Metric aspects of noncommutative homogeneous spaces

For a closed cocompact subgroup Γ of a locally compact group G, given a compact abelian subgroup K of G and a homomorphism ρ:Kˆ→G satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations C∗(Gˆ/Γ,ρ) of the homogeneous space G/Γ, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when G is a Lie group and G/Γ is connected, given any norm on the Lie algebra of G, the seminorm on C∗(Gˆ/Γ,ρ) 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 ρ continuously, with respect to quantum Gromov-Hausdorff distances.