# Workshop Classification and dynamical systems I: C*-algebras

February 22 - 26, 2016

Program Committee: Marius Dadarlat, Ilan Hirshberg, Yasuhiko Sato, Wilhelm Winter

**Monday Febuary 22**

9:30-10:15 Yasuhiko Sato

Title: Automorphisms of Nuclear C*-algebras

Abstract:In the recent progress of the classi fication theorem of C*-algebras, we have seen connections with the regularity properties and associated conditions in the theory of injective von Neumann algebras, which is shown by A. Connes and U. Haagerup. In particular, it is known that the proof with Connes' approach is based on his results on automorphisms of injective factors. In this talk, along the recent evolution of Elliott's program, I will revisit classifi cation theorems of automorphisms on nuclear C*-algebras and discuss the connections between them.

10:45-11:30 Kengo Matsumoto

Title: Orbit equivalence, zeta functions of Markov shifts and classification of gauge actions on Cuntz--Krieger algebras

Abstract: The first half of my talk is based on joint work with Hiroki Matui.

Isomorphism classes of Cuntz--Krieger algebras are closely related to continuous orbit equivalence classes of one-sided topological Markov shifts.

For two-sided topological Markov shifts, Boyle--Handelman (1996 Israel J.) have shown that orbit equivalence, ordered cohomology groups, flow equivalence and Ruelle's dynamical zeta functions are closely related to each other.

In this talk, I would like to show that they have deep connections with continuous orbit equivalence of one-sided topological Markov shifts and classification of gauge actions on Cuntz--Krieger algebras.

11:45-12:30 Eduardo Scarparo

Title: Dynamical characterizations of paradoxicality for groups

Abstract: After Rosenblatt, a group is said to be supramenable if it has no paradoxical subsets. In the first part of the talk, we characterize supramenability in terms of existence of tracial states on partial crossed products.

In the second part, we show that a group is locally finite if and only if its Roe algebra is finite. We also discuss the problem of characterizing the group C*-algebra of locally finite groups.

15:00-15:45 Christian Skau

Title: Ordered Bratteli diagrams and Cantor minimal systems.

Abstract: Simple dimension groups (G,G+,u),with distinguished order unit u,can be defined in three equivalent ways,and we list them in the order they were historically introduced:

(i) Via Bratteli diagrams.

(ii) Abstractly,as (unperforated) ordered abelian groups satisfying the

Riesz interpolation property.

(iii)Dynamically,via Cantor minimal systems.

It is well known that simple dimension groups appear as complete isomorphism invariants for (simple) AF-algebras as well as for C*-crossed products associated to Cantor minimal systems.Furthermore,simple dimension groups also appear as complete invariants for orbit equivalence, respectively,strong orbit equivalence,of Cantor minimal systems. In this talk we will mention some fairly recent results how change of the ordering of a given Bratteli diagram yield entirely different Cantor minimal systems,while the systems themselves are orbit equivalent, respectively,strong orbit equivalent.

16:15-17:00 Gabor Szabo

Title: Strongly self-absorbing C*-dynamical systems

Abstract: We discuss a generalization of the notion of strongly self-absorbing C*-algebras to the setting of C*-dynamical systems. The main result is an equivariant McDuff-type theorem that characterizes exactly when an action of a locally compact group on a separable C*-algebra absorbs a given strongly self-absorbing action tensorially up to cocycle conjugacy. I then demonstrate what kind of (equivariant) permanence properties carry over in this context, similar to how D-stability is closed under various C*-algebraic operations. If time permits, we also discuss some natural examples and/or a non-trivial application to actions on Kirchberg algebras.

**Tuesday Febuary 23**

9:30-10:15 Dominic Enders

Title: On a KK-theoretic description of semiprojectivity

Abstract: Recently, we showed that Kirchberg algebras satisfying the UCT are semiprojective if and only if their K-groups are finitely generated. Here we will discuss how one can avoid the UCT-assumption and obtain a characterization of semiprojectivity for Kirchberg algebras purely in KK-theoretic terms.

10:45-11:30 Ilan Hirshberg

Title: Nuclear dimension of C*-algebras of homeomorphisms.

Abstract: Suppose X is a compact metrizable space with finite covering dimension, and h a homeomorphism of X. Let A be the crossed product of C(X) by the induced automorphism. It was shown first by Toms and Winter, and in a different way by the speaker, Winter and Zacharias, that if h is a minimal homeomorphism then A has finite nuclear dimension. Szabo then showed that it suffices to assume that h is free.

In this talk, I'll discuss a recent preprint which settles the issue for arbitrary homeomorphisms. As a special case, we show that group C*-algebras of certain non-nilpotent groups have finite nuclear dimension.

This is joint work with Jianchao Wu.

11:45-12:30 Leonel Robert

Title: On the normal subgroups of invertibles and unitaries of a C*-algebra.

Abstract: I will discuss a number of results on the structure of the normal subgroups of the invertibles and the unitaries in the connected component of the identity in a C*-algebra.

15:00-15:45 Wilhelm Winter

Title: QDQ vs. UCT

Asbtract: I will outline the interplay of quasidiagonality and the Universal Coefficient Theorem in the recent classification result for tracial, separable, unital, simple, nuclear C*-algebras with finite nuclear dimension.

16:15-17:00 Jianchao Wu

Title: The amenability dimension for topological and C*-dynamics

Abstract: Developments in the classification and the structure theory of C*-algebras in the past decade have highlighted the importance of an assortment of regularity properties, one of the most prominent of them being the property of having finite nuclear dimension. This has spurred growing interests in the advances of noncommutative dimension theories, for which a focal challenge is to find ways of bounding nuclear dimension for crossed product C*-algebras. To this end, various dimensions of dynamical nature have been developed, including Rokhlin dimension, dynamical asymptotic dimension, amenability dimension, etc. Roughly speaking, these dimensions measure the complexity of the topological or C*-dynamical system that gives rise to a given crossed product. We will discuss some of these concepts as well as their generalizations and applications. The talk is based on joint works with Ilan Hirshberg, Gabor Szabo, Wilhelm Winter and Joachim Zacharias.

**Wednesday Febuary 24**

9:30-10:15 David Kerr

Title: Tower decompositions for free actions of amenable groups

Abstract: Recently Downarowicz, Huczek, and Zhang proved that every discrete amenable group can be tiled by translates of finitely many Følner sets with prescribed approximate invariance. I will show how this can be used to strengthen the Rokhlin lemma of Ornstein and Weiss, with applications to topological dynamics and the classification program for simple separable nuclear C*-algebras.

10:45-11:30 Francesc Perera

Title: Near unperforation, almost unperforation, and almost algebraic order.

Abstract: In this talk I will present a general overview of structural properties of the category Cu of abstract Cuntz semigroups, focusing on the tensor product of a semigroup of a C*-algebra of real rank zero with the semigroup $Z$ of the Jiang-Su algebra $\mathcal Z$. In particular, a somewhat surprising connection between the condition of almost unperforation and the so-called axiom of almost algebraic order. The talk is based on joint work with R. Antoine, H.Thiel, and also R. Antoine, H. Petzka.

11:45-12:30 Luis Santiago

Title: A variant of the Cuntz semigroup

Abstract: The Cuntz semigroup of a C* -algebra is an analogue for positive elements of the semigroup of Murray-von Neumann equivalence classes of projections. The Cuntz semigroup is deeply connected to the classification program of C*-algebras. It has been used to classify certain classes of nonsimple C*-algebras as well as to distinguish simple unital C*-algebras that can not be classified using K-theory and traces. In general, the Cuntz semigroup contains the tracial information of the algebra. Also, it is known that for stably finite unital C*-algebras the K$_0$-group of the algebra can be recovered from this semigroup. This however does not hold in the projectionless case. In this talk I will introduce a variant of the Cuntz semigroup that fixes this problem. This semigroup was introduce by Leonel Robert in order to classify certain classes of (not necessary simple) inductive limits of 1-dimensional noncommutative CW-complexes. In this talk I will discuss properties of this semigroup and give some computations of it. In particular, I will show that for simple stably projectionless algebras that are $\mathcal Z$-absorbing, this semigroup together with the K$_1$-group contains the same information as the Elliott invariant of the algebra.

This is a joint work with Leonel Robert.

**Thursday Febuary 25**

9:30-10:15 Pere Ara

Title: Følner C*-algebras and related notions

Abstract: This talk is based on joint work with Kang Li, Fernando Lledó and Jianchao Wu.

I will survey some amenability notions arising in different contexts. Generalizing amenability of groups, Block and Weinberger introduced the concept of (coarse) amenability for metric spaces with bounded geometry. We study also the concept of amenability for general algebras, introduced by Gromov, and we obtain a dichotomy result in this context, generalizing a result of Elek.

Finally F\o lner nets for C*-algebras of operators will be introduced, and all the notions will be related through the consideration of the uniform Roe algebra of a metric space with bounded geometry.

10:45-11:30 Selcuk Barlak

Title: Cartan subalgebras and the UCT

Abstract: A masa of a C*-algebra is said to be a Cartan subalgebra if it is the image of a faithful conditional expectation and if it is regular in the sense that its normalizer generates the ambient C*-algebra. By a remarkable result of Renault, a C*-algebra that admits a Cartan subalgebra can be realized as the reduced twisted groupoid C*-algebra of an étale, locally compact, Hausdorff groupoid. Applying this and Tu's striking results and techniques used in the proof of the Baum-Connes Conjecture for amenable groupoids, we will show that a separable, nuclear C*-algebra possessing a Cartan subalgebra satisfies the UCT. As an application, we shall see how the UCT for separable, nuclear C*-algebras KK-equivalent to their tensorial CAR-algebra stabilization relates to Cartan subalgebras and order two automorphisms of the Cuntz algebra O_2. This is joint work with Xin Li.

11:45-12:30 Joachim Zacharias

Title: A bivariant version of the Cuntz Semigroup

Abstract: We propose a bivariant version of the Cuntz Semigroup based on equivalence classes of order zero maps rather than positive elements.

The resulting theory contains the ordinary Cunz Semigroup as special case similarly to KK-theory containing K-theory and admits a composition product and a number of other useful properties. We explain a couple of examples and indicate how this bivariant Cuntz Semigroup can be used to classify stably finite algebras in analogy to the Kirchberg-Phillips classification of simple purely infinite algebras via KK-theory.

15:00-15:45 Joan Bosa

Title: Comparison Properties on the Cuntz Semigroup, with application on C*-algebras

Abstract: We study different comparison properties at the category $\Cu$ framework aiming to lift this information to the C*-algebraic setting. In particular, we give analogous characterizations for the so-called Corona Factorization property and the $\omega$-comparison property. These help to both determine whenever a C*-algebra has CFP and narrow the range of the category $\Cu$ as invariant for C*-algebras. In particular, we show that the well-known C*-algebra described by Rordam with a finite and an infinite projection does not enjoy CFP and that the so-called Elementary $\Cu$-semigroups can not derive from a C*-algebra. It is joint work with H. Petzka.

16:15-17:00 Sara Arklint

Title: The K-theoretical range of all Cuntz-Krieger algebras

Abstract: This summer, Eilers-Restorff-Ruiz-Sørensen showed that the class of Cuntz-Krieger algebras (including those not purely infinite) is classified up to stable isomorphism by reduced filtered K-theory with ordered K_0-groups. I will describe the range of their invariant.

This is joint work with Rasmus Bentmann.

**Friday Febuary 26**

9:30-10:15 George Nadareishvili

Title: Homological algebra in Kasparov categories of C*-algebras

Abstract: I will try to demonstrate how the machinery of homological algebra can be set up for categories of C*-algebras (such as Kasparov categories of C*-algebras over a topological space or of dynamical systems), and in fact is the right framework for deriving results like UCT or classification of subcategories.

10:45-11:30 James Gabe

Title: Purely infinite C*-algebras of real rank zero have nuclear dimension 1

Abstract: We prove that separable, nuclear, purely infinite C*-algebras with the ideal property (in particular of real rank zero) have nuclear dimension 1. These C*-algebras are not assumed to be simple, but in the special case of simple C*-algebras, we obtain a new and short proof of the fact that Kirchberg algebras have nuclear dimension 1.

11:45-12:30 Chris Phillips

Title: Mean dimension and radius of comparison

Abstract: We discuss some results and some things we think are close to being proved about the conjecture that the radius of comparison of the crossed product by a minimal homeomorphism is equal to half the mean dimension of the homeomorphism.