# isomorphism of Chern character in kk-theory

Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is isomorphism in K-theory for those algebras (after tensoring with complex numbers). Does that mean that Chern character is also isomorphism in K-homology (after tensoring with complex numbers)?

• Usually, what you probably refer to, Chern character is an isomorphism (after tensoring with complex numbers) with one side being K-homology (- not K-theory). To answer your question: unlikely. K-homology = Hom(A,C). K-theory = Hom(C,A). Why should they coincide? Chern character is also defined as a functor from the category $kk_0$ to $HP^0$. So your question seems unprecise. – hänsel Mar 12 '16 at 17:13