Past Functional Analysis Seminar

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.

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.

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.

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.

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.

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.

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.

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.

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.

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(??2), due to Kulish–Reshetikhin 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(??2) 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.

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.

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.

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 John’s 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.

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.

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.

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.

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.

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).

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.

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.

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.