Trieste, September 29 - October 2 2015

Hilbert modules and index theory


Paolo Antonini: "KK theory with R/Z coefficients, discrete groups actions and rho invariants"

Abstract: Based on a recent joint work with S. Azzali and G. Skandalis we present a model of equivariant KK theory with R and R/Z coefficients which is particularly suited to study the group actions on C^* algebras in relation to the properties of the KK_R elements defined by the group traces. Indeed a notion of free and proper action in K theory (KFP) will be introduced. It generalizes the action of the fundamental group on the universal cover of a topological space. In this case there is a strong relation with index theory: the Atiyah’s L^2 index theorem, the triviality of the Mishchenko bundle with coefficients a II_1 factor and the triviality of the group trace on the index of elliptic operators are all equivalent statements. The triviality of flat von Neumann bundles plays a crucial role in the construction of secondary invariants of type rho. For every \Gamma algebra A which is KFP and a unitary representation of \Gamma we give a canonical construction of a non—commutative rho invariant in KK^\Gamma with R/Z coefficients measuring the non triviality of the representation. It generalizes the Atiyah—Patodi—Singer invariant. We discuss examples of KFP algebras, in the torsion free case, if \Gamma satisfies the KK^\Gamma form of the Baum—Connes conjecture every algebra is KFP.


Tomasz Brzezinski: "K-theory of quantum lens and weighted projective spaces"

Abstract: In this talk we present results obtained jointly with Simon Fairfax and Wojciech Szymanski. Extending earlier results of Hong and Szymanski we interpret all quantum lens spaces as graph C*-algebras and thus design an efficient method for calculating K-groups of quantum lens spaces. Some special cases (not studied earlier) serve as an illustration. Using the freeness of the actions of the circle group on quantum lens spaces that define quantum weighted projective spaces we compute the K-groups of the latter.


Alan Carey: "Invariants from KK Theory" (Mini-Course)

Lecture 1: Review of KK Theory.

Lecture 2: (a) Models in condensed matter theory.

(b) Application to bulk-edge correspondence in the integer quantum Hall effect. <\p>

Lecture 3. Topological phases and $KKO$.


Fabio Cipriani: "Amenability of von Neumann algebras and spectrum of Dirichlet forms"

Abstract: The discussion will concern an applications of Noncommutative Potential Theory to approximation properties of von Neumann algebras: criteria for amenability of von Neumann algebras and relative amenability of finite inclusions of von Neumann algebras in terms of spectral properties of Dirichlet forms will be provided. A pivotal role will be played the notion of Connes' correspondance.


Alain Connes: "Geometry and the Quantum"

I will present in my talk the joint work with Chamseddine and Mukhanov, in which we introduce an equation on operators in Hilbert space whose solutions yield 4-dimensional manifolds and where the spectral action gives gravity coupled with the standard model. The picture that emerges is that the euclidean space-time unfolds to macroscopic size from the product of two 4-spheres of Planckian size.


Daniele Guido:"Endomorphisms, Semifinite spectral triples, and Crossed Products"


Debashish Goswami: "Quantum isometry of classical and noncommutative spaces"

Abstract: In this talk, we give an overview of the concept of quantum group of isometries in its various avatars and give some examples. In particular, we touch upon the no-go results of genuine quantum isometry (and more general quantum symmetries in some cases) for connected smooth classical manifolds and the formulation of a quantum isometry group in the purely metric space framework. The exposition will perhaps mention results obtained by joint works done with several mathematicians over a period of last five years or so: Bhowmick, Das, Joardar, Skalski, Banica, Etingof, Walton and Mandal.


Piotr M. Hajac: "Pulling back associated noncommutative vector bundles"

For any group-like element g in the C*-algebra H of a compact quantum group acting freely on a unital C*-algebra A, we can form an associated finitely generated projective module Ag over the fixed-point subalgebra B for this action. The group-like element g determines a one-dimensional representation of the compact quantum group, and Ag is the section module of the associated noncommutative line bundle. Given an H-equvariant C*-homomorphism f:A->A', we get the induced K-theory map f_*:K_0(B)->K_0(B'), where B' is the fixed-point subalgebra of A'. Using Chern-Galois theory, we prove that f_*([Ag])=[A'g]. As an application, we combine this formula with higher-rank graph C*-algebra technology and index pairing computations to prove that the noncommutative line bundles associated via the diagonal U(1)-action on the multi-pullback quantum odd-dimensional spheres are pairwise stably non-isomorphic. In particular, we conclude that the tautological line bundles over the multi-pullback quantum complex projective spaces are stably non-trivial. The same reasoning and conclusions hold for noncommutative line bundles associated to Vaksman-Soibelman quantum spheres. Based on joint work with D. Pask, A. Sims and B. Zielinski.


Michel Hilsum: "Analytic K-homology and complex spaces"

Abstract : In an article published at Acta Matematica (1976) Baum, Fulton and Mac Pherson establied a Grothendieck-Riemmann-Roch theorem for complex algebraic manifold, with value in the « topological » K-homology. In his talk we shall address the existence of this functor with instead analytic K-homology. This question entails a generalization of Block-Weinberger theorem on the birationnel invariance of the Todd class.


Jens Kaad: "Unbounded Kasparov products by differentiable Hilbert C*-modules"

Abstract: In this talk I will present a version of the unbounded Kasparov product between differentiable modules and unbounded cycles of a very general kind that includes all unbounded Kasparov modules and hence also all spectral triples. The assumptions on the differentiable module and the unbounded cycle are as minimalistic as possible and we do in particular not require that our module satisfies any kind of (smooth) projectivity condition nor do we require that our unbounded cycle is Lipschitz regular. The algebras that we work with are furthermore not required to possess a (smooth) approximate identity. If time permits I will illustrate the theory by showing how to restrict a spectral triple to a hereditary subalgebra generated by countably many elements (e.g. a countable union of open intervals of the real line).


Bram Mesland: "Shift-tail equivalence and the extension class of certain Cuntz-Pimsner algebras"

Abstract: In this talk I will discuss how the extension defining the Cuntz-Pimsner algebra of a C*-bimodule (satisfying some mild technical assumptions) can be represented by an explicit unbounded KK-cycle. The construction is inspired by the case of Cuntz-Krieger algebras, in which the underlying shift dynamics encodes the relevant K-homological information. In earlier work of Rennie-Robertson-Sims a bounded representative of the extension class is defined on a C*-module over the base. By creating an abstract "dynamical picture" of this C*-module, its fine structure can be uncovered, leading to a selfadjoint regular operator representing the extension class.This is joint work with M.Goffeng and A.Rennie.


Ryszard Nest:"Non-commutative differential calculus" (Mini-Course)

Abstract: The term "non-commutative calculus" stands for the theory that generalizes classical algebraic structures arising in differential calculus on manifolds to make them valid for any associative algebra (or, more generally, any differential graded category) instead of the algebra of functions on a manifold. The role of differential forms and multi-vector fields in this new theory is played by the Hochschild complexes of our algebra. In these lectures we will explain basic notions and properties of the calculus and give some applications, in particular to index theorems.


Paolo Piazza: "Index, eta and rho invarians on stratified spaces"

Abstract: Primary (i.e. index theoretic) and secondary invariants of Dirac-type operators have played a central role in the applications of non commutative geometry to the study of geometric properties of smooth manifolds. In this talk I will tackle analogous problems on stratified spaces, with emphasis on Witt spaces. This talk is based on joint work with Albin, Leichtnam and Mazzeo and also on a very recent work (still in progress) with Boris Vertman.


Martin Schlichenmaier: "Some naturally defined star products for Kähler manifolds"

Abstract: We give for the Kähler manifold case an overview of the constructions of some naturally defined star products. In particular, the Berezin-Toeplitz, Berezin, Geometric Quantization,Bordemann-Waldmann, and Karbegov standard star productare introduced. With the exception of the Geometric Quantization case they are of separation of variables type.The classifying Karabegov forms and the Deligne-Fedosov classes are given. Besides the Bordemann-Waldmann star product they are all equivalent. We recall that the Berezin-Toeplitz quantization also gives a continous field of C*-algebras.


Stéphane Vassout: "Convolution of distributions on Lie groupoids"

Abstract: This is joint work with Jean-Marie Lescure and Dominique Manchon. We extend the convolution product on a Lie groupoid G to a large class of distributions. We obtain a convolution algebra and show that $G$-operators are all convolution operators. We explain how the symplectic groupoid T^*G of Costes-Dazord-Weinstein appears naturally when one analyses the wave front set of the convolution of two distributions on $G$. Following this idea, we apply the Hörmander's theory of Lagrangian distributions to develop a calculus of Fourier integral operators on Lie groupoids.


Andrzej Zuk: "L^2 Betti Numbers"