C3

The reduced twisted C*-algebra A of an étale groupoid G has a canonical abelian subalgebra D: functions on G's unit space. When G has no non-trivial abelian subgroupoids (i.e., G is principal), then D is in fact maximal abelian. Remarkable work by Kumjian shows that the tuple (A,D) allows us to reconstruct the underlying groupoid G and its twist uniquely; this uses that D is not only masa but even what is called a C*-diagonal. In this talk, I show that twisted C*-algebras of non-principal groupoids can also have such C*-diagonal subalgebras, arising from non-trivial abelian subgroupoids, and I will discuss the reconstructed principal twisted groupoid of Kumjian for such pairs of algebras.

The classification program of C*-algebras aims to classify simple, separable, nuclear C*-algebras by their K-theory and traces, inspired by analogous results obtained for von Neumann algebras. A landmark result in this project was obtained in 2015, building upon the work of numerous researchers over the past 20 years. More recently, Carrión, Gabe, Schafhauser, Tikuisis, and White developed a new, more abstract approach to classification, which connects more explicitly to the von Neumann algebraic classification results. In their paper, they carry out this approach in the stably finite setting, while for the purely infinite case, they refer to the original result obtained by Kirchberg and Phillips. In this talk, I provide an overview of how the new approach can be adapted to classify purely infinite C*-algebras, recovering the Kirchberg-Phillips classification by K-theory and obtaining Kirchberg's absorption theorems as corollaries of classification rather than (pivotal) ingredients. This is joint work with Jamie Gabe.

The (Local) Lifting Property ((L)LP) is introduced by Kirchberg and deals with lifting completely positive maps. We will discuss various examples, characterizations, and closure properties of the (L)LP and, if time permits, connections with some other lifting properties of C*-algebras.  Joint work with Dominic Enders.

I will give a light introduction to the concept of a quantum expander, which is an analogue of an expander graph that arises in quantum information theory.  Most examples of quantum expanders that appear in the quantum information literature are obtained by random matrix techniques.  I will explain another, more algebraic approach to constructing quantum expanders, which is based on using actions and representations of discrete quantum groups with Kazhdan's property (T).  This is joint work with Eric Culf (U Waterloo) and Matthijs Vernooij (TU Delft).   

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, and still somewhat mysterious notion, with 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.

A C*-correspondence between two C*-algebras is a generalization of a *-homomorphism. Laca and Neshveyev showed that, like a *-homomorphism, there is an induced map between traces of the algebras. Given sufficient regularity conditions, the map defines a bounded operator between the spaces of (bounded) tracial linear functionals. 

This operator can be of independent interest - a special case of correspondence gives Ruelle's operator associated to a non-invertible discrete-time dynamical system, and the study of Ruelle's operator is the basis of his thermodynamic formalism. Moreover, by the work of Laca and Neshveyev, the operator's positive eigenvectors determine the KMS states of the gauge action on the Cuntz-Pimsner algebra of the correspondence.

Given a C*-correspondence from a C*-algebra to itself, we will present a sufficient condition on the C*-correspondence that implies the operator on traces has a unique positive eigenvector, and moreover a spectral gap. This result recovers the Perron-Frobenius theorem, aspects of Ruelle's thermodynamic formalism, and unique KMS state results for a variety of constructions of Cuntz-Pimsner algebras, including the C*-algebras associated to self-similar groupoids. The talk is based on work in progress.

The stable uniqueness theorem for KK-theory asserts that a Cuntz-pair of *-homomorphisms between separable C*-algebras gives the zero element in KK if and only if the *-homomorphisms are stably homotopic through a unitary path, in a specific sense. This result, along with its group equivariant analogue, has been crucial in the classification theory of C*-algebras and C*-dynamics. In this talk, I will present a unitary tensor category analogue of the stable uniqueness theorem and explore its application to a duality in tensor category equivariant KK-theory. To make the talk approachable even for those unfamiliar with actions of unitary tensor categories or KK-theory, I will introduce the relevant definitions and concepts, drawing comparisons with the case of group actions. This is joint work with Kan Kitamura and Robert Neagu.

The notion of semi-uniform stability of a strongly continuous semi-group refers to the stability of classical solutions of a linear evolution equation, and this has analogues with the classical Katznelson-Tzafriri theorem. The co-generator of a strongly continuous semigroup is a bounded linear operator that comes from a particular discrete approximation to the semigroup. After reviewing some background on (quantified) stability theory for semigroups and the Katznelson-Tzafriri theorem, I will present some results relating the stability of a strongly continuous semigroup with that of its cogenerator. This talk is based on joint work with David Seifert.

Roe type algebras are operator algebras designed to catch the large-scale behaviour of metric spaces. This talk focuses on the following question: if two Roe type algebras associated to spaces (X,d_X) and (Y,d_Y) are isomorphic, how similar are X and Y? We provide positive results proved in the last 5 years, and, if time allows it, we show that sometimes answers to this question are subject to set theoretic considerations