Back in the late 1990’s, Shuzhou Wang initiated the study of quantum symmetry groups of finite spaces using the language of operator algebras and non-commutative geometry. In stark contrast to the case of ordinary symmetry groups of finite structures (e.g., finite sets, graphs, metric spaces and so on) - which are always finite groups - it turns out that these same finite structures admit quantum symmetry groups which can be quite large, unwieldy, and highly interesting as operator algebraic objects. In this talk I will survey some recent developments in our understanding of quantum symmetry groups of finite graphs and also of finite dimensional C*-algebras. I will try and emphasize how recent ideas from quantum information theory have proved to be quite useful in studying the operator algebraic structure of these quantum groups.

We pull back paradoxical dynamical systems (e.g. hyperbolic groups acting on their Gromov boundary), to paradoxical decompositions of the acting group itself. This allows us to show that whenever such groups admit a minimal amenable topologically free action on a compact Hausdorff space, the system has dynamical comparison and the attached crossed product is a purely infinite classifiable C*-algebra.
This is joint work with Eusebio Gardella, Julian Kranz, and Petr Naryshkin.

A (non-selfadjoint) operator algebra is said to be residually finite dimensional (RFD) if it embeds into a product of matrix algebras. This notion is well studied in the context of C*-algebras. In particular, a theorem of Exel and Loring characterizes RFD C*-algebras in terms of the state space and in terms of a finite-dimensional approximation property for representations.
I will talk about a non-selfadjoint version of the Exel-Loring theorem. Moreover, I will discuss how this is related to a question of Clouatre, Dor-On and Ramsay about the RFD property of the maximal C*-algebra of a non-selfadjoint operator algebra.

Given an irreducible lattice Γ in the product of higher rank simple Lie groups, we prove: (i) every Γ-invariant von Neumann subalgebra of L(Γ) is generated by a normal subgroup; and (ii) given a non-amenable unitary representation π of Γ, every Γ-equivariant conditional expectation on C^*_π(Γ) is the canonical conditional expectation onto the C*-subalgebra generated by a normal subgroup. This is joint work with Nikolaos Panagopoulos.

I will discuss recent work characterizing the ideal intersection property for essential C*-algebras of étale groupoids with locally compact Hausdorff space of units. For Hausdorff groupoids, this C*-algebra coincides with the reduced C*-algebra. In the minimal case, the ideal intersection property is equivalent to simplicity, so as a consequence we obtain a characterization of étale groupoids that are "C*-simple".
This is joint work with Sam Kim, Xin Li, Sven Raum and Dan Ursu.

Tileability conditions in ergodic theory and topological dynamics have long been associated, via the crossed product construction, with finite-dimensional approximation in operator algebras. I will survey recent progress in understanding the scope of this relationship in the context of actions of amenable groups on compact spaces and the C*-algebraic property of \(\mathcal{Z}\)-stability.

After a brief introduction to C*-dynamical systems from number theory I will give a couple of examples to illustrate how their equilibrium and symmetries can encode central questions in mathematics. I will then focus on recent joint work with Astrid an Huef and Iain Raeburn, where we study a system based on the right Toeplitz algebra of the ax+b semigroup of the natural numbers. We describe the ideal structure, we show that the crystalline phases -low temperature equilibrium- resemble those of the left Toeplitz algebra, and we also establish that something unexpected happens at critical temperature. To finish, I will briefly discuss current joint work with Tyler Schulz, in which we show that the right Toeplitz system has a very interesting supercritical phase transition.

Factorizable quantum channels, introduced by C. Anantharaman-Delaroche within the framework of operator algebras, have recently found important applications in the analysis of quantum information theory, revealing new infinite dimensional phenomena, and leading also to reformulations of the celebrated Connes Embedding Problem. In recent work with M. Rørdam, we show that (infinite dimensional) von Neumann algebras are needed to describe such channels. Along the way, I will explain some of the several facets of the Connes Embedding Problem, with particular emphasis on the related interplay between operator algebras and quantum information theory.

A standard tool in the study of hyperbolic dynamical systems is the notion of a Markov partition, which allows a complicated topological system to be coded by combinatorial one. The simplest example of this is decimal (or binary) expansion of real numbers. In this talk, I will show how the process can be 'reverse engineered': starting from a shift of finite type and the idea of binary expansion, one can produce interesting hyperbolic systems. I will also discuss the implications for operator algebras associated with such systems. No prior knowledge of dynamics will be necessary.

There are several naturally occurring interesting examples of inclusions of simple C*-algebras with the property that all intermediate C*-algebras likewise are simple. Moreover, in many cases one even has a Galois type classification of intermediate C*-algebras of such inclusions. By analogy with von Neumann algebras, we refer to such inclusions as being C*-irreducible. We give an intrinsic characterization of C*-irreducible inclusions, and use this characterization to exhibit (and revisit) such inclusions, both known ones and new ones, arising from groups and dynamical systems. By a theorem of Popa, an inclusion of II₁-factors is C*-irreducible if and only if it is irreducible with finite Jones index. In a recent joint work with Echterhoff we show when inclusions of the form \(A^H \subseteq A \rtimes G\) are C*-irreducible, where G and H are groups acting on a C*-algebra A, and use this to exhibit new C*-irreducible inclusions with interesting properties.

Many properties of a C*-algebra A can be witnessed in von Neumann algebra completions of A. When A has a unique trace, it is naturally to consider the tracial von Neumann algebra completion of A given by the strong closure of A in the GNS representation of the trace. For C*-algebras with several traces, it is sometimes necessary to consider the tracial von Neumann algebra completions of A with respect to each of the traces and to study these von Neumann algebras in a uniform way over the trace space of A. These “uniform tracial completions” have become an important tool in work on the Toms-Winter Conjecture in recent years.

I will talk about crossed products of commutative C*-algebras by Hilbert bimodules, a special case of the Cuntz–Pimsner construction. When the Hilbert bimodule comes from a 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 inductive limits of 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.

Finite nuclear dimension, Z-stability, and strict comparison are regularity properties which play a decisive role in the structure and classification theory of simple nuclear C*-algebras.
In this talk I will give an overview of how these properties can be interpreted at the level of topological dynamical systems, and how the two viewpoints can be unified in the presence of C*-diagonals. For the latter, I will focus in particular on the notion of diagonal dimension, developed in joint work with Kang Li and Hung-Chang Liao.

Given a countable group G and a base space X, the Bernoulli action is the translation action of G on the product space X^G, equipped with a product of probability measures μ_g on X. I will present a survey of recent results on the ergodicity and Krieger type of such nonsingular Bernoulli actions. I will in particular present a joint work with Tey Berendschot providing a complete answer when G=Z is the group of integers. It turns out that most, but not all injective factors can be written as the crossed product by a nonsingular Bernoulli action of Z, since we prove that the associated Krieger flow must satisfy a divisibility property.

Cuntz-Pimsner algebras were introduced by Pimsner in the '90s, as generalization of both Cuntz-Krieger algebras and crossed products by the integers. The underlying mathematical object behind Pimsner's construction is a C*-correspondence over a C*-algebra. Of particular interest are Cuntz-Pimsner algebras arising from full, minimal, non-periodic, and finitely generated projective C*-correspondence over commutative C*-algebras. A large class of such examples is obtained by considering the set Γ(V,α) of continuous sections of a complex vector bundle V on a compact metric space X, where left multiplication is given by a twist by a homeomorphism α:X→X. The talk will focus on the structural properties and classification of Cuntz-Pimsner algebras associated with this class of C*-correspondences, in the case when α is a minimal homeomorphism.
In the case of crossed products by minimal homeomorphisms, the orbit-breaking subalgebra, defined by I. Putnam, is a large subalgebra in the sense of N. C. Phillips. Orbit-breaking subalgebras can also be defined in the context of Cunts-Pimsner algebras associated to correspondences of the form Γ(V,α). We show that when V is a line bundle, the orbit-breaking subalgebra is a centrally large subalgebra. Moreover, when X has finite covering dimension, the orbit-breaking subalgebra is classifiable. This is joint work with M. S. Adamo, D. Archey, M. Georgescu, M. Forough, J. A Jeong, and K. Strung.

A self-similar graph is a pair (P,E), where P is a semigroup acting on a directed graph E self-similarly. We investigate the Nica-covariant representations of (P,E) and the associated operator algebras. For a class of self-similar graphs, we show that every Nica-covariant representation of (P,E) is a direct sum of four special ones. Then we explicitly describe the dilation theory for this class. This facilitates us to find the C*-envelope of the `tensor algebra' of (P,E).
This is based on ongoing joint work with Boyu Li.

A landmark result for the tracial structure of reduced C*-algebras of discrete groups is a group dynamical description of the unique trace property. It states that the unique trace property is equivalent to faithfulness of the Furstenberg boundary action, and was achieved by Breillard, Kalantar, Kennedy, and Ozawa '17. A related characterization of C*-simplicity, achieved by Kalantar and Kennedy '14, sheds light on the relationship between C*-simplicity and the unique trace property. These characterizations show that the unique trace property is implied by C*-simplicity. We will talk about a generalization of these results for discrete quantum groups, which entails a discussion on quantum subgroups, and a duality result for (co)-amenability of quantum subgroups.

The radius of comparison, introduced by Toms, is a numerical invariant for unital C*-algebras which extends the theory of covering dimension to non-commutative spaces. The dynamical Cuntz semigroup and a dynamical version of strict comparison, inspired by comparison for topological dynamics, have been recently introduced by Bosa, Perera, Wu, and Zacharias in work in preparation to study an extension of Kerr’s notion of almost finiteness to actions of discrete groups on non-commutative C*-algebras. In this talk, I will present work in progress concerning some preliminary results related to a radius of comparison associated with the dynamical Cuntz semigroup for group actions on C*-algebras which are not Jiang-Su stable. Part of the talk is joint work with N. Christopher Phillips.

Each algebraic semigroup action naturally gives rise to a concrete C*-algebra. I will explain a method for finding groupoid models for such C*-algebras, and then present results characterizing certain properties of these groupoids in terms of the initial algebraic semigroup action. As an application, we obtain structural results for the C*-algebras arising from several large classes of algebraic semigroup actions. This is joint work with Xin Li (University of Glasgow).

The program to classify certain simple C*-algebras has now reached a very advanced state, for complex C*-algebras. The situation for real C*-algebras is much less clear. We will give a survey of structure and classification results for real C*-algebras.

In this talk, I will consider a special class of singular foliations, in the sense of Androulidakis and Skandalis. There will only be one singular leaf, a hypersurface. It turns out the holonomy around a loop in the singular leaf makes sense at the level of jets. The resulting holonomy data can be used to give a complete classification of these foliations. The reduction of the holonomy groupoid to a small transversal is an interesting blowup space and, in the transversely-oriented case, can be used to define the fundamental class of the foliation in K-theory.

Let A be a separable (not necessarily unital) simple C*-algebra with strict comparison. We show that if A has tracial approximate oscillation zero then A has stable rank one and the canonical map Γ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds.
As a by-product, we find that a separable simple C*-algebra, which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map Γ is surjective.
This is a joint work with Huaxin Lin.

In this talk I will discuss anomalous actions on simple C*-algebras. I will start by introducing the notion of an anomalous action and talk about the existence question, in this part I will discuss K-theoretic obstructions that occur, one that follows from considering the algebraic K1 group and one that appears from considering the K0 group. I will also shortly discuss the classification question of these.

The dual of an operator space is naturally an operator space, but when is the dual of a (unital) operator system again an operator system? The dual space has both matrix norm and order structure, but usually lacks an order unit. Call an operator system "dualizable" if its dual embeds into B(H).
Recently C.K. Ng obtained a nice theory of dualizability for operator systems, but it requires us to permit non-unital operator systems. Kennedy, Kim, and Manor showed that non-unital operator systems are categorically dual to pointed noncommutative convex sets. I will give a crash course on non-unital operator systems, and re-interpret dualizability in terms of geometric conditions using the language of noncommutative convexity. This is joint work with Matthew Kennedy and Nicholas Manor.

First I will review the concept of Cu-nuclear and some of its properties. Then I will review recent work by Toms where he constructed homotopies of positive elements with constant Cuntz class for C*-algebras which are unital, separable, exact, approximately divisible and of real rank zero. Building on Toms's result, I argue that a similar statement is true for simple AI-algebras. This last statement is a work in progress.

From the AdS/CFT correspondence, we have a holographic isometric map arising between the local operator algebras of the bulk theory and the boundary conformal field theory. I will explain how operator algebras can naturally be used for understanding spacetime theories in this physical context to unveil some structures of quantum gravity. In particular, I will focus on building the formalism on the bulk reconstruction from the boundary operators to the bulk operators and explain how quantum extremal surfaces aid in studying the relative entropy of the bulk and the boundary. I will further describe how we can understand the formulation in low-dimensions to describe the topology changes of the bulk.

Quantum Markov Semigroups (QMSs) model time evolution of open quantum systems under dissipative interactions with their environments. Modified logarithmic-Sobolev inequalities (MLSIs) lower bound exponential relative entropy decay rates to invariant subspaces. Recent developments have shown a tensor-stable version of MLSI known as complete MLSI (CMLSI) for all finite-dimensional QMSs having a form of detailed balance. In the absence of detailed balance, many questions have remained open. I show finite-dimensional counterexamples to universal (C)MLSI. Furthermore, I examine a surprising conflict between semigroup-induced decay and generalizations of the quantum Zeno effect: when dissipative time evolution co-occurs with Hamiltonian time evolution, the overall decay rate may depend inversely on that of the dissipative process. I prove upper and lower bounds on decay rates in these circumstances.

A self-similar graph encodes the intertwining relations between a directed graph and a semigroup. We consider covariant representations of these self-similar graphs and derive their Wold decompositions. This generalize an earlier result on the Wold decomposition of Nica covariant representations of the odometer semigroups. We will also present several examples using atomic representations. This is a joint work with Dilian Yang.

Groupoid C*-algebras and twisted groupoid C*-algebras were introduced by Renault in the late ’70. Twisted groupoid C*-algebras have since proved extremely important in the study of structural properties for large classes of C*-algebras. Moreover, Kumjian and Renault developed the C*-algebraic theory of Cartan pairs which arise from twisted groupoid algebras of effective etale groupoids, and the groupoid and twist are uniquely (up to equivalence) determined. I will introduce a purely algebraic analogue of Cartan pairs and give a correspondence between such pairs and twists over effective etale groupoids.
It is a joint work with Becky Armstrong, Gilles G. de Castro, Lisa Orloff Clark, Kristin Courtney, Kathryn McCormick, Jacqui Ramagge, Aidan Sims, and Benjamin Steinberg.

The quantum graphs are a non-commutative analogue of classical graphs and recently developed in the interactions between theories of operator algebras, quantum information, non-commutative geometry, quantum groups, etc. It is well-known that the spectrum of the adjacency matrix can characterize some properties of a (regular) classical graph, for example, connectedness, bipartiteness, and expanders. It is natural to expect that quantum graphs have similar characterizations, and indeed Ganesan (2021) shows that such a spectral approach is valid for the chromatic numbers of quantum graphs. Similarly to the classical case, the degree of a regular quantum graph is shown to be the spectral radius of the adjacency matrix. Thus it makes sense to consider the behavior of the spectrum in [-d,d] for d-regular undirected quantum graphs. We introduce bipartiteness and connectedness for quantum graphs in terms of graph homomorphisms, and we give their spectral characterizations for regular quantum graphs.

Consider a dynamical system, and let us study the structure of the corresponding crossed product C*-algebra, in particular on the classifiability, comparison, and stable rank. More precisely, let us introduce a uniform Rokhlin property and a relative comparison property (these two properties hold for all free and minimal Z^d actions). With these two properties, the crossed product C*-algebra is shown to always have stable rank one, to satisfy the Toms-Winter conjecture, and that the comparison radius is dominated by half of the mean dimension of the dynamical system.

Given two separable C* algebras B and D and an ideal I of D. We can equip the space of -homomorphism from B to D/I, Hom (B, D/I), with the point-norm topology. B is l-open (or l-closed) if for any separable C​ algebras D and Ideal I of D, the set of *-homomorphism from B to D/I, which lifts to *-homomorphism from B to D, is always open (or closed) in Hom (A, B/I). Bruce Blackadar showed that semiprojective C*-algebras are l-open and l-closed. In this talk, we will characterize l-open C*-algebras and deduce that l-open C* algebras are l-closed as conjectured by Blackadar. Moreover, we will show that the notion of l-open and semiprojectivity coincide in the category of commutative unital C* algebras.

In this talk I will review some recent progress on the semiclassical analysis of Schroedinger operators on noncommutative tori. Results include semiclassical Weyl’s laws and analogues of the Lieb-Thirring inequalities on noncommutative tori. This partly based on joint work with Edward McDonald.

n joint work with Martin Mathieu (Queen's University Belfast), we study relative cohomological dimension for operator modules over operator algebras. We give an overview on results for two important classes of operator space modules. Although these classes appear quite different, we show that the conditions for having associated cohomological dimension equal to zero coincide. The main new and unifying tool is the use of exact structures on categories of operator modules.

Zero-range processes are interacting particle systems where particles hop between the lattice sites with rates that depend solely on the number of particles of the departure site. The behaviour on the long wavelength and time-scale of zero-range processes have been extensively studied, and asymptotic results such as hydrodynamic scaling limit, central limit theorem, or large deviations of the empirical distribution of particles have been established. A specific zero-range process on the 1-dimensional infinite lattice, the q-Boson system, was introduced by Sasamoto and Wadati. The q-Boson specifies that a single particle leaves a site at a rate equal to [n], the q-integer of the site occupancy n. Notably, the q-Boson was shown to be integrable in the sense that a class of eigenfunctions can be constructed for the Hamiltonian of the process. Also, several authors (Korhonen and Lee, 2013, Borodin, Corwin, Petrov, and Sasamoto 2014) showed that the transition probabilities can be computed exactly . Later the q-Boson was generalized by Takeyama to a q-Block using the algebra structure generated by the multiplication and divided-difference operators of Lascoux and Schützenberger. The q-Block is totally asymmetric, exhibits zero-range interaction, and has the novel feature that any number of available particles can leave the site. We show that the system of Takeyama can be enhanced to allow the movement of particles to both left and right and remain integrable. Also, we discuss the attractiveness and propose that the hydrodynamic scaling limit of the system is a first-order quasilinear partial differential equation.

The KMS structures 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 a 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 the 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 these measures using classical functions of this order from number theory. Joint work with Marcelo Laca.

Important classes of C*-algebras can be described as partial crossed products. Although a partial action of a discrete group on a C*-algebra is in an appropriate sense always equivalent to a global action, the commutativity of the underlying C*-algebra may be lost under this correspondence. I will report on joint work with A. Buss and D. Ferraro, in which we generalise a result by Matsumura for ordinary actions by showing that the partial crossed product of a commutative C*-algebra by an exact discrete group is nuclear whenever the full and reduced partial crossed products coincide.

When Voiculescu introduced the notion of bi-free independence, he envisioned that the theory would model the transition between past and future states. As such, the objects under consideration were called “pairs of faces” after the Roman god of transitions Janus who is often depicted with two faces, one looking to the past and one to the future. In this talk, we will discuss how the transition operators for non-commutative stochastic processes can be modelled using technology from bi-free probability. Several important examples are recovered with this approach and new formulae are obtained for processes with free increments. The benefits of this approach are also discussed.

I will discuss characterizations of property (T) for II₁ factors by weak spectral gap in inclusions into tracial von Neumann algebras. I will explain how this is related to the non-weakly-mixing property of the bimodules containing almost central vectors, from which we also obtain a characterization of property (T).

Powers' averaging property has played an important role in answering when the reduced group C*-algebra \(C^*_r(G)\) of a discrete group G is simple, starting with Powers' original result in 1975 that this is true for \(\mathbb{F}_2\), the free group on two generators. It was recently independently shown by Kennedy and Haagerup that the averaging property considered by Powers is in fact equivalent to C*-simplicity of the group. The property can be summarized as follows: G has Powers' averaging property if and only if the closed convex G-orbit of any element \(a \in C^*_r(G)\) contains \(\tau_0(a)\), where \(\tau_0 : C^*_r(G) \to \mathbb{C}\) is the canonical trace.
In joint work with Tattwamasi Amrutam, we consider the case of the reduced crossed product \(C(X) \rtimes_r G\), and show that the crossed product being simple is equivalent to a generalized version of Powers' averaging property. This has some interesting consequences, including generalized versions of the unique stationarity results obtained by Hartman and Kalantar, and also the fact that any C*-algebra lying between two simple crossed products \(C(X) \rtimes_r G\) and \(C(Y) \rtimes_r G\) (where \(C(X) \subseteq C(Y)\)) is itself simple.

For a sufficiently nice 2 dimensional shape, we define its approximating matrix (or patterned matrix) as a random matrix with iid entries arranged according to the given pattern. For large approximating matrices, we observe that the eigenvalues roughly follow an underlying distribution. This phenomenon is similar to the classical observation on Wigner matrices. We prove that the moments of such matrices converge asymptotically as the size increases and equals to the integral of a combinatorially-defined function which counts certain paths on a finite grid.

We study the eigenvalue distributions for sums of independent rank-one k-fold tensor products of large n-dimensional vectors. Previous results in the literature assume that k=o(n) and show that the eigenvalue distributions converge to the celebrated Marčenko-Pastur law under appropriate moment conditions on the base vectors. In this paper, motivated by quantum information theory, we study the regime where k grows faster, namely k=O(n). We show that the moment sequences of the eigenvalue distributions have a limit, which is different from the Marčenko-Pastur law. As a byproduct, we show that the Marčenko-Pastur law limit holds if and only if k=o(n) for this tensor model. The approach is based on the method of moments. This is a joint work with Benoit Collins and Jianfeng Yao.

AF-embeddability, ie the question whether a given C*-algebra A can be realised as a subalgebra of an AF-algebra has been studied for a long time with prominent early results by Pimsner and Voicuescu who constructed such embeddings for irrational rotation algebras in 1978. Since then many constructive but many also non-constructive AF-embeddability results have been obtained which typically assume the UCT to hold for A.
A classical result of Alexandrov, Hausdorff and Uryson says that every compact metrisable space is a quotient of the Cantor set which says that commutative separable unital C*-algebras are AF-embeddable. By mimicking this proof we construct AF-embeddings for separable C*-algebras of finite decomposition rank without assuming the UCT for A.