Let $(\pi,H,F)$ be a Fredholm module: here $\pi:A \to B(H)$ is a representation of an algebra on the Hilbert space $H$ and $F$ is a self adjoint operator with square one such that for each $a \in A$ we have $[F,\pi(a)]$ is compact. Let us assume that there is some $p$ such that $[F,\pi(a)]$ lies in the Schatten ideal $S^p$: we call such Fredholm module to be $p$-summable. In the even case we assume that there is additionally a grading $\gamma$ i.e. $\gamma^2=1$ such that $\pi$ is even and $F$ is odd with respect to this grading. For $p$-summable Fredholm module let us define $$Ch^n(a_0,a_1,...,a_n):=c_n Tr(\epsilon F[F,a_0][F,a_1]...[F,a_n])$$ where $c_n$ are apriopriate normalizing constants, $\epsilon$ is either the identity or the grading $\gamma$ if our Fredholm module is odd or even respectively and $n$ is sufficiently big natural number. This is called Chern character for the Fredholm modules. One shows that this formula defines a cyclic cocycle over $A$ and this is compatibile with the periodicity operator $S$. Therefore this formula defines also a class in the periodic cyclic cohomology.
Suppose that for two Fredholm modules with the same parity one has that their Chern characters define the same classes in cyclic cohomology. What can be sayed about the Fredholm modules?
