Rieffel Deformation of Homogeneous Spaces

TL;DR

This paper demonstrates that Rieffel deformation preserves the structure of homogeneous spaces, showing an isomorphism between deformed quotient spaces and deformed function algebras, with implications for quantum subgroup theory.

Contribution

It establishes the isomorphism between the Rieffel deformation of the quotient space and the quotient of the deformed algebra, extending understanding of quantum homogeneous spaces.

Findings

G(f)/H(f) is isomorphic to C*X(f) as G(f)-C*-algebras

C*X(f) is a G(f)-simple object when H(f) cannot be constructed

Rieffel deformation preserves the structure of homogeneous spaces

Abstract

Let H be a closed subgroup of a locally compact group G and let X=G/H be the quotient space of left cosets. Let C*X be the corresponding G-C*-algebra of continuous functions on X, vanishing at infinity. Suppose that L is a closed abelian subgroup of H and let f be a 2-cocycle on the dual group of L. Let G(f) be the Rieffel deformation of G. Using these data we may construct G(f)-C*-algebra C*X(f) - the Rieffel deformation of C*X. On the other hand we may perform the Rieffel deformation of the subgroup H obtaining the closed quantum subgroup H(f) of G(f) which in turn, by the results of Vaes, leads to the G(f)-C*-algebra G(f)/H(f). In this paper we show that G(f)/H(f) and C*X(f) are isomorphic G(f)-C*-algebras. We also consider the case where L is a subgroup of G but not of H, for which we cannot construct the subgroup H(f). Then C*X(f) cannot be identified with a quantum quotient. What may be shown is that it is a G(f)-simple object in the category of G(f)-C*-algebras.