Abstract: I will discuss the notion of an $L^p$-operator algebra, which are Banach algebras which act on $L^p$ spaces, but approached by analogy to $C^*$ and von Neumann algebras; the $p=2$ case is of course that of non-self-adjoint operator algebras, in full generality.  I will discuss what happens when $p=1$, and will argue that this should be thought of as a special case.  I will also discuss other notions of $\ell^1$-operator algebras, which are different from algebras acting on $L^1$-spaces, but somehow related.  This will be through two families of examples: work with Bence Horvath on analogues of the Cuntz algebras, and work in progress on Generic algebras and Hecke algebras.

Abstract: We discuss some results on the simplicity of inverse semigroup algebras, and the Steinberg algebras of étale groupoids. In particular, we discuss characterizations of simplicity, and consider the algebraic analogues of the essential algebra and the singular ideal. We show that the natural action of an inverse semigroups $S$ on its tight characters space $\hat E_T$ always induces a representation of the essential algebra on the vector space $K\hat E_T$, and if $S$ satisfies some additional conditions, this representation is faithful.

This work is joint with Benjamin Steinberg and Chris Bruce.

Abstract: A C*-algebra is residually finite-dimensional (RFD) if it has a separating family of finite-dimensional representations. The property of a C*-algebra of being RFD is central in C*-algebra theory and has connections with other important notions and problems.

The topic of this talk will be the RFD property in dynamical context, namely we will discuss the RFD property of crossed products by amenable actions

and, if time permits,  of C*-algebras of amenable etale groupoids. We will present consequences of our results to residual properties of groups and to approximations of representations in spirit of Exel and Loring, and we will discuss examples. Joint work with Adam Skalski.

Abstract: Central sequences play a fundamental role in operator algebras (property Gamma, the MacDuff property, Z-stability for example). Given the deep link between operator algebras and dynamics, it is natural to look for dynamical analogues of central sequence properties. This turns out to be subtle. I will discuss a body of work related to this idea. This is joint work with Grigorios Kopsacheilis, Hung-Chang Liao, and Andrea Vaccaro.

Abstract: A dynamical characterization of C*-simplicity in terms of boundary actions was recently discovered by Breuillard,  Kalantar, Kennedy, and Ozawa. We will explore this characterization with two case studies: the free group acting on its Gromov boundary, and the action of the modular group on the projective line.

Abstract: In this talk, I will discuss the notion of the nuclear dimension of a C*-algebra and with selected examples, its application to graph C*-algebras. I will then explain how both of these topics are connected to my current research. 

Non-Hausdorff etale groupoids naturally occur as an intermediate step in the construction of C*-algebras from combinatorial and algebraic data. A paper of Kwasnieski and Meyer presents a quotient of the reduced C*-algebra, namely the essential C*-algebra, whose structure contains fruitful information about the underlying groupoid, matching that already known for reduced C*-algebras in the Hausdorff case. In this short talk, we give a vague description of the essential C*-algebra, and consider whether a well-known uniqueness theorem for the reduced C*-algebra for Hausdorff etale groupoids extends to this setting.

Abstract: Given a group $G$, equipped with a probability measure $\mu$, there is an associated random walk on the group, where each step is a random variable with distribution $\mu$. The Poisson boundary is a measure space which encodes how such a random walk behaves at infinity. Various group properties - including growth and amenability - can be studied, and sometimes characterised, using Poisson boundaries.

Abstract: A brief overview of the work I have done together with my PhD supervisor Dr. Ulrich Pennig. This work achieves its C*-algebra flavour from its connection to Dixmier-Douady classes.

I will introduce the idea of exponential functors and discuss how they induce maps between certain spaces which in turn induce (not quite) natural transformations of cohomology theories.

In our quest to understand these induced maps, we require an understanding of the Weyl map W: SU(n)/T x T -> SU(n), the class of this map in K-theory, and the tautological line bundles over SU(n). Homotopies and extraordinary cohomology theories abound!

Abstract: In this talk I'll supplement Jeremy Hume's talk by giving a very broad overview of the research direction that he'll speak about. Of course, this is joint work with Jeremy Hume.

Abstract: In 2008 Ruy Exel introduced tight C*-algebras associated to inverse semigroups, that generalize group C*-algebras but also work

well with Cuntz algebras, graph C*-algebras,  self-similar actions, etc. I will present how to generalize these constructions and results

to Banach algebras based on the theory of groupoid Banach algebras that we developed recently together with Bartosz Kwaśniewski and Andrew McKee.

Abstract: A classical result of Dixmier and Douady enables us to classify locally trivial bundles of C*-algebras with compact operators as fibres via methods in homotopy theory. Dadarlat and Pennig have shown that this generalises to the much larger family of bundles of stabilised strongly self-absorbing C*-algebras, which are classified by the first group of the cohomology theory associated to the units of complex topological K-theory. Building on work of Evans and Pennig I consider Z/pZ-equivariant C*-algebra bundles over Z/pZ-spaces. The fibres of these bundles are infinite tensor products of the endomorphism algebra of a Z/pZ-representation. In joint work with Pennig we show that the theory refines completely to this equivariant setting. In particular, we prove a full classification of the C*-algebra bundles via equivariant stable homotopy theory.

Abstract: We begin by discussing the noncommutative (C*-algebraic) variant of peak interpolation sets in function theory, on which several profound results in that theory rely.  We find appropriate ‘quantized’ versions of some of these classical facts. Through a delicate generalization of a theorem of Varopoulos, we show that, roughly speaking, sufficiently regular interpolation projections are peak precisely when their atomic parts are. As an application, we give alternative proofs and sharpenings of some recent peak-interpolation results of Davidson and Hartz for algebras on Hilbert function spaces. In another direction, given a convex subset of the state space, we study the associated Riesz projection.  This is then applied to various important topics in noncommutative function theory, such as the F.& M. Riesz property, the existence of Lebesgue decompositions, the description of Henkin functionals, and Arveson’s noncommutative Hardy spaces (maximal subdiagonal algebras). Joint work with Raphael Clouatre.

Abstract: A locally compact group is called “type I” if all of its unitary representations generate type I von Neumann algebras. It turns out that type I locally compact groups are essentially the class of locally compact groups whose unitary representations are tractable to classify. Classically, the problem of determining which locally compact groups are type I was studied heavily in the context of connected Lie groups and algebraic groups over local fields, and we have a good understanding of the type I groups in these classes. Modern research in locally compact group theory is largely concerned with dealing with the totally disconnected groups (abbreviated as “tdlc groups”), many of which are non-linear and not algebraic groups. There has been ongoing progress over the past 30 years in understanding which non-amenable tdlc groups are type I, but relatively little work has been done on the amenable case. In this talk I will discuss a recent project that focuses on understanding which amenable tdlc groups are type I.

Abstract: Amenability of groups is a very well-studied topic in the context of dynamics. We shall be looking at one particular generalisation of the notion: amenability of an action, along the lines of R. J. Zimmer and C. Anantharaman-

Delaroche. This is essentially a vector-valued version of amenability of groups. We shall also briefly be looking at orbit equivalence relations (some of the results by Connes, Feldman, and Weiss in Ornstein-Weiss theory).

Abstract: Regularity theory in free probability has proven to be a powerful technique for extracting structural properties of von Neumann algebras from information about a set of generators. One particularly useful property is Jung's notion of strong 1-boundedness, which places a harsh limit on the availability of matricial approximations of the law of a generating tuple. It can, however, be challenging to establish.

 For a large class of sofic groups with vanishing first $\ell^2$ Betti number, Shlyakhtenko has shown that strong 1-boundedness holds for the group von Neumann algebra. This critically uses the fact that the generators can be approximated by permutation matrices, which have entries with very nice algebraic properties (i.e., being integers) and therefore one can control the Fuglede-Kadison determinant of integer polynomials in the generators can be controlled. I will speak about joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson where we extend these ideas to algebras with generators admitting "Galois-bounded microstates", which are matricial approximations with somewhat nice algebraic properties. This allows us to show that strong 1-boundedness of graph products of matrix algebras is implied by vanishing first $\ell^2$ Betti number.

Abstract: Product systems represent powerful contemporary tools in the study of mathematical structures. A major success in the theory came from Katsura (2007), who provided a complete description of the gauge-invariant ideals of many important C*-algebras arising from product systems over Z+. This result recaptures existing results from the literature, illustrating the versatility of product system theory. The question now becomes whether or not Katsura's result can be bolstered to product systems over semigroups other than Z+ and, if so, what applications do we obtain? An answer has been elusive, owing to the more pathological nature of product systems over general semigroups. However, recent strides by Dor-On and Kakariadis (2018)

supply a more tractable subclass of product systems that still includes the important cases of C*-dynamics, row-finite higher-rank graphs and regular product systems. In this talk we will provide a parametrisation of the gauge-invariant ideals, thus extending Katsura's result, explore

the higher-rank subtleties that are not witnessed in the low-rank case, and comment on the applications.

Abstract: A classical result of Malcev is that not all semigroups are embeddable into groups. By playing around with this, one quickly sees that cancellativity is necessary and perhaps conjecture is sufficient - alas there exist cancellative monoids which are not embeddable. Indeed, Malcev gave an (infinitely long) list of conditions required for embeddability. In a recent paper of Brix, Bruce and Dor-On, a question of Li regarding C*-envelopes is answered in the positive for groupoid-embeddable categories, but crucially also for right LCM / 1-aligned cancellative monoids – in particular, these monoids need not be group-embeddable! We show that Malcev's conditions may be used to construct a plethora of non group-embeddable cancellative monoids, and we may show that these monoids are right LCM. This truly demonstrates the second generalisation of the result in Brix et al. We also give an example of non group-embeddable, cancellative and 2-aligned monoid (which is not 1-aligned), prompting us to ask Li's question for finitely aligned monoids. This is joint work with Milo Edwardes (University of Manchester).

Abstract: A C*-algebra is quasi-diagonal if there is a unital embedding into the ultraproduct of matrices. The Tikusis-White-Winter theorem shows that any stably finite, simple, separable, nuclear, unital C*-algebra satisfying the UCT will be quasidiagonal. Thus an interesting question is to understand how unique are such embeddings given the existence.

We are able to answer this question for simple C*-algebras, by obtaining a KK-uniqueness theorem. The major difficulty appears since the codomain of the map of interest does not separably absorb the Jiang-Su algebra. We will explain how to tackle the difficulty by looking at K-theoretical properties of a special C*-algebra, the Paschke Dual algebra associated to the map.

Abstract: We investigate the properties of a certain class of self-similar groupoid actions on graphs (in the sense of Laca-Raeburn-Rammage-Whittaker) defined from two integer valued matrices A, B, which we call Katsura groupoid actions. By results of Katsura and Exel-Pardo, the C*-algebras of these actions exhaust all Kirchberg algebras. We show a recent class of dynamical systems studied by Putnam can be realized as a sub-class of the limit space dynamical systems associated to Katsura groupoid actions. We prove these limit spaces embed into the plane, answering a question of Putnam. This is joint work with Mike Whittaker.

Abstract: I will discuss how to construct a $C^*$-algebra from a vector bundle and a partial action of the integers on the base space of the bundle, using the machinery of Cuntz–Pimsner algebras. This generalizes partial crossed products by the integers, as well as the homeomorphisms twisted by vector bundles recently introduced by Adamo–Archey–Forough–Georgescu–Jeong–Strung–Viola. I will briefly address how to show classifiability of these $C^*$-algebras.

Abstract: Whenever a locally compact group acts on von Neumann algebras $M,N$, it gives a canonical diagonal action on their tensor product $M\bar{\otimes}N$. This is no longer true if we consider actions of locally compact quantum groups. Nonetheless, not all is lost. If on von Neumann algebras $M,N$ acts not only quantum group $\mathbb{G}$ but also its dual $\widehat{\mathbb{G}}$, and these actions are compatible (satisfy Yetter-Drinfeld condition), then one can form a twisted version of tensor product, called the braided tensor product $M\overline{\boxtimes}N$. This is a new von Neumann algebra which contains $M,N$ as subalgebras and which carry a canonical action of $\mathbb{G}$. More generally, one can consider actions of quasi-triangular quantum groups. I will discuss construction of $M\overline{\boxtimes} N$ and some results concerning (non)existence of braided tensor product of maps. Based on a joint work with Kenny De Commer.

Abstract: After a reminder about classical (commutative) Stone duality,

I will survey some of the results that have been obtained as non-commutative generalizations.

The best known and most relevant to operator algebra theorists is the connection between a class of inverse monoids and a class of \'etale groupoids,

but I will also show how \'etale categories and domain-\'etale categories arise in this context from natural generalizations of inverse monoids.

The talk will survey my own work with Daniel Lenz and Ganna Kudryavtseva as well as work by 

Pedro Resende and, most recently, by Richard Garner which shows that these results are of wider interest than might be thought.

Abstract: In topological dynamics, a classical idea is to construct covers with better properties to study the original dynamical system. This talk is about the construction of what we call minimal covers which improve properties of transfer operators in topological dynamics. For subshifts, our construction gives a conceptual explanation for constructions due to Krieger, Fischer, Matsumoto and Carlsen. For groupoids, our construction produces etale groupoid models whenever we restrict an etale groupoid to an arbitrary closed subset of its unit space, explaining previous work of Thomsen. This is joint work with Kevin Brix and Jeremy Hume.

Abstract: Groupoid C*-algebras have been playing an ever-increasing role in C*-algebra theory since 80’s. Given a C*-algebra A and a commutative subalgebra B, Renault showed that the pair (A, B) is Cartan (or B is a Cartan subalgebra of A) if and only if there is a twisted effective groupoid such that A can be realised by a reduced twisted groupoid C*-algebra. In fact, the groupoid can be directly constructed from the normaliser semigroup N(B) of A. In my talk, I will first introduce the notion of Cartan semigroups of a C*-algebra and its associated semi-Cartan subalgebra. I will then show that with these Cartan semigroups we can get exactly the same result as Renault’s even for non-effective groupoids.

This is joint work with Tristan Bice, Lisa Orloff Clark, and Kathryn McCormick

Abstract: In 1970s, Tingley proposed a problem in geometry of Banach spaces, and asked whether one can extend a bijective isometry between the unit spheres of two real Banach spaces to a real linear bijective isometry on the whole space. Peralta asked an analogue question in 2018 in the setting of operator algebras: whether one can extend a bijective isometry between the positive parts of the unit spheres of two C*-algebras to a real linear bijective isometry. This can be seen as an order type Tingley's problem. In this talk, we give a positive answer of this problem for type I finite von Neumann algebras that have bounded dimensions of irreducible representations.

Abstract: This is a talk about the K-theory of étale groupoid C*-algebras with the purpose of emphasising two key points. The first is the ubiquity and usefulness of a kind of groupoid morphism called étale correspondences. The second point is the close connection between groupoid homology and operator K-theory for ample groupoids.

Unlike continuous homomorphisms of étale groupoids, étale correspondences play nicely with C*-algebras, their K-theory and groupoid homology groups. We use correspondences to strengthen the link between homology and K-theory, resulting in a better understanding of the K-theory of étale groupoid C*-algebras. In particular, we can for a new class of inverse semigroups compute the K-theory of their left regular C*-algebras.

Abstract: The real rank of C*-algebras was introduced by Brown and Pedersen as a non-commutative analogue of topological dimension. Particularly, the notion of real rank zero distinguished itself due to the fact that many interesting classes of C*-algebras were shown to have this property. In this talk, we will introduce a notion of real rank zero for inclusions of C*-algebras. We will offer a complete description of the commutative case and prove that a large class of full infinite inclusions have real rank zero. Furthermore, interesting inclusions arising from dynamics are shown to have this property. This is joint work with James Gabe.

Abstract: Periodic cyclic homology is a variation of cyclic homology and it is a fundamental tool in noncommutative geometry because it plays the same role as de Rham cohomology in the commutative framework. 

This theory has been widely studied and also extended in the equivariant context. 

In this talk we aim to go beyond the group case, introducing the category of modules over the convolution algebra of an ample groupoid.

Then we will present all the objects needed to define equivariant periodic cyclic homology for algebras in this category.

Abstract: A G-kernel is a group homomorphism from a (discrete) group G to Out(A), the outer automorphism group of a C*-algebra A. There are cohomological obstructions to lifting such a G-kernel to a group action. In the setting of von Neumann algebras, these have been almost completely understood through deep results of Connes, Jones and Ocneanu. In the talk I will explain how G-kernels on C*-algebras and the lifting obstructions can be interpreted in terms cohomology with coefficients in crossed modules (or strict 2-groups in the language of category theory). The fact that 2-groups have classifying spaces makes some open conjectures about G-kernels amenable to methods from homotopy theory. This is a joint project with S. Giron Pacheco and M. Izumi.

Abstract: I will give a short introduction to several related dynamical properties and their role in the Elliot classification.  These include the small boundary property, topological small boundary property, almost finiteness, and almost finiteness in measure.  I will go on to sketch how these properties relate to sharp bounds for the dynamic asymptotic dimension.  

Abstract: Associated to every shift space, the Cuntz-Krieger algebra (C-K algebra for abbreviation) is an invariant of conjugacy defined and developed by K. Matsumoto, S. Eilers, T. Carlsen and many of their collaborators in the last decade. In particular, Carlsen defined the C-K algebra to be the full groupoid C*-algebra of the “cover”, which is a topological system consisting of a surjective local homeomorphism on a zero-dimensional space induced by the shift space. 

In 2022, K. Brix proved that the C-K algebra of the Sturmian shift has finite nuclear dimension, where the Sturmian shift is the (unique) minimal shift space with the smallest complexity function: p_X(n)=n+1. In  recent results (joint with Z. He), we show that for any minimal shift space with finitely many left special elements, its C-K algebra always have finite nuclear dimension. In fact, this can be further applied to the class of aperiodic shift spaces with non-superlinear growth complexity.

Abstract: Let G be a discrete group. For p greater than or equal to 1, we define CVp(G) to be the algebra of convolution operators on lp(G); when p=2 we recover the group von Neumann algebra. In this talk we discuss the space of weak*-closed ideals of CVp(G). In particular, we shall discuss recent work that shows that CVp(G) is weak*-simple if and only if G is an ICC group, generalising a basic fact about the group von Neumann algebra. 

Abstract: Winter and Zacharias’ nuclear dimension is a non-commutative covering dimension which has strong connections to classification.  One key feature is that (unlike its forerunner the decomposition rank) an extension E of A by I has finite nuclear dimension if and only if both A and I do, While there are bounds on the nuclear dimension of E in terms of that of A and I, exact computations are harder to come by.  I’ll discuss some of the known results in this direction, and add one more (which is joint work with Evington, Ng and Sims).

Abstract: The Hilbert transform H is a basic example of Fourier multipliers. Its behaviour on Fourier series is the following:

\sum_{n∈Z} a_n exp(inx) → \sum_{n∈Z} m(n)a_n exp(inx),

with m(n) = −i sgn(n). Riesz and Cotlar proved that H is a bounded operator on L_p(T) for all 1 < p < ∞. In this talk, we will give a definition for Hilbert transforms on SL(2,Z). We know that SL(2,Z) is a lattice of SL(2,R), we will then discuss how Hilbert transforms on SL(2,Z) can be extended to an L_p-bounded Fourier multiplier on SL(2,R) using a transference method.

Based on a joint work with A. González-Pérez and J. Parcet, and an ongoing project with Simeng Wang and Gan Yao.

Abstract: We will establish a hypertrace characterization of property (T) for II_1 factors: Given a II_1 factors $M$, $M$ does not have property (T) if and only if there exists a von Neumann algebra $A$ with $M \subset A$ such that $A$ admits a $M$-hypertrace but no normal hypertrace. For $M$ without property (T), such an inclusion $M \subset A$ also admits vanishing Furstenberg entropy. With the same construction of $M \subset A$, we also establish similar characterizations of Haagerup property for II_1 factors