# UK Virtual Operator Algebras Seminar

Due to the Coronavirus crisis we have created an online seminar on the Zoom platform meeting fortnightly on Thursdays at 4pm. The **Zoom coordinates** are announced using our **mailing list**, please contact one of the seminar organisers to have your name added to the mailing list.

Our aim is to host talks which are more expository than a typical research talk might be. We plan to invite experts in their field to present introductions to their areas of research, expositions of key proof techniques or constructions, or perhaps sketching the background required to appreciate recent breakthroughs. We hope talks will be widely accessible to graduate students and postdocs, as well as established researchers.

**Upcoming talks:**

**Upcoming talks:**

All times are UK time, which is currently British Summer Time (UTC+1).

**Thursday June 24, 2021**

We take a break. You may be interested in the Operator Algebras minisymposia at the ECM.

We then take a break for the summer.

**Previous talks:**

**Previous talks:**

**Thursday June 10, 2021****Speaker:**Adam Dor On (Copenhagen / Illinois)**Time:**4:00pm**Title:**Toeplitz quotient C*-algebras and ratio-limits for random walks (copy of the slides)**Abstract:**We showcase some newly emerging connections between the theory of random walks and operator algebras, obtained by associating concrete algebras of operators to random walks. The C*-algebras we obtain give rise to new and interesting notions of ratio limit space and boundary, which are computed by appealing to various works on the asymptotic behavior of transition probabilities for random walks. Our results are leveraged to shed light on a question of Viselter on symmetry-unique quotients of Toeplitz C*-algebras of subproduct systems arising from random walks.**Assumed knowledge:**Basics of operator theory, and some familiarity with universal and concrete algebras of operators. The necessary theory on random walks will be explained in the talk.**Thursday June 10, 2021****Speaker:**Cornelia Drutu (Oxford)**Time:**4:45pm**Title:**C* algebras and Geometric Group Theory (copy of the slides)**Abstract:**In this talk I will discuss a number of topics at the interface between C* algebras and Geometric Group Theory, with an emphasis on Kazhdan projections, various versions of amenability and their connection to the geometry of groups. This is based on joint work with P. Nowak and J. Mackay.**Thursday May 27, 2021**Emily Peters (Loyola University, Chicago)

Speaker:**Time:**4:00pm**Title:**Jones Index for Subfactors**Abstract:**In this talk I will explain how a subfactor (ie an inclusion of type II_1 factors) give rise to a diagrammatic algebra called the Temperley-Lieb-Jones algebra. We will observe the connection between the index of the subfactor, and the TLJ algebra. In the TLJ algebra setting, we will observe that indices below four are discrete, while any number above four can be an index.**Assumed knowledge**: von Neumann algebras, factors and their type classification**Thursday May 27, 2021**SĂžren Eilers (University of Copenhagen)

Speaker:**Time:**4:45pm**Title:**C*-equivalence of directed graphs (copy of slides)**Abstract:**The graph C*-algebra construction associates a unital C*-algebra to any directed graph with finitely many vertices and countably many edges in a way which generalizes the fundamental construction by Cuntz and Krieger. We say that two such graphs are C*-equivalent when they define isomorphic C*-algebras, and give a description of this relation as the smallest equivalence relation generated by a number of "moves" on the graph that leave the C*-algebras unchanged. The talk is based on recent work with Arklint and Ruiz, but most of these moves have a long history that I intend to present in some detail.**Assumed knowledge:**Basic knowledge of C*-algebras, in particular universal such. There is a strong undercurrent of classification by K-theory in the work that I intend to suppress completely.**Thursday April 29, 2021**Andreas Thom (TU Dresden)

Speaker:**Time:**4:00pm**Title:**On the isometrisability of group actions on p-spaces**Abstract:**In this talk we consider a p-isometrisability property of discrete groups. If p=2 this property is equivalent to unitarisability. We prove that any group containing a non-abelian free subgroup is not p-isometrisable for any pâ(1,â). We also discuss some open questions and possible relations of p-isometrisability with the recently introduced Littlewood exponent Lit(Î).**Assumed knowledge**: The talk will be gentle and no prior knowledge other than some familiarity with infinite groups and functional analysis is assumed.**Thursday April 29, 2021**Bram Mesland (Leiden)

Speaker:**Time:**4:45pm**Title:**The KK-theory perspective on noncommutative geometry (copy of slides)**Abstract:**The observation that the Dirac operator on a spin manifold encodes both the Riemannian metric as well as the fundamental class in K-homology leads to the paradigm of noncommutative geometry: the viewpoint that spectral triples generalise Riemannian manifolds. To encode maps between Riemannian manifolds, one is naturally led to consider the unbounded picture of Kasparov's KK-theory. In this talk I will explain how smooth cycles in KK-theory give a natural notion of noncommutative fibration, encoding morphisms noncommutative geometry in manner compatible with index theory.**Assumed knowledge**: Some familiarity with the K-theory of C*-algebras, basics of manifolds. Knowledge of K-homology or KK-theory is not required.**Thursday April 15, 2021**Shirly Geffen (Ben-Gurion University of the Negev and University of MĂŒnster)

Speaker:**Time:**4:00pm**Title:**Nuclear dimension of crossed products associated to partial homeomorphisms (copy of the slides)**Abstract:**The concept of a C*-algebraic partial automorphism, namely an isomorphism between two ideals of a C*-algebra, was introduced by Exel in the 1990s. Many important C*-algebras that cannot be written as a crossed product by a (global) automorphism, have a description as a crossed product by a partial automorphism. I will review the known results regarding the nuclear dimension of crossed products associated to dynamical systems, in connection to the classification program. I will discuss some cases in which crossed products associated to partial homeomorphisms on finite dimensional metrizable spaces, have finite nuclear dimension. This extends work on global systems by Hirshbreg and Wu.**Assumed knowledge**: Basic knowledge of C*-algebras including crossed products; some knowledge of Elliott's classification program might be useful.**Thursday April 15, 2021**Adam Skalski (IMPAN, Warsaw)

Speaker:**Time:**4:45pm**Title:**Classifying C*-algebras with the K-theory: (Hecke) algebras of Coxeter groups (copy of the slides)**Abstract:**I will first discuss in very general terms how one can attempt to distinguish unital C*-algebras using their K-theoretic invariants. Then I will present a specific example of group C*-algebras of Coxeter groups and their deformations, showing both the power and limitations of K-theoretic techniques in a concrete context.

Based on joint work with Sven Raum.**Assumed knowledge**: K_0 and K_1 - groups for unital C*-algebras, (amalgamated) free product of C*-algebras, group C*-algebras**Thursday April 1, 2021**Ying-Fen Lin (Queen's University Belfast)

Speaker:**Time:**4:00pm**Title:**Twisted Steinberg algebras (copy of the slides)**Abstract:**Groupoid C*-algebras and twisted groupoid C*-algebras are introduced by Renault in the late â70, where twisted groupoid C*-algebras are generalisation of groupoid C*-algebras in which multiplication and involution are twisted by a 2-cocycle on the groupoid. Twisted groupoid C*-algebras have since proved extremely important in the study of structural properties for large classes of C*-algebras. Steinberg algebras are introduced independently by Steinberg and Clark, Farthing, Sims and Tomforde around 2010 which are a purely algebraic analogue of groupoid C*-algebras. Steinberg algebras not only provide useful insight into the analytic theory of groupoid C*-algebras, they also give rise to interesting examples of *-algebras; for example, all Leavitt path algebras and Kumjian-Pask algebras can be realised as Steinberg algebras. In this talk, I will first recall some relevant background on topological groupoids and twisted groupoid C*-algebras, then I will introduce twisted Steinberg algebras which generalise the Steinberg algebras and provide a purely algebraic analogue of twisted groupoid C*-algebras.

This is joint work with Becky Armstrong, Lisa O. Clark, Kristin Courtney, Kathryn McCormick and Jacqui Ramagge.**Assumed knowledge**: Basic C*-algebra theory, including Ă©tale groupoid C*-algebras and twisted groupoids.**Thursday April 1, 2021**Mehrdad Kalantar (University of Houston)

Speaker:**Time:**4:45pm**Title:**On ideal and trace structures of C*-algebras of quasi-regular representations**Abstract:**The talk is concerned with the general problems of determining ideal and trace structures of C*-algebras generated by quasi-regular representations of discrete groups. For certain classes of subgroups, we obtain complete description of traces and characterization of simplicity of the C*-algebras of the corresponding quasi-regular representations. These results are consequences of a more fundamental uniqueness property of a class of morphisms in the category of C*-dynamical systems, called `noncommutative boundary maps'. The main technical tool in proving the uniqueness of these maps is the notion of boundary actions in the sense of Furstenberg. We will explain the general/categorical ideas behind the proofs, and give applications in concrete examples.

This is joint work with Eduardo Scarparo.**Assumed knowledge**: basics of C*-algebras, group actions and unitary representations.**Thursday March18, 2021**Mark Lawson (Heriot-Watt University)

Speaker:**Time:**4:00pm**Title:**The universal Boolean inverse semigroup presented by the abstract Cuntz-Krieger relations (copy of the slides)**Abstract:**We describe the non-commutative duality that exists between Boolean inverse semigroups and what we call Boolean groupoids (etale groupoids whose space of identities is a Boolean space) and then apply this theory to study what we term the*tight completion*of an inverse semigroup. This will lead us naturally to the Cuntz-Krieger relations. This is joint work with Alina Vdovina. A paper on this work is available at arXiv:1902.02583v3.**Assumed knowledge**: I shall explain everything from scratch but an exposure to the basics of inverse semigroup theory would be helpful. You can find what you need here: Section 2 of arXiv:2006.01628v1 is more than enough. For background on etale groupoids the first few pages of*Lectures on Ă©tale groupoids, inverse semigroups and quantales,**Lecture Notes**for the GAMAP IP Meeting, Antwerp, 4-18 September, 2006, 115 pp.***Thursday March 18, 2021**Judith Packer (University of Colorado, Boulder)

Speaker:**Time:**4:45pm**Title:**Cocycles associated to certain groupoids associated to commuting k-tuples of local homeomorpisms acting on a compact metric space (copy of the slides)**Abstract:**Suppose we are given k commuting surjective local homeomorphisms acting on a compact metric space X. It is well-known that we can construct an etale locally compact Hausdorff groupoid G from this data. This talk will go over the construction of both the groupoid G and its associated groupoid C*-algebra C*(G). We review the continuous 1-cocycles in the groupoid G taking on values in a locally compact abelian group, and provide a characterization of these, given in terms of*k*-tuples of continuous functions on the unit space of G satisfying certain canonical identities. When the locally compact abelian group being considered is the additive group of real numbers, we discuss the construction of a one-parameter automorphism group acting on C*(G) corresponding to the continuous 1-cocycle on G. Depending on time constraints, we might discuss the question of the existence of KMS (Kubo-Martin-Schwinger) states corresponding to these one-parameter automorphism groups.

The work discussed is joint with C. Farsi, L. Huang, and A. Kumjian.**Assumed knowledge**: Basic C*-algebra theory.**Thursday March 4, 2021**Kevin Brix

Speaker:**Time:**4:00pm**Title:**The conjugacy problem for subshifts and Williams' conjecture (copy of the slides)**Abstract:**Symbolic dynamics and the conjugacy problem for subshifts has had profound influence on ideas and constructions in C*-algebra theory (e.g. Cuntz-Krieger algebras, graph algebras, and C*-classification). I will discuss how this problem and Williams' conjecture continue to inspire new results and deeper investigation into C*-algebras arising from dynamical systems. Can we as operator algebraists repay the debt?**Assumed knowledge**: Basic C*-algebra theory. Might contain traces of groupoids, cohomology, and C*-classification.**Thursday March 4, 2021**Sylvie Paycha (UniversitĂ€t Potsdam)

Speaker:**Time:**4:45pm**Title:**Regularised traces and Getzlerâs rescaling revisited**Abstract:**Inspired by Gilkey's invariance theory, Connes' deformation to the normal cone and Getzler's rescaling method, we single out a class of geometric operators among pseudodifferential operators acting on sections of a class of natural vector bundles, to which we attach a rescaling degree. This degree is then used to express regularised traces of geometric operators in terms of a rescaled limit of Wodzicki residues. When applied to complex powers of the square of a Dirac operator, this amounts to expressing the index of a Dirac operator in terms of a local residue involving the Getzler rescaled limit of its square.

This is joint work with Georges Habib.**Thursday February 18, 2021**Sam Kim (Glasgow)

Speaker:**Time:**4pm**Title:**A duality theorem for non-unital operator systems**Abstract:**The recent work on nc convex sets of Davidson, Kennedy, and Shamovich show that there is a rich interplay between the category of operator systems and the category of compact nc convex sets, leading to new insights even in the case of C*-algebras. The category of nc convex sets are a generalization of the usual notion of a compact convex set that provides meaningful connections between convex theoretic notions and notions in operator system theory. In this talk, we present a duality theorem for norm closed self-adjoint subspaces of B(H) that do not necessarily contain the unit. Using this duality, we will describe various C*-algebraic and operator system theoretic notions such as simplicity and subkernels in terms of their convex structure. This is joint work with Matthew Kennedy and Nicholas Manor.**Assumed knowledge**: Basic C*-algebra theory**Thursday February 18, 2021**Evgenios Kakariadis (Newcastle)

Speaker:**Time:**4:45pm**Title:**Co-universal C*-algebras for product systems (copy of the slides)**Abstract:**Continuous product systems were introduced and studied by Arveson in the late 1980s. The study of their discrete analogues started with the work of Dinh in the 1990âs and it was formalized by Fowler in 2002. Discrete product systems are semigroup versions of C*-correspondences, that allow for a joint study of many fundamental C*-algebras, including those which come from C*-correspondences, higher rank graphs and elsewhere.

Katsuraâs covariant relations have been proven to give the correct Cuntz-type C*-algebra for a single C*-correspondence X. One of the great advantages of Katsura's Cuntz-Pimsner C*-algebra is its co-universality for the class of gauge-compatible injective representations of X. In the late 2000s Carlsen-Larsen-Sims-Vittadello raised the question of existence of such a co-universal object in the context of product systems. In their work, Carlsen-Larsen-Sims-Vittadello provided an affirmative answer for quasi-lattices, with additional injectivity assumptions on X. The general case has remained open and will be addressed in these talk using tools from non-selfadjoint operator algebra theory.

This is joint work with Adam Dor-On, Elias Katsoulis, Marcelo Laca and Xin Li.**Assumed knowledge**: Basic C*-algebras theory (and probably some familiarity with examples like graph C*-algebras and C*-algebras of discrete groups).**Thursday December 17, 2020**

Christmas Party! The festivities will begin with 3 minute talks by PhD students and beginning postdocs. We will split the audience randomly into one of three breakout rooms, which will feature the following speakers, with a break at 16:25:**Room 1**

16:05 Antje Dabeler

16:10 M. Ali Asadi-Vasfi

16:15 Amudhan Krishnaswamy Usha

16:20 Ali Raad

16:30 Gerrit Vos

16:35 Mary Angelica Tursi

16:40 Mitch Haslehurst**Room 2**

16:05 Mario Klisse

16:10 Diego Martinez

16:15 Alistair Miller

16:20 Srivatsav Kunnawalkam Elayavalli

16:30 Jeroen Winkel

16:35 Chian Yeong Chuah

16:40 Dimitrios Andreou**Room 3**

16:05 Gabriel Favre

16:10 Bat-Od Battseren

16:15 Shen Lu

16:20 MarĂa de Nazaret Cueto Avellaneda

16:30 Dimitris Gerontogiannis

16:35 Sergio Giron Pacheco

16:40 Joseph Eisner

For talk titles and one sentence abstracts, please see the PDF of abstracts.

After the talks, there will be some time for chit-chat and then a couple rounds of operator algebras Pictionary/charades. We encourage you to bring a snack and drink for the chit-chat / Pictionary / charades part of the party (Christmas jumpers welcome).**Thursday December 3, 2020****Speaker:**Vern Paulsen (Waterloo)**Time:**4:00pm**Title:**Algebras and Games (copy of the slides)**Abstract:**There are many constructions that yield C*-algebras. For example, we build them from groups, quantum groups, dynamical systems, and graphs. In this talk we look at C*-algebras that arise from a certain type of game. It turns out that the properties of the underlying game gives us very strong information about existence of traces of various types on the game algebra. The recent solution of the Connes Embedding Problem arises from a game whose algebra has a trace but no hyperlinear trace.**Assumed knowledge:**Familiarity with tensor products of Hilbert spaces, the algebra of a discrete group, and free products of groups.**Thursday December 3, 2020****Speaker:**Anna Duwenig (University of Wollongong)**Time:**4:45pm**Title:**Cartan subalgebras for non-principal twisted groupoid C*-algebras (copy of the slides)**Abstract:**The reduced C*-algebra of a topologically principal twisted groupoid has a canonical Cartan subalgebra: functions on its unit space. The remarkable Weyl construction, due to Renault, asserts the converse: If a C*-algebra A admits a Cartan subalgebra, there exists such a groupoid whose C*-algebra is isomorphic to A in a Cartan-preserving way. In this talk, I will explain how to identify subgroupoids of (not necessarily topologically principal!) groupoids that also give rise to Cartan subalgebras. If time permits, we will have a look at the associated Weyl groupoid and twist. This talk is based on joint works with Gillaspy, Norton, Reznikoff, and Wright, and with Gillaspy and Norton.**Assumed knowledge:**Basic C*-algebra theory, including Ă©tale groupoid C*-algebras. Knowledge about (2-cocyle-)twisted groupoids and about transformation/action groupoids is helpful, but not required.**Thursday November 19, 2020**Yemon Choi

Speaker:**(Lancaster)**4:00pm

Time:**Title:**An introduction to non-amenability of B(E) (copy of the slides)**Abstract:**Amenability for Banach algebras was introduced in the 1970s by Johnson, and one of the earliest questions was to decide if there are any infinite-dimensional Banach spaces E such that B(E) is amenable. For over 30 years no such E was known, yet very few cases were known where non-amenability could be proved. In this talk I will give an introduction to this problem, starting with the definition of amenability and some examples, before mentioning some of the dramatic progress since 2000 due to various authors (Read, Ozawa, Runde, Argyros and Haydon, ...) Time permitting, I will talk briefly about a new proof that B(L_1) is not amenable, using some old results on the ideal of representable operators.**Assumed knowledge**: Basic knowledge of Banach algebras and linear functional analysis, such as the definitions of $\ell_p$ and $L_p$. Some exposure to tensor products in the purely algebraic sense would be helpful for following certain statements. No prior knowledge of amenability is required.**Thursday November 19, 2020**Simon Schmidt (

Speaker:**Saarland University**)**Time:**4:45pm**Title:**Quantum automorphism groups of finite graphs (copy of the slides)**Abstract:**To capture the symmetry of a graph one studies its automorphism group. We will talk about a generalization of automorphism groups of finite graphs in the framework of Woronowiczâs compact matrix quantum groups. An important task is to see whether or not a graph has quantum symmetry, i.e. whether or not its quantum automorphism group is commutative. We will discuss a criterion for graphs to have quantum symmetry and also look at tools that allow us to show that a graph does not have quantum symmetry. Finally, we will discuss a link between quantum automorphism groups and non-local games.**Assumed knowledge**: Basic knowledge of C*-algebras**Thursday November 5, 2020**Caleb Eckhardt

Speaker:**(Miami University)**4:00pm

Time:**Title:**Virtually polycyclic groups and their C*-algebras (copy of the slides)**Abstract:**Polycyclic groups form an interesting and well-studied class of groups that properly contain the finitely generated nilpotent groups. I will discuss the C*-algebras associated with virtually polycyclic groups, their maximal quotients and recent work with Jianchao Wu showing that they have finite nuclear dimension.**Assumed knowledge**: Basic knowledge of C*-algebras.**Thursday November 5, 2020**Chris Bruce

Speaker:**(Queen Mary University London and University of Glasgow)**4:45pm

Time:**Title:****Abstract:**I will give an introduction to semigroup C*-algebras of ax+b-semigroups over rings of algebraic integers in algebraic number fields, a class of C*-algebras that was introduced by Cuntz, Deninger, and Laca. After explaining the construction, I will briefly discuss the state-of-the-art for this example class: These C*-algebras are unital, separable, nuclear, strongly purely infinite, and have computable primitive ideal spaces. In many cases, e.g., for Galois extensions, they completely characterise the underlying algebraic number field.**Assumed knowledge**: Intermediate theory of C*-algebras; basic algebraic number theory (helpful, but not required).**Thursday October 22, 2020**Lauren Ruth

Speaker:**(Mercy College, NY, USA)**3:30pm

Time:**Title:**Von Neumann algebras and equivalences between groups**Abstract:**We have various ways of describing the extent to which two countably infinite groups are "the same." Are they isomorphic? If not, are they commensurable? Measure equivalent? Quasi-isometric? Orbit equivalent? W*-equivalent? Von Neumann equivalent? In this expository talk, we will define these notions of equivalence, discuss the known relationships between them, and work out some examples. Along the way, we will describe recent joint work with Ishan Ishan and Jesse Peterson.**Assumed knowledge**: Elementary theory of von Neumann algebras; crossed product construction**Thursday October 22, 2020**Robin Deeley

Speaker:**(University of Colorado, Boulder)**4:15pm

Time:**Title:**The C*-algebras associated to a Wieler solenoid (copy of the slides)**Abstract:**Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Through examples I will discuss how this allows one to compute the K-theory of the stable algebra, S, and the stable Ruelle algebra, S\rtimes Z. These computations involve writing S as a stationary inductive limit and S\rtimes Z as a Cuntz-Pimsner algebra. These constructions reemphasize the view point that Smale space C*-algebras are higher dimensional generalizations of Cuntz-Krieger algebras. The main results are joint work with Magnus Goffeng and Allan Yashinski.**Assumed knowledge**: Basic C*-algebra theory including etale groupoid C*-algebras as discussed in previous talks by, for example, Nadia Larsen and Karen Strung. No knowledge of Smale spaces or Wieler solenoids is required.**Thursday October 8, 2020**Gandalf Lechner

Speaker:**(Cardiff)**16:00

Time:**Title:**Yang-Baxter representations of the infinite braid group and subfactors (copy of the slides)**Abstract:**Unitary solutions of the Yang-Baxter equation ("R-matrices") play a prominent role in several fields, such as quantum field theory and topological quantum computing, but are difficult to find directly and remain somewhat mysterious. In this talk I want to explain how one can use subfactor techniques to learn something about unitary R-matrices, and a research programme aiming at the classification of unitary R-matrices up to a natural equivalence relation. This talk is based on joint work with Roberto Conti, Ulrich Pennig, and Simon Wood.**Assumed knowledge**: Elementary von Neumann algebra theory, ideally including the hyperfinite II_1 factor and trace-preserving conditional expectations. It would be helpful, but not necessary, to know the definition of the braid groups B_n and the Yang-Baxter equation.**Thursday October 8, 2020**Jamie Gabe

Speaker:**(University of Southern Denmark)**16:45

Time:**Title:**Purely infinite C*-algebras and their classification (copy of the slides)**Abstract:**Cuntz introduced pure infiniteness for simple C*-algebras as a C*-algebraic analogue of type III von Neumann factors. Notable examples include the Calkin algebra B(H)/K(H), the Cuntz algebras O_n, simple Cuntz-Krieger algebras, and other C*-algebras you would encounter in the wild. The separable, nuclear ones were classified in celebrated work by Kirchberg and Phillips in the mid 90s. I will talk about these topics including the non-simple case if time permits.**Assumed knowledge**: basic C*-algbra knowledge including tensor products.**Thursday September 24, 2020**Nadia Larsen (University of Oslo)

Speaker:**Time**: 16:00**Title**: Groupoid C*-algebras and ground states (copy of the slides)**Abstract**: C*-algebras associated to etale groupoids appear as a versatile construction in many contexts. For instance, groupoid C*-algebras allow for implementation of natural one-parameter groups of automorphisms obtained from continuous cocycles. This provides a path to quantum statistical mechanical systems, where one studies equilibrium states and ground states. The early characterisations of ground states and equilibrium states for groupoid C*-algebras due to Renault have seen remarkable refinements. It is possible to characterise in great generality all ground states of etale groupoid C*-algebras in terms of a boundary groupoid of the cocycle (joint work with Laca and Neshveyev). The steps in the proof employ important constructions for groupoid C*-algebras due to Renault.**Assumed knowledge**: Basic C*-algebra theory.**Thursday September 24, 2020**David Kyed (University of Southern Denmark)

Speaker:**Time**: 16:45**Title**: An introduction to compact quantum metric spaces (copy of the slides)**Abstract**: The Gelfand correspondence between compact Hausdorff spaces and unital C*-algebras justifies the slogan that C*-algebras are to be thought of as "non-commutative topological spaces", and Rieffel's theory of compact quantum metric spaces provides, in the same vein, a non-commutative counterpart to the theory of compact metric spaces. The aim of my talk is to introduce the basics of the theory and explain how the classical Gromov-Hausdorff distance between compact metric spaces can be generalized to the quantum setting. If time permits, I will touch upon some recent results obtained in joint work with Jens Kaad and Thomas Gotfredsen.**Assumed knowledge**: Elementary C*-algebra theory**Thursday September 10, 2020**Makoto Yamashita (Oslo)

Speaker:**Time**: 16:00**Title**: Compact quantum Lie groups (copy of the slides)**Abstract:**Quantum groups, which has been a major overarching theme across various branches of mathematics since late 20th century, appear in many ways. Deformation of compact Lie groups is a particularly fruitful paradigm that sits in the intersection between operator algebraic approach to quantized spaces on the one hand, and more algebraic one arising from study of quantum integrable systems on the other.

On the side of operator algebra, Woronowicz defined the C*-bialgebra representing quantized SU(2) based on his theory of pseudospaces. This gives a (noncommutative) C*-algebra of "continuous functions" on the quantized group SUq(2).

Its algebraic counterpart is the quantized universal enveloping algebra Uq(đđ2), due to KulishâReshetikhin and Sklyanin, coming from a search of algebraic structures on solutions of the Yang-Baxter equation. This is (an essentially unique) deformation of the universal enveloping algebra U(đđ2) as a Hopf algebra.

These structures are in certain duality, and have far-reaching generalization to compact simple Lie groups like SU(n). The interaction of ideas from both fields led to interesting results beyond original settings of these theories.

In this introductory talk, I will explain the basic quantization scheme underlying this "q-deformation", and basic properties of the associated C*-algebras. As part of more recent and advanced topics, I also want to explain an interesting relation to complex simple Lie groups through the idea of quantum double.**Assumed Knowledge**: Basic familiarity with C*-algebras and Lie algebras / Lie groups. It helps to know the ideas around GelfandâNaimark duality principle and K-theory for C*-algebras. On the side of representation theory, it helps to have some familiarity in structure and representations of simple complex/compact Lie groups.**Thursday September 10, 2020**GĂĄbor SzabĂł (KU Leuven)

Speaker:**Time**: 16:45**Title**: A peek into the classification of C*-dynamics (copy of the slides)**Abstract**: In the structure theory of operator algebras, it has been a reliable theme that a classification of interesting classes of objects is usually followed by a classification of group actions on said objects. A good example for this is the complete classification of amenable group actions on injective factors, which complemented the famous work of Connes-Haagerup. On the C*-algebra side, progress in the Elliott classification program has likewise given impulse to the classification of C*-dynamics. Although C*-dynamical systems are not yet understood at a comparable level, there are some sophisticated tools in the literature that yield satisfactory partial results. In this introductory talk I will outline the (known) classification of finite group actions with the Rokhlin property, and in the process highlight some themes that are still relevant in today's state-of-the-art.**Assumed Knowledge**: Elementary C*-algebra theory. Some passing familiarity with elementary Elliott intertwining, for example as treated in Chapter 2 of RĂžrdam's book (in particular Corollary 2.3.4), is also assumed. Some prior knowledge on the general Elliott classification program (as treated in prior talks) is helpful but not necessary.**Thursday August 27, 2020**

Summer holiday**Thursday August 13, 2020****Speaker:**Ulrich Pennig (Cardiff)**Time**: 16:00**Title**: Bundles of C*-algebras: An Introduction to Dixmier-Douady theory (copy of the slides)**Abstract**: A bundle of C*-algebras is a collection of algebras continuously parametrised by a topological space. There are (at least) two different definitions in operator algebras that make this intuition precise: Continuous C(X)-algebras provide a flexible analytic point of view, while locally trivial C*-algebra bundles allow a classification via homotopy theory. The section algebra of a bundle in the topological sense is a C(X)-algebra, but the converse is not true. In this talk I will compare these two notions using the classical work of Dixmier and Douady on bundles with fibres isomorphic to the compacts as a guideline. I will then explain joint work with Marius Dadarlat, in which we showed that the theorems of Dixmier and Douady can be generalized to bundles with fibers isomorphic to stabilized strongly self-absorbing C*-algebras. An important feature of the theory is the appearance of higher analogues of the Dixmier-Douady class.**Assumed Knowledge**: basic familiarity with C*-algebras, some background in topology, in particular cohomology, might be useful, but is not required, similarly for strongly self-absorbing C*-algebras.**Thursday August 13, 2020****Speaker:**Amine Marrakchi (Lyon)**Time**: 16:45**Title**: The Gaussian functor and its applications (copy of the slides)**Abstract**: The Gaussian functor associates to every real hilbert space a canonical probability space called the Gaussian probability space. This construction is an important tool with a wide range of applications in various domains : ergodic theory, probability theory, operator algebras, representation theory and more recently geometric group theory. In this talk, I will recall this construction and present several applications.**Thursday July 30, 2020****Speaker:**Veronique Fischer (University of Bath)**Time**: 4:00pm**Title**: Quantum Limits. (copy of the slides)**Abstract**: Quantum limits are objects describing the limit of quadratic quantities (Af_n,f_n) where (f_n) is a sequence of unit vectors in a Hilbert space and A ranges over an algebra of bounded operators. We will discuss the motivation underlying this notion with some important examples from quantum mechanics and from analysis.**Assumed Knowledge**: Elementary C*-algebra theory. Some exposure to quantum mechanics, partial differential equations (PDE) and pseudo-differential operators might be useful.**Thursday July 16, 2020****Speaker:**Karen Strung (Czech Academy of Sciences)**Time**: 4:00pm**Title**: Groupoids for C*-algebras (copy of the slides)**Abstract**: Groupoids are a generalisation of groups where multiplication is only partially defined. When equipped with a so-called âĂ©taleâ topology, they can be considered as generalisations of discrete groups. In his thesis, Jean Renault initiated the study of the C*-algebras of groupoids. The construction of a C*-algebra from an Ă©tale groupoid generalizes not only the construction of a discrete group C*-algebra, but also many well-known constructions such as crossed products of commutative C*-algebras, Cuntz--Krieger C*-algebras, as well as other constructions related to dynamical systems such as C*-algebras from Smale spaces and tiling C*-algebras. I will discuss the basics of Ă©tale groupoid C*-algebras as well as the role that groupoids have playedâand continue to playâas models for various classes of C*-algebras, particularly those related to the classification programmeâthose that are separable, simple, nuclear and Jiang-Su stable.**Assumed Knowledge**: Elementary C*-algebra theory. Some exposure to topological dynamics might be useful.**Thursday July 16, 2020**Sven Raum (Stockholm University)

Speaker:**Time**: 4:45pm**Title**: C*-superrigidity (copy of the slides)**Abstract**: A discrete group is called C*-superrigid, if it can be recovered from its reduced group C*-algebra. This notation parallels W*-superrigidty for von Neumann algebras. Motivated by Higman's unit conjecture for complex group rings, one can ask whether every torsion-free group is C*-superrigid. Surprisingly until 2017 the only known examples of torsion-free C*-superrigid groups were abelian.

In this talk, I will introduce you to the concept and motivation of C*-superrigidity. Subsequently, I will describe developments around this topic since 2017, and exemplify relevant ideas through my joint work with Caleb Eckhardt, which shows that all finitely generated, torsion-free, 2-step nilpotent groups are C*-superrigid. I will end the talk by sharing my perspective on the future of the topic.**Assumed Knowledge**: Basics of operator algebras.**Thursday July 2, 2020**: Nigel Higson (Pennsylvania State University, USA)

Speaker**Time**: 4:00pm**Title**: John Roe and Coarse Geometry (Slides 1, slides 2)**Abstract**: John Roe was a much admired figure in topology and noncommutative geometry, and the creator of the C*-algebraic approach to coarse geometry. John died in 2018 at the age of 58. My aim in the first part of the lecture will be to explain in very general terms the major themes in Johnâs work, and illustrate them by presenting one of his best-known results, the partitioned manifold index theorem. After the break I shall describe a later result, about relative eta invariants, that has inspired an area of research that is still very active.**Assumed Knowledge**: First part: basic familiarity with C*-algebras, plus a little topology. Second part: basic familiarity with K-theory for C*-algebras.**Thursday June 18, 2020****Speaker**: Ivan Todorov (Queen's University Belfast)**Time**: 4:00pm**Title**: Non-local games: operator algebraic approaches (copy of the slides)**Abstract**: The study of non-local games has involved fruitful interactions between operator algebra theory and quantum physics, with a starting point the link between the Connes Embedding Problem and the Tsirelson Problem, uncovered by Junge et al (2011) and Ozawa (2013). Particular instances of non-local games, such as binary constraint system games and synchronous games, have played an important role in the pursuit of the resolution of these problems. In this talk, I will summarise part of the operator algebraic toolkit that has proved useful in the study of non-local games and of their perfect strategies, highlighting the role C*-algebras and operator systems play in their mathematical understanding.**Assumed Knowledge**: Basic operator algebra theory**Thursday June 18, 2020**: Matthew Kennedy (University of Waterloo, Canada)

Speaker**Time**: 4:45pm**Title**: The algebraic structure of C*-algebras associated to groups (copy of the slides)**Abstract**: Since the work of von Neumann, the theory of operator algebras has been closely linked to the theory of groups. On the one hand, operator algebras constructed from groups provide an important source of examples and insight. On the other hand, many problems about groups are most naturally studied within an operator-algebraic framework. In this talk I will give an overview of some problems relating the structure of a group to the structure of a corresponding C*-algebra. I will discuss recent results and some possible future directions.**Assumed Knowledge**: Basics of operator algebras.**Further reading:**On C*-simplicity: Sven Ruam's Bourbaki Seminar**Thursday June 4, 2020****Speaker**: Ana Khukhro (University of Cambridge)**Time**: 4:00pm**Title**: Expanders and generalisations (copy of the slides)**Abstract**: After recalling some motivation for studying highly-connected graphs in the context of operator algebras and large-scale geometry, we will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang. We will explore some applications of this definition, hopefully culminating in joint work with Li, Vigolo and Zhang.**Assumed Knowledge**: basic operator-algebraic concepts, some geometric intuition**Thursday June 4, 2020****Speaker**: Hannes Thiel (University of MĂŒnster, Germany)**Time**: 4:45pm**Title**: Cuntz semigroups (copy of the slides)**Abstract**: The Cuntz semigroup is a geometric refinement of K-theory that plays an important role in the structure theory of C*-algebras. It is defined analogously to the Murray-von Neumann semigroup by using equivalence classes of positive elements instead of projections.

Starting with the definition of the Cuntz semigroup of a C*-algebra, we will look at some of its classical applications. I will then talk about the recent breakthroughs in the structure theory of Cuntz semigroups and some of the consequences.**Assumed Knowledge**: Basic C*-algebra theory, including comparison theory for projections.**Thursday May 21, 2020**: Kristin Courtney (University of MĂŒnster, Germany)

Speaker**Time**: 4:00pm**Title**: Kirchbergâs QWEP Conjecture: Between Connesâ and Tsirelsonâs Problems (copy of slides)**Abstract**: In January of this year, a solution to Connes' Embedding Problem was announced on arXiv. The paper itself deals firmly in the realm of information theory and relies on a vast network of implications built by many hands over many years to get from an efficient reduction of the so-called Halting problem back to the existence of finite von Neumann algebras that lack nice finite-dimensional approximations. The seminal link in this chain was forged by astonishing results of Kirchberg which showed that Connes' Embedding Problem is equivalent to what is now known as Kirchberg's QWEP Conjecture. In this talk, I aim to introduce Kirchberg's conjecture and to touch on some of the many deep insights in the theory surrounding it.**Assumed Knowledge**: Basic von Neumann and C*-theory**Thursday May 21, 2020**: Alain Valette (University of NeuchĂątel, Switzerland)

Speaker**Time**: 4:45pm**Title**: Some examples of the Baum-Connes assembly map (copy of the slides)**Abstract**: We will introduce the Baum-Connes conjecture without coefficients, in the setting of discrete groups, and try to explain why it is interesting for operator algebraists. We will give some idea of the LHS and the RHS of the conjecture, without being too formal, and rather than trying to define the assembly map, we will explain what it does for finite groups, for the integers, for free groups, and finally for wreath products of a finite group with the integers (the latter result is joint work with R. Flores and S. Pooya; it raises a few open questions about classifying the corresponding group C*-algebras up to isomorphism).**Assumed Knowledge**: groups, group C*-algebras, low-dimensional homology.**Thursday May 14, 2020**Francesca Arici (Universiteit Leiden)

Speaker:**Time:**4:00pm**Title:**An introduction to Cuntz--Pimsner algebras (copy of the slides)**Abstract:**In 1997 Pimsner described how to construct two universal C*-algebras associated with an injective C*-correspondence, now known as the Toeplitz--Pimsner and Cuntz--Pimsner algebras. In this talk I will recall their construction, focusing for simplicity on the case of a finitely generated projective correspondence. I will describe the associated six-term exact sequence in K(K)-theory and explain how these can be used in practice for computational purposes. Finally, I will describe how, in the case of a self-Morita equivalence, these exact sequences can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles.**Assumed Knowledge:**Elementary C*-algebra theory.**Thursday May 14, 2020**Christopher Schafhauser (University of Nebraska, Lincoln)

Speaker:**Time:**4:45pm**Title:**Approximating traces: what, why and how. (copy of the slides)**Abstract:**Traces have played a fundamental role in operator algebras every since the beginning of the subject starting with Murray and von Neumannâs study of finite von Neumann algebras and the classification of projections in such algebras via the centre-valued trace. In Connesâs study of injective II_1-factors, he considered a certain finite-dimensional approximation property of the trace (now known as amenability) and showed a II_1-factor is injective if and only if the trace is amenable â this equivalence is one of the major steps in his proof that injective implies hyperfinite. Several related approximation conditions were introduced and studied by N. Brown in the early 2000âs. His notion of a quasidiagonal trace, which is motivated both by Connesâs notion of an amenable trace and by Halmosâs notion of a quasidiagonal operator, has proven to be especially important in the structure of simple nuclear C*-algebras and plays a key role in the classification of such algebras through the work of Elliott, Gong, Lin, and Niu. I will discuss approximation properties of traces (in particular amenability and quasidiagonality) and the relations between them.**Assumed Knowledge:**Basics of C*-algebras.**Thursday April 30, 2020****Speaker**: Scott Atkinson (University of California, Riverside)**Time**: 4:00pm**Title**: Amenability via ultraproduct embeddings for II_1 factors (copy of the slides)**Abstract**: The property of amenability is a cornerstone in the study and classification of II_1 factor von Neumann algebras. Likewise, ultraproduct analysis is an essential tool in the subject. We will discuss the history, recent results, and open questions on characterizations of amenability for separable II_1 factors in terms of embeddings into ultraproducts.**Assumed Knowledge**: Casual familiarity with II_1 factors (definition and go-to examples). Fundamental operator algebraic notions like ucp maps and unitary equivalence. The tracial ultraproduct construction will be reviewed in the talk.**Thursday April 30, 2020****Speaker**: Christian BĂ¶nicke (University of Glasgow)**Time**: 4:45pm**Title**: Extensions of C*-algebras (copy of the slides)**Abstract**: Having its roots in classical operator theoretic questions, the theory of extensions of C*-algebras is now a powerful tool with applications in geometry and topology and of course within the theory of C*-algebras itself. In this talk I will give a gentle introduction to the topic highlighting some classical results and more recent applications and questions.**Assumed Knowledge**: Elementary C*-algebra theory. Some exposure to operator K-theory is helpful but not strictly necessary.**Thursday April 16, 2020****Speaker**: David Seifert (Newcastle University)**Time**: 4:00pm**Title**: Introduction to C_0 semigroups (copy of the slides)**Abstract**: This talk will introduce some of the basic notions and results in the theory of C_0-semigroups, including generation theorems, growth and spectral bounds. If time permits, I will also try to discuss one or two classical results in the asymptotic theory of C_0-semigroups.**Assumed Knowledge**: Only basic operator theory, ideally including some awareness of closed (but not necessarily bounded) linear operators on Banach spaces.**Thursday April 16, 2020****Speaker**: Runlian Xia (University of Glasgow)**Time**: 4:45pm**Title**: Introduction to non-commutative L_p-spaces (copy of the slides)**Abstract**: This talk will give an easy introduction to non-commutative L_p spaces associated with a tracial von Neumann algebra. Then I will focus on non-commutative L_p spaces associated to locally compact groups and talk about some interesting completely bounded multipliers on them.**Assumed Knowledge**: Basics of von Neumann algebras**Thursday April 2, 2020****Speaker**: Sam Evington (University of Oxford)**Time**: 4pm**Title**: The Jiang-Su Algebra (copy of the slides)**Abstract**: In this talk, I will give an elementary introduction to the Jiang-Su algebra and outline its role in the theory of C*-algebras.**Assumed knowledge**: Basics of C*-algebras and K_0