2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170133Using similarities between topological $K$-theory and periodic cyclic homology we show that, after tensoring with $\mathbb C$, for certain Fréchet algebras the Chern character provides an isomorphism between these functors. This is applied to prove that the Hecke algebra and the Schwartz algebra of a reductive $p$-adic group have isomorphic periodic cyclic homology. The main theorem in the first version was incorrect for algebras related to noncompact manifolds. This has no effect on the results concerning p-adic groups. In the appendix we show that an analogous cohomological result does hold in the noncompact case.17 pages, in the third version an appendix was addedK-Theory and Homology19D55, 19L10, 22E50, 46L80text