# Operator Algebras Seminar

The Operator Algebra Seminar will be held on Tuesdays starting at 1:30 pm EST in CL50 Room 129 unless specified otherwise. Additional details about a specific talk can be uncovered by clicking on the speaker’s name.

## Current Semester Schedule

## August 29 - Rolando de Santiago

Title: Algebraic Soficity and graph products

Abstract: We show that graph products of non trivial finite dimensional von Neumann algebras are strongly 1-bounded when the underlying *-algebra has vanishing first L2-Betti number. We use a combination of the following two new ideas to obtain lower bounds on the Fuglede Kadison determinant of matrix polynomials in a generating set: a notion called 'algebraic soficity' for *-algebras allowing for the existence of Galois bounded microstates with asymptotically constant diagonals; a new probabilistic construction of permutation models for graph independence over the diagonal.

## September 5 - Connor Thompson, Purdue University

Title: Properly Proximal von Neumann Algebras

Abstract: I will introduce definitions for properly proximal groups and von Neumann algebras as well as explain the relation between them. Then, following a paper by Changying Ding, I will discuss various results relating to properly proximal von Neumann algebras.

## September 12 - Dave Penneys, The Ohio State University (Zoom)

Title: An operator algebraic axiomatization of local topological orders

Abstract: Topological order is a notion in theoretical condensed matter physics describing new phases of matter beyond Landau's symmetry breaking paradigm. Bravyi, Hastings, and Michalakis introduced certain topological quantum order (TQO) axioms to ensure gap stability of a commuting projector local Hamiltonian and stabilize the ground state space with respect to local operators in a quantum spin system.

In joint work with Corey Jones, Pieter Naaijkens, and Daniel Wallick (arXiv:2307.12552), we study nets of finite dimensional C*-algebras on a \(\mathbb{Z}^{k}\) lattice equipped with a net of projections as an abstract version of a quantum spin system equipped with a local Hamiltonian. We introduce a set of local topological order (LTO) axioms which imply the TQO conditions of Bravyi-Hastings-Michalakis in the frustration free commuting projector setting, and we show our LTO axioms are satisfied by known 2D examples, including Kitaev's toric code and Levin-Wen string net models associated to unitary fusion categories (UFCs).

From the LTO axioms, we can produce a canonical net of algebras on a codimension 1 sublattice \(\mathbb{Z}^{k-1}\) which we call the net of boundary algebras. We get a canonical state on the boundary net of algebras, and we calculate this canonical state for both the toric code and Levin-Wen string net models. Surprisingly, for the Levin-Wen model, this state is a trace on the boundary net exactly when the UFC is pointed, i.e., all quantum dimensions are equal to 1. Moreover, the boundary net of algebras for Levin-Wen are bounded-spread isomorphic to nets of algebras arising directly from the UFC; for these latter nets, Corey Jones' category of DHR bimodules recovers the Drinfeld center, leading to a bulk-boundary correspondence where the bulk topological order is described by representations of the boundary net.

## September 19 - March Boedihardjo, MSU (Zoom)

Title: Spectral norm and strong freeness

Abstract: We give a non-asymptotic estimate for the spectral norm of a large class of random matrices that is sharp in many cases. We also obtain strong asymptotic freeness for certain sparse Gaussian matrices. Joint work with Afonso Bandeira and Ramon van Handel.

## September 28 (Thursday Room Math 731 at 1:30 pm) - Hui Tan, UCSD

Title: Existential closedeness and the structure of bimodules of II\(_1\) factors

Abstract: A separable II\(_1\) factor is existentially closed if any larger II\(_1\) factor can be realized as an intermediate subfactor of the inclusion into one of its ultrapowers. I will talk about the consequence existential closedness has on the structure of bimodules for a II\(_1\) factor. I will also discuss the notion of existential closedness within the class of full factors. This is joint work with Adrian Ioana.

## October 3 - Nicholas LaRacuente, Indiana University

Title: Entropy-Producing Quantum Dynamics in von Neumann Algebras

Abstract: Finite-dimensional information or complexity theoretic analogies could explain phenomena usually modeled via quantum field theories in von Neumann algebras of type II or III. Hence a broad question is how quantum dynamics depend on their underlying operator algebra. Araki's definition of quantum relative entropy via the relative modular operator suggests that this notion of information and corresponding non-invertibility applies widely across von Neumann algebras. I will discuss recent results that hold across types, conjectures on what other phenomena might, and techniques that transfer estimates across types. Particular emphasis is on dynamics modeled by quantum Markov semigroups, channels, and interactions between subalgebras. This talk will include joint work with Li Gao, Marius Junge, and Haojian Li.

## October 24 - Jenny Pi, UCI (joint with Model Theory Seminar)

Title: A Peek at Model Theory for Tracial von Neumann Algebras

Abstract: Since the work of Farah, Hart, and Sherman in "Model Theory of Operator Algebras" I, II, and III in the early 2010's, there has been a large body of results stemming from the application of model theory to studying operator algebras (and in the past few years, the converse as well). In this talk, I will give a bit of history and survey a few of these results in the case of tracial von Neumann algebras. Notably, I will highlight some structural results related to the classification of von Neumann algebras and their first-order theories. Both operator algebraists and logicians are welcome!

## October 31 - Christopher Felder, IU

Title: Forward Operator Monoids

Abstract: This talk will introduce a forward operator monoid— a discrete collection of bounded linear operators acting on a separable Hilbert space, which contains the identity, and has a strict monoid structure. We will then discuss generalized inner and cyclic vectors for these monoids, with a focus on vectors which are both inner a cyclic, which we will call units. Time permitting, we will connect this to some interesting open problems in analysis, including some completeness and approximation problems.

## November 14 - Eli Bashwinger, University of Albany SUNY

Title: on Neumann Algebras of Thompson-like Groups

Abstract: In this talk, I will discuss some results concerning the von Neumann algebras of Thompson-like groups arising from \(d\)-ary cloning systems (for \(d \ge 2\)). This talk will essentially be a survey of results coming from three papers I have written so far on this topic, the first two of which I wrote with Matthew Zaremsky, who invented cloning systems with Stefan Witzel in 2018. Given a \(d\)-ary cloning system on a sequence \((G_n)_{n \in \mathbb{N}}\) of groups, we can form a Thompson-like group denoted by \(\mathscr{T}_d(G_*)\), and this group canonically contains \(F_d\), the smallest of the Higman-Thompson groups. The group inclusion \(F_d \le \mathscr{T}_d(G_*)\) translates to an inclusion of their group von Neumann algebras \(L(F_d) \subseteq L(\mathscr{T}_d(G_*))\). Concerning this inclusion, I was able to prove it satisfies the weak asymptotic homomorphisms property (WAHP), which is equivalent to \(L(F_d)\) being a weakly mixing subfactor of \(L(\mathscr{T}_d(G_*))\). That the inclusion satisfies the WAHP will have a number of consequences which I will discuss. Another main result is that many of these Thompson-like groups yield McDuff factors, a considerable generalization of Jolissaint's result that \(L(F)\) is a McDuff factor, where \(F = F_2\) is one of the "classical" Thompson's groups. Using cloning systems, I constructed a machine which takes in any group and produces a Thompson-like group yielding a McDuff factor. Modifying this construction, I was also able to construct another machine which takes in any finite group and any other group and produces an infinite index singular inclusion of \(II_1\) factors without the WAHP, the first examples of their kind. Using cloning systems and character rigidity, I was also able to prove the Higman-Thompson groups \(F_d\) are McDuff (in the sense of Vaes-Deprez) and that, in some cases, a certain canonical subgroup of \(\mathscr{T}_d(G_*)\) yields a Cartan subalgebra in these Thompson-like group factors. If time permits, I will also mention some preliminary results on the \(C^*\)-algebra side of Thompson-like groups arising from cloning systems.

## November 28 - Alon Dogon, Weizmann Institute of Science (Zoom)

Title: Hilbert-Schmidt stability versus Connes Embeddability of groups

Abstract: In his seminal paper from 1976, A. Connes singled out an important approximation property for tracial von Neumann algebras, and asked whether all separable II1 factors satisfy it. Consequently, F. Radulescu asked the question of whether all group von Neumann algebras are Connes embeddable. This question became of major interest to group theory, through its connection with the notion of sofic groups. On the other side of the story, several new robustness properties of groups, colloquially referred to as stability, have gathered considerable attention in recent years. Among them is Flexible Hilbert Schmidt stability: A group G is flexibly HS-stable if any approximate finite dimensional unitary representation of G is close to a compression of a genuine representation of slightly larger dimension. In this talk, we will give conditional statements of the form "If G is flexibly HS-stable, then there exists a non Connes embeddable group". This statement is shown to hold for many property (T) groups, including Gromov random groups, and lattices in non-simply connected simple Lie groups. Time permitting, we will comment on the operator-algebraic proof.

## Previous Semester Seminar Schedules

## Spring 2023

## January 17th, Marius Dadarlat, Purdue University

Title: Uniform epsilon-representations which are not pointwise close to genuine representation

Abstract: Kazhdan showed that a surface group of genus greater than one admits uniform epsilon-representations which cannot be approximated by genuine representations in the point-norm topology. We shall present other classes of hyperbolic groups with the same property.

## January 24 - Daniel Wallick, The Ohio State University (Zoom)

Title: An algebraic quantum field theoretic approach to toric code with gapped boundary

Abstract: Topologically ordered quantum spin systems have become an area of great interest, as they may provide a fault-tolerant means of quantum computation. One of the simplest examples of such a spin system is Kitaev's toric code. Naaijkens made mathematically rigorous the treatment of toric code on an infinite planar lattice (the thermodynamic limit), using an operator algebraic approach via algebraic quantum field theory. We adapt his methods to study the case of toric code with gapped boundary. In particular, we recover the condensation results described in Kitaev and Kong and show that the boundary theory is a module tensor category over the bulk, as expected.

Additional Resources:- Main paper with result: ArXiV Link
- Pieter Naaijkens’ paper on the sector theory for toric code(no boundary). This paper kickstarted this whole field of research, and much of the work in my paper is based on Naaijkens’ work in this paper: ArXiV Link
- Alexei Kitaev and Liang Kong’s paper on boundary theories and defects. This paper describes the physics results that I recovered in my work, specifically the condensation results and the module tensor category structure: ArXiV Link

## January 31 - Lara Ismert, Embry-Riddle Aeronautical University in Prescott, AZ (Zoom)

Title: Analytic Vectors for Derivations on C\(^*\)-algebras

Abstract: Given a (possibly unbounded) symmetric operator D on a Hilbert space \(H\), Nelson’s Analytic Vector Theorem states that \(D\) is in fact (essentially) self-adjoint (which is a stronger condition than being symmetric when \(D\) is unbounded) if and only if \(D\) has a dense set of analytic vectors in \(H\). In this talk, we will review the difference between symmetric and self-adjoint unbounded operators on Hilbert space, and we will discuss the definition of an analytic vector for such an operator. Then, we will consider the definition of an analytic vector for a derivation on C\(^*\)-algebra which arises via commutation with a self-adjoint operator on a Hilbert space. We will finish by showing that assuming our derivation is weak operator topology continuous, there is a weak operator topology dense set of analytic vectors for that derivation, which is an analogous theorem to Nelson’s but in the case of derivations on C\(^*\)-algebras.

Additional Resources:## February 7 - Angel Roman, Washington University St. Louis

Title: C\(^*\)-Algebraic Mackey Analogy of Reductive Groups

Abstract: In the 1970's, George Mackey proposed an analogy between some unitary representations of a semisimple Lie group and unitary representations of its associated semidirect product group, known as the Cartan motion group. Recently, Nigel Higson and Alexandre Afgoustidis made this analogy precise between equivalence classes of irreducible tempered representations of a reductive groups and equivalence classes of irreducible unitary representations of the associated Cartan group. First, I will introduce tempered unitary representations of a reductive Lie group to develop the Mackey analogy. The goal of this talk is to show that the reduced group C\(^*\)-algebra of the Cartan motion group can be embedded into the reduced group C\(^*\)-algebra of the reductive group. To accomplish this, I will show the construction of a continuous fields of C\(^*\)-algebras generated from a Lie groupoid known as deformation to the normal cone. I will also discuss the Fourier transform picture.

Additional resoureces## February 21 - Mitch Hamidi, Embry-Riddle Aeronautical University in Prescott, AZ (Zoom)

Title: Quantum edge spaces and C*-algebras associated to quantum graphs

Abstract: Quantum graphs are noncommutative generalizations of finite graphs that have been of significant interest in recent years. In this talk, I will introduce the edge space of a quantum graph and discuss connections between two C*-algebras one can associate to a quantum graph.

Additional Resources:- The paper on which this talk is based: Quantum edge correspondences and quantum Cuntz—Krieger algebras
- The paper that introduced the motivation for the class of C*-algebras associated to quantum graphs relevant to this talk: Quantum Cuntz-Krieger algebras

## February 28 - Aldo Garcia Guinto, Michigan State University

Title: Free Stein dimension of crossed products by finite group

Abstract: In this talk, we will discuss a free probabilistic quantity called free Stein dimension and compute it for a crossed product by a finite group. The free Stein dimension is the Murray-von Neumann dimension of a particular subspace of derivations. Charlesworth and Nelson defined this quantity in the hope of finding a von Neumann algebra invariant. While it is still not known to be a von Neumann algebra invariant, it is an invariant for finitely generated unital tracial *-algebras and algebraic methods have been more successful than analytic ones in studying it. Our result continuous this trend and reveals a formula for the free Stein dimension of a crossed product by a finite group that is reminiscent of the Schreier formula for a finite index subgroup of free groups.

## Mar 7 - Changying Ding, Vanderbilt University

Title: First \(\ell^2 \)-Betti numbers and proper proximality

Abstract: Properly proximal groups were introduced by Boutonnet, Ioana, and Peterson, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will show that exact groups with positive first \(\ell^2\)-Betti numbers are properly proximal. I will also describe an OE-superrgidity result of Bernoulli shifts of nonamenable non-properly proximal exact groups.

Additional Resources:## Mar 21 - Iason Moutzouris, Purdue University

Title: When do reduced group C\(^∗\)-algebras of amenable groups not have real rank zero?

Abstract: For every torsion free, discrete and amenable group \(G\), the Kadison-Kaplansky conjecture has been verified, so C\(^∗ _r (G)\) has no nontrivial projections. On the other hand, every torsion element \(g \in G\), of order \(n\), gives rise to a projection \( \frac{1+g+\cdots +g^{n-1}}{n} \in C^∗_r (G)\). Actually, if \(G\) is locally finite, then \(C^∗_r (G)\) is an AF-algebra, so it has an abundance of projections. So, it natural to ask what happens on the intermediate cases. In 2017, Scarparo proved that for every discrete, infinite, finitely generated elementary amenable group \(G\), \(C^∗_r (G)\) cannot have real rank zero. Roughly speaking, having real rank zero is a weaker property than being an AF-algebra and is characterized by having many projections. In this talk, we will prove that if \(G\) is discrete, amenable and has a normal subgroup that is not locally finite but is elementary amenable with finite Hirsch length, then \(C^∗_r (G)\) does not have real rank zero.

## April 4 - Roy Araiza, UIUC

Title: Operator systems generated by projections

Abstract: During this lecture I will discuss recent work describing operator systems and k-AOU spaces generated by a finite number of (abstract) projections, satisfying a finite number of linear relations. These families are constructed as inductive limits of particular operator systems, and particular k-AOU spaces, respectively. We will discuss how if we impose the nonsignalling relations on our projections, then we obtain a hierarchy of ordered spaces, dual to that of correlation sets.

## April 10 - Srivatsav Kunnawalkam Elayavalli, IPAM (Math 731, 4:30 pm)

Title: Conjugacy for almost homomorphisms of sofic groups.

Abstract: I will discuss recent joint work with Hayes wherein we show that any sofic group G that is initially sub-amenable (a limit of amenable groups in Grigorchuk's space of marked groups) admits two embeddings into the universal sofic group S that are not conjugate by any automorphism of S. Time permintting, I will also characterise precisely when two almost homomorphisms of an amenable group G are conjugate, in terms of certain IRS's associated to the two actions of G. One of the applications of this is to recover the result of Becker-Lubotzky-Thom around permutation stability for amenable groups. The main novelty of our work is the usage of von Neumann algebraic techniques in a crucial way to obtain group theoretic consequences.

## April 11 - Sayan Das, UC Riverside (Zoom)

Title: Poisson boundaries of finite von Neumann algebras

Abstract: The study of noncommutative Poisson boundaries for finite von Neumann algebras was initiated by Prof. Jesse Peterson and the speaker in a recent work. In this talk, I will describe the construction of noncommutative Poisson boundaries, and present a double ergodicity theorem. I will also show how the double ergodicity theorem can be used to prove that every II_1 factor satisfies Popa’s Mean-Value property, thereby answering a question he posed in 2019. If time permits, I will present some new results on the structure of noncommutative Poisson boundaries. This talk is partly based on a joint work with Prof. Jesse Peterson.

## April 25 - Eric Roon, Univeristy of Arizona (Zoom)

Title: Ergodic Quantum Processes in Finite von Neumann algebras

Abstract: In 1997, H. Hennion used a non-standard metric to show a kind of multiplicative ergodic theorem for the convergence of an infinite product of positive random matrices. Recently Movassagh and Schenker proved a quantum-channel version of Hennion's ergodic theorem. We will discuss the necessary background to understand the generalization of Hennion's metric to the state space of a tracial von Neumann algebra $(M,\tau)$, and a characterization of contraction mappings in this metric. We will discuss generalizations of the theorems of Movassagh and Schenker to everywhere-defined, positive maps on the noncommutative $L^1$ space. Time permitting, we will sketch the proof of our characterization of contraction mappings and discuss its applications to locally normal states on the spin chain. This is part of work in progress in collaboration with Brent Nelson.

## Fall 2022

## September 13 - Krishnendu Khan, Purdue University

Title: TBD

Abstract: TBD

## September 20 - Forrest Glebe, Purdue University

Title: Frobenius Non-Stability of Nilpotent Groups

Abstract: A countable discrete group is said to be Frobenius stable if every function from the group to unitary matrices that is "almost multiplicative" in Frobenius norm is "close" to a unitary representation in Frobenius norm. The purpose of this talk is to outline a proof of my recent result that torsion free finitely generated nilpotent groups, other than the integers and the trivial group, are not Frobenius stable.

## October 4 - Julia Plavnik, Indiana University

Title: On the classification of modular categories

Abstract: Modular categories are intricate organizing algebraic structures appearing in a variety of mathematical subjects including topological quantum field theory, conformal field theory, representation theory of quantum groups, von Neumann algebras, and vertex operator algebras. They are fusion categories with additional braiding and pivotal structures satisfying a non-degeneracy condition. The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories, and we will also give some concrete examples to have a better understanding of their structures. I will give an overview of the current situation of the classification program for modular categories and mention some open directions to explore.

## October 18 - Lydia de Wolf, Kansas State University (Zoom)

Title: Beyond K-graphs

Abstract: In this talk, we will present discrete Conduche fibrations and Kumjian-Pask fibrations, which were studied as a generalization of \(k\)-graphs. We will briefly summarize the basic results about their C\(^*\)-algebras and path groupoids in general, and then demonstrate examples of KPfs in which specific choices allow for more exploration than the fully general case.

## November 1 - Andrew Toms, Purdue University

Title: Homotopy groups of Cuntz classes

Abstract: TBD

## November 10 - Priyanga Ganesan, UCSD (Location BRNG B238)

Title: Quantum Non-Local Games

Abstract: In recent years, nonlocal games have received significant attention in operator algebras and resulted in highly fruitful interactions, including the recent resolution of the Connes Embedding Problem. A nonlocal game involves two non-communicating players (Alice and Bob) who cooperatively play to win against a referee. In this talk, I will provide an introduction to the theory of non-local games and quantum correlation classes. We will discuss the role of C\(^*\)-algebras and operator systems in the study of their perfect strategies and observe that mathematical structures arising from entanglement-assisted strategies for nonlocal games can be naturally interpreted and studied using tools from operator algebras and quantum groups. An alternate formulation of quantum correlations in terms of quantum channels will be presented. This will be used to introduce a more general framework of non local games involving quantum inputs and quantum outputs.

## December 6 - Rolando de Santiago

Bounds on Quantum Chromatic Numbers for Products of Graphs

Abstract:

Additional Resources:## Spring 2022

## January 25 - Iason Moutzouris, Purdue University

Title: Extensions of quasidiagonal \(C^*\)-algebras.

Abstract: It is known that a quasidiagonal \(C^*\)-algebra is stably finite. While the converse is not true (one counterexample is \(C_r(\mathbb{F}_2))\), Blackadar and Kirchberg conjectured that it holds for separable and nuclear \(C^*\)-algebras. In my talk, I will examine the validity of this conjecture for \(C^*\)-algebras that arise as extensions of separable, nuclear and quasidiagonal \(C^*\)-algebras that satisfy the UCT. By the work of Brown and Dadarlat, this problem is equivalent to proving a K-theoretic embedding property, called \(K_0\)-embedding property. Using this approach, I will solve the problem in the case that the ideal is locally approximated by algebras in a class that contains all approximately subhomogeneous algebras, as well as certain crossed products by the integers.

## March 22 - Hui Tan, UCSD

Title: Spectral gap characterizations of property (T) for II\(_1\) factors.

Abstract: For property (T) II\(_1\) factors, any inclusion into a tracial von Neumann algebra has spectral gap, and therefore weak spectral gap. I will discuss characterizations of property (T) for II\(_1\) factors by weak spectral gap in inclusions. I will explain how this is related to the non-weakly-mixing property of the bimodules containing almost central vectors, from which we also obtain a characterization of property (T).

Additional Resources- Primary reference for main result Spectral Gap Chacterization of Property (T) for II Factors
- Additional Reading Related to Property (T) Kazhdan’s Property (T) by Bekka, de la Harpe, and Valette.
- Additional Reading Related to II\(_1\) Factors An Introduction to II_1 factors by Anantharaman-Delaroche and Popa

## March 29 - Andrew Toms, Purdue University (Room BRNG B212)

Title: Homotopies of Constant Cuntz Class

Abstract: Let \(A\) be a unital simple separable exact C\(^*\)-algebra which is approximately divisible and of real rank zero. We prove that the set of positive elements in \(A\) with a fixed Cuntz class is path connected. This result applies in particular to irrational rotation algebras and AF algebras.

## April 5 - Rolando de Santiago, Purdue University (Room BRNG B248)

Title: Group von Neumann Algebras

Abstract: A foundational question of von Neumann algebras is “does the group von Neumann algebra retain information about the underlying group?” While in general the answer to this question is no, making classification of group von Neumann algebras a challenging questions. In this talk we will show how one applies group theoretic information to distinguish certain classes of von Neumann algebras.

## April 12 - Forrest Glebe, Purdue University (Room BRNG B248)

Title: A Constructive Proof that Many Groups are Not Matricially Stable

Abstract: A discrete group is matricially stable if every function from the group to a complex unitary group that is ``almost multiplicative'' in the point-operator norm topology is ``close'' to a genuine unitary representation in the point-operator norm topology. It follows from a recent result due to Dadarlat shows that all polycyclic groups with non-torsion integral 2-cohomology are not matricially stable, but the proof does not lead to explicit examples of asymptotic representations that are not perturbable to genuine representations. In this talk I will construct explicit examples of asymptotic representations that are not perturbable to genuine representations for a class of groups that contains all finitely generated groups with a non-torsion 2-cohomology class that corresponds to a central extension where the middle group is residually finite.

## April 19 - Thomas Sinclair, Purdue University (Room BRNG B248)

Title: A notion of index for inclusions of operator systems

Abstract: Inspired by a well-known characterization of the index of an inclusion of II\(_1\) factors due to Pimsner and Popa, we define an index-type invariant for inclusions of operator systems. We compute examples of this invariant, show that it is multiplicative under minimal tensor products, and explain how it generalizes the Lov\'asz theta invariant to general matricial systems in a manner that is closely related to the quantum Lov\'asz thetha invariant defined by Duan, Severini, and Winter. This is joint work with Roy Araiza and Colton Griffin.

## April 26 - Krishnendu Khan, University of Iowa(Room BRNG B248)

Title: Primeness results for von Neumann algebras associated to groups acting on trees

Abstract: A type II\(_1\) factor is called prime if it cannot be decomposed as the tensor product of two type II\(_1\) factors. In this talk, I will show primeness of group von Neumann algebras associated to a class of groups acting non elementarily on tree with weakly malnormal edge stabilizer. This talk is based on a Joint work with I. Chifan and S. Kunnawalkam Elayavalli.

## Fall 2021

## September 21st - Srivatsav Kunnuwalkam Elayavalli, Vanderbilt (Room BRNG B222 @ 1:30 pm)

Title: Proper proximality for groups acting on trees.

Abstract: In joint work with Changying Ding, we obtained recently new examples of properly proximal groups arising from actions on trees. These include amalgamated products whose amalgams satisfy a weak form of malnormality. I will discuss these examples and show you why they are properly proximal. Time permitting, I will also discuss a complete classification result for graph products of groups, in terms of proper proximality. There are several interesting open questions that arise, and I will mention a few of them.

Additional Resources:- The main paper where proper proximality is introduced Proper Proximality
- The paper on which this talk is based Proper proximality
- The lecture given by Jesse Peterson about proper proximality in YMC*A: Lectures 3 and 4.
- A previous talk by Srivatsav K.E. on this subject: Item 1.

## October 19th - Priyanga Ganesan, Texas A & M

Title: Spectral bounds for chromatic number of quantum graphs

ABSTRACT: Quantum graphs are an operator space generalization of classical graphs that have appeared in different branches of mathematics including operator systems theory, non-commutative topology and quantum information theory. In this talk, I will review the different perspectives to quantum graphs and introduce a chromatic number for quantum graphs using a non-local game with quantum inputs and classical outputs. I will then show that many spectral lower bounds for chromatic numbers in the classical case (such as Hoffman’s bound) also hold in the setting of quantum graphs. This is achieved using an algebraic formulation of quantum graph coloring and tools from linear algebra.

Additional Resources:- Runyao Duan, Simone Severini, and Andreas Winter, Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lov ́asz number, IEEE Trans. Inform. Theory, 59(2):1164–1174, 2013.
- Benjamin Musto, David Reutter, and Dominic Verdon., A compositional approach to quantum functions., J. Math. Phys., 59(8):081706, 42, 2018.
- Nik Weaver, Quantum graphs as quantum relations, The Journal of Geometric Analysis, pages 1–23, 2021.
- Dan Stahlke, Quantum zero-error source-channel coding and non-commutative graph theory.,IEEE Trans. Inform. Theory, 62(1):554–577, 2016.
- Michael Brannan, Priyanga Ganesan, and Samuel J. Harris, The quantum-to-classical graph homomorphism game, [ arXiv:2009.07229 ].
- Priyanga Ganesan, Spectral bounds for chromatic number of quantum graphs, 2021. [ will be available on arxiv soon]

## October 26th - Sami Brooker, ASU

Title: Relative Toeplitz graph algebras: subgraphs, ideals, quotients, and pullbacks

Abstract: Abstract: Part of the appeal of graph \(C^*\)-algebras is that we can get information about them by looking at the structure of their underlying directed graphs. For instance, the quotient of \(C^*(E)\) by a gauge-invariant ideal is naturally isomorphic to the graph algebra of a certain subgraph of \(E\) – well, sometimes; if \(E\) is row-finite then this is always the case. Otherwise, the quotient \(C^*\)-algebra may be realized as the algebra of a graph that can be built from \(E\), according to data encoded in the ideal (Bates, et al. 2002), and, for certain ideals, the result will in fact be a subgraph of \(E\). We can elide some of this fussiness if we look instead at the relative Toeplitz algebra \(\mathcal{T}C^*(E,A)\) given by some set \(A \subseteq \text{reg}(E)\). We choose to take this approach, and introduce a corresponding category of relative directed graphs. We describe our algebras using groupoids built from the paths in the graph. In our main result, we characterize when the commuting square of relative Toeplitz algebras given by a pushout diagram of relative graphs is, in fact, a pullback diagram of \(C^*\)-algebras, generalizing results of Hajac et al. (2020). This is joint work with Jack Spielber

## Nov 2nd - Elizabeth Gillaspy, University of Montana

Title: The Weyl construction for dynamical Cartan subalgebras

Abstract: uilding on earlier work of Kumjian, Renault proved in 2008 that a C*-algebra \(A\) has a Cartan subalgebra \(B\) iff \(A\) is the C*-algebra of a twist over a topologically principal groupoid \(W\) (the Weyl groupoid of the Cartan pair \((B, A))\). However, one can build C*-algebras out of much more general groupoids. Do those have Cartan subalgebras, and if so, what is the relationship between the Weyl groupoid and the original groupoid? In joint work with A. Duwenig, R. Norton, S. Reznikoff, and S. Wright, we identified situations when a subgroupoid S of a non-principal groupoid G will give rise to a Cartan subalgebra \(B = C*(S)\) of \(A = C*(G)\). Subsequent work, joint with A. Duwenig and R. Norton, revealed that the Weyl groupoid \(W\) of the pair \((B, A)\) is a semidirect product: \(W = G/S \ltimes \widehat{S}\). We also describe the Weyl twist explicitly in the situation where there is a continuous section \(G/S \to G\). If you're still mostly lost after reading this abstract, never fear! The talk will not assume familiarity with groupoids, their C*-algebras, or Cartan subalgebras for C*-algebras, and should (I hope) be more comprehensible.

## November 16th - Ionut Chifan, The University of Iowa @ 1:15 PM Eastern

Title: On Connes Rigidity Conjecture

Abstract: In this talk I will introduce a new class of groups, which we call *wreath-like products*. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have strong fixed point properties including Kazhdan's property (T). In this paper, we establish several new rigidity results for von Neumann algebras of wreath-like products. In particular, we obtain the first continuum of property (T) groups whose von Neumann algebras satisfy Connes' rigidity conjecture and the first examples of W\(^*\)-superrigid groups with property (T). We also compute automorphism groups of von Neumann algebras of a wide class of wreath-like products; as an application, we show that for every finitely presented group \(Q\), there exists a property (T) group \(G\) such that \(Out(L(G))\cong Q\).
This is based on new joint work with Adrian Ioana, Denis Osin and Bin Sun.

- I. Chifan, A. Ioana, D. Osin, and B. Sun,
*Wreath-like product groups and rigidity of their von Neumann algebras*, https://arxiv.org/pdf/2111.04708.pdf.