  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} $.
