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 c_n*(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.