# Barcelona-Kiel Operator Algebra Seminar (BKOAS)

## Introduction

The Barcelona-Kiel Operator Algebra Seminar (BKOAS) is an online seminar organized by Ramon Antoine and Francesc Perera in Barcelona, and Hannes Thiel in Kiel.

The seminar will run in a fortnightly basis starting November 30, 2021 and will focus on regularity properties of C*-Algebras and Dynamical Systems.

## List of talks

### Upcoming

July 19, 2022. 15:00-16:00 CEST
Speaker: Eduard Vilalta (UAB Barcelona)
– Title: The Global Glimm Problem
Abstract: A C*-algebra is nowhere scattered (that is, it has no nonzero elementary ideal-quotients) whenever it satisfies the Global Glimm Property. The Global Glimm Problem asks if the converse holds.
I will first recall both the notion of nowhere scatteredness and the Global Glimm Property. Then, I will introduce two new conditions on the Cuntz semigroup that capture precisely what a nowhere scattered C*-algebra needs to have the Global Glimm Property. By studying when these conditions hold, one recovers known solutions to the Global Glimm Problem and can also provide some new ones.
The talk is based on joint work with Hannes Thiel.

### Previous

November 30, 2021. 15:00-16:00 CET
Speaker: David Kerr (WWU Münster)
– Title: Dynamical tilings and elementary amenability
Abstract: I will discuss the notion of almost finiteness as a topological analogue of the Ornstein-Weiss tiling theorem for measure-preserving actions of amenable groups and show that this property holds for every free continuous action of a countably infinite elementary amenable group on a finite-dimensional compact metrizable space. As a consequence, the crossed products of such minimal actions are $\mathcal{Z}$-stable and classified by their Elliott invariant. This is joint work with Petr Naryshkin.

December 14, 2021. 15:00-16:00 CET
Speaker: Jamie Gabe (University of Southern Denmark)
– Title: Semifinite tracial ultraproducts and quasidiagonality
Abstract: I will introduce tracial ultraproducts for C*-algebras equipped with a lower semicontinuous tracial weight. In contrast to the usual ultraproducts using tracial states, this tracial ultraproduct is not necessarily a finite W*-algebra. Instead, its multiplier algebra is a semifinite W*-algebra. As an application, I extend the Tikuisis-White-Winter theorem to tracial weights, and use this to show that locally compact unimodular groups that are amenable, type I, or almost connected, give rise to quasidiagonal (left regular) C*-algebras.

January 11, 2022. 15:00-16:00 CET
Speaker: Shirly Geffen (KU Leuven)
– Title: Purely infinite crossed products by amenable actions
Abstract: We pull back paradoxical dynamical systems (e.g. hyperbolic groups acting on their Gromov boundary), to paradoxical decompositions of the acting group itself.
This allows to show that whenever such groups admit a minimal amenable topologically free action on a compact Hausdorff space, the attached crossed product is a purely infinite classifiable C*-algebra.

January 25, 2022. 15:00-16:00 CET
Speaker: Jianchao Wu (Fudan University)
– Title: Three shades of dynamical strict comparison
Abstract: The notion of strict comparison of a C*-algebra was inspired by the theory of II_1 factors and has played an instrumental role in the Elliott classification program. Recently, the study around a dynamical analog of strict comparison has been gaining interest, pioneered by Kerr and his coauthors. In this talk, I will focus on a part of my joint work with Bosa, Perera, and Zacharias, which takes a more systematic look at the formulation of dynamical strict comparison and, as a result, produces three different kinds of dynamical strict comparison for a topological dynamical system. The weakest of the three agrees with Kerr’s dynamical strict comparison, while the other two are properties of dynamical analogs of the Cuntz semigroup (one of the semigroups turns out to be isomorphic to Xin Ma’s generalized type semigroup). Contrary to previous speculations, we now know for many topological dynamical systems that Kerr’s dynamical strict comparison holds without the presence of other regularity properties such as mean dimension zero and strict comparison of the crossed products. This somewhat unexpected mismatch may be remedied by replacing Kerr’s dynamical strict comparison with the strongest of our three dynamical strict comparison properties. As an application, our work widens the scope of topological dynamical systems whose crossed products are classifiable.

February 8, 2022. 15:00-16:00 CET
Speaker: Stuart White (University of Oxford)
– Title: Tracially complete C*-algebras
Abstract: I’ll discuss the class of tracially complete C*-algebras, which lie somewhere between a C*-algebra and a tracial von Neumann algebra.  These provide a generalisation of Ozawa’s notion of W*-bundles, and provide tools for tackling problems in the structure and classification of amenable C*-algebras whose trace space is big.  In this talk, I’ll introduce tracially complete C*-agebras and discuss concepts such as amenability, property Gamma and look at classification results in the spirit of Connes theorem these tracially complete C*-algebras. This is joint work with Carrión, Castillejos, Evington, Gabe, Schafhauser and Tikuisis.

March 8, 2022. 15:00-16:00 CET
Speaker: Karen Strung (Czech Academy of Science)
– Title: Constructions from C*-correspondences over commutative C*-algebras
Abstract: In this talk I will show how one constructs C*-algebras from C*-correspondences over a commutative C*-algebra C(X). A particularly tractable type of correspondence comes from the module of sections of a vector bundle where multiplication on one side of the module is given by composition by a homeomorphism \alpha: X\to X. When X is an infinite compact metric space with finite covering dimension and \alpha is minimal, the resulting C*-algebras are classifiable by the Elliott invariant. I will discuss this and related results, which is based on joint work with Adamo, Archey, Georgescu, Jeong, Strung and Viola, as well as work in progress with Jeong and Forough.

April 26, 2022. 15:00-16:00 CEST
Speaker: Mikael Rørdam (University of Copenhagen)
– Title: Traces and quasi-traces on C*-algebras
Abstract: We give an overview of older results about the existence of traces, respectively, quasi-
traces on C*-algebras, and Haagerup’s proof that quasi-traces on exact C*-algebras are
traces (in the version of Haagerup-Thorbjoersen). We discuss stability properties of
obstructions to having traces, respectively, quasi-traces, and use this to describe when
ultra-powers of a sequence of C*-algebras admit a (quasi-)trace. We give an example
of an ultra-power of a sequence of simple unital C*-algebras neither of which admit
a quasi-trace, but where the ultra-power does. This example illustrates a theorem of
Ozawa describing traces on ultra-power of C*-algebras. We have not yet been able to
decide if the ultra-power in our construction in fact admits a trace. This is a join work
with my PhD student Henning Milhøj.

May 10, 2022. 15:00-16:00 CEST
Speaker: Jorge Castillejos (UNAM Mexico)
– Title: The non-unital Toms-Winter conjecture
Abstract: The Toms-Winter conjecture predicts that three regularity conditions of different flavours are equivalent for separable, simple, unital, nuclear, and non-elementary C*-algebras. In this talk, I will discuss the current state of the Toms-Winter conjecture in the non-unital case. This is joint work with Sam Evington.

May 24, 2022. 15:00-16:00 CEST
Speaker: Tatiana Shulman (University of Gothenburg)
– Title: On almost commuting matrices
Abstract: Questions of whether almost commuting matrices are necessarily close to commuting ones are old. They are closely related with C*-algebra theory and have somewhat topological nature. We investigate which relations for families of commuting matrices are stable under small perturbations and give applications to lifting problems for commutative C*-algebras, in particular to liftings from the Calkin algebra. Joint work with Dominic Enders.

June 7, 2022. 15:00-16:00 CEST
Speaker: Kang Li (FAU Erlangen-Nürnberg)
– Title: Rigidity for Roe algebras and measured asymptotic expanders
Abstract: (Uniform) Roe algebras are C*-algebras associated to metric spaces, which reflect coarse properties of the underlying metric spaces. It is well-known that if X and Y are coarsely equivalent metric spaces with bounded geometry, then their (uniform) Roe algebras are (stably) *-isomorphic. The rigidity problem refers to the converse implication. The first result in this direction was provided by Ján Špakula and Rufus Willett, who showed that the rigidity problem has a positive answer if the underlying metric spaces have Yu’s property A. I will in this talk review all previously existing results in literature, and then report on the latest development in the rigidity problem by means of measured asymptotic expanders. This is joint work with Ján Špakula and Jiawen Zhang.

June 22, 2022. 15:00-16:00 CEST (Note the irregular switch to Wednesday)
Speaker: Tristan Bice (Czech Academy of Science)
– Title: Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras
Abstract: A natural problem that has plagued C*-algebra theory since its inception has been to find an appropriate noncommutative extension of the classic Gelfand duality, thus truly justifying the viewpoint of C-algebra theory being “noncommutative topology”. Here we outline our work to unify two of the more successful such extensions, namely the C*-bundle construction of Dauns and Hofmann from the late 60’s and the more recent Weyl groupoid construction of Kumjian and Renault. We do this via ultrafilters, inspired by a somewhat forgotten paper of Milgram from the late 40’s, allowing for a more direct construction of the required groupoid. Our construction is also functorial with respect to a general class of morphisms between Fell bundles akin to those considered by Varela between C*-bundles back in the 70’s.

July 5, 2022. 15:00-16:00 CEST
Speaker: Jeffrey Im (University of Toronto)
– Title: Coloured isomorphism of classifiable C*-algebras
Abstract: Coloured equivalence of order zero maps is a notion which appeared in the Z-stable implies finite nuclear dimension part of the Toms-Winter conjecture. A related notion which uses unitaries rather than contractions was used by Jorge Castillejos in his thesis to bring the relation of maps to the level of algebras (much in the same way homotopies of maps are involved in defining a homotopy equivalence of spaces), which he called n-coloured isomorphism. Among other things, Castillejos shows that when two classifiable C*-algebras both have no trace or both have a unique trace, then they are n-coloured isomorphic. In a joint work with George Elliott, we consider a notion at the level of algebras using coloured equivalence, which we call coloured isomorphism. The main result is that two classifiable C*-algebras are coloured isomorphic if, and only if, their tracial simplices are isomorphic. Since much of our approach follows the outline given by Castillejos in his thesis, we will explain his set-up and some of the obstacles moving past the case of at most one trace. A key step is the construction of a c.p.c. order zero map from an AF algebra into a classifiable C*-algebra realizing prescribed tracial data. This reduces to showing that a faithful trace on the half-open interval induces a functional which preserves compact containment.