TITLE: Atiyah-Singer index theorem, pairing between K-homology and K-theory and Chern character
QUESTION [12 upvotes]: 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.

REPLY [3 votes]: No.
What you describe is purely analytic (the definitions of the groups, of the pairings, and of the Chern-Connes characters), but the Atiyah-Singer index theorem has also a topological part.
By the way, I wouldn't call what you describe an "index theorem" (because, as I said, you are missing completely the topological part). What you have is just the compatibility of the Chern-Connes characters with the pairings.