Scandinavian Operator Algebra Workshop (ScaOAW)

Abstract: 

The motivation comes from the spectacular breakthrough in the Elliott classification program for simple nuclear C*-algebras: the class of all separable, simple, finite nuclear dimensional C*-algebras satisfying the UCT is classified by their Elliott invariants. Shortly after, Xin Li proved that those classifiable C*-algebras have a twisted étale groupoid model (G, Σ). A natural question is which twisted étale groupoid C*-algebras have finite nuclear dimension. Very recently, Bönicke and I have extended the previous results to show that their nuclear dimensions are bounded by the dynamic asymptotic dimension of the underlying groupoid G and the covering dimension of its unit space G0, and are actually independent of Σ. The essential flaw is that dynamic asymptotic dimension cannot be consistent with nuclear dimension for simple C*-algebras because every simple C*-algebra with finite nuclear dimension has nuclear dimension either zero or one. Therefore, we (together with Liao and Winter) introduced the so-called diagonal dimension for an inclusion (D ⊆ A) of C*-algebras. In this talk, I will explain how the diagonal dimension of (C0(G0)⊆ Cr*(G,Σ)) is indeed consistent with dynamic asymptotic dimension of G and the covering dimension of G0. Moreover, we compute the diagonal dimension and the dynamic asymptotic dimension for Xin Li's groupoid model. This is joint work with Christian Bönicke, as well as Zehong Huang and Hang Wang.

Abstract: 

We consider bundles of tracial von Neumann algebras over compact Hausdorff spaces, first studied by Ozawa.  We establish C*-algebraic properties such as stable rank one, real rank zero, Dixmier averaging, and strict comparison for a class of W*-bundles that includes all locally trivial bundles.  Inspired by Popa's theorem on approximate free independence and Robert's notion of selflessness for C*-algebras, we introduce the class of selfless W*-bundles for which there exist elements in M  that are, in each fiber, approximately freely independent from a given finite subset of M  with the respect to the trace, with uniform error bounds over all the fibers.  This allows us to apply results from free probability to the bundle M , in a uniform way over the fibers.

For instance, to prove stable rank one (that invertible elements are dense), we use that any x in a von Neumann algebra can be perturbed to have trivial kernel by adding a freely independent circular operator.  In a selfless W*-bundle M , we can carry this perturbation out approximately, and argue that operators that have trivial kernel are generic, and such operators admit a unitary polar decomposition in M .

Abstract: 

The triangulated category whose objects are C*-algebras over a locally compact space X, and morphism spaces are Kasparov’s representable KK-groups, can be upgraded to a stable infinity category. The map spaces from this construction form a spectrum-valued sheaf on X. This fits into the six-functor formalism for étale groupoids G when we take X to be the nerves of G. When G is a torsion-free étale groupoid satisfying the Baum-Connes conjecture with coefficients, this correspondence leads to a rational isomorphism between the K-groups of crossed product C*-algebras A x G and the groupoid hyperhomology over G (a variation of Matui's “HK conjecture”). This specializes to the classical Chern character isomorphism if G = X, a space as a groupoid, and A = C_0(X). Based on a joint work in progress with Valerio Proietti.

Abstract: 

It is well known that Woronowics quantum SU(2), as a C*-algebra, is isomorphic to a graph C*-algebra, for a fixed directed graph of two vertex and three arrows.

This talk concerns a generalization of this result for quantum SU(3). However, higher dimensional quantized compact semi-simple Lie groups does not admit a description as a Graph-C*-algebra. Therefore, Instead of a directed graph, we use a higher dimensional generalization introduced by Pask and Kumijan, called a higher rank graph (or n-graph), and its associated C*-algebra. 

In the case of SU(3), the graph will have dimension 2.  A description of the 2-graph whose C*-algebra is isomorphic to SU(3) will be concretely given, and we’ll discuss aspects of it’s construction. We will further give an outline of what the appropriate generalization should be for quantum SU(n) (the graph will be of dimension n-1), as well as the framework and technical obstacles for  proving these types of results. This result is related to similar research by M. Matassa and R. Yuncken.

Abstract: 

Motivated by results of Ken Dykema, Mikael Rørdam, and Sorin Popa, Leonel Robert introduced the notion of selfless C*-algebras. These C*-algebras contain a copy of themselves in their ultrapower that is free from the diagonal copy with respect to a fixed state. Apart from being an interesting property on its own right, selflessness yields a number of regularity properties, such as strict comparison of positive elements (in the tracial case). In this talk I will present recent work with Felipe Flores, Mario Klisse, and Micheal Ó Cobhthaigh, in which we show that reduced free products satisfying the so called Avitzour condition, as well as certain reduced graph products, are selfless. The preprint of our article can be found on arxiv.

Abstract:  

Twenty years after the appearance of twisted spectral triples in a preprint of Connes and Moscovici, I will discuss the new framework of 'conformally generated cycles', which incorporates every known example of twisted spectral triple with nontrivial index theory, while, at the same time, extending in a well-defined way the unbounded picture of KK-theory. Distinctive features of this framework are the use of conformal factors instead of twists and the appearance of ternary rings of operators. I will also mention the notion of 'matched' operator on a Hilbert module, relying on the Pedersen ideal, which gives conformally generated cycles their dynamical quality, reminiscent of Woronowicz's C*-algebras generated by unbounded elements. In considering descent for conformally group- and quantum-group-equivariant spectral triples, we obtain a number of novel examples.

This is joint work with Adam Rennie (arXiv:2412.17220).