Type III Noncommutative Geometry and KK-theory

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Noncommutative Geometry seeks to analyze C*-algebras associated to various geometric situations, or situations in which one has a dynamical system, like a complicated group action by symmetries of a geometric space, by adapting the methods of geometric analysis on manifolds to work for C*-algebras. The broad idea is that to many of these situations we know how to construct a C*-algebra, and this in turn can be analyzed topologically (as if it were a space), using K-theory, and also geometrically, using the idea of a `spectral pair', consisting of a representation of the C*-algebra on a Hilbert space, and an unbounded operator D, playing the role of the Dirac operator on a manifold, in the classical case. A spectral pair produces a map on K-theory for which the Local Index Formula of A. Connes and co-authors provides a formula. This formula describes the K-theory map in terms of residues of certain zeta functions associated to the triple, in an extremely interesting `local', highly geometric manner, philosophically analogous to the way one integrates a differential form over a smooth manifold. For a spectral pair one requires that the operator D commutes with the C*-algebra A up to lower order terms, in a certain sense. But many important situations, like the boundary action of a hyperbolic group, produce C*-algebras with a kind of fractal nature (they are purely infinite) for which this notion is unsuitable, because spectral pairs properly defined induce densely defined traces, and these examples admit no traces. In this Proposal we aim to follow a more recent idea of A. Connes for rectifying this: we aim to use some ideas from quantum statistical mechanics to study a variation of the idea of a spectral pair, to now allow two actions of A on the Hilbert space H, one only defined for a dense subalgebra of A, but the operator is now only required to be equivariant as a map from H with the original action, to H with the twisted action, up to lower order operators. It turns out that this twisting of the definition has no effect cohomologically (on the Chern character), but the obstruction (failure of traces to exist for purely infinite algebras) to extending `integration' no longer exists, because instead, it becomes a kind of twisted integration, corresponding to a KMS state, a concept from quantum thermodynamics -- KMS states in these examples do exist, and there is an extremely interesting and rich theory of them. My goal is to construct twisted spectral triples in connection with boundary actions of hyperbolic groups, systems I have already studied extensively, and in several other examples and families of examples, to connect them to K-theory, study the corresponding index maps, and more broadly, investigate the connection between KMS states and K-theory which seems to be implied by the framework of twisted spectral triples.