Abstract: We proof a homotopy invariance theorem for algebraic K-theory of locally convex algebras stabilized by Schatten ideals. This gives some insight in the possible interplay between algebra and functional analysis and allows to compute the coefficients of a certain bivariant homology theory. I report about joint work with Joachim Cuntz.

Abstract: In this talk, first we give conditions under which the product and the sum of regular operators on C*-module is a regular operator. Then, we give a simple criteria so that an unbounded operator d on a C*-module is regular and so that the operator d+d* has compact resolvant. Finally, we show that these results permit to associate an element of K-theory to the signature operator on a Lipschitz manifold with group action.

Abstract: We study the Lip-normed C*-algebras introduced by M. Rieffel, showing that the family of equivalence classes up to isomorphism preserving the Lip-seminorm is not complete w.r.t. the complete quantum Gromov-Hausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely isomorphic to any C*-algebra. Conditions ensuring the existence of a C*-structure on the limit are considered, making use of the notion of ultraproduct. More precisely, necessary and sufficient conditions are given for the existence, on the limiting operator system, of a C*-product structure inherited from the approximating C*-algebras. Such conditions can be considered as a generalisation of the $f$-Leibnitz conditions considered by Li and Kerr. Furthermore, it is shown that our conditions are not necessary for the existence of a C*-structure tout court, namely there are cases in which the limit is a C*-algebra, but the C*-structure is not inherited from the approximating C*-algebras.

Abstract: Cyclic homology and cohomology were discovered by Connes and Tsygan in the early 1980's, and since then have come to play a central role in Alain Connes' formulation of noncommutative differential geometry, representing an extension of de Rham cohomology to various categories of noncommutative algebras. The very simplest formulation of cyclic cohomology arises by considering linear functionals (abstract integrals) on noncommutative differential calculi - the analogue of the graded algebra of differential forms on a manifold.
Quantum groups also appeared in various guises from the early 1980's onwards, with the first example of a "compact quantum group" in the C*-algebraic setting being Woronowicz's "quantum SU(2)". For compact quantum groups, the appropriate differential calculi to study are covariant under the natural action of the quantum group, and "integrals" now give rise to cocycles twisted by an automorphism of the algebra. This was developed into the theory of twisted cyclic cohomology by Kustermans, Murphy and Tuset.
In joint work with Ulrich Kraehmer (Humboldt University, Berlin), we calculated twisted Hochschild and cyclic homology for the quantum SL(2) group. This calculation puts known examples of differential calculi and twisted cocycles in a general framework, and very strikingly shows that the "dimension drop" phenomenom in Hochschild homology widely encountered when passing from the classical to the quantum situation can be overcome by twisting via a specific family of automorphisms arising very naturally from the canonical Haar state on the associated compact quantum SU(2) group.
I will give a thorough introduction to all this material, and then discuss extensions to larger classes of quantum groups, in particular the q-deformed coordinate algebras of semisimple Lie groups, and compact matrix quantum groups in the sense of Woronowicz.

Abstract: We introduce the notion of the first homotopy group of a partially ordered set (poset) and show how it is connected with the net-cohomolgy of the poset. In the case that the poset is a basis for a topological space ordered under inclusion, the fundamental group of the poset and that of the underlying topological space are isomorphic. This allow us to give an answer to the existence of topological obstructions to a kind of triviality of 1-cocycles. Furthermore, we established the equivalence, in the categorical sense, of net-cohomolgy of different basis for the topological space.

Abstract: It is demonstrated how subfactor theory and modular theory control the precise "holographic" relationship between local QFT on a halfspace and non-local QFT on the boundary.

Abstract: I examine some alternative possibilities for an action functional for $\kappa$-Minkowski noncommutative-spacetime.
Action functionals present in literature are structured in such a way to reflect $kappa$-Poincaré invariance, renouncing to invariance under cyclic permutations of the arguments of the action functional.
I show the construction of a cyclic action functional by introducing an integration measure. I discuss the symmetries of the cyclicity-inducing measure. I also show that such a cyclicity-inducing measure can be derived using a map which connects the $\kappa$-Minkowski coordinates and the coordinates of a canonical spacetime, with coordinate-independent commutators.

Abstract: I shall give a brief account of Kumjian and Pask's generalisation of Robertson and Steger's construction of a higher-rank Cuntz-Krieger C*-algebra from a k-tuple of special 0-1 matrices, and show how in certain cases their K-theory can be described.

Abstract: Connes has demonstrated that for a compact spin Riemannian manifold, the geodesic metric is determined by the Dirac operator. The concepts involved in this construction may be generalized to any unital C*-algebra. The problem is then to define a good candidate for "the Dirac operator", or rather how can we, for a given unital C*-algebra, construct a "good" spectral triple ?
Based on Connes' construction of a spectral triple for the reduced C*-algebra of a discrete group with a length function, there is a natural candidate for a spectral triple for a unital AF C*-algebra.
Cristina Antonescu and I have shown that this candidate works and that the eigenvalues of the Dirac operator can be chosen quite freely without any upper limit. In particular an AF C*-algebra has only zero as a lower bound for it's dimension as a toplogical space. It is easy to demonstrate this for the Cantor Set, which is the spectrum of an Abelian AF C*-algebra, and a totally disconnected compact space.
This freedom in the choice of eigenvalues for the Dirac operator is a way of expressing that unital AF C*-algebras are totally disconnected non commutative compact spaces. We can support this point of veiw by a theorem which states that if for a certain Dirac operator on a unital C*-algebra, there is no upper limit for the eigenvalues, while keeping the eigen spaces, then the C*-algebra is AF.

Abstract: We show that for any non-exact C*-algebra, A, there is a continuous bundle, B, of C*-algebras on [0,1] with constant fibre such that A\otimes B is discontinuous at 1. This has various consequences:
1. For any non-exact C*-algebra A, infinite locally compact Hausdorff space X and limit point x₀ of X there is a continuous C*-bundle on B on X with constant fibre such that A\otimes B is discontinuous at x₀.
2. For any non-exact C*-algebra A and nonempty closed subset S of [0,1] there is a continuous C*-bundle B on [0,1] with constant fibre such that A\otimes B is discontinuous at all points of S and continuous on S^c. In particular, there is a B such that A\otimes B is discontinuous at all points of [0,1].
3. It was shown by Kirchberg and Wassermann [Operations on continuous bundles of C*-algebras. Math. Ann. 303 (1995), 677--697] that a given C*-algebra A is exact if and only if for any continuous bundle B= {B, Ň, π_n:B-->B_n} of C*-algebras over the one-point compactification Ň of N (the integers) the minimal tensor product bundle A\otimes B is continuous. The analogous condition with the base space Ň replaced by any infinite locally compact Hausdorff space X holds.
The techniques used to prove these results also yield an example of a non-exact C*-algebra A and a continuous bundle B of C*-algebras on [0,1] with constant, simple exact fibre such that A\otimes B is discontinuous on any prescribed closed subset of [0,1].

Abstract: Over the last few years, the speaker in collaboration with U. Haagerup and H. Schultz have used random matrix techniques in connection with problems in C*-algebra theory. In particular, Haagerup and the speaker proved in 2002 the existence of non-invertible elements in the extension semi-group of the reduced C*-algebra associated to the free group on two generators. This result settled in the negative a problem which had remained open since the work of J. Anderson from 1978.
In the talk I shall outline the proof of the result on the extension semi-group mentioned above, and I will discuss other results on C*-algebras which can be proved by virtue of random matrices.

Abstract: We discuss which invariant cones on infinite dimensional symmetric domains stem from an embedding into the space of bounded operators on Hilbert space.

Abstract: The construction of a C*-algebra by a groupoid was introduced by A. Connes and it plays an important role in K-theory, as well as deformation quantization. In this talk we give an overview of the results on the existence of a groupoid whose orbits are exactly the leaves of a singular foliation.

Abstract: We will prove that if a group acts cocompactly on some spaces and if the stablizers have the property of Rapid Decay with respect to the induced lengths then the group has the property of Rapid Decay.

Abstract: The principles of Quantum Mechanics and of Classical General Relativity imply Uncertainty Relations between the different spacetime coordinates of the events, which yield to a basic model of Quantum Minkowski Space, having the full (classical) Poincare' group as group of symmetries.
The formulations of interactions between quantum fields and of gauge theories on Quantum Spacetime will be discussed. The various approaches to interactions, equivalent to one another on the Minkowski background, yield to different schemes on Quantum Spacetime, with the common feature of a breakdown of Lorentz invariance due to interactions. In particular one of these schemes will be discussed and motivated, leading also to fully ultraviolet-finite theories.
Quantum fields will depend on the quantum coordinates, but, in presence of Gravity, the commutators of the coordinates might in turn depend on the quantum fields, giving rise to a quantum texture where fields and spacetime coordinates cannot be separated.

Abstract: I will talk about C*-algebras associated with actions of inverse semigroups on sets (without topology or any other structure). These C*-algebras can be described as universal C*-algebras generated by partial isometries subject to conditions given by the inverse semigroup and a Boolean algebra. I will describe how Cuntz-Krieger algebras (both for finite and infinite matrices), C*-algebras associated to shift spaces and C*-algebras of higher-rank graphs in a very natural way can be constructed as C*-algebras of actions of inverse semigroups. This allow us to make natural generalizations of these C*-algebras.

Abstract: Coactions generalize group actions in the same way that quantum groups generalize locally compact groups. Like for actions, one may define reduced and full crossed products of coactions. The two types of crossed products are related by the abstract notion of `normalization'. For regular quantum groups, it can be used to state and study generalized Takesaki-Takai duality theorems, which also appear in reduced and full versions. Normal coactions are exactly those who satisfy reduced duality. Every coaction has a normalization, i.e.~a biggest normal quotient. On the other hand, maximal coactions are by definition the ones satisfying full duality, and the `maximalization' of a coaction is the smallest maximal cover. It exist for every coaction precisely if the quantum group is `strongly regular', which means that the trivial coaction on ${\mathbb C}$ is maximal. In this situation, the method used by Echterhoff, Kalizewski and Quigg for group coactions can be applied for the construction of maximalizations.

Abstract: we give a prediction from one-loop perturbation theory of the existence of a first order phase transition in noncommutative gauge theories on the fuzzy sphere. In particular we show that the fuzzy sphere is unstable and decays when we cross to the matrix phase. This result is verified in Monte Carlo simulations and thus one can conclude that the one-loop correction of the theory becomes exact in the large N continuum limit. We define an alternative limit of the model where the fuzzy sphere phase dominates the phase space. We will also discuss the UV-IR mixing in this model and contemplate on its relation to the phase transition.

Abstract: Motivated by examples arising in the setting of Hilbert bimodules, we introduce the notions of weak symmetry and generalized determinant in the context of tensor C*-categories. We prove that every symmetric tensor C*-category with conjugates is a continuous bundle of tensor C*-categories (in a suitable sense), and give some duality results. We give examples of tensor C*-categories which are duals of non-isomorphic groups; the reason of such a fact is the non-unicity of the embedding into the category of vector bundles.

Abstract: My lecture will be an introductory lecture on noncommutative entropy. The definition of Connes and myself as it appears in the literature, is quite technical and seems ad hoc. I?ll try to show that it is a natural extension from the classical abelian case and compare it with the definitions of Voiculescu.