Classification of the spaces $C_p^{\ast }(X)$ within the Borel-Wadge hierarchy for a projective space $X$

We study the complexity of the space $C_p^{\ast }(X)$ of bounded continuous functions with the topology of pointwise convergence. We are allowed to use descriptive set theoretical methods, since for a separable metrizable space $X$, the measurable space of Borel sets in $C_p^{\ast }(X)$ (and also in the space $C_p(X)$ of all continuous functions) is known to be isomorphic to a subspace of a standard Borel space. It was proved by A. Andretta and A. Marcone that if $X$ is a $\sigma$-compact metrizable space, then the measurable spaces $C_p(X)$ and $C_p^{\ast }(X)$ are standard Borel and if $X$ is a metrizable analytic space which is not $\sigma$-compact then the spaces of continuous functions are Borel-${\mathrm{\Pi }}_1^1$-complete. They also determined under the assumption of projective determinacy (PD) the complexity of $C_p(X)$ for any projective space $X$ and asked whether a similar result holds for $C_p^{\ast }(X)$. We provide a positive answer, i.e. assuming PD we prove, that if $n\ge 2$ and if $X$ is a separable metrizable space which is in ${\mathrm{\Sigma }}_n^1$ but not in ${\mathrm{\Sigma }}_{n-1}^1$ then the measurable space $C_p^{\ast }(X)$ is Borel-${\mathrm{\Pi }}_n^1$-complete. This completes under the assumption of PD the classification of Borel-Wadge complexity of $C_p^{\ast }(X)$ for $X$ projective.