Schedule for: 19w5073 - Topology and Measure in Dynamics and Operator Algebras

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

It can be argued that the Lin-Dadarlat-Eilers stable uniqueness theorem is one of the main driving forces behind several recent landmark results related to the classification program for nuclear C*-algebras. In a nutshell, the theorem strengthens the Cuntz picture of bivariant K-theory, and translates a KK-theoretic assumption into a rather strong statement involving (stable) asymptotic unitary equivalence of *-homomorphisms, which becomes immensely useful for extracting the role of K-theory in classification. In this talk I will present a generalization of the stable uniqueness theorem to the setting of C*-dynamical systems over a given locally compact group. I will also explain why this should be expected to be important in the context of classifying C*-dynamics up to cocycle conjugacy. This is joint work with James Gabe.

We report on a joint project with J. Gabe, C. Schafhauser, A. Tikuisis, and S. White that develops a new approach to the classification of C*-algebras and the *-homomorphisms between them.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Consider a free and minimal topological dynamical system and the corresponding crossed-product C*-algebra. We show that, under an assumption of Rokhlin property and an assumption of Cuntz comparison of open sets, the radius of comparison of the C*-algebra is at most the half of the mean (topological) dimension of the dynamical system. Moreover, still under these two assumptions, if the mean dimension is zero, then the C*-algebra is Jiang-Su stable or finite dimensional. This includes all free and minimal actions by Z^d and some other systems.

A group and the corresponding groupoid of germs are naturally associated with every locally expanding self-covering of a compact space (such as, for example, hyperbolic complex rational functions when restricted to their Julia sets). We will show that the dynamical asymptotic dimension of this groupoid is equal to the topological (covering) dimension of the space. (In particular, it is finite.) The K-theory of the corresponding C*-algebras can be explicitly computed for the case of complex rational functions. Such groupoids are particular cases of groupoids naturally associated with a Ruelle-Smale dynamical system. One can show that in this general case the dynamical asymptotic dimension is also finite, but a more precise statement is still a conjecture.

In general orbit equivalence between free measure-preserving actions of countably infinite groups on standard probability measure spaces may not preserve entropy. A few years ago Tim Austin showed that integrable orbit equivalence between actions of finitely generated amenable groups does preserve entropy. I will introduce a notion of Shannon orbit equivalence, weaker than integrable orbit equivalence, and a property SC for actions. The Shannon orbit equivalence between actions of sofic groups with the property SC preserve the maximal sofic entropy. If a group G has a w-normal subgroup H such that H is amenable and not locally virtually cyclic, then every action of G has the property SC. In particular, if two Bernoulli shifts of such a sofic group are Shannon orbit equivalent, then they are conjugate. This is joint work with David Kerr.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Higher-rank graphs (k-graphs) are a combinatorial model for C*-algebras; indeed, much of the structure of k-graph C*-algebras is encoded in the underlying combinatorial data of the k-graph. However, different k-graphs can give rise to isomorphic or Morita equivalent C*-algebras. In this talk, we present several ways to modify the structure of a k-graph which preserve the Morita equivalence class of the associated C*-algebra. Our constructions are inspired by the analogous work for graph C*-algebras of Bates and Pask, as well as by the textile system approach to describing k-graphs. This is joint work with C. Eckhardt, K. Fieldhouse, D. Gent, I. Gonzales, and D. Pask.

I will discuss the new concepts of complemented partitions of unity (CPoU) and uniform property gamma, introduced in joint work with Castillejos, Tikuisis, White and Winter.

An $L^p$-operator algebra is a Banach algebra that admits an isometric representation on some $L^p$-space ($p$ not 2). Given such an algebra $A$, we show that it contains a unique maximal sub-C*-algebra, which we call its C*-core. The C*-core is automatically abelian, and its spectrum is naturally equipped with an inverse semigroup of partial homeomorphisms. We call the associated groupoid of germs the Weyl groupoid of $A$.

We introduce several equivalence relations on the set of subgroups of a countable group G, defined in terms of the quasi-regular representations, and present some rigidity results in terms of those equivalence relations for certain classes of subgroups. Furthermore, we give some results concerning the ideal structure of the C*-algebras generated by the quasi-regular representations. This is joint work with Bachir Bekka.

I’ll discuss developments in computing nuclear dimension in the presence of enough $O_\infty$ stability.

I will discuss applications of complemented partitions of unity and uniform property $\Gamma$ in the Toms-Winter conjecture and the structure theory of uniform tracial completions.

We give conditions on a potentially non-Hausdorff étale groupoid which guarantee that its associated C*-algebra is simple. As a key source of examples, work of Nekrashevych and Exel-Pardo describes a class of C*-algebras arising from the action of a group on a finite alphabet (or more generally, a finite graph). The above authors described these as groupoid C*-algebras and gave conditions which guaranteed their simplicity, usually starting from assumptions which imply the groupoid is Hausdorff. These groupoids need not be Hausdorff, notably for the self-similar action associated to the Grigorchuk group, so it was an open question whether the C*-algebra of the Grigorchuk group action was simple or not. We answer this question in the affirmative. We also discuss simplicity criteria for the associated Steinberg algebras.

The small boundary property for topological dynamical systems was introduced by Lindenstrauss to construct small entropy factors. A recent work of Kerr and Szabo showed that this dynamical property is closely related to uniform property Gamma of the crossed product, hence plays an important role in classification of crossed products. In this talk we discuss how to strengthen the connection by considering a relative version of uniform property Gamma. This is a joint work with Aaron Tikuisis.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.