Eventually non-decreasing codimensions of ∗-identities

Let A be a PI-algebra. If A is an associative algebra, the sequence of codimensions c n ( A ) , n = 1 , 2 , … , of A is asymptotically non-decreasing. For the non-associative case, there are examples of PI-algebras whose sequence of codimensions is not eventually non-decreasing. For a associative PI-algebra A with involution ∗ : A → A , it was recently shown that its sequence of ∗ -codimensions c n ∗ ( A ) , n = 1 , 2 , … , is also asymptotically non-decreasing. In the present paper, we construct a non-associative algebra whose sequence of ∗ -codimensions is not eventually non-decreasing.