Past Functional Analysis Seminar

This is a continuation of https://www.maths.ox.ac.uk/node/61240

In the operator algebraic approach to quantum field theory, the DHR category is a braided tensor category describing topological point defects of a theory with at least 1 (+1) dimensions. A single von Neumann algebra with no extra structure can be thought of as a 0 (+1) dimensional quantum field theory. In this case, we would not expect a braided tensor category of point defects since there are not enough dimensions to implement a braiding. We show, however, that one can think of central sequence algebras as operators localized ``at infinity", and apply the DHR recipe to obtain a braided tensor category of bimodules of a von Neumann algebra M, which is a Morita invariant. When M is a II_1 factor, the braided subcategory of automorphic objects recovers Connes' chi(M) and Jones' kappa(M). We compute this for II_1 factors arising naturally from subfactor theory and show that any Drinfeld center of a fusion category can be realized. Based on joint work with Quan Chen and Dave Penneys.

In view of Takesaki-Takai duality, we can go back and forth between C*-dynamical systems of an abelian group and ones of its Pontryagin dual by taking crossed products. In this talk, I present a similar duality between actions on C*-algebras of two constructions of locally compact quantum groups: one is the bicrossed product due to Vaes-Vainerman, and the other is the double crossed product due to Baaj-Vaes. I will explain the situation by illustrating the example coming from groups. If time permits, I will also discuss its consequences in the case of quantum doubles.

K-homology is the dual theory to K-theory for C*-algebras.  I will show how under appropriate quasi-diagonality and countability assumptions K-homology (more generally, KK-theory) can be realized by completely positive and contractive, and approximately multiplicative, maps to matrix algebras modulo an appropriate equivalence relation.  I’ll briefly explain some connections to manifold topology and existence / uniqueness theorems in C*-algebra classification theory (due to Dadarlat and Eilers).

Some of this is based on joint work with Guoliang Yu, and some is work in progress

Since Segal's formulation of axioms for 2d CFTs in the 80s, it has remained a major problem to construct examples of CFTs that satisfy the axioms.

I will report on ongoing joint work with James Tener in that direction.

A coarse structure is a way of talking about "large-scale" properties.  It is encoded in a family of relations that often, but not always, come from a metric.  A coarse structure naturally gives rise to Hilbert space operators that in turn generate a so-called uniform Roe algebra.

In work with Bruno Braga and Joe Eisner, we use ideas of Weaver to construct "quantum" coarse structures and uniform Roe algebras in which the underlying set is replaced with an arbitrary represented von Neumann algebra.  The general theory immediately applies to quantum metrics (suitably defined), but it is much richer.  We explain another source based on measure instead of metric, leading to the new, large, and easy-to-understand class of support expansion C*-algebras.

I will present the big picture: where uniform Roe algebras come from, how Weaver's framework facilitates our definitions.  I will focus on a few illustrative examples and will not presume familiarity with coarse structures or von Neumann algebras.

It is well known that two non-isomorphic groups (groupoids) can produce isomorphic C*-algebras. That is, group (groupoid) C*-algebras are not rigid. This is not the case of the L^p-operator algebras associated to locally compact groups ( effective groupoids) where the isomorphic class of the group (groupoid) uniquely determines up to isometric isomorphism the associated L^p-algebras. Thus, L^p-operator algebras are rigid.  Liao and Yu introduced a class of Banach *-algebras associated to locally compact groups. We will see that this family of Banach *-algebras are also rigid.  

It is well known that if a group von Neumann algebra of a (nontrivial) discrete group is a factor, then it is a factor of type II_1. During the talk, I will answer the following question: which types appear as types of injective factors being group von Neumann algebras of discrete quantum groups (or looking from the dual perspective - von Neumann algebras of bounded functions on compact quantum groups)? An important object in our work is the subgroup of real numbers t for which the scaling automorphism tau_t is inner. This is joint work with Piotr Sołtan.

We are surrounded by systems that involve many elements and the interactions between them: the air we breathe, the galaxies we watch, herds of animals roaming the African planes and even us – trying to decide on whom to vote for.

As common as such systems are, their mathematical investigation is far from simple. Motivated by the realisation that in most cases we are not truly interested in the individual behaviour of each and every element of the system but in the average behaviour of the ensemble and its elements, a new approach emerged in the late 1950s - the so-called mean field limits approach. The idea behind this approach is fairly intuitive: most systems we encounter in real life have some underlying pattern – a correlation relation between its elements. Giving a mathematical interpretation to a given phenomenon and its emerging pattern with an appropriate master/Liouville equation, together with such correlation relation, and taking into account the large number of elements in the system results in a limit equation that describes the behaviour of an average limit element of the system. With such equation, one hopes, we could try and understand better the original ensemble.

In our talk we will give the background to the formation of the ideas governing the mean field limit approach and focus on one of the original models that motivated the birth of the field – Kac’s particle system. We intend to introduce Kac’s model and its associated (asymptotic) correlation relation, chaos, and explore attempts to infer information from it to its mean field limit – The Boltzmann-Kac equation.

In this talk, I will present a noncommutative analogue of Margulis’ factor theorem for higher rank lattices. More precisely, I will give a complete description of all intermediate von Neumann subalgebras sitting between the von Neumann algebra of the lattice and the von Neumann algebra of the action of the lattice on the Furstenberg-Poisson boundary. As an application, we infer that the rank of the semisimple Lie group is an invariant of the pair of von Neumann algebras. I will explain the relevance of this result regarding Connes’ rigidity conjecture.

The classification by K-theory and traces of the category of simple, separable, nuclear, Z-stable C*-algebras satisfying the UCT is an extraordinary feat of mathematics. What's more, it provides powerful machinery for the analysis of the internal structure of these regular C*-algebras. In this talk, I will explain one such application of classification: In the subclass of classifiable C*-algebras consisting of those for which the simplex of tracial states is nonempty, with extremal boundary that is compact and has the structure of a connected topological manifold, automorphisms can be shown to be generically tracially chaotic. Using similar ideas, I will also show how certain stably projectionless C*-algebras can be described as crossed products.

 I will start this talk with a brief introduction and summary of the outcome of a joint work with James Gabe. An important special case of the main result is that for any countable discrete amenable group G, any two outer G-actions on stable Kirchberg algebras are cocycle conjugate precisely when they are equivariantly KK-equivalent. In the main body of the talk, I will outline the key arguments toward a special case of the 'uniqueness theorem', which is one of the fundamental ingredients in our theory: Suppose we have two G-actions on A and B such that B is a stable Kirchberg algebra and the action on B is outer and equivariantly O_2-absorbing. Then any two cocycle embeddings from A to B are approximately unitarily equivalent. If time permits, I will provide a (very rough) sketch of how this leads to the dynamical O_2-embedding theorem, which implies that such cocycle embeddings always exist in the first place.

Equivariant Jiang-Su stability is an important regularity property for group actions on C*-algebras.  In this talk, I will explain this property and how it arises naturally in the context of the classification of C*-algebras and their actions. Depending on the time, I will then explain a bit more about the nature of equivariant Jiang- Su stability and the kind of techniques that are used to study it, including a recent result of Gábor Szabó and myself establishing an equivalence with equivariant property Gamma under certain conditions.
 

In this talk, I will discuss the notion of quantum limits from different viewpoints: Cordes' work on the Gelfand theory for pseudo-differential operators dating from the 70’s as well as the micro-local defect measures and semi-classical measures of the 90’s. I will also explain my motivation and strategy to obtain similar notions in subRiemannian or subelliptic settings. 

I will introduce a K-theoretic complete invariant of inductive limits of finite dimensional actions of fusion categories on unital AF-algebras. This framework encompasses all such actions by finite groups on AF-algebras. Our classification result essentially follows from applying Elliott's Intertwining Argument adapted to this equivariant context, combined with tensor categorical techniques.

Our invariant roughly consists of a finite list of pre-ordered abelian groups and positive homomorphisms, which can be computed in principle. Under certain conditions this can be done in full detail. For example, using our classification theorem, we can show torsion-free fusion categories admit a unique AF-action on certain AF-algebras.

Connecting with subfactors, inspired by Popa’s classification of finite-depth hyperfinite subfactors by their standard invariant, we study unital inclusions of AF-algebras with trivial centers, as natural analogues of hyperfinite II_1 subfactors. We introduce the notion of strongly AF-inclusions and an Extended Standard Invariant, which characterizes them up to equivalence.

Since the completion of the Elliott classification programme it is an important question to ask which C*-algebras satisfy the assumptions of the classification theorem. We will ask this question for the case of crossed-product C*-algebras associated to actions of nonamenable groups and focus on two extreme cases: Actions on commutative C*-algebras and actions on simple C*-algebras. It turns out that for a large class of nonamenable groups, classifiability of the crossed product is automatic under the minimal assumptions on the action. This is joint work with E. Gardella, S. Geffen, P. Naryshkin and A. Vaccaro. 

One of the most important constructions in operator algebras is the tracial ultrapower for a tracial state on a C*-algebra. This tracial ultrapower is a finite von Neumann algebra, and it appears in seminal work of McDuff, Connes, and more recently by Matui-Sato and many others for studying the structure and classification of nuclear C*-algebras. I will talk about how to generalise this to unbounded traces (such as the standard trace on B(H)). Here the induced tracial ultrapower is not a finite von Neumann algebra, but its multiplier algebra is a semifinite von Neumann algebra.

 I will discuss ongoing work with Toke Carlsen and Aidan Sims on ideal structure of C*-algebras of commuting local homeomorphisms. This is one aspect of a general attempt to bridge C*-algebras with multidimensional (symbolic) dynamics.

Hirschman-Widder densities may be viewed as the probability density functions of positive linear combinations of independent and identically distributed exponential random variables. They also arise naturally in the study of Pólya frequency functions, which are integrable functions that give rise to totally positive Toeplitz kernels. This talk will introduce the class of Hirschman-Widder densities and discuss some of its properties. We will demonstrate connections to Schur polynomials and to orbital integrals. We will conclude by describing the rigidity of this class under composition with polynomial functions.

 This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).

The string 2-group is a fundamental object in string geometry, which is a refinement of spin geometry required to describe the spinning string. While many models for the string 2-group exist, the construction of a representation for it is new. In this talk, I will recall the notion of strict 2-group, and then give two examples: the automorphism 2-group of a von Neumann algebra, and the string 2-group. I will then describe the representation of the string 2-group on the hyperfinite III_1 factor, which is a functor from the string 2-group to the automorphism 2-group of the hyperfinite III_1 factor.

Groupoid C*-algebras and twisted groupoid C*-algebras are introduced by Renault in the late ’70. Twisted groupoid C*-algebras have since proved extremely important in the study of structural properties for large classes of C*-algebras. On the other hand, 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 provide useful insight into the analytic theory of groupoid C*-algebras and give rise to interesting examples of *-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. If I have enough time, I will further introduce pair of algebras which consist of a Steinberg algebra and an algebra of locally constant functions on the unit space, it is an algebraic analogue of Cartan pairs

A few years back, Smale spaces were shown to exhibit noncommutative Poincaré duality (with Jerry Kaminker and Ian Putnam). The fundamental class was represented as an extension by the compacts. In current work we describe a Fredholm module representation of the fundamental class. The proof uses delicate approximations of the Smale space arising from a refining sequence of (open) Markov partition covers. I hope to explain all these notions in an elementary manner. This is joint work with Dimitris Gerontogiannis and Joachim Zacharias.

A Hausdorff and etale groupoid is said to be C*-simple if its reduced groupoid C*-algebra is simple. Work on C*-simplicity goes back to the work of Kalantar and Kennedy in 2014, who classified the C*-simplicity of discrete groups by associating to the group a dynamical system. Since then, the study of C*-simplicity has received interest from group theorists and operator algebraists alike. More recently, the works of Kawabe and Borys demonstrate that the groupoid case may be tractible to such dynamical characterizations. In this talk, we present the dynamical characterization of when a groupoid is C*-simple and work out some basic examples. This is joint work with Xin Li, Matt Kennedy, Sven Raum, and Dan Ursu. No previous knowledge of groupoids will be assumed.

In the late 1980s, Berger and Coburn showed that the Hankel operator $H_f$ on the Segal-Bargmann space of Gaussian square-integrable entire functions is compact if and only if $H_{\bar f}$ is compact using C*-algebra and Hilbert space techniques. I will briefly discuss this and three other proofs, and then consider the question of whether an analogous phenomenon holds for Schatten class Hankel operators.