Schedule for: 24w5174 - Group Operator Algebras: Classification, Structure and Rigidity

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

Meet and Greet at BIRS Lounge (Professional Development Centre Building, 2nd floor)

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.

A countable group $G$ is said to be W$^*$-superrigid if $G$ can be entirely recovered from its ambient group von Neumann algebra $L(G)$. In this talk, I will present a joint work with Milan Donvil in which we establish the following new degree of W$^*$-superrigidity for certain wreath product groups $G$: if $L(G)$ is virtually isomorphic, in the sense of admitting a bifinite bimodule, with any other group von Neumann algebra $L(H)$, then the groups $G$ and $H$ must be virtually isomorphic. Moreover, we allow both group von Neumann algebras to be twisted by an arbitrary 2-cocycle. At the end, I will also present an ongoing joint work with Milan Donvil on W$^*$-superrigidity within the broader class of discrete quantum groups.

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 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!

For bounded linear operators acting on (the \ell_2 space of) a uniformly locally finite metric space, there are two notions of localness, finite-propagation and quasi-locality. The distinction of these two is similar to that of compactly supported functions and functions vanishing at infinity. John Roe has asked whether quasi-local operators are approximately finite-propagation. It has been proved over the time that this is the case provided that the underlying space is sufficiently nice. On the other hand, I recently found the first example of a quasi-local operator that is not approximately finite-propagation. This is done by looking at embeddability of matrix algebras into the uniform Roe C*-algebra consisting of approximately finite-propagation operators and the quasi-local version of it.

In a recent joint work with Stefaan Vaes we proved the first W*-superrigidity theorem for twisted group von Neumann algebras. Building on this result, we proof a W*-superrigidity theorem for certain central extensions of the groups in our previous work to obtain the first examples of W*-superrigid groups with infinite center.

We introduce the first examples of groups $G$ with infinite center which are completely recognizable from their von Neumann algebras, $\mathcal{L}(G)$. Specifically, assume that $G=A\times W$, where $A$ is an infinite abelian group and $W$ is an ICC wreath-like product group \cite{cios22,amcos23} with property (T) and trivial abelianization. Then whenever $H$ is an arbitrary countable group such that $\mathcal{L}(G)$ is $\ast$-isomorphic to $\mathcal L(H)$, it must be the case that $H= B \times H_0$ where $B$ is infinite abelian and $H_0$ is isomorphic to $W$. Moreover, we completely describe the $\ast$-isomorphism between $\mathcal L(G)$ and $\mathcal L(H)$. This yields new applications to the classification of group C$^*$-algebras, including examples of non-amenable groups which are recoverable from their reduced C$^*$-algebras but not from their von Neumann algebras. This is joint work with Ionu\c{t} Chifan and Hui Tan.

As an analogue of topological boundary of discrete groups $\Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M,\tau)$ and apply it to generalize a recent result by Amrutam-Hartman-Oppelmayer, showing that for a trace preserving action $\Gamma \curvearrowright (A,\tau_A)$ on an amenable tracial von Neumann algebra, a $\Gamma$-invariant probability measure $\mu$ supported on amenable intermediate subalgebras between $A$ and $\Gamma\ltimes A$ is necessarily supported on subalgebras of $\mathrm{Rad}(\Gamma) \ltimes A$. By taking $(A,\tau_A)=L^\infty(X,\nu_X)$ for a free p.m.p. action $\Gamma \curvearrowright (X,\nu_X)$, we obtain a similar result for invariant random subequivalence relations of $\mathcal{R}_{\Gamma \curvearrowright X}$.

Given a C*-dynamical system, a fruitful avenue to study its properties is to study the dynamics on its injective envelope. This approach relies on the result of Kalantar and Kennedy (2017), who show that C*-simplicity can be characterized via the Furstenberg boundary using injective envelope techniques. In this talk, we will discuss consequences of this idea for partial C*-dynamical systems. In particular, we will introduce injective envelopes in this setting and discuss the ideal intersection property for partial C*-dynamical systems. This is joint work with Matthew Kennedy and Camila Sehnem.

In this short talk I will showcase various applications of sequential commutation in the theory of von Neumann algebras. We will see connections to entropy, model theory, and classification of von Neumann algebras.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

Generalized inductive sequence constructions promise a new perspective for viewing and building (nuclear) C*-algebras. In this construction, one replaces the *-homomorphisms from classic inductive sequences with cpc maps, which somehow still yield a C*-algebra in the limit. To guarantee a C*-structure on the limit, one must place conditions on the cpc maps in these systems: asymptotic multiplicativity, asymptotic orthogonality preserving, or C*-encoding. The first two conditions, though structurally interesting, are often too stringent to allow for many natural and/or constructive examples. On the other hand, the C*-encoding criteria is easier (and in fact necessary) to satisfy, opening up the possibility of relatively hands-on inductive limit constructions of broad classes of nuclear C*-algebras. To demonstrate this, I will describe how one can use Følner(-like) sequences to build inductive limit constructions of C*-algebras arising from amenable group (actions).

We discuss cohomological obstructions to group stability with respect to the operator norm, with an emphasis on uniform-to-local stability, touching on recent joint work with Forrest Glebe.

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

Using property (T) together with a separability argument, Ozawa showed in 2004 that there does not exist a separable II_1 factor that every other separable II_1 factor embeds into. We give a new proof of this result by introducing a hierarchy of Haagerup type approximation properties for groups and II_1 factors indexed by the ordinals. We show that for each countable ordinal $\alpha$, the $\alpha$-Haagerup property passes to subgroups/subfactors, and we construct groups/II_1 factors that have the $\alpha$-Haagerup property but do not have the $\beta$-Haagerup property for any $\beta < \alpha$. The class of II_1 factors we consider is disjoint than the class considered by Ozawa in that the II_1 factors we consider do not contain property (T) II_1 subfactors. This is joint work with Fabian Salinas.

In joint work with Matthew Kennedy, we consider the problem of characterizing when the subalgebras of a reduced crossed product $A \rtimes_r G$ are canonical, restricting our attention to intermediate subalgebras of the form $A \subseteq B \subseteq A \rtimes_r G$. The ``canonical'' subalgebras to consider arise from partial subactions of $G \curvearrowright A$, and can be thought of as generalizations of subalgebras of the form $A \rtimes_r H$ for $H \leq G$. We obtain a nearly complete, two-way characterization on when all such subalgebras are of this form, modulo some mild assumptions. In the case of commutative $A = C(X)$, this condition ends up being freeness of the action of $G$ on $X$. For the noncommutative setting, we needed to identify the ``correct'' notion of freeness of an action of $G$ on $A$, of which several already exist in the literature. The techniques involved are also quite different from those used in earlier results in the literature, and are more akin to those involved in studying simplicity of $A \rtimes_r G$. In particular, we make heavy use of the dynamics on injective envelope $I(A)$, a sort of ``noncommutative boundary'' of $A$, and also rely on some of the noncommutative convexity results of Magajna.

A group G is called Hilbert-Schmidt stable if every finite-dimensional unitary almost-representation of G is close in the Hilbert-Schmidt metric to an actual such representation. A powerful criterion of Hadwin and Shulman relates Hilbert-Schmidt stability of amenable groups to Thoma characters. We explore this connection for a wide range of groups: nilpotent, polycyclic and metabelian ones, among solvable groups, as well as diagonal products and the classical family of B.H. Neumann groups. This approach has two aspects: character classification and character approximation. For certain groups, we are able to translate the character approximation question to an interesting problem in dynamics. The talk will be based on joint works with Alon Dogon and Itamar Vigdorovich.

During the 60s and the 70s, Furstenberg developed two parallel theories regarding a group's boundaries of different flavours: the Furstenberg boundary and the Furstenberg-Poisson boundary. Both have universal properties (injectivity\Zimmer amenability) and so, their structure reveals certain groups' properties. It is no surprise that the research of these two theories and their connections with Operator Algebra Theory and rigidity phenomena of higher-rank lattices is still very active. These theories are known to share some common components, but the reason for that is not so apparent. I'll argue that the reason is that they share the same driving force, potentially suggesting direct connections between these objects. We will develop one categorical machinery to produce them both simultaneously, despite their different nature - two Boundary Theories for the price of one! Joint with Mehrdad Kalantar.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

In this talk we discuss groups acting on rooted trees with self-similarity. One can study length contraction of sections under the recursion down the tree. Such contractions are closely related to volume growth, amenability, and other properties of the group. We will explain various measurements of complexity and connections to the conformal dimension of the associated limit space. Based on joint works with Bartholdi, Matte Bon, and Nekrashevych.

Connes’ Rigidity Conjecture predicts that ICC property (T) groups can be completely recognized from their group von Neumann algebras. We investigate a natural version of Connes’ Rigidity Conjecture for central extensions, and provide examples of property (T) groups with infinite center where this holds. This is joint work with Ionuţ Chifan, Adriana Fernández Quero and Denis Osin.

(Based on joint work with T. Shulman). A group G is Hilbert-Schmidt (HS) stable if every approximate homomorphism from G to unitary matrices is close to a true homomorphism (the terms "approximate" and "close" refer to the normalized Hilbert-Schmidt norm on matrices). For amenable groups, Hadwin and Shulman provided an equivalent definition of HS-stability that makes it easier (at least for me) to use operator algebraic techniques to study these groups. I will discuss how we used some classic "uniqueness of trace" results to provide examples of amenable HS-stable groups that are not permutation stable and other results about HS-stability of amenable groups.

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

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

Studying groups via their actions on Gromov hyperbolic spaces has been a recurrent theme in geometric group theory over the past three decades. Of particular interest in this approach are actions of general type, i.e., non-elementary actions without fixed points at infinity. For a given group G, it is natural to ask whether it is possible to classify all general type actions of G on hyperbolic spaces. In a joint paper with K. Oyakawa, we propose a formalization of this question based on the complexity theory of Borel equivalence relations. Our main result is the following dichotomy: for every countable group G, general type actions of G on hyperbolic spaces can either be classified by an explicit invariant ranging in the infinite-dimensional projective space or are unclassifiable by countable structures. Special linear groups over countable fields provide examples satisfying the former alternative, while every non-elementary hyperbolic group satisfies the latter.

We will explore the notion of almost finiteness, as introduced by Kerr, in the setting of essentially free actions. This notion is one of the main tools in establishing classifiability of crossed product C*-algebras of actions of countable amenable groups on compact, metrizable spaces. Very recently, Joseph produced the first examples of minimal actions of amenable groups which are topologically free and not essentially free. While our general machinery does not give any information for his examples, as those are not almost finite,we develop ad-hoc methods to show that his actions have classifiable crossed products. This is joint work with Eusebio Gardella, Rafaela Gesing, Grigoris Kopsacheilis, and Petr Naryshkin.

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

Let X be an infinite compact metric space and \alpha : X \to X a homeomorphism. A classic construction in C*-algebras gives us the crossed product of C(X) by the integers. Adding in additional data allows us to construct more C*-algebras: Given a closed subspace Y \subset X, we can construct a so-called orbit-breaking groupoid and its groupoid C*-algebra. Given a vector bundle V over X, we can construct a C*-correspondence and its Cuntz–Pimnser algebra, and if we include both Y and V, we can build an orbit-breaking C*-correspondence and its Cuntz–Pimsner algebra. In the case the V is a line bundle, these Cuntz–Pimnser algebras can be viewed as twisted groupoid C*-algebras over the transformation groupoid and orbit-breaking groupoid, respectively. In this talk I will discuss the relationship between these C*-algebras. In particular I will talk about properties––such as simplicity, Jiang–Su stability and stable rank—are shared by all four, and how they differ. This is based on ongoing joint works with Forough and Jeong.

Property (TE) is a rigidity property concerning how a group may act on Banach spaces belonging to a class, E. I will define Property (TE), a weaker relative of it, and their generalization to Banach algebras, and give an overview of how these notions of rigidity are related.

We will construct (nonseparable) II$_1$ factors with the property that all their Haar unitaries are conjugated to each other by unitaries, and which are not the classical examples of ultraproduct II$_1$ factors. These factors may be assumed to have certain additional properties, for example being existentially closed. We will observe some consequences of this unique unitary conjugacy orbit property, some of which follow from earlier works of Popa. A new observation that these factors cannot be group algebras of groups with a certain conjugation property on group elements will also be made. This is joint work with Srivatsav Kunnawalkam Elayavalli, Gregory Patchell, and Hui Tan.

Results of Archbold and Spielberg, and Kalantar and Kennedy assert that a discrete group admits a topologically free boundary if and only if the reduced crossed product of continuous functions on its Furstenberg boundary by the group is simple. In this talk, I will show a similar result for totally disconnected locally compact groups.

In the past two decades there has been major progress in producing $W^*$-superrigid groups (groups $G$ that can be completely recovered from the group von Neumann algebra $\mathcal{L}(G)$). On the other hand, there are many classes of groups that were already known to not be $W^*$-superrigid. In particular, if $\mathcal{L}(G)$ is a McDuff factor, then $G$ cannot be $W^*$-superrigid as $\mathcal{L}(G) \cong \mathcal{L}(G\times A)$ for any icc amenable group $A$. In our work, we introduce the first examples of groups whose lack of $W^*$-superrigidity can be completely characterized. Specifically, we introduce the notion of, and construct, groups that are McDuff $W^*$-superrigid, that is groups $G$ such that if $\mathcal{L}(G) = \mathcal{L}(H)$ (for an arbitrary group $H$), then we must have $H = G \times A$ for some icc amenable group $A$. We do this by combining geometric group theory methods to construct wreath-like product groups with a 2-cocycle with uniformly bounded support, and deformation/rigidity methods (via the interplay of two types of deformations) to prove these groups possess the infinite product rigidity property. This is ongoing work with Ionu\c{t} Chifan, Denis Osin and Bin Sun.

We will present some ideas from an ongoing joint work with Itamar Vigdorovich. These arise in an attempt to find new spectral gap properties of higher rank lattices without property (T) in order to complete the missing "property (T) half" of character rigidity. Somewhat surprisingly, we show that character rigidity is equivalent to a weak form of Hilbert--Schmidt stability for such groups.

In this talk, I will introduce a graph for graph product of groups, which I call the extension graph. This new object enables us to exploit the geometry of a defining graph to study properties of graph product of groups beyond the case of finite defining graphs. As an application in operator algebras, I present strong solidity and relative bi-exactness of graph product of finite groups whose defining graph is hyperbolic.

We give a new complete description theorem of the intermediate operator algebras, which unifies the discrete Galois correspondence results and the crossed product splitting results, and involves 2-cocycles. As an application, we obtain a Galois’s type result for Bisch--Haagerup type inclusions arising from isometrically shift-absorbing actions of compact-by-discrete groups. Based on my preprint arxiv:2406.00304

For free *-algebras, free group algebras, and related algebras, it is possible to decide if an element is positive (in all representations) using results of Helton, Bakonyi-Timotin, Helton-McCullough, and others. In this talk, I'll discuss joint work with Arthur Mehta and Yuming Zhao showing that this problem becomes undecidable for tensor products of this algebras. I'll also discuss how results of this type could be aided by having a Higman embedding theorem for algebras with states, as well as work in progress on this question.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, 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.

In this talk I will report on a recent result on the ideal intersection property for group actions on noncommutative C*-algebras. This is joint work with M. Kennedy and L. Kroell.

A state on a C*-algebra (or a positive definite function on a group) is called a trace if it is ad-invariant. It is called a stationary-trace if it is ad-invariant on average (with respect to a fixed probability measure on the unitaries). Every trace is, of course, a stationary trace. When the converse also holds (also known as stiffness), powerful consequences arise. I will present some general questions on this topic, followed by a result with Uri Bader concerning (non-semisimple) arithmetic groups.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

A conjecture of Popa states that the ${\rm II}_1$ factor arising from a Bernoulli action of a nonamenable group has a unique (group measure space) Cartan subalgebra, up to unitary conjugacy. In this talk, I will discuss this conjecture and show that it holds for weakly amenable groups with constant $1$ among algebraic actions. The proof involves the notion of properly proximal groups introduced by Boutonnet, Ioana, and Peterson.

A group is said to be matricially stable if every function from the group to unitary matrices that is "almost multiplicative" in the point-operator norm topology is "close," in the same topology, to a genuine representation. A result of Dadarlat shows that even cohomology obstructs matricial stability. The obstruction in his proof can be realized as follows. To each almost-representation, we may associate a vector bundle. This vector bundle has topological invariants, called Chern characters which lie in the even cohomology of the group. If any of these invariants are nonzero, the almost-representation is far from a genuine representation. Previous work of mine sheds light on how one can make explicit examples of almost representations that observe the nonvanishing of the obstruction in 2-cohomology. In this talk, I will describe examples from upcoming joint work with Marius Dadarlat that observe the obstructions in higher cohomology. These examples come about by exploiting the multiplicative structure of the cohomology ring.