Spectral metrics on quantum projective spaces

TL;DR

The work proves that the Connes metric associated with the D'Andrea–Dąbrowski spectral triple on quantum projective spaces metrizes the weak-$*$ topology on their state spaces, thereby making quantum CP$_q^oldell$ a spectral metric space. The authors develop a robust framework based on Lipschitz algebras, the coaction of quantum $SU(N)$, and finite-dimensional approximations to relate global metric properties to the quantized interval $I_q$. A central extension theorem shows that compactness of the metric structure on $I_q$ implies compactness on CP$_q^oldell$, with a detailed analysis of weak-$*$ convergence to the counit and Monge–Kantorovich convergence. This approach unifies the noncommutative differential geometry of quantum flag manifolds with compact quantum metric space theory and recovers the Podleś-sphere results as a special case. The results have implications for quantum Gromov–Hausdorff theory and highlight how spectral data on quantum groups feed into metric structure on quantum homogeneous 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.

Theorems & Definitions (64)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Theorem 2.5
• Definition 2.6
• Theorem 2.7
• proof
• Lemma 3.1
• proof