  A generalization of Connes-Thom isomorphism is given for stable, homotopy
invariant, and split exact functors on separable $C^*$-algebras. As examples of
these functors, we concentrate on asymptotic and local cyclic cohomology and
the result is applied to improve some formulas in asymptotic and local cyclic
cohomology of $C^*$-algebras. As an other application, it is shown that these
cyclic theories are rigid after Rieffel's deformation quantizations.
