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These notes are based on a lecture course given by the first author in the Sedano Winter School on K theory held in Sedano, Spain, on January 22,27th of 2007. They aim at introducing K theory of C*algebras, equivariant K homology and KK-theory in the context of the Baum Connes conjecture.
This paper applies K-homology to solve the index problem for a class of hypoelliptic (but not elliptic) operators on contact manifolds. K-homology is the dual theory to K-theory. We explicitly calculate the K-cycle (i.e., the element in geometric K-homology) determined by any hypoelliptic Fredholm operator in the Heisenberg calculus. The index theorem of this paper precisely indicates how the analytic...
We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C*-algebras. Then we employ this result to show that the K-groups of our family of...
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