The backbone of Connes' approach to noncommutative geometry is the Chern--Connes pairing. By discovering the cyclic homology of an algebra and then pairing it the $K$-theory of that algebra, Connes showed how to find numerical invariants of $K$-theory classes. This is analaogous to the way in which the Chern character map gives numerical invariants of vector bundles. I would like to know what has the Chern--Connes pairing been used for? Besides its intrinsic interest as a noncommutative version of a classical phenomenon, why is it important?
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The Selberg principle in representation theory of semisimple $p$-adic groups $G$, says that the orbital integral of a coefficient of supercuspidal representation of $G$, on a hyperbolic conjugacy class in $G$, is $0$. Connes suggested around 1982 that this might be interpreted by means of the Chern-Connes pairing: indeed a coefficient of supercuspidal representation is, up to a scalar factor, an idempotent in the convolution algebra $C_c^\infty(G)$, while an orbital orbital gives a trace over $C_c^\infty(G)$; hence the interpretation of the Selberg principle by saying that certain traces give the zero pairing on the $K$-theory $K_0(C_c^\infty(G))$. This program was implemented for rank 1 groups in this paper: http://www.theta.ro/jot/archive/1986-016-002/1986-016-002-007.pdf
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I guess you named already one if the biggest points yourself: getting invariants! So more specifically, Connes obtained (with Moscovici and others) invariants of pretty ugly foliations, it helps in the classifying certain $C^*$-algebras etc. There are also more algebraic versions in deformation quantization, where people have formulated index theorems using this pairing. In some sense, it is also a new way of understanding index theorems in general, though I must admit that I'm not really an expert here...
answered Apr 29, 2011 at 13:25