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 $C^{*}$-algebras) extends to a cochain map from the universal calculus on $M_{n}(\mathbb{C} )$ to the holomorphic $\bar{\partial}$ calculus on $\mathbb{C} \mathbb{P}^{n-1}$. The method uses connections on Hilbert $C^{*}$-bimodules. This implies the existence of various functors, including one from $M_{n}(\mathbb{C})$ modules to holomorphic bundles on $\mathbb{C} \mathbb{P}^{n-1}$.