Involution codimensions of finite dimensional algebras and exponential growth

Let F be a field of characteristic zero and let A be a finite dimensional a lgebra with involution * over F. We study the asymptotic behavior of the se quence of *-codimensions c(n)(A, *) of A and we show that Exp(A, *) = lim(n -->infinity) (n)root c(n)(A, *) exists and is an integer. We give an expli cit way for computing Exp(A, *) and as a consequence we obtain the followin g characterization of *-simple algebras: A is *-simple if and only if Exp(A , *) = dim(F) A. (C) 2000 Academic Press.