Star-algebras and almost polynomial growth of central polynomials

Abstract

Let A be an algebra with involution ⁎ over a field of characteristic zero. There are three numerical sequences attached to the ⁎-polynomial identities Id⁎(A) satisfied by A: the sequence of codimensions cn⁎(A), the sequence of central codimensions cn⁎.z(A) and the sequence of proper central codimensions cn⁎,δ(A) n=1,2,…. They give a measure of the ⁎-polynomial identities, the central ⁎-polynomials and the proper central ⁎-polynomials of the algebra A. It has been proved ([9], [18]) that when A satisfies a non-trivial identity, the three sequences either grow exponentially or are polynomially bounded. Now, an algebra A has almost polynomial growth of the codimensions if cn⁎(A) grows exponentially and for any algebra B such that Id⁎(B)⊋Id⁎(A), cn⁎(B) is polynomially bounded. Similarly we have the definitions of almost polynomial growth of the central codimensions cn⁎.z(A) and of the proper central codimensions cn⁎,δ(A). We aim to classify, up to ⁎-PI-equivalence, the algebras with almost polynomial growth of one of the above codimensions. This has already been done in [8] regarding the codimensions cn⁎(A). Here we classify, up to ⁎-PI-equivalence, the algebras having almost polynomial growth of the central ⁎-polynomials by exhibiting three subalgebras of upper triangular matrices. We also construct three more finite dimensional algebras giving the classification of almost polynomial growth of the proper central ⁎-polynomials.