rate of growth exp⁎(A)exp⁎(A) of any PI-algebra A   with involution. It turns out that exp⁎(A)exp⁎(A) is an integer and, in case the base field is algebraically closed, it coincides with the dimension of an admissible subalgebra of maximal dimension of B.