Zimmer amenability is a group dynamical property that generalizes amenability: every action of an amenable locally compact group on a von Neumann algebra is amenable. This notion manifests across various disciplines by its deep connections to boundary theory, representation, entropy, and operator algebras. Less than a decade ago, Zimmer amenability was introduced and then studied by various authors for locally compact quantum groups. The quantization of Zimmer amenability has been successful and productive, with much of the theory successfully passing to the quantum setting. There is a striking and surprising caveat: De Commer and De Ro showed that a certain natural set of examples of compact quantum groups (note that compactness is stronger than amenability) act non-amenably on certain von Neumann algebras. Such examples came in the form of coideals, which are quantizations of groups acting on their quotient spaces. Later, De Ro gave a satisfying characterization of the coideals of amenable discrete quantum groups with amenable actions, leading to many natural examples both with and without a Zimmer amenable action. Despite the interesting theoretical results on the subject, producing examples of compact quantum groups with non-Zimmer amenable actions remains a hard problem. In this talk we will introduce the notion of a relatively inner amenable action as a tool for studying Zimmer amenability of the actions of locally compact quantum groups on their coideals. In particular and loosely speaking, we show that this notion is "dual" to Zimmer amenability. In the setting of compact quantum groups acting on coideals, we will demonstrate its use for building examples of compact quantum groups both with non-Zimmer and Zimmer amenable actions.
This talk is based on joint work-in-progress with Joeri De Ro.

Although the classical classification of well-behaved simple C*-algebras may by now be a little too classical, it seems natural to begin by reviewing this.

A glorious assembly of these algebras was recently discovered by Chun Guang Li, Zhuang Niu, Jianguo Zhang, and me inside certain Roe algebras. In the (non-separable) AF case, these Roe algebras were classified by Kang Li and Hung-Chang Liao. Can they be classified more generally?

Can the partial classification of stable rank one Villadsen algebras (using the Toms radius of comparison and generalizations of this, Cuntz semigroup information) obtained by Chun Guang, Zhuang and me be extended?

The Fourier algebra A(G) of a locally compact group G is a Banach algebra that captures both the topology and the group structure of G through its unitary representation theory. An important structural property of Banach algebras is the notion of amenability. Various amenability properties of the Fourier algebra, and their connections to the underlying group structure, have been extensively studied over the past few decades. In this talk, we discuss a quantitative invariant for amenability known as the amenability constant. For Fourier algebras, explicit formulas for the amenability constant of A(G) are known only in limited settings. In particular, B.E. Johnson computed the finite-group case. We present new upper bounds for the amenability constant in the case of discrete groups, and present new classes of examples where the amenability constant can be explicitly computed.

The irreducibility of Poisson boundary representations, as conjectured by Bader and Muchnik, is known to hold for a wide range of geometric groups, but fails in general. In this talk, we discuss the nature of this failure, and propose a refined conjecture, which we interpret as an L²-version of the Strong Approximately Transitive (SAT) property. This appears to be the robust dynamical core of the original conjecture. We prove this for the very example where irreducibility fails. This is joint work with Yair Hartman.

The mathematical theory of majorization was introduced in order to formalize the intuitive idea of one set of numbers being more "spread out" than another. The theory has since become a major component of convexity theory, with applications throughout mathematics, economics and, more recently, quantum information theory. In this talk, I will introduce a theory of noncommutative majorization, where sets of numbers are replaced by sets of (not necessarily commuting) operators. I will discuss some applications to operator algebras and quantum information theory. This is joint work with Paul Skoufranis.

Stable rank one is established for the reduced group C*-algebras of the C*-simple topological full groups and dynamical alternating groups constructed by Tucker-Drob and myself. The proof relies on both type II₁ and type III phenomena, in the first case via the use of Følner towers and in the second via Ozawa’s recent results on selflessness as applied to direct products of nonelementary free products. Joint work with Spyros Petrakos.

This research was originally motivated by the following open problem: Is it true that a separable amenable simple C*-algebra has tracial rank zero if and only if it has real rank zero and is Z-stable? N. Brown gave an example of a unital separable simple C-algebra with real rank zero, stable rank one, and strict comparison that does not have tracial rank zero. Z. Niu and Y. Wang showed that there exist separable simple C*-algebras of tracial rank zero that are not Z-stable. We will give an affirmative answer to the question. We will introduce a weaker condition of tracial approximation by finite-dimensional C*-algebras. We find that a separable amenable simple C*-algebra is Z-stable if and only if it satisfies this condition of tracial approximation by finite-dimensional C*-algebras.

The classical Rokhlin lemma states that an aperiodic measure preserving dynamical system can be decomposed to an arbitrary high tower of measurable sets and a remainder of arbitrarily small measure. Then let us say that a topological dynamical system has the uniform Rokhlin property if it can be decomposed to arbitrary high towers of open sets and a remainder which is uniformly small with respect to all invariant probability measures. Any free and minimal Zⁿ action have the uniform Rokhlin property. In the talk, I’ll discuss the applications of the uniform Rokhlin property to the structure of the crossed product C*-algebras such as comparison radius, stable rank, and Jiang-Su stability, and I’ll also talk about a recent application to the Roe algebra of a discrete amenable group. The talk is partially based on the joint work with George Elliott, Chun Guang Li, and Jianguo Zhang.

We introduce the notion of uniform exactness, or uniform amenability at infinity, for discrete groups and prove it for a wide class of groups containing free groups and their limit groups. This shows a novel strong convergence phenomenon that any convergent sequence of such groups in the space of marked groups converges strongly in the operator algebraic sense. In particular, convergence of the spectral radius formula is uniform over probability measures on such groups whose supports have a fixed cardinality.

I will introduce the notion of selfless C*-probability space and then discuss several extensions of this concept in some new directions: to operator-valued C*-probability spaces, to cp maps, and to C*-correspondences with a real structure. Joint work with Gao, Junge, Kunnawalkam Elayavalli, and Patchell.

The Namioka-Phelps tensor product of two convex compact comes in two variants: the minimal and the maximal tensor product, where the former is contained in the latter. They show that the two tensor products agree if one of the two convex compact sets is a Choquet simplex. It remains an open problem, known as Barker’s conjecture, if the converse also holds. Barker’s conjecture was recently verified by Aubrun–Lami–Palazuelos–Plavala in the finite dimensional case, and was verified by Namioka and Phelps when one of the two convex compact sets is the square. We show that Barker’s conjecture holds when the compact convex sets are state spaces of C*-algebras, and we describe the two Namioka-Phelps tensor products in this case. The minimal tensor product is precisely the set of entangled states in the (minimal) tensor product of the C*-algebras, while the maximal tensor product is more elusive, but can be described in terms of positive maps. We identify the trace space of the tensor product of C*-algebras as the Namioka-Phelps tensor product of the trace spaces and use this to say when the trace simplex of a tensor product of C*-algebras is the Poulsen simplex. Joint work with Magdalena Musat.

In this talk I will consider the ideal structure of reduced crossed products associated to noncommutative C*-dynamical systems. I will report on joint work with M. Kennedy and L. Kroell in which we characterize primality for reduced crossed products for actions of discrete groups. For a class of groups containing finitely generated groups of polynomial growth, we can further characterize the ideal intersection property.

Orbit-breaking in topological dynamical systems was introduced by Ian Putnam in his study of the C*-algebras associated to Cantor minimal systems. By breaking an orbit at a single point, Putnam was able to construct all simple unital AF algebras as subalgebras by way of a Cantor minimal system. Since then, orbit-breaking algebras have been used to construct dynamical models for many stably finite “classifiable” C*-algebras, most notably the Jiang–Su algebra. In this talk I will highlight these results and also discuss recent work-in-progress with Robin Deeley and Ian Putnam, where we generalize orbit-breaking for a homeomorphism to the setting of a continuous, surjective, local homeomorphism, allowing for purely infinite constructions.

(Paired with Turowska.) In these talks, based on a joint work with G. Baziotis and A. Chatzinikolaou, we provide a setup for no-signalling correlations that allow to describe the strategies of infinite parallel repetition of non-local games as a one-shot entity. Much as in the finite case, the resulting Cantor correlations have differentiated types, and their operator-algebraic description is provided via inductive limit operator systems. We show that in the graded case, synchronous and bisynchronous Cantor correlations can be characterised via traces on appropriate C*-algebras, and apply our results to provide examples of factorisable maps in infinite dimensions and quantum isomorphisms of Cantor graphs.

In 2001, Whyte proved that all "higher" Baumslag-Solitar groups, i.e. BS(p,q) with |p|, |q|≠1 and |p|≠|q|, are quasi-isometric, thereby completing the quasi-isometry classification of all Baumslag-Solitar groups initiated by Farb and Mosher. Since then, the question of their measure equivalence has remained an intriguing open problem. In joint work with D. Gaboriau, A. Poulin, R. Tucker-Drob, and K. Wróbel, we solve this problem, establishing the measure equivalence analogue of Whyte's theorem and completing the measure equivalence classification of all Baumslag-Solitar groups. We do so by first reducing the problem to the measure equivalence of the automorphism groups of the Bass-Serre trees of the higher Baumslag-Solitar groups, which are nonunimodular locally compact groups. This shift from discrete (unimodular) groups to nonunimodular groups naturally changes the setting from probability-measure-preserving (type II₁) to merely measure-class-preserving (type III), where we leverage recent advances in descriptive set theory alongside new ideas.

(Paired with Todorov.) In these talks, based on a joint work with G. Baziotis and A. Chatzinikolaou, we provide a setup for no-signalling correlations that allow to describe the strategies of infinite parallel repetition of non-local games as a one-shot entity. Much as in the finite case, the resulting Cantor correlations have differentiated types, and their operator-algebraic description is provided via inductive limit operator systems. We show that in the graded case, synchronous and bisynchronous Cantor correlations can be characterised via traces on appropriate C*-algebras, and apply our results to provide examples of factorisable maps in infinite dimensions and quantum isomorphisms of Cantor graphs.

Generalizing a notion of amenability defined by Connes for von Neumann algebras, Runde introduced the notion of Connes-amenability as a weak*-version of Banach algebraic amenability for dual Banach algebras in 2001 and subsequently showed that the measure algebra M(G) of a locally compact group is Connes-amenable if and only if G is amenable in 2003. By analog, one might guess that the Fourier-Stieltjes algebra B(G) is operator Connes-amenable if and only if G is amenable, but this is not the case since it fails for G=\(\mathbb F_2\) (Runde-Spronk, 2004). In this talk, we will describe conditions that imply the failure of operator Connes-amenability for B(G). This provides the first examples of groups where B(G) fails to be operator Connes-amenable. We will also discuss some open problems and a possible characterization of operator Connes amenability of B(G) when G is discrete.

This is based on joint work with V. Runde and N. Spronk.

The 2-adic ring C*-algebra \(\mathcal{Q}_2\) is the universal C*-algebra generated by a unitary and a copy of the Cuntz algebra \(\mathcal{O}_2\) satisfying some relations. It has several realizations in the literature due to its importance. In this talk, I will report an ongoing joint work with Dolapo Oyetunbi on the structure related to its endomorphisms. In particular, our results answer some open questions on endomorphisms of \(\mathcal{Q}_2\).

We present a characterization of when a closed subgroup is amenable in terms of a certain G-C*-algebra being G-injective. This generalizes the fact that a group G is amenable if and only if the complex numbers are G-injective. We also introduce the notion of proximally irreducible extensions for affine G-flows and prove a similar characterization for when a closed subgroup is strongly amenable.

I will discuss a notion of minimality for noncommutative dynamical systems differing from the usual one. I will explain how this concept of minimality relates to C*-irreducibility and go through some examples. This is based on joint work with Erik Séguin.

I will discuss recent joint work with Julian Kranz on the problem of realizing étale groupoids as transformation groupoids arising from partial actions of discrete groups. We develop obstructions showing that several important classes of groupoids, including Higson–Lafforgue–Skandalis groupoids and many Deaconu–Renault groupoids, do not admit such models. As a consequence, we obtain the first explicit examples of étale groupoids that are not inner amenable. I will also explain recent positive results showing that every unital Kirchberg algebra can nevertheless be realized as a partial crossed product by a discrete group.

We recall the definitions of homology and cohomology for ample groupoids \(\mathcal G\) using \(\mathcal G\)-modules and resolutions. For \(\mathcal G'\subseteq \mathcal G\) an open subgroupoid with the same unit space and \(M\) a \(\mathcal G\)-module, we define the relative homology \(H_*(\mathcal G, \mathcal G';M)\) and the relative cohomology \(H^*(\mathcal G, \mathcal G';M)\). We prove long exact sequences relating \(H_*(\mathcal G, \mathcal G';M)\) and \(H^*(\mathcal G,\mathcal G';M)\) to the homology and cohomology of \(\mathcal G\) and \(\mathcal G'\). We prove that \(H^*(\mathcal G,\mathcal G';M)\) is isomorphic to the relative sheaf cohomology \(H^*_c(\mathcal G, \mathcal G';\mathcal M)\) defined with cocycles vanishing on \(\mathcal G'\).

I will discuss progress on the following conjecture: The stable and unstable C*-algebras associated to a mixing Smale space are isomorphic after tensoring with the universal UHF-algebra. Smale spaces are a class of hyperbolic dynamical systems. These systems and their C*-algebras will be introduced in the talk, so no previous knowledge of Smale spaces is required. Special cases of Smale spaces (such as shifts of finite type and certain solenoids) where the conjecture holds will be the main focus.

Consider the CW complex on the line. Associated to this complex are the Poincare duality operator \(S\) and the boundary operator \(b\). During the talk I will discuss a representation of the \(C^*(S, b)\) as a subalgebra of \(M_2(C(S^1))\). I will also discuss how the C*-algebra arises as a symbol algebra for the 2-Toeplitz operators.

During this talk, we will discuss recent results on von Neumann algebras associated with discrete groupoids. Indeed, the work discussed involves defining the Gaussian deformation for discrete p.m.p. groupoids and using the techniques of Popa's deformation/rigidity theory to obtain results related to primeness, fullness and unique prime factorizations for these algebras. This talk is based on joint work with James Harbour.

This talk will report on an application of operator-theoretic methods to the field of aperiodic order. Roughly stated, the motivating question is: if a structure has many almost periods in mean (with respect to a given averaging process) must it contain a substructure well-approximated in mean by Bohr almost periodic functions? In the joint work arXiv:2601.05327 with C. Ramsey and N. Strungaru, we give an affirmative answer using spectral methods and Choquet theory.

Suppose that \(G\) is a finitely generated virtually abelian group, and \(\sigma\) is a 2-cocycle of the group that takes values in the roots of unity. In this talk, I will explain why the nuclear dimension of \(C^*(G;\sigma)\) is equal to the rank of the finite-index abelian subgroup of \(G\). Based on joint work with Pradyut Karmakar and Iason Moutzouris.

The Paterson compactification extends the classical Tychonoff theorem from compact spaces to the setting of locally compact, Hausdorff, second countable spaces, by providing a natural compactification of the corresponding product space. In recent work, this construction is used as a basic tool in the study of Deaconu–Renault systems. In this talk, I will give an introduction to the Paterson compactification and describe a convenient basis for its topology. I will then specialize to the case of a discrete alphabet and explain how the resulting compactification is naturally identified with the OTW one-sided subshift. Finally, using this point of view, I will discuss subshift algebras and prove that the spectrum of a natural commutative subalgebra is precisely the OTW subshift. This gives a concrete operator-algebraic realization of the Paterson compactification in the symbolic setting.

We consider the action of an inverse semigroup \(\mathcal{S}\) by partial left translations on its Stone-Čech compactification, \(\beta\mathcal{S}\). Unlike the situation for groups, the resulting action groupoid \(\mathcal{G} = \mathcal{S}\ltimes\beta\mathcal{S}\) is not always Hausdorff or principal. We show that each of these properties are equivalent to the realizability of Lledó and Martínez's uniform Roe algebra of \(\mathcal{S}\) as the reduced groupoid \(\textup{C}^*\)-algebra of \(\mathcal{G}\).

A C*-algebra is said to have Property (SP) if each of its hereditary subalgebras contains a nonzero projection. This property is closely related to, though weaker than, the more familiar notion of real rank zero. Motivated by recent work of Gabe and Neagu (2025) on inclusions of real rank zero, we introduce a corresponding notion of Property (SP) for inclusions of C*-algebras and begin its systematic study. Extending classical work of Osaka (2001), we also show that, under suitable hypotheses, Property (SP) for C*-algebras is preserved under irreducible C*-discrete inclusions. This is ongoing joint work with Roberto Hernández Palomares.

There are two notions of \(G\)-equivariant pseudodifferential operators on a Riemannian symmetric space of non-compact type \(G/K\). One is the traditional Hörmander class of pseudodifferential operators defined using the Fourier inversion formula on \(R^n\) that are \(G\)-equivariant. These operators do not have a well-defined complete symbol function. The other is the class of \(G\)-equivariant multiplier operators defined using the Helgason Fourier transform on \(G/K\) satisfying the appropriate symbol-type estimates. I will call them Harish-Chandra pseudodifferential operators. We will see that these two notions coincide identically for operators whose kernel have an appropriate rapid off-diagonal decay property. In the process, we will show that the \(G\)-equivariant pseudodifferential operator algebra also provides a \(C^*\)-algebra extension of the \(K\times K\)-invariant part of \(C_r^*(G)\). To compare the two symbols - Hörmander and Harish-Chandra, we will use the Mackey-Higson deformation of the group \(G\) to its Cartan Motion Group \(G_0\), and it will turn out that the Hörmander principal symbol will be an appropriate limit of the Harish-Chandra symbol.

The way-below relation (or compact containment) for Hilbert C*-modules provides a noncommutative analogue of the topological notion. This talk presents recent results on three key aspects of this relation. First, in joint work with N. Oleny, we prove the equivalence of two important noncommutative definitions: one using compact self-adjoint endomorphisms and the other via order-theoretic compact containment in directed unions. Second, in collaboration with H. Labrecque, we establish stability under tensor products. Finally, in ongoing joint work with M. Frank, we study regularity properties and show that if M is a Hilbert submodule of N so that M has trivial orthogonal complement relative to N, then every self-adjoint bounded A-linear operator on N vanishing on M must be zero. One aim of these results is to unify geometric intuition with operator-algebraic structure.

A group G is matricial field (MF) if it admits a strongly convergent sequence of approximate homomorphisms into matrices, meaning that polynomial norms converge to those in the left regular representation. It is purely MF (PMF) if the maps can be chosen to be genuine homomorphisms, and purely finite field (PFF) if, moreover, each homomorphism has finite image. These properties have applications to C*- and von Neumann algebras, spectral geometry, random walks and graphs, spectral gaps of hyperbolic manifolds, minimal surface theory, and related areas of applied mathematics.
Using Toeplitz-Pimsner algebras, we construct new classes of MF, PMF, and PFF groups, including certain group doubles and graph products. In particular, we prove that right-angled Artin groups are PFF, vastly generalizing a result of Magee and Thomas, with potential geometric applications via Song's random-matrix approach to minimal surfaces.
Joint work with David Gao, Srivatsav Kunnawalkam Elayavalli, and Gregory Patchell.

An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the entanglement-assisted zero-error capacity under some restrictions is uncomputable; indicative of the fact that quantum channel capacities may generally be undecidable.

Selflessness, introduced by Robert and strengthened by Ozawa through the notion of complete selflessness, provides a C*-algebraic analogue of II_1-factor phenomena in the tracial setting. In this talk, I will present recent work establishing complete selflessness for large classes of reduced free products and reduced graph products of C*-algebras under a natural Avitzour-type condition, without assuming previous rapid decay criteria.

The problem of determining which Kirchberg algebras admit Cartan subalgebras with the pure extension property remains an active area of research. Given etale groupoids G and H equipped with compatible cocycle and an automorphism of H, one can construct an associated twisted product groupoid. In this talk, we establish conditions under which the C*-algebra of this twisted product groupoid is a Kirchberg algebra in the UCT class. We also present several constructions illustrating how this framework produces new examples of Kirchberg algebras with C*-diagonals.

In the framework of Connes’ noncommutative geometry the role of the integral is played by traces on the weak trace class. In this talk, I will explain how to extend Connes integration to Lorentz ideals. This motivated by Connes’ approach to Riemann hypothesis and the various occurrence of so-called “nonclassical” Weyl laws.

The Fourier algebra A(G) of a locally compact group was introduced by P. Eymard in 1967. A year later, H. Leptin proved that G is amenable if and only if A(G) has a bounded approximate identity. As shown by B. E. Forrest and myself, A(G) is amenable in the Banach algebraic sense if and only if G is finite-by-abelian. As the predual of the group von Neumann algebra, A(G) carries a natural operator spaces structure turning it into a completely contractive Banach algebra. Every function in A(G) induces a completely bounded operator on A(G) through multiplication. This induces the cb-multiplier norm on A(G). The completion \(A_{M_{cb}}(G)\) of A(G) with respect to this norm is again a completely contractive Banach algebra, which coincides with A(G) if and only if G is amenable. The characterization of those G for which \(A_{M_{cb}}(G)\) is an amenable Banach algebra has been open for quite some time. I will give an overview of what is known and point out possible lines for future attacks.

After providing a summary of how bi-free probability extends free probability, we will demonstrate how bi-free probability can be utilized to solve problems in non-commutative probability theory.

We characterise the stable finiteness and pure infiniteness of the essential crossed product of a C*-algebra by an action of an inverse semigroup. Under additional assumptions, we prove a stably finite / purely infinite dichotomy. Our main technique is the development of a "dynamical Cuntz semigroup'' that is a quotient of the usual Cuntz semigroup by an induced action. We prove that the essential crossed product is stably finite / purely infinite if and only if the dynamical Cuntz semigroup admits / does not admit a nontrivial state. Joint work with Becky Armstrong, Lisa Clark, Astrid an Huef and Diego Martı́nez.

Generally speaking, operator-algebraic quantum error correction has primarily focused on the correction of errors induced by channels over algebras of bounded observables. However, many of the most physically important observables such as position, momentum, and quantum field observables are unbounded. In some cases, there is passage from algebras of unbounded observables via spectral theory to affiliated algebras of bounded operators. In the context of quantum field theory, however, this is not always a valid assumption; thus, a treatment of quantum error correction at the level of unbounded algebras/quantum fields, wherever possible, is warranted from a mathematical physics perspective. In this talk, we go over some basic definitions, examples, and structure pertaining to these so-called 'unbounded algebras' and discuss generalizations of well-known results from operator algebras. On the concrete side of things, we shall look at how these objects arise naturally in the setting of axiomatic quantum field theory as well as develop a generalized theory of quantum error correction for channels over unbounded algebras. Finally, we will look at potential application of generalized error correction to the conditions for holographic error correction in the AdS/CFT correspondence at the level of quantum fields.