Deformation of C⁎⁎-algebras by cocycles on locally compact quantum groups

Given a C⁎⁎-algebra A with a left action of a locally compact quantum group G on it and a unitary 2-cocycle Ω   on Gˆ, we define a deformation AΩAΩ of A. The construction behaves well under certain additional technical assumptions on Ω  , the most important of which is regularity, meaning that C0(G)Ω⋊GC0(G)Ω⋊G is isomorphic to the algebra of compact operators on some Hilbert space. In particular, then AΩAΩ is stably isomorphic to the iterated twisted crossed product Gˆop⋉ΩG⋉A. Also, in good situations, the C⁎⁎-algebra AΩAΩ carries a left action of the deformed quantum group GΩGΩ and we have an isomorphism GΩ⋉AΩ≅G⋉AGΩ⋉AΩ≅G⋉A. When G is a genuine locally compact group, we show that the action of G   on C0(G)Ω=Cr⁎(Gˆ;Ω) is always integrable. Stronger assumptions of properness and saturation of the action imply regularity. As an example, we make a preliminary analysis of the cocycles on the duals of some solvable Lie groups recently constructed by Bieliavsky et al., and discuss the relation of our construction to that of Bieliavsky and Gayral.