Rieffel deformation of homogeneous spaces

Let G1 ⊂ G be a closed subgroup of a locally compact group G and let X = G/G1 be the quotient space of left cosets. Let X = (C0(X), X) be the corresponding G-C∗-algebra where G = (C0(G), ). Suppose that Γ is a closed abelian subgroup of G1 and let Ψ be a 2-cocycle on the dual group ˆ Γ . Let GΨ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct GΨ -C∗- algebra XΨ – the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G1 obtaining the closed quantum subgroup GΨ1 ⊂ GΨ , which in turn, by the results of S. Vaes, leads to the GΨ -C∗-algebra GΨ /GΨ1 . In this paper we show that GΨ /GΨ1 ∼= XΨ .We also consider the case where Γ ⊂ G is not a subgroup of G1, for which we cannot construct the subgroup GΨ1 . Then generically XΨ cannot be identified with a quantum quotient. What may be shown is that it is a GΨ -simple object in the category of GΨ -C∗-algebras. © 2010 Elsevier Inc. All rights reserved