Polynomial growth and star-varieties

Abstract

Let V be a variety of associative algebras with involution over a field F of characteristic zero and let cn⁎(V), n=1,2,…, be its ⁎-codimension sequence. Such a sequence is polynomially bounded if and only if V does not contain the commutative algebra F⊕F, endowed with the exchange involution, and M, a suitable 4-dimensional subalgebra of the algebra of 4×4 upper triangular matrices. Such algebras generate the only varieties of ⁎-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all subvarieties of the ⁎-varieties of almost polynomial growth by giving a complete list of finite dimensional ⁎-algebras generating them.