Metric aspects of noncommutative homogeneous spaces

TL;DR

This paper extends the theory of noncommutative deformations of homogeneous spaces by showing that certain seminorms induce compact quantum metrics and that these metrics depend continuously on deformation parameters.

Contribution

It demonstrates that for Lie groups, the induced seminorms define compact quantum metrics and depend continuously on the deformation homomorphism, generalizing previous constructions.

Findings

Seminorms induce compact quantum metrics on deformed spaces

Metrics depend continuously on the deformation parameter ho

Extension of Rieffel's quantum Heisenberg manifolds construction

Abstract

For a closed cocompact subgroup $\Gamma$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $\rho:\hat{K}\to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations $C^*(\hat{G}/\Gamma, \rho)$ of the homogeneous space $G/\Gamma$, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when $G$ is a Lie group and $G/\Gamma$ is connected, given any norm on the Lie algebra of $G$, the seminorm on $C^*(\hat{G}/\Gamma, \rho)$ induced by the derivation map of the canonical $G$-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on $\rho$ continuously, with respect to quantum Gromov-Hausdorff distances.