Some characterizations of algebras with involution with polynomial growth of their codimensions

Let A be an associative algebra endowed with an involution of the first kind and let denote the sequence of -codimensions of A. In this paper, we are interested in algebras with involution such that the -codimension sequence is polynomially bounded. We shall prove that A is of this kind if and only if it satisfies the same identities of a finite direct sum of finite dimensional algebras with involution , each of which with Jacobson radical of codimension less than or equal to one in . We shall also relate the condition of having polynomial codimension growth with the sequence of cocharacters and with the sequence of colengths. Along the way, we shall show that the multiplicities of the irreducible characters in the decomposition of the cocharacters are eventually constants. Finally, we shall give a classification of the algebras with involution whose -codimensions are at most of linear growth.