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

Abstract

Given a C*-algebra A with a left action of a locally compact quantum group G on it and a unitary 2-cocycle Omega on \hat G, we define a deformation A_Omega of A. The construction behaves well under certain additional technical assumptions on Omega, the most important of which is regularity, meaning that C_0(G)_Omega\rtimes G is isomorphic to the algebra of compact operators on some Hilbert space. In particular, then A_\Omega is stably isomorphic to the iterated twisted crossed product \hat G^{op}\ltimes_\Omega G\ltimes A. Also, in good situations, the C*-algebra A_\Omega carries a left action of the deformed quantum group G_\Omega and we have an isomorphism G_\Omega\ltimes A_\Omega\cong G\ltimes A. When G is a genuine locally compact group, we show that the action of G on C_0(G)_Omega=C*_r(\hat G;Omega) 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.