On almost polynomial growth of proper central polynomials

To any associative algebra A A is associated a numerical sequence c n δ (A) cₙ^ (A), n ≥ 1 n 1, called the sequence of proper central codimensions of A A. It gives information on the growth of the proper central polynomials of the algebra. If A A is a PI-algebra over a field of characteristic zero it has been recently shown that such a sequence either grows exponentially or is polynomially bounded. Here we classify, up to PI-equivalence, the algebras A A for which the sequence c n δ (A) cₙ^ (A), n ≥ 1 n 1, has almost polynomial growth. Then we face a similar problem in the setting of group-graded algebras and we obtain a classification also in this case when the corresponding sequence of proper central codimensions has almost polynomial growth.