A characterization of algebras with polynomial growth of the codimensions

Let A be an associative algebras over a field of characteristic zero. We pr ove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decompositi on into suitable subalgebras; we also give a decomposition of the n-th coch aracter of A into suitable S-n-characters. We give similar characterizations of finite dimensional algebras with invol ution whose *-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group.