Let G be the set of real points of a (complex) algebraic group defined over the real numbers and let K be a maximal compact subgroup of G. We assume that G acts smoothly on a suitable topological algebra A. We show that the localizations of the periodic cyclic homologies of the crossed products of A by G and by K are isomorphic (up to a shift equal to the dimension of G/K). The localizations of the cyclic homology of the crossed product by the maximal compact subgroup K, in turn, identify as the Weyl-group invariant part of the corresponding localizations for the crossed product by a maximal torus. Examples of groups G include SL_n(R) and compact Lie groups. The choice of algebraic groups is justified by the fact that the orbit spaces of algebraic actions are not too singular. The methods used in the real case work also for p-adic groups.