*-Graded Capelli Polynomials and their Asymptotic

Let $F\langle Y \cup Z, \ast \rangle$ be the free $\ast$-superalgebra over a field $F$ of characteristic zero and let $\Gamma^\ast_{M^{\pm}, L^{\pm}}$ be the $T^\ast_{\mathbb{Z}_2}$-ideal generated by the set of the $\ast$-graded Capelli polynomials $Cap^{(\mathbb{Z}_2, \ast)}_{M^+} [Y^+,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{M^-} [Y^-,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{L^+} [Z^+,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{L^-} [Z^-,X]$ alternating on $M^+$ symmetric variables of homogeneous degree zero, on $M^-$ skew variables of homogeneous degree zero, on $L^+$ symmetric variables of homogeneous degree one and on $L^-$ skew variables of homogeneous degree one, respectively. We study the asymptotic behavior of the sequence of $\ast$-graded codimensions of $\Gamma^\ast_{M^{\pm}, L^{\pm}}.$ In particular we prove that the $\ast$-graded codimensions of the finite dimensional simple $\ast$-superalgebras are asymptotically equal to the $\ast$-graded codimensions of $\Gamma^\ast_{M^{\pm}, L^{\pm}}$, for some fixed natural numbers $M^+, M^-, L^+$ and $L^-$.