  Assume A is a Frechet algebra equipped with a smooth isometric action of a
vector group V, and consider Rieffel's deformation A_J of A. We construct an
explicit isomorphism between the smooth crossed products V\ltimes\A_J and
V\ltimes\A. When combined with the Elliott-Natsume-Nest isomorphism, this
immediately implies that the periodic cyclic cohomology is invariant under
deformation. Specializing to the case of smooth subalgebras of C*-algebras, we
also get a simple proof of equivalence of Rieffel's and Kasprzak's approaches
to deformation.
