Past seminars

Title: Domains of quantum metrics on AF-algebras

Abstract: Given a compact quantum metric space (A, L), we prove that the domain of L coincides with A if and only if A is finite-dimensional. Intuitively, this should allow for different quantum metrics with distinct domains when A is infinite-dimensional, and we show how to explicitly build such quantum metrics. Lastly, we also provide a strategy for the calculation of distance between certain states in these quantum metrics in the special cases of the quantized interval and the Cantor space. This is joint work with Konrad Aguilar and Frederic Latremoliere.

Title: The Levi-Civita connection on noncommutative differential forms

Abstract: I will discuss the main results and consequences of recent work with Bram Mesland. Using mostly algebraic techniques, with just a little Hilbert module analysis, we present necessary and sufficient conditions for the existence of Hermitian torsion-free connections on noncommutative  one-forms, such as those arising from spectral triples. Various consequences and examples will be discussed also.

Title: Classification of simple C*-algebras from singular graphs

Abstract: There is a rich classification theory for unital simple C*-algebras associated to finite graphs with no sinks, allowing to decide by invariants when two such C*-algebras are isomorphic, not just in their own right, but equipped with their natural diagonals (Cartan subalgebras) or their natural circle actions. All of these results were proved by involving results from symbolic dynamics, even though some now are special cases of much more general results.

When one allows singular vertices — sinks or infinite emitters — the connection to the rich theory of shifts of finite type is lost, but the classical classification results have natural generalizations which one may aspire to show by other means. I will discuss and compare different notions of sameness of such graph C*-algebras; all fully understood in the regular case, both only half resolved in general. All work presented is joint with Efren Ruiz, and some also with Aidan Sims.

Title: Pure C*-algebras and *-homomorphisms

Abstract: The notion of (m,n)-pure C*-algebras was introduced by Winter in his seminal work on separable, simple, unital C*-algebras of finite nuclear dimension. Although a lot of effort has been put on understanding (0,0)-pureness (often simply called pureness), much less is known about the apparently weaker notion of (m,n)-pureness for m,n>0. This is especially the case in the non-simple setting.

I will begin the talk by recalling the Toms-Winter conjecture and the importance of pureness in its study. I will then discuss results from two different ongoing projects: Pure *-homomorphisms and their properties (joint with J. Bosa), and (m,n)-pure C*-algebras (joint with R. Antoine, F. Perera, and H. Thiel).

Title: Dynamic asymptotic dimension and applications

Abstract: Dynamic asymptotic dimension is a dimension theory for topological dynamical systems introduced by Guentner, Willett, and Yu in 2017. Since its inception it has found numerous applications to the structure theory of C*-algebras as well as homology and K-theory. In this talk I will discuss some aspects of this dimension theory and its applications to C*-Algebras.

Title: Stein’s groups as topological full groups

Abstract: Topological full groups are a framework which allow us to understand simple groups with finiteness properties through ample groupoids. In particular, purely infinite groupoids give us interesting generalisations of Thompson’s group V. In this talk, I will describe how I have used this framework to better understand a class of groups introduced by Melanie Stein in 1992. 

Title: The dynamical Kirchberg-Phillips theorem

Abstract:  I will talk about the problem of classifying C*-dynamical systems for which the underlying C*-algebra is purely infinite and simple, such as the Cuntz algebras O_n. I will try to make the talk accessible with focus on some elementary examples. This is joint work with Gabor Szabo.

Title: A spectral gap for twisted Dolbeault–Dirac operators over irreducible quantum flag manifolds

Abstract: In this talk we present a geometric approach to understanding the analytic properties of twisted Dolbeault–Dirac operators D_{\overline{\partial}} over the irreducible quantum flag manifolds \mathcal{O}_q(G/L_S). We deduce from a noncommutative Akizuki–Nakano identity that twisting a Dolbeault–Dirac operator by a positive line bundle results in a spectral gap for these operators. We then conclude that each positively twisted D_{\overline{\partial}} is an (unbounded) Fredholm operator, and calculate its index using the recently established noncommutative Borel–Weil theorem. (Joint work with Biswarup Das and Petr Somberg)

Title: Admissible KMS bundles on classifiable C*-algebras

Abstract: Named after mathematical physicists Kubo, Martin, and Schwinger, KMS states are a special class of states on any C*-algebra admitting a continuous action of the real numbers. The collection of KMS states for a given flow on a C*-algebra can be quite intricate. In this talk, I will explain what abstract properties these simplices have and show how one can realise such a simplex on various classes of simple C*-algebras.

Title: Classification of crossed product C*-algebras arising from essentially free actions

Abstract: Essentially free actions play an important role in ergodic theory and group measure space II1-factors. In this talk, I will explain why (uniform) essential freeness is a better prerequisite than topological freeness in the classification of crossed product C*-algebras, and provide a first general result on uniform property Γ and Z-stability, assuming only topological freeness, in this direction. 

Title: The primitive-ideal spaces of some groupoid C*-algebras

Abstract: The primitive-ideal space of the universal C*-algebra of a dynamical system is an important bit of structural information, but very hard to compute. For reversible, abelian dynamics it can be computed using powerful results going back to Mackey, Green and Rieffel. For rank-1 irreversible dynamics, it can be computed using recent results Katsura, building on earlier work of an Huef-Raeburn and then Hong-Szymanski, that rely on reducing key components of the dynamics to reversible subsystems. For rank two or more, such reductions are not available in general. I’ll discuss an approach using groupoids that applies to C*-algebras of large classes of actions of free abelian monoids, including all 2-graph C*-algebras. This is joint work with Kevin Brix and Toke Carlsen.

Title: C*-envelopes of semigroup operator algebras

Abstract: I will present on joint work with Kevin Aguyar Brix and Adam Dor-On in which we prove that every normal coaction of a discrete group on an operator algebra extends to a (unique) normal coaction on the C*-envelope of the operator algebra. As an application, we identify the C*-envelopes operator algebras of many (e.g., groupoid-embeddable) small categories with the boundary quotient C*-algebra of the category, answering a question posed by Xin Li. I will focus on the case where the category is a monoid, where we obtain the first results outside of the group-embeddable case.

Title: Pure largeness and the Elliott-Kucerovsky theorem

Abstract: The concept of absorbing extensions is a ubiquitous tool in Brown-Douglas-Filmore theory of extensions, with applications in (modern) C*-classification results and structure of C*-algebras. In their 2001 paper, Elliott and Kucerovsky characterize this abstract and a priori extrinsic property by an intrinsic condition which they call being "purely large". During this talk, we will introduce both concepts, explore them in greater detail, and highlight some of the main ideas of the proof. If time permits, we will also briefly discuss its connection with modern C*-classification theory, in the framework of separably inheritable properties.

Title: Quantum metric Choquet simplices

Abstract: I will describe some work in progress on a special class of compact quantum metric spaces. Abstractly, the additional requirement on the underlying order unit spaces is the Riesz interpolation property. In practice, I am interested in Lipschitz structures associated to metrics on the *trace* spaces of separable, unital C*-algebras. This added structure is designed for witnessing metric features of morphisms into the category of classifiable C*-algebras. I will give some examples and explain how to build (continuous fields of) new ones by forming 'quantum crossed products' associated to dynamical systems on compact, connected metric spaces.

Title: NC Convexity and Operator System Duality

Abstract: Operator systems are linear spaces of bounded operators on a fixed Hilbert space that is closed under the involution operation. If the operators in this operator system commute, this is what is known in the literature as a function system. A foundational result of Kadison demonstrates that function systems are categorically dual to the category of compact convex sets. Here the function system can be thought of as the scalar valued affine functions over this convex set. We may therefore think of operator systems in general as a kind of non-commutative function system.

NC convex sets were introduced first in the matricial case by Webster--Winkler in 2008, and in a more general unital setting by Davidson--Kennedy in 2019, and finally in general by myself, Matt Kennedy, and Nicholas Manor. These nc convex sets are geometric objects closed under a certain matricial analogue of convex combinations. Surprisingly, Kadison's orginal duality theorem goes through completely in this far more general setting, allowing us use intuition that we have for convex sets in this new category. In this talk we will explore this duality in more detail, and explain how one may use commutative intuition to derive facts about operator systems using this nc convex algebraic geometry. 

Title: Ideals in groupoid C*-algebras and their isotropy fibers 

Abstract: In this talk I will report on a joint project with Sergey Neshveyev where we investigate how any ideal in a locally compact étale groupoid C*-algebra defines a family of ideals in the group C*-algebras of isotropy groups and to which extent these families determine the ideal. I will illustrate how this question is connected to the study of certain (non) exotic norms on the isotropy groups of the groupoid. As an application I will among other things describe the maximal ideals of any étale groupoid C*-algebra and classify the primitive ideals of a class of graded groupoids. 

Title: Embezzlement and the classification of type III factors

Abstract: In quantum information science, embezzlement refers to the counterintuitive possibility of extracting arbitrary entangled states from a reference state (the "embezzler") via local unitary operations while hardly perturbing the latter. Basic conservation laws imply that an embezzler has infinite entanglement.

To study this phenomenon, we consider a pair (M, M’) of a von Neumann algebra and its commutant in standard form as the bipartite system and study how well a given pure state, i.e., a unit vector Ω in the positive cone, performs at the task of embezzling arbitrary entangled states.

We find a connection to the flow of weights on M: Ω is good at embezzling if and only if the dual state ω̂ is an approximate fixed point of the flow of weights. In particular, embezzling states correspond to fixed points of the flow of weights, and the type III_1 factor can be uniquely characterized by the property that all states are embezzling. For type III_λ factors, 0<λ<=1, the value of λ can be recovered from the worst-case embezzlement performance of all bipartite pure states. The λ=0 is open.

This is joint work with A. Stottmeister, H. Wilming, and R.F. Werner.

Title: Generalized inductive systems, convergence of dynamics, and renormalization

Abstract: I will discuss a generalized notion of inductive systems of Banach spaces, coined soft inductive systems, that can serve as a flexible tool to describe the limits of physical theories, e.g., in the context of the thermodynamic limit and phase transitions or renormalization of quantum field theories.

General criteria for the convergence of dynamics can be obtained in this setting, which I will illustrate using the example of renormalization of free fermion theories on a one-dimensional spatial lattice. Specifically, this example can be used to prove the conformal invariance of various correlation functions of the classical Ising model using operator algebraic methods.

This talk is based on work with L. van Luijk, T.J. Osborne, and R.F. Werner.

Title: A type III_1 factor with the smallest outer automorphism group

Abstract: Outer automorphism groups have played a crucial role as invariants in understanding II_1 factors for a long time. After Connes showed that property (T) factors have a countable outer automorphism group in 1980, it remained open for many years to determine if there is a II_1 factor with a trivial outer automorphism group. Finally, in 2004, such a II_1 factor was shown to exist by Ioana, Peterson and Popa and then explicitly constructed by Popa and Vaes in 2008. In the case of type III factors, the modular automorphisms unavoidably intersect the outer automorphism group and the natural analogous question is: can one construct a factor with the `smallest' outer automorphism group? In this talk, we shall give an explicit ergodic theoretic construction of a full type III factor M with Out(M) equal to the real line.

Title: Local Hilbert-Schmidt stability

Abstract: We will introduce a local notion of Hilbert-Schmidt stability (HS-stability), partially motivated by the recent introduction of local permutation stability by Bradford. We will discuss some basic properties, and then establish a local character criterion for local HS-stability of amenable groups, by analogy with the character criterion for HS-stability of Hadwin and Shulman. We will then discuss further examples of (flexible versions of) local HS-stability. Finally, we show that infinite sofic (resp. hyperlinear) property (T) groups are never locally permutation (resp. HS-) stable, answering a question by Lubotzky. This is based on joint work with Francesco Fournier-Facio and Maria Gerasimova.

Title: The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

Abstract: The Laplace-Beltrami operator is a fundamental tool in the study of compact Riemannian manifolds. In this talk, I will introduce the logarithmic analogue of this operator on Ahlfors regular spaces. These are metric-measure spaces that might lack any differential or algebraic structure. Examples are compact Riemannian manifolds, several fractals, self-similar Smale spaces and limit sets of hyperbolic isometry groups. Further, this new operator is intrinsically defined, has properties analogous to those of elliptic pseudo-differential operators on manifolds and exhibits compatibility with non-isometric actions in the sense of noncommutative geometry. This is joint work with Bram Mesland (Leiden). 

Title: Crossed products as compact quantum metric spaces

Abstract: In 2013 Hawkins, Skalski, White and Zacharias constructed and investigated certain spectral triples on crossed product C*-algebras by actions of discrete groups which are in a natural sense equicontinuous. Following Connes, one of the ingredients of their construction are certain multiplication operators associated with length functions on the group. In their article they further formulated the question for whether their triples turn the corresponding crossed product C*-algebras into compact quantum metric spaces. The aim of this talk is to give a short introduction into Rieffel's theory of compact quantum metric spaces. By combining his ideas on horofunction boundaries of groups with results from metric geometry, I will further answer the question of Hawkins, Skalski, White and Zacharias in the affirmative in the case of virtually abelian groups, equipped with suitable length functions.

Title: Groupoid models of semigroup C*-algebras

Abstract: For a left-cancellative semigroup, $S$, one always has an associated reduced C*-algebra $C_r^*(S)$,

but defining a universal counterpart is usually challenging. One approach is to find an étale groupoid, $G(S)$,

such that $C_r^*(S)$ is isomorphic to $C_r^*(G(S))$- the reduced C*-algebra of $G(S)$. Then $G(S)$ is called a

groupoid model for $S$ and once we have it, it makes sense to define the universal C*-algebra of $S$ as the

universal C*-algebra of $G(S)$. In this talk we go through the construction of a so called reduced Paterson

groupoid associated to $S$, $G(S)$, and give rather weak conditions on $S$ which guarantees that

$C_r^*(S)\simeq C_r^*(G(S))$. We also present a groupoid model for the boundary quotient of $S$, which is a

quotient of $C_r^*(S)$. This is based on joint work with Sergey Neshveyev. 

Title: Nuclearity and Inductive Limits

Abstract: Inductive limits are a central construction in operator algebras because they allow one to use well-understood building blocks to naturally construct more complicated objects whose properties remain tractable. In the classical setting, few nuclear C*-algebras arise as inductive limits of finite-dimensional C*-algebras. However, by generalizing our notion of an inductive system, Blackadar and Kirchberg were able to characterize quasidiagonal nuclear C*-algebras as those arising as (generalized) inductive limits of finite dimensional C*-algebras. In this talk, I will describe new notions of inductive systems which allow us to realize any nuclear C*-algebra as the limit of a system of finite dimensional C*-algebras. Though seemingly abstract, these systems correspond naturally to completely positive approximations of nuclear C*-algebras. This is based in part on joint work with Wilhelm Winter.

Title: From directed graphs of groups to Kirchberg algebras 

Abstract: Directed graph algebras have long been studied as tractable examples of C*-algebras, but they are limited by their inability to have torsion in their K_1 group. Graphs of groups, which are famed in geometric group theory because of their intimate connection with group actions on trees, are a more recent addition to the C*-algebra scene. In this talk, I will introduce the child of these two concepts – directed graphs of groups – and describe how their algebras inherit the best properties of its parents’, with a view to outlining how we can use these algebras to model a class of C*-algebras (stable UCT Kirchberg algebras) which is classified completely by K-theory. 

Title: Generalized dynamics yielding nuclear crossed products

Abstract: Given a (generalized) dynamical system defined on a commutative C*-algebra, one can construct a suitable notion of reduced crossed product. This crossed product, however, may fail to admit a conditional expectation onto the original C*-algebra, but it does admit a weak variant. In this talk we will define these systems, and construct their associated C*-algebras. We will then give a sufficient condition for the nuclearity of these algebras that generalizes amenability for group actions on C*-algebras and twisted étale groupoids. This talk will be based on joint work with Alcides Buss.

Title: Finite Approximations of Dynamical Systems

Abstract:  I will give an introduction to residually finite and quasidiagonal group actions first defined by Kerr and Nowak.  These are group actions which admit finite (dimensional) approximations and, as a result, have reduced crossed products which can be approximated by finite dimensional algebras whenever the reduced algebra of the acting group can be.  We will see that isometric actions on compact metric spaces have especially nice approximation properties.  Toward the end, there will be some discussion of connections to approximate representations and semiprojective C^*-algebras. 

Title: Passing Z-stability in finite index inclusions

Abstract: The question of under what conditions a crossed product of a Z-stable C*-algebra by a group action remains Z-stable has been the focus of much recent research, with many techniques developed to address this. In this talk, I will discuss actions of tensor categories on C*-algebras and how key concepts and techniques from the group action setting can be adapted to this broader context. Through these adaptations, we aim to address the question of what conditions ensure that Z-stability is preserved in inclusions of finite index. This talk is based on joint work with Samuel Evington, Corey Jones and Stuart White.

Title: Towards noncommutative dynamics

Abstract:  Starting with work of Muhly, Pask, and Tomforde fundamental notions (such as Strong Shift Equivalence) from symbolic dynamics were imported into C*-correspondences and their Cuntz-Pimsner algebras. The analogy goes deeper: C*-correspondences may be viewed as noncommutative symbolic systems and I will describe how to perform state splittings that give (all?) examples of Strong Shift Equivalences. Joint with Alex Mundey and Adam Rennie.

Title: Morphisms between étale groupoids that lift to C*-algebras

Abstract: The obvious choice of arrow between two groupoids is a homomorphism (or functor). This works well enough until one wants to induce morphisms of groupoid C*-algebras. The problem that arises is that homomorphisms of groupoids would be covariant on the source and range fibres of the groupoid, but contravariant on the unit space, so a *-homomorphism of the C*-algebras would have to face both directions.

Buneci and Stachura introduce a different morphism of groupoids by considering actions of groupoids on other groupoids. They show that such morphisms covariantly induce *-homomorphisms of the groupoid C*-algebras. Meyer and Zhu call these morphisms 'actors' and build a categorical framework around the category of groupoids with actors as morphisms.

In this talk, I will define actors for étale (twists over) étale groupoids and show that any *-homomorphism between Cartan pairs which entwines all of the Cartan structure must come from an actor between the underlying groupoids. In this way an equivalence of categories is realised.

Title: Tracially amenable actions and purely infinite crossed products

Abstract: Amenable group actions on C*-algebras are actions that behave like actions of amenable groups. I will review the theory of amenable actions and report on applications to the classification of C*-algebras. I will also discuss how the search for new examples resulted in the concept of tracially amenable actions and how these give rise to new examples of purely infinite crossed products. This is joint work with E. Gardella, S. Geffen, P. Naryshkin and A. Vaccaro. 

Title: Equivariant Jiang-Su stability and equivariant property Gamma

Abstract: Equivariant Jiang-Su stability is an important regularity property for group actions on C*-algebras.  In this talk, I will explain what this property is and how it arises naturally in the context of the classification of C*-algebras and their actions. By work of Matui-Sato, it turns out that equivariant Jiang-Su stability can be captured in terms of uniform tracial data. In the rest of the talk I will explain how one can use this to establish equivariant Jiang-Su stability for certain actions, and how this leads to the recent result of Gábor Szabó and myself showing that equivariant Jiang-Su stability is equivalent to equivariant property Gamma under certain conditions.

Title: Simple AF embeddability for unimodular group C*-algebras

Abstract: For any locally compact group G, the left regular (unitary) representation generates a C*-algebra of bounded operators on the Hilbert space L^2(G). J. Rosenberg proved in the 80's that a discrete group G is amenable provided its induced C*-algebra forms a quasidiagonal set of operators on L^2(G), and he conjectured that the converse also holds. The conjecture was confirmed in 2015 by Tikuisis, White, and Winter, and using methods of Ozawa, Rørdam, and Sato for elementary amenable groups, they showed the stronger result that such group C*-algebras embed into a simple approximately finite-dimensional (AF) C*-algebra. I will report on some developements on how to extend this result to unimodular groups.

Title: A groupoid approach to sampling and interpolation in unimodular groups

Abstract: In this talk I will show how groupoids associated to point sets in unimodular locally compact groups and their von Neumann algebras can be used to approach sampling and interpolation problems in harmonic analysis. The talk is based on joint work with Sven Raum.

Title:  K-theory of noncommutative Bernoulli shifts

Abstract: For any discrete group G, for any G-set Z and for any unital C*-algebra A, the Bernoulli shifts G-action on the (minimal) tensor product of A over Z can be defined naturally. In a joint work with Sayan Chakraborty, Siegfried Echterhoff and Julian Kranz, we developed a new technique in operator K-theory to study the K-theory of the reduced crossed product of the Bernoulli shifts G-actions, assuming the Baum-Connes conjecture with coefficients for G. In this talk, I aim to describe this technique and some of our main results. I also plan to briefly describe the analogous results for algebraic K-theory. The latter is a joint work in progress with Julian Kranz.

Title: Finitely generated simple groups of homeomorphisms of R

Abstract: I will give an introduction to Thompson's groups F and T and then build and discuss three families of examples of finitely generated simple groups of homeomorphisms of the real line. This represents joint work with Yash Lodha and Cristobal Rivas.

Title:  The bundle of KMS states for flows on simple C*-algebras.   

Abstract: I will describe what is presently known about the structure of the collection of KMS states for flows on a simple separable unital C*-algebra, and I will try to sketch both the history behind and the proofs of the latest results. 

Title: Equivariant isomorphisms of quantum lens spaces

Abstract: In the study of noncommutative geometry many classical spaces have been given a quantum analogue. An example is quantum lens spaces, which are defined as fixed point algebras of the quantum sphere by Vaksman and Soibelman under the actions of finite cyclic groups. Quantum lens spaces have been given a graph $C^*$-algebraic description which has made it possible to work on classifying them.

Every quantum lens space comes with a natural circle action, leading to an equivariant isomorphisms problem. In this talk, I will present some recent work on the existence and construction of equivariant isomorphisms of low dimensional quantum lens spaces. Contrary to the isomorphism problem, we can no longer use the graph $C^*$-algebraic description to solve the equivariant isomorphism problem. This is due to the fact that full corners of graph $C^*$-algebras are not always equivariantly isomorphic to graph $C^*$-algebras themselves.

This is based on joint work with Søren Eilers.

Title: Homology of ample groupoids

Abstract: I will give a brief overview of the history of ample groupoid homology as used in operator algebras. I’ll introduce ample groupoid homology with the module approach, and sketch a cute proof that the homology is preserved by Morita equivalences of groupoids.

Title: A comparison of two quantum distances

Abstract:  Rieffel’s theory of compact quantum metric spaces comes equipped with the so-called quantum Gromov-Hausdorff distance, which allows one to measure the distance between two compact quantum metric spaces. In fact, there exist several competing versions of the quantum Gromov-Hausdorff distance, whose internal relationships are reasonably well understood, but in a recent work by van Suijlekom, a new candidate entered the scene.  Although this version is very closely related to Rieffel’s original definition,  it turns out that that the two are indeed different and, in my talk, I will explain a proof of this fact.  The talk is based on a joint paper with Jens Kaad. 

Title: An index pairing in scattering theory

Abstract: In this talk I will begin by introducing some concepts from mathematical scattering theory, including the wave and scattering operators and Levinson's theorem. I will review some recent work of Kellendonk and Richard, relating scattering theory to index theory and discuss how we can interpret Levinson's Theorem as an index pairing between the class of the scattering operator and an appropriate spectral triple.