Relation between noncommutative geometry and functional analysis

Question

Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of operator algebras and that together with many other tools/techniques are used to study geometric or topological problems. I couldn't help but ask (perhaps naively) whether things can go the other way, i.e. using tools in noncommutative geometry to study operator algebras (or even other objects that one would associate to functional analysis, such as operator spaces etc.). If anyone knows of such an approach, I would appreciate some descriptions or references.

Answer 1

One instance where an operator theory problem was solved via "noncommutative topology" (actually it is the introduction of that viewpoint into operator theory) is the Brown-Douglas-Fillmore theory for classifying essentially normal operators. They found that, in addition to the essential spectrum, the Fredholm index was the key ingredient for classification and it can be placed in the context of a cohomology theory, called $Ext$, for $C^*$-algebras.

Another case is the classification of separable nuclear $C^*$-algebras by invariants built using $K$-theory, which is a cohomology theory that survives, more or less intact, the passage from topological space to $C^*$-algebras.

Both $K$-theory and $Ext$ are instances of $KK$-theory, a cohomology-like bivariant functor from the category of $C^*$-algebras to the category of abelian groups. It is due to Kasparov.

An introduction to BDF theory and some basic classification results can be found in $C^*$-algebras by Example by Davidson, who's an operator theorist.

The other seminal result in this direction is the Atiyah-Singer index theorem. An introduction that takes the $C^*$-algebraic route is at the link folk.uio.no/rognes/higson/Book.pdf‎. (I high recommend anything written by Nigel Higson.)

Also, there was an AMS Notices What is ... article a while back by J. Cuntz, on noncommutative topology.

As for operator spaces, they have a fairly established theory at this point and there is not much geometry/topology in it. One might call that noncommutative functional analysis as opposed to topology.

You also may want to post this to MathOverflow where there might be more noncommutative geometers around.