# Atiyah-Singer index theorem, pairing between K-homology and K-theory and Chern character

There is a general (abstract) index theorem in noncommutative geometry: you take a K-theory class and K-homology class (which is represented by a triple $(A,H,F)$) and you pair them together. This pairing is computed as an index of certain operator. There is a notion of (noncommutaive analog of) Chern character in this context which takes values in cyclic cohomology and cyclic homology. Therefore you can apply this Chern character to both: K-theory class and K-homology class obtaining two classes, in cyclic cohomology and homology. In this context you have a natural pairing between cohomology and homology. The remarkable result is that this equal to the previous pairing between K-theory and K-homology (I would like to omit all technical details involving precise definitions: detailed discussion can be found in the book "Basic Noncommutative Geometry" by Masoud Khalkhali). My question is:

Can one deduce the 'usual' Atiyah Singer theorem from this abstract index theorem?

If so, how to proceed? For example, one problem is that in the data defining K-homology cycle the operator $F$ is bounded which is not the case for order $>0$ differential operators.

• You can replace an unbounded Fredholm operator $D$ by a bounded one of the same index $D\circ(1+D^*D)^{-1/2}$. – Sebastian Goette Mar 8 '16 at 17:42