# Workshop Classification and set theory

March 21 - 24, 2016

Program Committee: Aaron Tikuisis, Asger Törnquist

### Monday March 21

9:30-10:30 Damien Gaboriau

Title: Direct products and approximations in measured orbit equivalence

Abstract: Hyperfiniteness (which plays a central role in orbit equivalence), consists in approximability by finite subrelations.

In a joint work with Robin Tucker-Drob we initiate a general study of the notion of \textit{approximation} for countable standard probability measure preserving equivalence \ relations. We shall concentrate on non-approximability for actions of non-amenable direct products.

11:00-12:00 Narutaka Ozawa

Title: C*-norms on B(\ell_2) \otimes B(\ell_2) and related topics

Abstract: M. Junge and G. Pisier [GAFA 1995] have shown that there are at least two distinct C*-norms on B(\ell_2) \otimes B(\ell_2), which disproved E. Kirchberg's conjecture that the minimal tensor norm on tensor products of C*-algebras can be characterized by the injectivity property. Actually how many C*-norms exist on B(\ell_2) \otimes B(\ell_2) remains an open problem since then.

This talk is based on the joint work with G. Pisier [arXiv:1404.7088] which shows that there are at least c distinct C*-norms, where c denotes the cardinal of the continuum. It is plausible that there are exactly 2^c distinct C*-norms, and probably logicians can help us.

14:00-15:00 Todor Tsankov

Title: On metrizable universal minimal flows

Abstract: To every topological group, one can associate a unique universal minimal flow (UMF): a flow that maps onto every minimal flow of the group. For some groups (for example, the locally compact ones), this flow is not metrizable and does not admit a concrete description. However, for many "large" Polish groups, the UMF is metrizable, can be computed, and carries interesting combinatorial information. The talk will concentrate on some new results that give a characterization of metrizable UMFs of Polish groups. It is based on two papers, one joint with I. Ben Yaacov and J. Melleray, and the other with J. Melleray and L. Nguyen Van Thé.

15:15-15:45 Asger Törnquist

Title: The descriptive view of classification in operator algebras

Abstract: I will give an overview of the achievements of the program to study the complexity of classification problems in operator algebras from the descriptive set-theoretic point of view. I will also discuss some of the remaining open problems.

16:15-16:45 George Elliott

Title: A (one-eyed) bird's eye view of classification theory

Abstract: A brief overview is given of two opposite extremes of functorial classification theory---the general abstract functor that covers all separable C*-algebras, and the pared-down, concretized, version of this that covers the suitably well-behaved unital simple case.

17:00-17:30 Eusebio Gardella

Title: Classifiability of non-selfadjoint UHF-algebras

Abstract: It is a classical result due to Glimm that UHF (uniformly hyperfinite) C*-algebras are classified, by K-theory. In particular, they are classified by countable structures. A UHF operator algebra is, roughly speaking, an operator algebra obtained as an infinite tensor product of matrix algebras, where these matrix algebras are given a norm making them into an operator algebra (but not necessarily a C*-algebra). Such algebras have been studied by N. C. Phillips, who, among other things, exhibited uncountably many non-isomorphic UHF operator algebras. In this talk, which is based on joint work with Martino Lupini, we will show that UHF operator algebras are not classifiable by countable structures (even if they are assumed to have a very special and tractable form).

Our proof relies on Borel complexity theory, and in particular Hjorth's theory of turbulence. Our methods also allow us to treat the case of (non-spatial) UHF $L^p$- operator algebras, for $1\leq p<\infty$. These are defined by letting the matrix algebras act on an $L^p$-space, and giving them the corresponding operator norm.

### Tuesday March 22

9:30-10:30 David Kyed

Title: Topologizing Lie algebra cohomology

Abstract: I will explain how Lie algebra cohomology can be topologized in a way such that classical results, such as the van Est isomorphism, extend to the augmented context. Along the way I will define (most of) the objects involved, and the talk does not require prior knowledge about Lie algebra cohomology.

11:00-12:00 Mikael Rørdam

Title: Just infinite C*-algebras

Abstract: There is a well-established notion of just infinite groups, i.e., infinite groups for which all proper quotients are finite. The residually finite just infinite groups are particularly interesting. They are either branch groups (e.g., Grigorchuk's group of intermediate growth) or hereditarily just infinite groups (eg. Z, the infinite dihedral group, and SL_n(Z)). It is natural to consider the analogous notion for C*-algebras, whereby a C*-algebra is just infinite if it is infinite dimensional and all its proper quotients are finite dimensional. The study of these C*-algebras was motivated by a question of Grigorchuk if the group C*-algebra associated with his group might have this property. We give a classification of just infinite C*-algebras in terms of their primitive ideal space. We will discuss examples and properties of the residually finite dimensional just infinite C*-algebras; and we will also discuss the question of Grigorchuk.

This is joint work with R. Grigorchuk and M. Musat.

14:00-15:00 David Kerr

Title: Actions of amenable groups on the Cantor set

Abstract: This talk will be an invitation to study the decriptive set theory of the space of actions of a countable amenable group on the Cantor set. The motivation comes from the effort to classify the crossed products of such actions, and specifically to determine when these crossed products are Z-stable, an issue which I will discuss in detail.

15:15-15:45 David Schrittesser

Title: Absoluteness

Abstract: Suppose brilliant mathematician Klaupaucius, despite all his efforts, finds he is unable to come up with a proof for his favorite conjecture and starts to suspect it might be logically independent of the axioms. So he consults his friend Trurl the logician, who, after asking a number of questions, declares that the statement in question is absolute (due to its simple logical structure) and hence cannot be shown to be independent with the standard methods.

In my talk I will give a short introduction to the notion of "absoluteness" alluded to in this little story.

16:15-16:45 Hannes Thiel

Title: Cuntz semigroups of ultraproducts

Abstract: The Cuntz semigroup is an invariant for C*-algebras that is constructed analogously to K_0-theory by using positive elements in place of projections. The Cuntz semigroup of a C*-algebra is an element in a category Cu of abstract Cuntz semigroups, introduced by Coward-Elliott-Ivanescu. We show that the category Cu admits products and ultraproducts and that the functor from C*-algebras to Cu preserves these. In other words, the Cuntz semigroup of an (ultra)product of C*-algebras is the (ultra)product of the Cuntz semigroups of the C*-algebras.

This is joint work with Ramon Antoine and Francesc Perera.

17:00-17:30 Jiangchao Wu

Title: Furstenberg's x2, x3 problem and unitary representations of a metabelian group

Abstract: In 1967, Furstenberg raised the question whether the (normalized) Lebesgue measure is the only non-atomic probabilistic measure on the unit circle that is ergodic under the x2 and x3 endomorphisms. Since then this problem has attracted much interest, but despite many impressive results, it stays open in general. I will introduce this problem and propose an operator algebraic approach that involves the group C*-algebra of a semidirect product of Z[1/6] by Z^2.

This is joint work with Huichi Huang.

### Wednesday March 23

9:30-10:30 Julien Melleray

Title: Sets of invariant measures of minimal homeomorphisms of a Cantor space

Abstract: Given a compact set K of probability measures on a Cantor space X, one might ask when there exists a minimal homeomorphism g of X such that K is the set of all g-invariant measures. I will present an answer to that question, discuss an open problem related to that answer, as well as some questions about full groups of minimal homeomorphisms. This is joint work with Tomas Ibarlucia.

10:45-11:45 Anush Tserunyan

Title: Finite index pairs of equivalence relations and treeability

Abstract: I will give an overview of an open question of Jackson--Kechris--Louveau from the late 90s as to whether finite index extensions of countable treeable equivalence relations are themselves treeable. We will consider this question in both Borel and measurable settings and discuss relevant results and possible approaches in each.

12:00-12:30 Alessandro Vignati

Title: Set theory and automorphisms of C*-algebras

Abstract: After the key results on the structure of the automorphisms group of the Calkin algebra, Farah and Coskey conjectured that under the Continuum Hypothesis there are wild automorphisms of corona algebras, while under the Proper Forcing Axiom the situation is conjectured to be quite rigid. McKenney and I recently veriﬁed the conjecture is presence of forcing axioms for a large class of algebras, although very much remains open. We present the current situation and then focus on some of the diﬃculties stopping from expanding our results, with, possibly exploring connections to perturbation theory.

### Thursday March 24

09:30-10:30 Chris Phillips

Title: Simple nuclear C*-algebras not equivariantly isomorphic to their opposites

Abstract: We exhibit examples of simple separable nuclear C*-algebras, along with actions of the circle group and outer actions of the integers, which are not equivariantly isomorphic to their opposite algebras. In fact, the fixed point subalgebras are not isomorphic to their opposites. The C*-algebras we exhibit are well behaved from the perspective of structure and classification of nuclear C*-algebras: they are unital C*-algebras in the UCT class, with finite nuclear dimension. One is an AH-algebra with unique tracial state and absorbs the CAR algebra tensorially. The other is a Kirchberg algebra.

This is joint work with Marius Dadarlat and Ilan Hirshberg.

11:00-12:00 Hiroshi Ando

Title: Descriptive analysis of self-adjoint operators and the Weyl-von Neumann equivalence relation

Abstract: In a previous work, we studied the space $SA(H)$ of all (possibly unbounded) self-adjoint operators on a separable Hilbert space $H$ and in particular showed that the relation of unitary equivalence modulo compacts (Weyl-von Neumann equivalence) is unclassifiable by countable structures. Moreover, for a closed set $F$ of the real line, there may or may not be self-adjoint operators $A,B$ for which the essential spectrum coincide with $F$ but $A,B$ are not Weyl-von Neumann equivalent.

In this talk, we characterize which $F$ satisfies the conclusion of Weyl-von Neumann theorem. We also discuss the Borel complexity of Schatten class perturbations of unbounded self-adjoint operators, and see that they are all Borel reducible to the universal essentially K_{\ sigma} equivalence relation $\ell^{\infty}$.

This is joint work with Yasumichi Matsuzawa (Shinshu University).

14:00-15:00 Thomas Sinclair

Title: Model theory of C*-algebras as operator systems

Abstract: I will discuss various aspects of the model-theoretic structure of the class of C*-algebras within the class of operator systems. This is based on joint work with Isaac Goldbring.

15:15-15:45 Adam Sierakowski

Title: Dimension theory for k-graph C*-algebras

Abstract: I will present at least one reason why P-graphs are interesting. Using P-graphs one can characterise when a k-graph C*-algebra has topological dimension zero in terms of the underlying graph. This is joint work with D. Pask and A. Sims.

16:00-16:30 Martino Lupini

Title: The noncommutative Poulsen simplex

Abstract: In my talk, based on a joint work with Ken Davidson and Matt Kennedy, I will give an overview of the recent developments in the theory of noncommutative simplices. I will then explain how one can define in this setting a canonical object, which can be regarded as the noncommutative analog of the classical Poulsen simplex. Such a noncommutative Poulsen simplex plays, for the theory of noncommutative simplices, a similar role as the Cuntz algebra O_2 plays for the theory of nuclear C*-algebras.