On deformations of crossed products

Let A∗ΓA∗Γ be a crossed product algebra, where AA is semisimple, finitely generated over its center and ΓΓ is a finite group. We give a necessary and sufficient condition in terms of the outer action of ΓΓ on AA for the existence of a multi-parametric semisimple deformation of the form A((t1,…,tn))∗ΓA((t1,…,tn))∗Γ (with the induced outer action). The main tool in the proof is the solution of the so-called twisting problem. We also give an example which shows that the condition is not sufficient if one drops the condition on the finite generation of AA over its center.