  To each algebra over the complex numbers we associate a sequence of abelian
groups in a contravariant functorial way. In degree (m-1) we have the
m-summable Fredholm modules over the algebra modulo stable m-summable
perturbations. These new finitely summable K-homology groups pair with cyclic
homology and algebraic K-theory. In the case of cyclic homology the pairing is
induced by the Chern-Connes character. The pairing between algebraic K-theory
and finitely summable K-homology is induced by the Connes-Karoubi
multiplicative character.
