Copenhagen
Groups and Operator Algebras Seminar

Abstract:  The Boone-Higman conjecture asserts that every countable group with solvable word problem embeds into a finitely presented simple group. In this talk, I will survey some recent progress on this conjecture using embeddings into topological full groups of certain étale groupoids. The resulting finitely presented simple groups can be viewed as generalisations of Thompson's group V. This is joint work with Collin Bleak, James Hyde, Francesco Matucci, and Matthew Zaremsky.

Abstract:  We review different extensions of Grothendieck inequalities and a conjecture by Blecher on matrix extensions of Grothendieck's original inequality, and Pisier, Haagerup, Shlyakhtenko's extension. Using ideas from quantum games, we give qualitative bounds showing how much a commuting set of contractions fails to admit a dilation to commuting unitaries, and disprove the open conjectures by Blecher, Shlyakhtenko and Pisier.

Abstract:  From Zeno's paradoxes to quantum physics, the question of the continuous nature of our world has been prominent and remains unanswered. Does space-time really exist, or is it just a good model for an enormous, but finite number of elementary particles?

Discrete structures behave quite differently from continuous ones. The great success story of mathematics in the 18-th and 19-th centuries was the development of analysis, with extremely powerful tools such as differential equations or Fourier series, and with by now very standard methods like the famous (infamous?) epsilon-delta technique. Discrete mathematics had a later start, but for importance of its applications it is catching up. Its proof techniques are different, such as enumeration or induction. In the continuous world, algorithms are mostly computations, with numerical analysis at the center. In the discrete world, algorithmic ideas are more diverse, including searching, recurrence, and (yes!) pulling in methods from continuous mathematics.

I will argue that these worlds are not as far apart as they seem. The use of computers forces us to approximate continuous structures by finite ones; but perhaps more surprisingly, very large finite structures can be very well approximated by continuous structures, and this approximation gets rid of inconvenient and unnecessary details. Many fundamental questions of mathematics, probability, or physics can be asked in both settings, and their approaches cross-fertilize each other.

Abstract:   I will discuss quantum no-signaling correlations introduced by Duan and Winter and its different subclasses (quantum commuting, quantum and local). They will appear as strategies of non-local games with quantum inputs and quantum outputs. I will then introduce an analogue of bisynchronous correlations and characterise them by tracial states on the universal C*-algebra of the projective free unitary group, showing that in the quantum input/output setup, quantum permutations of finite sets must be replaced by quantum automorphisms of matrix algebras. As an application, I will discuss quantum graph isomorphisms by giving their non-local game interpretation, and compare our approach with the existing algebraic notions of quantum graph isomorphisms. In the case of classical graphs our operational notion of quantum isomorphism leads to new quantum symmetries.

This is a joint work with Michael Brannan, Sam Harris and Ivan Todorov.

Abstract:  Given a sequence of groups G_n with inclusions G_n -> G_{n+1}, an important question in group (co)homology is whether there is homological stability. That is, if for a given integer j, there is some m such that for all n>m, the map H_j(G_n) -> H_j(G_{n+1}) is an isomorphism. To detect this behaviour one usually constructs a family of highly connected simplicial complexes K_n on which the groups G_n naturally act. For example, for the linear groups GL_n or SL_n, K_n can be the Tits building or the complex of unimodular sequences, while for the automorphism group of the free groups F_n one can take the complex of free factors.

In this talk, we discuss a categorical framework that describes these constructions in a unified way. More precisely, for an initial symmetric monoidal category C, we take an object X and consider the poset of subobjects of X. From this bounded poset, we take only those subobjects which are complemented, i.e. x \vee y = 1  and  x \wedge y = 0, and the join operation coincides with the monoidal product. The monoidal product should be interpreted as the "expected" coproduct of the category. Thus, for the free product in the category of groups, if we start with a free group of finite rank then the complemented subobject poset is exactly the poset of free factors, and for the category of vector spaces with the direct sum we obtain the subspace poset. From this construction, we define related combinatorial structures, such as the poset of (partial) decompositions or the complex of partial bases, and establish general properties and connections among these posets. Finally, we specialise these constructions to matroids, modules over rings, and vector spaces with non-degenerate forms, where there are still many open questions.

Abstract:  Connes’ reconstruction theorem says that you can reconstruct a manifold from a (nice enough) spectral triple. Renault’s theorem says that you can reconstruct a twist over an étale groupoid from a Cartan pair of C*-algebras. In this talk I will discuss how to combine these theorems to recover a Lie twist over an étale Lie groupoid from suitable functional-analytic data. This is joint work with Anna Duwenig.

Abstract:  Bounded cohomology of groups was defined in the 1970s by Johnson and Trauber, and has emerged as an area with several connections to classical topics in group theory and topology. I will provide a gentle introduction to this topic, and describe some connections with notions such as amenability and stable commutator length. I will then describe a new algebraic criterion for the vanishing of bounded cohomology (for a certain large class of coefficients) which is recent joint with Campagnolo, Fournier-Facio, and Moraschini.

Abstract:  I will discuss the factoriality, non-injectivity and fullness questions of q-Araki-Woods von Neumann algebras. These algebras are non-tracial counterparts of q-Gaussian algebras combining q-deformations of Bozejko-Speicher and quasi free deformations of Shlyakhtenko. This is joint work with Adam Skalski and Mateusz Wasilewski, and with Simeng Wang.

Abstract:  In 1983 Voiculescu came up with an example of a sequence of pairs of unitary matrices that commute asymptotically in operator norm but remain far, in operator norm, from any commuting pair of unitaries. An elegant proof, called the "winding number argument," that these matrices are far from commuting matrices was developed by Kazhdan and independently by Excel and Loring. More generally, the argument may be used to show that a function from a group to matrices that is "almost multiplicative" (in the point operator norm topology) is "far" from a genuine representation. In this talk, I explain how to reinterpret this argument as a pairing between an almost representation and 2-homology class of the group. I will explain how this interpretation has led to a systematic way of making almost representations that are far from genuine representations and showing that finitely generated nilpotent groups are stable in the Frobenius norm if and only if they are virtually cyclic.

Abstract:  In the last 5 decades, there has been a pronounced tendency to obtain classification results for C*-algebras by first classifying *-homomorphisms between them. I will briefly describe some of these classification results for *-homomorphisms and show how they can be used to study regularity properties of inclusions such as finite nuclear dimension and real rank zero. Moreover, I will explain how classification of *-homomorphisms can be used to examine the possible bundles of KMS states for large classes of C*-algebras. If time permits, we will define morphisms in other categories and explore some future goals around classifying actions of C*-tensor categories on C*-algebras.

Abstract:  A C*-algebra is said to be residually finite-dimensional (RFD) when it has 'sufficiently many' finite-dimensional representations. The RFD property is an important, natural, and still somewhat mysterious notion, admitting several equivalent descriptions and having subtle connections to residual finiteness properties of groups. In this talk I will present certain characterisations of the RFD property for C*-algebras of amenable étale groupoids and for C*-algebraic crossed products by amenable actions of discrete groups, extending (and inspired by) earlier results of Bekka, Exel and Loring. I will also explain the role of the amenability assumption and describe several consequences of our main theorems. Finally I will discuss some examples, notably these related to semidirect products of groups, and outstanding open problems. Based on the joint work with Tatiana Shulman.

Abstract:  If $H$ is a reproducing kernel Hilbert space and $Mult(H)$ denotes the space of pointwise multipliers of that space, we say that  $f\in H$ is cyclic if $Mult(H)f$ is dense in $H$. The first important example of cyclic functions are the so called outer functions which emerge from Beurling's famous theorem about invariant subspaces of the unilateral shift operator on the Hardy space $H^2$. Later, the work of Korenblum, Brown and Shields  revealed that in smaller spaces of analytic functions in  the disc, cyclicity of an outer function also depends on the size of its zero-set on the boundary. In general, a complete characterization of such functions is lacking, but the area is rich with deep results.
Much less is known in the setting of spaces of functions on the unit ball in several complex variables and as is to expect, the situation is quite complicated. For example, there are polynomials without zeros in the ball which are not cyclic in the standard Drury-Arveson space. The purpose of the talk is to present some recent results in this direction and the material is based on joint work with K.M. Perfekt, S. Richter, C. Sundberg and J. Sunkes.

Abstract:  In 1984 Weiss' asked whether the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space is automatically hyperfinite. In this talk we investigate the analogous question in the setting of amenable actions of hyperbolic groups. In particular, we give a new short proof of Marquis and Sabok' result that the orbit equivalence relation induced by the action of a finitely generated hyperbolic group on its Gromov boundary is hyperfinite. We expand moreover their result and show that any such action has finite Borel asymptotic dimension.

Abstract:  Entropy was brought into ergodic theory by Kolmogorov and Sinai in the 1950s, not long after Shannon used it to lay the foundations of information theory. By now it has been shown to have a remarkable range of consequences for the structure and behaviour of measure-preserving dynamical systems, such as Ornstein's celebrated result that two stationary stochastic processes with independent coordinates are ergodic-theoretically isomorphic if and only if they have the same entropy.

This talk will survey some recent developments in this story, emphasizing the ergodic theory of actions of non-amenable groups such as free groups. In that setting, work by Lewis Bowen and others has revealed whole new vistas in the last twenty years, but large gaps in our understanding remain, and many new phenomena seem to be waiting for us. If time allows, I will build up to a description of a simple, single new example in this setting which shows that several of the main older results for single measure-preserving transformations are false once one considers actions of a `large enough' group. This part of the talk is based on a current joint project with Lewis Bowen and Christopher Shriver.

This talk will assume familiarity with measure theory and the basic language of probability theory (such as `events' and `random variables'). Prior experience with ergodic theory is not essential.

Abstract:  I will discuss some examples of finitely generated and finitely presented simple groups of orientation preserving homeomorphism of the real line with an emphasis on finiteness conditions.

Abstract: In the first part of the talk I will report on a recent non-isomorphism result for q-Gaussian von Neumann algebras (and C*-algebras). The main result (due to Caspers) is that the q-Gaussian von Neumann algebra is not isomorphic to the free group factor in the case of infinitely many generators. I will also discuss a version of this non-isomorphism for the corresponding C*-algebras (due to Borst, Caspers, Klisse, and myself).

In the second part of the talk I will present a recent complete solution to the factoriality problem of q-Araki-Woods algebras, the non-tracial version of the q-Gaussian algebras, obtained in collaboration with Kumar and Skalski. Our approach combines results of Miyagawa and Speicher about conjugate variables for q-Gaussian variables and abstract von Neumann algebraic results of Nelson.

Abstract: In this talk we will discuss some recent work on groups which connect self-similar and Higman-Thompson groups to big mapping class groups via "braiding". We will explain some results on the topological finiteness properties of the resulting groups, which are topological generalizations of the algebraic properties of being finitely generated and finitely presented. The talk will involve recent joint works with Xiaolei Wu (Fudan) and Matthew Zaremsky (Albany).

Abstract: Two II_1 factors are elementarily equivalent if they admit isomorphic ultrapowers with respect to some ultrafilters on arbitrary sets. I will present a construction of a II_1 factor which does not have property Gamma and is not elementarily equivalent to a free group factor. This provides the first explicit example of two non-elementarily equivalent II_1 factors without property Gamma. Moreover, the construction also provides the first explicit example of a II_1 factor without property Gamma that is not elementarily equivalent to an ultraproduct of matrix algebras. This is joint work with Ionut Chifan and Srivatsav Kunnawalkam Elayavalli.

Abstract: The non-commutative Marcinkiewicz theorem for maximal functions by Junge and Xu is now a fundamental tool for lots of applications (martingales, ergodic averages etc). We tried to get a better understanding of it, and ended up with a rather elementary proof that is almost commutative, and which also provides some small improvements. This is a joint work with L. Cadilhac. 

Abstract: Groups of surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called Beauville structure. Gul and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, that act on the p-adic tree, admit Beauville structures. Such groups acting on p-adic trees (i.e. rooted trees) first appeared in the context of the Burnside problem, where they delivered the first explicit examples of finitely generated infinite torsion groups. Since then, groups acting on rooted trees have gone on to play a key role in group theory and beyond. In particular, Steinberg and Szakacs have recently given necessary and sufficient conditions for the Nekrashevych algebra of certain GGS-type groups to be simple. 

We extend the result of Gul and Uria-Albizuri by showing that quotients of infinite periodic GGS-groups, that act on the p^n-adic tree, also admit Beauville structures for all primes p and positive integers n. This is joint work with Elena Di Domenico and Şükran Gül.

Abstract: The notion of a highly proximal extension of a flow generalizes the one of an almost one-to-one extension (injective on a dense G_delta set), which is an important tool in topological dynamics. The existence of maximal such extensions was proved by Auslander and Glasner in the 70s for minimal flows using an abstract argument, and a concrete construction using near-ultrafilters was recently given by Zucker for arbitrary flows. When the acting group is discrete, the MHP extension is nothing but the Stone space of the Boolean algebra of the regular open sets of the space. We give yet another construction of the MHP extension for arbitrary topological groups and prove that for MHP flows of a locally compact group G, the stabilizer map x -> G_x is continuous (for general flows, this map is only semi-continuous). This is a common generalization of a theorem of Frolík that the set of fixed points of a homeomorphism of a compact, extremally disconnected space is open and a theorem of Veech that the action of a locally compact group on its greatest ambit is free. This is joint work with Adrien Le Boudec.

Abstract: A group G of homeomorphisms of the Cantor set is vigorous if for any non-empty clopen set A and non-empty proper clopen subsets B and C of A there exists an element g of G whose support is a subset of A and such that (B)g is a subset of C. We will use the space of marked groups to sketch a proof that all vigorous simple finitely generated groups of homeomorphisms of the Cantor are 2-generated and give some examples. This represents joint work with Collin Bleak and Luke Elliott.

Abstract: Inspired by Rørdam-Sierakowski's work in the context of crossed products, Matui introduced a sufficient condition on a groupoid to have a purely infinite C*-algebra. Later in the context of group theory, Bleak-Elliot-Hyde introduced a sufficient condition for a simple, infinite group to be 2-generated, which they call vigorousness. In this talk, I will explain how, surprisingly, the framework of topological full groups renders these two conditions in some sense equivalent, and explain how we can think of the topological full groups of purely infinite groupoids as generalizations of Thompson's group V, with an emphasis on examples.

Abstract: In quantum information theory, the bisynchronous correlations form an interesting class of bipartite no-signaling correlations where Alice and Bob's joint input-output behaviour is correlated in such a way that it looks as though Alice and Bob are both using a common injective function to prescribe outputs to inputs.  In this talk, I will review some interesting operator algebraic characterizations of bisynchronous correlations, and then I will move on to talk about what it means for a correlation to be bisynchronous when we allow more general bipartite quantum states as inputs and outputs.  In this more general setup, it turns out that out that the quantum permutation matrices (that play such a big role in the theory of bisynchronous correlations) need to be replaced by quantum automorphisms of matrix algebras.   I'll explain how these considerations in the context of graph isomorphism games give rise to a seemingly mysterious new class of quantum groups which act as generalized (or ``fuzzy'') quantum automorphisms of the underlying graphs. This is joint work with S. Harris, I. Todorov, and  L. Turowska.

Abstract: The Deroin space of a finitely generated group G is a compact and metrizable topological space that (morally) parametrizes the space of actions of G on the real line, up to semi-conjugacy. In this talk I will present some general aspects of these spaces, and then focus on the solvable case.

The talk will be based on joint work with Nicolás Matte Bon, Cristóbal Rivas and Michele Triestino.

Abstract: We give a characterization of automorphisms on a C*-algebra satisfying a Dixmier type averaging condition, introduced by Popa. We demonstrate how this property of automorphisms, or actions, leads to greatly simplified proofs – as well as extensions of – some known results about crossed product C*-algebras.

Abstract: I will discuss the role of topological full groups in the interaction between topological dynamics and C*-algebras. Topological full groups provide examples of groups with important new properties but they can be quite difficult to compute (and to distinguish!). If time permits, I will sketch some new directions I find interesting.