Schedule for: 19w5134 - Classification Problems in von Neumann Algebras

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will discuss several conjectures related to Connes' embedding problem. One the main conjectures is an extension of a certain map defined on positive definite functions which is an analog of an intertwining operator for almost commuting representations of free groups.

A countable group is said to have the Schmidt property if it admits an ergodic free p.m.p. action such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. It was originally asked by Klaus Schmidt whether any inner amenable group has this property. We show that any countable group with infinite FC-center has the Schmidt property, and as its consequence, any inner amenable, property (T) group has the Schmidt property. This is a joint work with Robin Tucker-Drob.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

It is well known that the universal and the reduced group C*-algebra of a locally compact group coincide if and only if the group is amenable. In general, there can be many C*-algebras, called exotic group C*-algebras, which lie between these two algebras. In a joint work with Timo Siebenand, we consider simple Lie groups $G$ with real rank one and investigate their exotic group C*-algebras $C^*_{L^{p+}}(G)$, which are defined through $L^p$-integrability properties of matrix coefficients of unitary representations of $G$. First, we show that the subset of equivalence classes of $L^{p+}$-representations forms a closed ideal of the unitary dual of the groups under consideration. This result holds more generally for Kunze-Stein groups. Second, for (almost) every connected simple Lie group with real rank one and finite center and $2 \leq q < p \leq \infty$, we determine when the canonical quotient map $C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G)$ has non-trivial kernel. To this end, it suffices to study the integrability properties of spherical functions of class one representations of $G$. Our results generalize recent results of Samei and Wiersma on exotic group C*-algebras of $\mathrm{SO}_{0}(n,1)$ and $\mathrm{SU}(n,1)$, with completely different methods.

We say that a groupoid or a group action on a $C^*$-algebra has the weak containment property (WCP) if the corresponding full and reduced $C^*$-algebras coincide. For a group it is well known that this property is equivalent to amenability. Whether WCP implies amenability in general is a quite subtle question that we will discuss in this talk.

We will consider q-gaussian analogues of free products with amalgamation. Our aim is to apply the Popa-Vaes-Ozawa program to these algebras, but the task turned out to be harder then we thought. I will explain why, and how ultraproduct technques come into play. This is joint work with Udrea.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A character of a discrete group G is a central positive definite function on G which is indecomposable. The set Char(G) of characters of G parametrizes the unitary representations of G which generate a finite factor, up to quasi equivalence. We will discuss the characters of the group G(k) of k-points of an algebraic group G defined over a field k. Under the assumption that G(k) is generated by its unipotent elements, a complete classification of Char(G(k)) will be given in two cases: 1) G is k-simple and k is an arbitrary infinite field; 2) G is an arbitrary algebraic group and k is a number field. The description obtained in the first case can be viewed as an extension of Tits' simplicity theorem. The description in the second case is based on Ratner's measure rigidity theorems.

I present a joint work with Michael Björklund and Zemer Kosloff on nonsingular Bernoulli actions. These are the translation actions of a discrete group $G$ on the product space $\{0,1\}^G$ equipped with the product of the probability measures $\mu_g$ on $\{0,1\}$. We prove in almost complete generality that such an action is either dissipative or weakly mixing, and we determine its Krieger type. In particular, we prove that the group of integers does not admit a Bernoulli action of type II$_\infty$. We prove that a group $G$ admits a Bernoulli action of type III$_1$ if and only if $G$ has nonzero first $L^2$-cohomology. We also prove that type III$_\lambda$ only arises when $G$ has more than one end.

Popa's intertwining theory is a very powerful tool to study inclusions of finite von Neumann algebras. This is in fact one of the most important technical ingredient in Popa's deformation/rigidity theory. Since this intertwining technique works only for finite von Neumann algebras, there have been many attempts to generalize it to inclusions of general von Neumann algebras. In this talk, I will give a survey of recent development for these attempts, which are mainly given by C. Houdayer, S. Vaes, Y. Ueda, A. Marrakchi, and myself. I also mention applications to the deformation/rigidity theory.

We will present examples of equivalence relations without the Jones-Schmidt property, answering a question from Jones and Schmidt from 1985. We will then discuss its connections with structural properties of central sequence algebras, and implications for unique McDuff decompositions. This is based on joint works with Adrian Ioana.

I will present recent joint work with Cyril Houdayer about a stationary version of characters on discrete groups. Our main result states that such stationary characters on lattices of semisimple Lie groups are actually genuine characters. I will present applications of this result to representation theory and operator algebras, as well as topological dynamics. I will also mention a result in non-commutative ergodic theory, which is the key to our approach and is of independent interest.

The free group factor $L(\mathbb{F}_d)$ can realized as the von Neumann algebra generated by $d$ freely independent semicircular varables $S_1, \ldots, S_d$, the free probabilistic analogue of an independent family of Gaussians. We consider non-commutative random variables $X_1, \ldots, X_m$ whose non-commutative distribution satisfy the integration-by-parts relation $\tau(D_{X_j} V(X) p(X)) = \tau \otimes \tau(\partial_{X_j} p(X))$, where $V$ is a suitably regular convex function. By studying "conditional transport of measure" for the associated $N \times N$ random matrix models in the large $N$ limit, we show that there is an isomorphism $W^*(X_1, \ldots, X_d) \to W^*(S_1, \ldots, S_d)$ such that the $W^*$-algebra of the first $k$ generators is mapped to the $W^*$-algebra of the first $k$ generators for every $k$. In particular, $W^*(X_1)$ is sent to the generator MASA in $L(F_d)$, so it is freely complemented. As an application, we deduce that for every non-commutative polynomial $p$, the algebra $W^*(S_1 + \epsilon p(S))$ is a freely complemented MASA in $L(F_d)$ for sufficiently small $\epsilon$.

In this talk we provide a natural complement to Monod and Shalom's orbit equivalence superrigidity theorem for irreducible actions of product groups by providing a large class of product actions whose orbit equivalence relation remember the product structure. More precisely, we show that if a product $\Gamma_1\times\dots\times\Gamma_n \curvearrowright X_1\times\dots\times X_n$ of measure preserving actions is stably orbit equivalent to a measure preserving action $\Lambda\curvearrowright Y$, then $\Lambda\curvearrowright Y$ is induced from an action $\Lambda_0\curvearrowright Y_0$ and there exists a direct product decomposition $\Lambda_0=\Lambda_1\times\dots\times\Lambda_n$ into $n$ infinite groups. Moreover, there exists a measure preserving action $\Lambda_i\curvearrowright Y_i$ that is stably orbit equivalent to $\Gamma_i\curvearrowright X_i$, for any $1\leq i\leq n$, and the product action $\Lambda_1\times\dots\times\Lambda_n\curvearrowright Y_1\times\dots\times Y_n$ is isomorphic to $\Lambda_0\curvearrowright Y_0$.

It is a classical problem to recover a discrete group from various rings or algebras associated with it, such as the integral group ring. By analogy, in an operator algebraic framework we want to recover torsion-free groups from certain topological completions of the complex group ring, such as the reduced group $C^*$-algebra. Groups for which this is possible are called $C^*$-superrigid. My talk will discuss how a group can be recovered from its group rings, before I introduce the reduced group $C^*$-algebras and describe the state-of-the-art in $C^*$-superrigidity. I will end with a short account on other kinds of superrigidity for group operator algebras putting the subject into a bigger perspective.

I will give an overview of some recent results on inner amenable groups, focusing on some joint work with P. Wesolek and B. Duchesne in which we obtain a complete characterization of inner amenableity for permutational wreath product groups. One of the main ingredients used for this is a general lemma which we call the location lemma, which allows us to “locate” conjugation invariant means on a group G relative to a given normal subgroup N of G. I'll give several further applications of the location lemma, and discuss some related open problems.

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the 2-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $PSL_2(\mathbb Z)$ on $P^1( \mathbb R)$.

Following work of Kavruk et al. on tensor products of operator systems, we discuss the tensor theory of matrix convex sets. Among applications, we obtain a new formulation of Connes' embedding problem via noncommutative Choquet theory. This is joint work with Roy Araiza and Adam Dor-On.

In this talk, I will introduce several games that one can play with II$_1$ factors and the connection between these games and other active areas of research, including the Connes Embedding Problem and the study of elementarily equivalent II$_1$ factors (that is, II$_1$ factors with isomorphic ultrapowers). Some of the work in this talk is joint with Thomas Sinclair.

In the first part of my talk I will discuss the problems of reconstructing a countable discrete group from its von Neumann algebra (W*-superrigidity) and its reduced C*-algebra (C*-superrigidity) and I will survey several recent results in this direction. In the second part, using and interplay between von Neumann algebraic and C*-algebraic methods, I will introduce a new class of C*-superrigid groups which appear as wreath products with non-amenable core. As an application we obtain complete calculations of the symmetry groups of various group C*-algebras---a problem barely touched in the literature. This is based on a recent joint work with Alec Diaz-Arias.

We introduce a new equivalence relation on groups, which we call von Neumann equivalence, and which is coarser than both measure equivalence and W*-equivalnce. We introduce a general procedure for inducing actions in this setting and use this to show that the class of properly proximal groups is closed in this equivalence relation. In particular, proper proximality is preserved under both measure equivalence and W*-equivalence, and from this we obtain examples of non-inner amenable groups which are not properly proximal. This is based on joint work with Ishan Ishan and Jesse Peterson.

The classification program for von Neumann algebras witnessed remarkable progress which is in large part due to Popa's deformation/rigidity theory. We expand on the implications of the existence of maximal rigid algebras, provide concrete examples in the group setting, and applications when considering families of deformations as in $L^{2}$ rigidity, in particular, further supporting the Peterson-Thom Conjecture. I will give some of the background that goes into our theorem, and Rolando will expand on the details in a follow up talk.

The classification program for von Neumann algebras witnessed remarkable progress which is in large part due to Popa's Deformation/Rigidity theory. Proceeding from where Ben Hayes ended in Part I, we expand on the implications of the existence of maximal rigid algebras, provide concrete examples in the group setting, and describe applications when considering families of deformations as in $L^2$ rigidity; the latter result lends further support to the Peterson-Thom Conjecture.

I will discuss $\varepsilon$-independence, which is an interpolation of classical and free independence originally studied by Motkowski and later by Speicher and Wysoczanski. To be $\varepsilon$-independent, a family of algebras in particular must satisfy pairwise classical or free independence relations prescribed by a $\{0, 1\}$-matrix $\varepsilon$, as well as more complicated higher order relations. I will discuss how matrix models for this independence may be constructed in a suitably-chosen tensor product of matrix algebras. This is joint work with Benoit Collins.

We consider a semi-direct product $G=\mathbb Z^2\rtimes F_2$, where $F_2$ is a free subgroup of rank 2 in $SL_2(\mathbb Z)$; by a result of M. Burger (1991), $G$ has relative property (T) with respect to $\mathbb Z^2$. It was known that, unlike property (T), relative property (T) should not be an obstruction for proving Baum-Connes: indeed $G$ is K-amenable (P. Julg and myself, 1983) and satisfies Baum-Connes with coefficients (H. Oyono-Oyono, 2001). We confirm this philosophy by computing explicitly the assembly map, when $F_2$ is generated by the matrices (1,2,0,1) and its transpose. We exploit the fact that $G$ has a classifying space of dimension 3, so that the left hand side of BC is computable, by results of G. Mislin. This is work in progress with my PhD student A. Zumbrunnen.

The separably acting hyperfinite/amenable II$_1$-factor $R$ has the property that any two embeddings into its own ultrapower are unitarily conjugate. Jung showed in 2009 that the converse holds modulo the Connes Embedding Problem. In this talk we will discuss the recent result that amenability for tracial von Neumann algebras satisfying the Connes Embedding Problem is characterized by the weaker property that any two embeddings into an ultrapower of R are conjugate by unital completely positive maps. Time permitting, we will discuss other recent characterizations of amenability within this context. This is based on joint work with Srivatsav Kunnawalkam Elayavalli.

Residual finite dimensionality is the $\mathrm{C}^*$-algebraic analogue for maximal almost periodicity and residual finiteness for groups. Just as with the analogous group-theoretic properties, there is significant interest in when residual finite dimensionality is preserved under standard constructions, in particular amalgamated free products. In general, this question is quite difficult; however the answer is known when the amalgam is finite dimensional or when the two $\mathrm{C}^*$-algebras are commutative. In moving beyond these cases, group theoretic restrictions suggest that we consider central amalgams. We generalize the commutative case to pairs of so-called ``strongly residually finite dimensional" $\mathrm{C}^*$-algebras amalgamated over a central subalgebra. Examples of strongly residually finite dimensional $\mathrm{C}^*$-algebras include group $\mathrm{C}^*$-algebras associated to virtually abelian groups, certain just-infinite groups, and Lie groups with only finite dimensional irreducible unitary representations. Though this property may seem restrictive, a recent result of Thom indicates that it is in fact necessary. This is joint work with Tatiana Shulman.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.