Differentiating the State Evaluation Map from Matrices to Functions on Projective Space

We show that the pure state evaluation map from $M_n(\mathbb{C})$ to $C(\mathbb{C}{\mathbb{P}}^{n-1})$ (a completely positive map of $C^{\ast }$-algebras) extends to a cochain map from the universal calculus on $M_n(\mathbb{C})$ to the holomorphic $\bar{\mathrm{\partial }}$ calculus on $\mathbb{C}{\mathbb{P}}^{n-1}$. The method uses connections on Hilbert $C^{\ast }$-bimodules. This implies the existence of various functors, including one from $M_n(\mathbb{C})$ modules to holomorphic bundles on $\mathbb{C}{\mathbb{P}}^{n-1}$.