Spectral metrics on quantum projective spaces

TL;DR

This paper demonstrates that the noncommutative geometry of quantum projective spaces aligns with Rieffel's quantum metric space theory, showing that the Connes metric induces the weak-* topology on the state space.

Contribution

It extends spectral metric results from quantum spheres to quantum projective spaces, establishing compatibility with Rieffel's framework.

Findings

Connes metric metrizes the weak-* topology on the state space.

The noncommutative differential geometry aligns with Rieffel's quantum metric space theory.

Generalizes previous spectral metric results from spheres to projective spaces.

Abstract

We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital spectral triple introduced by D'Andrea and Dabrowski. In particular, we establish that the Connes metric metrizes the weak-* topology on the state space of quantum projective space. This generalizes previous work by the second author and Aguilar regarding spectral metrics on the standard Podles spheres.