Metric aspects of noncommutative homogeneous spaces

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.