This is a survey of the relationship between C *-algebraic deformation quan-tization and the tangent groupoid in noncommutative geometry, emphasizing the role of index theory. We first explain how C *-algebraic versions of deformation quantization are related to the bivariant E-theory of Connes and Higson. With this background, we review how Weyl–Moyal quantization may be described using the ta...

We are interested in making sense of an integral over a singular space, for example the quotient of the torus T 2 by the lines of irrational slope α / ∈ Q. More generally, we would like to study the quotient space of the global unstable manifolds for a hyperbolic dynamical system. In this chapter we present an “easy” version of Connes’ noncommutative integration theory. We start by studying int...

We give a natural and complete description of Ecalle’s mould–comould formalism within a Hopf–algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes–Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogene...

In this paper, we use the KK-theory of Kasparov to prove exactness of sequences relating the K-theory of a real C∗ -algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, the complex Baum-Connes assembly map is an isomorphism if and only if the re...

For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined through an extension of Connes’ cyclic category Λ. We show that, in the case of module coalgebras, bivariant Hopf cyclic cohomology specializes to Hopf cyclic...