Rieffel Deformation via Crossed Products

We present a refinement of Rieffel Deformation of C∗-algebras. We start from Rieffel data (A,Ψ, ρ), where A is a C∗-algebra, ρ is an action of abelian group Γ on A and Ψ is a 2-cocycle on the dual group. Using Landstad theory of crossed product we get a deformed C∗-algebra A. In case of Γ = R we obtain a very simple proof of invariance of K-groups under the deformation. In any case we get a simple proof that nuclearity is preserved under the deformation. We show how Rieffel deformation leads to quantum groups and investigate the duality. The general theory is illustrated by an example of deformation of SL(2,C). The description of it, in terms of noncommutative coordinates α̂, β̂, γ̂, δ̂ is given.