Rieffel deformation via crossed products

Abstract

We start from Rieffel data ( A , Ψ , ρ ) , where A is a C ∗ -algebra, ρ is an action of an 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 Ψ . In the case of Γ = R n we obtain a very simple proof of invariance of K -groups under the deformation. In the general case we also get a very simple proof that nuclearity is preserved under the deformation. We show how our approach leads to quantum groups and investigate their duality. The general theory is illustrated by an example of the deformation of SL ( 2 , C ) . A description of it, in terms of noncommutative coordinates α ˆ , β ˆ , γ ˆ , δ ˆ , is given.