Titles and abstracts

Cuntz-Krieger algebras are given as universal C*-algebras based on certain combinatorial data which also define symbolic dynamical systems, and Cuntz and Krieger showed in their celebrated 1980 paper that the C*-algebras adorned with relevant additional structure are invariants of the symbolic dynamics in a natural way. The founding fathers offer little speculation to that effect in the paper, but one may naturally ask whether the opposite implications hold, leading to the conclusion that the C*-algebras remember the underlying dynamics. Such *rigidity results* have appeared in recent years, reflecting substantial progress in the understanding of classification and structure theory of C*-algebras since the inception of Cuntz-Krieger algebras.

I will try to provide the full picture of what is now known about rigidity, and what remains open. Assuming no prior knowledge on the dynamical side, I will emphasize the idea of a “graph move” which plays a key role in my own work, and in several cases can be used to provide a framework for a better understanding of the deep and beautiful ideas of the many contributors to this field.

David Kerr (University of Münster)- Dynamical tilings and Z-stability

I will discuss the topological-dynamical tiling property of almost finiteness and explain its relationship to dynamical comparison, the small boundary property, and the C*-algebraic concept of Z-stability.

**Volodymyr Nekrashevych (Texas A&M University) - C*-algebras, self-similar groups, and hyperbolic dynamics**

We will discuss algebraic objects associated with hyperbolic dynamical systems: self-similar groups (and bisets) and C*-algebras (and Hilbert bimodules). The former are naturally associated with expanding covering maps (of orbifolds). The latter are also associated with them, but also with more general hyperbolic dynamical systems (e.g., so-called Smale spaces). We will discuss some of their properties, in particular, a phenomenon of hyperbolic duality, and consider some examples of computation of K-theory of the C*-algebras associated with self-similar groups.

Invited Speakers

Anna Duwenig (University of Wollongong) - The Zappa–Szép product of a Fell bundle by a groupoid

The Zappa–Szép (ZS) product originated as a generalization of the semi-direct product of groups. Keeping in mind the relationship of such semi-direct products to crossed products of C*-algebras, we define an analogue of ZS products for operator algebras: if a groupoid H acts in a sufficiently nice way on a Fell bundle B over G, we construct a new Fell bundle over the ZS product of the groupoids G and H. For this new “ZS Fell bundle”, there is a natural relationship between its representations and those representations of B and H that are covariant in an appropriate sense. Furthermore, this ZS construction lends itself to generalizations of imprimitivity theorems à la Kaliszewski–Muhly–Quigg–Williams.

This talk is based on joint work with Boyu Li (University of Waterloo).

Becky Armstrong (University of Münster) - Twisted Steinberg algebras

Steinberg algebras are a purely algebraic analogue of groupoid C*-algebras that generalise both Leavitt path algebras and Kumjian–Pask algebras. Since their introduction in 2010, Steinberg algebras have served as a bridge to facilitate the transfer of various concepts and techniques between the algebraic and analytic settings. Given the importance of twisted groupoid C*-algebras in C*-algebraic research (for instance, Renault’s theorem that every C*-algebra with a Cartan subalgebra is a twisted groupoid C*-algebra), a natural question to ask is whether Steinberg algebras can be twisted in an analogous way. In this talk I will introduce twisted Steinberg algebras and various notions of algebraic Cartan pairs and will explain the important role these play in algebraic reconstruction theorems. (This is joint work with Gilles G. de Castro, Lisa Orloff Clark, Kristin Courtney, Ying-Fen Lin, Kathryn McCormick, Jacqui Ramagge, Aidan Sims, and Benjamin Steinberg.)

Camila Sehnem (University of Waterloo) - C-envelopes of tensor algebras of product systems and semigroup C-algebras

The C*-envelope of an operator algebra C is the smallest C*-algebra generated by a completely isometric copy of C . Muhly and Solel showed that the C*-envelope of the tensor algebra T (E )⁺ of a correspondence E is canonically isomorphic to the Cuntz--Pimsner algebra OE under certain assumptions on E , which were later removed by Katsoulis and Kribs. In this talk I will report on a generalisation of this result for a product system of correspondences over a submonoid of a group. I will discuss in more detail applications in the setting of Toeplitz algebras of semigroups and their boundary quotients.

Brita Nucinkis (Royal Holloway, University of London) - On generalisations of Thompson’s group V

In the 1960s R. Thompson defined three groups, F, T and V, all of which have shown to have surprising properties. For example, V was the first example finitely presented infinite simple group. In the early 2000s Nekrashevych and Matui realised these as topological full groups of some Cuntz-Algebras.

In this talk I will give an overview over some generalisations of V, especially those that are automorphism groups of Cantor-algebras. As it turns out, these can also be realised as topological full groups of certain groupoids. This is ongoing work with A. Vdovina.

Eduardo Scarparo (University of Glasgow) - Almost finiteness and the AH conjecture for Cantor minimal dihedral systems

Given an ample groupoid G, Matui made two conjectures about its homology: the HK conjecture relates it with the K-theory of the reduced C*-algebra of G, and the AH conjecture relates the low-dimensional homology with the abelianization of the topological full group of G. In connection with the AH conjecture, Matui also introduced a regularity property for ample groupoids called "almost finiteness".

The first counterexample to the HK conjecture came from a non-free minimal action of the infinite dihedral group on the Cantor set. In this talk, we will show that Cantor minimal dihedral systems are almost finite and satisfy the AH conjecture. This is based on joint work with Eduard Ortega.

Shirly Geffen (KU Leuven) - Z-stability and dynamical comparison

Z-stability of a crossed product A⋊G for an outer action of an amenable group G on a classifiable C*-algebra A is conjectured to hold in general, and is established under additional assumptions on the action. Extending the known cases, we prove, under certain freeness conditions on the induced action on the space of extreme traces of A, that A⋊G is classifiable (modulo the UCT). Similar to recent classification results in the commutative setting, we use the new notion of dynamical comparison, in its noncommutative version (due to Bosa-Perera-Zacharias-Wu).

This is joint work with Eusebio Gardella, Petr Naryshkin, and Andrea Vaccaro.

Christian Bönicke (Newcastle University) - Homology and K-theory of ample groupoids

Étale groupoids form a convenient framework to study various kinds of (topological) dynamical systems. They also play an increasingly important role in the abstract theory of C*-algebras and their classification. This talk will focus on homological and K-theoretical invariants of groupoids. In particular, I will give an overview on some recent progress on a conjecture posed by Hiroki Matui that relates the K-theory of a groupoid C*-algebra to the homology of the groupoid itself.

Karen Strung (Czech Academy of the Sciences) - Crossed products of commutative C*-algebras by Hilbert bimodules

Given a Hibert C(X)-bimodule, one can construct a crossed product which generalizes the usual crossed product by a homeomorphism. When the Hilbert bimodule comes from a minimal homeomorphism of mean dimension zero twisted by a line bundle, the resulting C*-algebra absorbs the Jiang–Su algebra. We can also classify their orbit-breaking subalgebras. In this case we furthermore have a nice description of them as locally subhomogeneous C*-algebras. With no assumptions on the mean dimension, the tensor product of two or more such C*-algebras also absorbs the Jiang–Su algebra. This entails their classification by the Elliott Invariant. This is based on joint work with Jeong and Forough as well as work with Adamo, Archey, Forough, Georgescu, Jeong, Viola.

Contributed Talks

Are Austad (University of Southern Denmark) - C*-uniqueness results for groupoids

Given a reduced ∗-algebra A admitting an enveloping C*-algebra, there are in general many non-isomorphic C*-completions of A. However, if the enveloping C*-algebra is the unique C*-completion of A up to isomorphism, we say A is C*-unique. In this talk we consider the question of C*-uniqueness for L¹ (G), where G is a second-countable locally compact Hausdorff étale groupoid. The property of C*-uniqueness is strictly stronger than that of weak containment, both for groups and groupoids. We find sufficient conditions for C*-uniqueness by considering the analogous question for the (discrete) groups appearing in the fibers of the interior of the isotropy subgroupoid ISO(G)^◦.

Ali Raad (KU Leuven) - Inductive Limits of Noncommutative Cartan Inclusions

We prove that an inductive limit of aperiodic noncommutative Cartan inclusions is a noncommutative Cartan inclusion whenever the connecting maps are injective, preserve normalisers and entwine conditional expectations. Consequently, we subsume the case where the building block Cartan subalgebras are commutative and provide a purely C*- algebraic proof of a theorem of Xin Li. This is based on joint work with Ralf Meyer and Jonathan Taylor.

Jonathan Taylor (University of Göttingen) - C*-inclusions arising from twists over effective groupoids

In 2008, Renault defined commutative Cartan pairs and showed that such pairs arise from twists over effective, étale, locally compact, Hausdorff, second countable groupoids. In particular, for such a twist, one considers the algebra of continuous functions on the unit space of the groupoid and includes it into the reduced twisted groupoid C*-algebra. Conversely, from a (commutative) Cartan pair, Renault constructs the Weyl groupoid and twist, and shows that this is unique among twists over groupoids with these adjectives giving rise to this C*-inclusion. However, if the groupoid is not globally Hausdorff, but rather only has Hausdorff unit space, the reduced twisted groupoid C*-algebra is in a certain sense “too large”. Kwaśniewski and Meyer define the essential groupoid C*-algebra to compensate for this, and we show inclusions with the essential groupoid C*-algebra give rise to Cartan-like pairs. We show that by weakening the definition of Cartan pairs (by removing certain restrictions on the conditional expectation), we gain a similar bijective correspondence between twists over effective étale groupoids with locally compact Hausdorff unit space and such essential Cartan pairs. The construction of the underlying Weyl groupoid and twist is the same as Renault defines, and we show that this is again unique among groupoids with this set of adjectives giving this C*-pair.

Jeremias Epperlein (Universitat Passau) - A new construction of Wagoner's complexes

Wagoner in the late 80s gave a construction which realizes the automorphism group of a subshift of finite type as the fundamental group of a certain CW-complex associated to the subshift. This classifying space was used in some of the deepest classification results for SFTs, e.g. in the disprove of Williams conjecture by Kim and Roush. We will sketch a new construction of this complex as the nerve of the groupoid of topological conjugacies between SFTs with regard to the generating set of elementary conjugacies. This significantly simplifies the construction and allows for generalizations to other groupoids.

Alex Mundey (University of Wollongong) - Iterated function systems with fat overlaps and equilibrium states for their C*-algebras

Iterated function systems (IFS) are a class of multi-map dynamical systems that lie at the heart of fractal geometry. Topological subtleties make modelling IFS dynamics with operator algebras difficult. One of the more successful C*-algebraic approaches⁠—due to Kajiwara and Watatani—uses the graph of the Hutchinson operator construct a Cuntz-Pimsner algebra. This algebra is sensitive to branches in the dynamics.

Building on the work of Izumi-Kajiwara-Watatani, we characterise the KMS-states of Kajiwara-Watatani algebras for IFS with fat overlaps. In the process we introduce a notion of Markov partition for IFS and modify a technique of Hutchinson to construct an invariant measure for the dynamics. This is ongoing work with Nicholas Seaton and Michael Mampusti.

Dimitris Gerontogiannis (Leiden University) - K-homological finiteness of Ruelle algebras

The stable and unstable Ruelle algebras are purely infinite C*-algebras from Smale spaces, considered as higher dimensional Cuntz–Krieger algebras. This talk provides a brief overview on the K-homological finiteness of Ruelle algebras exhibited at the level of bounded Fredholm modules. This finiteness condition is interesting since Ruelle algebras are purely infinite, but also because it allows to study index theory on Ruelle algebras through noncommutative geometry. This result follows from the KK-duality between the stable and unstable Ruelle algebras, and the construction of smooth stable and unstable subalgebras whose interaction encodes the fractal dimension of the underlying Smale space.

Krzysztof Swiecicki (Wrocław University of Science and Technology)- There is no equivariant coarse embedding of Lp into l p

In this talk I will present some history of embedding problems between L_p and ℓ_p spaces and present a new nonlinear result of this type. I will explain the motivation from analysis for studying equivariant coarse embeddings between Banach spaces and how they are connected to geometric group theory and algebraic topology.

David Jekel (University of California, San Diego)- Entropy for model-theoretic types in tracial W*-algebras

Voiculescu’s free entropy theory studies the "amount" of matrix approximations that exist for some tuple of operators in a tracial W*-algebra (or von Neumann algebra). The work of Jung and Hayes defined a version called 1-bounded entropy which has a closer analogy with dynamical entropy. In this work, we adapt the definition by modifying the notion of "matrix approximation" to include the model-theoretic type, which deals with formulas built from traces of polynomials with operations of supremum and infimum in auxiliary variables. We give a couple applications to the study of ultraproducts of matrix algebras, and embeddings of other tracial W*-algebras into such an ultraproduct.

Victor Wu (University of Sydney) - K-theory of crossed products associated to group actions on trees

An action of a countable discrete group on a locally finite tree induces an action of that group on a topological space called the boundary of the tree, and we can consider the reduced crossed product C*-algebra for that action. It turns out that the K-theory of such a crossed product fits neatly into a six-term exact sequence, and I will be talking about how we got there.

Tyler Schulz (University of Victoria) - Supercritical equilibrium states on a C*-algebra from number theory

The equilibrium structure of C*-dynamical systems arising from number theory have so far shared the property that there is at most one equilibrium state at temperatures above the critical point, but new considerations show that this is not always the case. I will describe the supercritical phase transition on the right Toeplitz algebra of the ax + b semigroup of the natural numbers in terms of a class of subconformal measures on the circle. The phase transition depends on the orders of points on the circle, and we provide explicit formulas for the measures using classical functions from number theory.