Rieffel Deformation of Group Coactions

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Let G be a locally compact group, \({\Gamma\subset G}\) an abelian subgroup and let Ψ be a continuous 2-cocycle on the dual group \({\Hat\Gamma}\). Let B be a C*-algebra and \({\Delta_B\in{\rm Mor}\,(B,B\otimes{\rm C}_0(G))}\) a continuous right coaction. Using Rieffel deformation, we can construct a quantum group \({({\rm C}_0(G)^{\tilde\Psi\otimes\Psi},\Delta^\Psi)}\) and the deformed C*-algebra B Ψ. The aim of this paper is to present a construction of the continuous coaction \({\Delta_B^\Psi}\) of the quantum group \({({\rm C}_0(G)^{\tilde\Psi\otimes\Psi},\Delta^\Psi)}\) on B Ψ. The transition from the coaction Δ B to its deformed counterpart \({\Delta_B^\Psi}\) is nontrivial in the sense that \({\Delta_B^\Psi}\) contains complete information about Δ B . In order to illustrate our construction we apply it to the action of the Lorentz group on the Minkowski space obtaining a C*-algebraic quantum Minkowski space.