A star-variety with almost polynomial growth

Let F be a field of characteristic zero. In this paper we construct a Finit e dimensional F-algebra with involution M and we study its *-polynomial ide ntities; on one hand we determine a generator of the corresponding T-ideal of the free algebra with involution and on the other we give a complete des cription of the multilinear *-identities through the representation theory of the hyperoctahedral group. As an outcome of this study we show that the *-variety generated by il M, var( M, *) has almost polynomial growth, i.e., the sequence of * -codimensions of M cannot be bounded by any polynomial f unction but any proper *-subvariety of var( M, *)has polynomial growth. If G(2) is the algebra constructed in Giambruno and Mishchenko (preprint), we next prove that M and C, are the only two finite dimensional algebras with involution generating *-varieties with almost polynomial growth. (C) 2000 A cademic Press.