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

TL;DR

This paper extends the pure state evaluation map from matrix algebras to continuous functions on projective space, establishing a cochain map that links algebraic and geometric structures via connections on Hilbert C*-bimodules.

Contribution

It introduces a novel extension of the state evaluation map to a cochain map connecting algebraic calculus with holomorphic calculus on projective space, using connections on Hilbert C*-bimodules.

Findings

Extension of the evaluation map to a cochain map

Existence of functors from modules to holomorphic bundles

Connections on Hilbert C*-bimodules facilitate the construction

Abstract

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} $.