# UCSD Functional Analysis Seminar

The UCSD Functional Analysis seminar, Fall 2022, happens on Tuesdays, usually at 11:00-11:50am Pacific time (US), from September 28 through November 30. It is organized by Priyanga Ganesan and myself. The seminar is open to attend for faculty and graduate students in operator algebras from any institution. Please email djekel@ucsd.edu for the meeting details since the Zoom link will not be posted publicly.

## Future Meetings

### November 22, 2022: Sayan Das (in person)

**Title:** Strong Approximate Transitivity

**Abstract:** The notion of Strong Approximate Transitivity (SAT) for group actions on probability measure spaces was introduced by Jaworski in the early 90's. A canonical example of an SAT group action is provided by a group acting on its Poisson boundary (with respect to some "nice" probability measure on the group). In this talk, I will discuss a noncommutative analogue of the SAT property, and its connection with noncommutative Poisson boundary inclusions.

### November 29, 2022: Runlian Xia (Zoom)

### November 29, 2022: Camille Horbez (2pm, in person)

**Joint Functional Analysis and Group Actions Seminar, APM 7321**

**Title:** Measure equivalence rigidity among the Higman groups

**Abstract:** The Higman groups were introduced in 1951 (by Higman) as the first examples of infinite finitely presented groups with no nontrivial finite quotient. They have a simple presentation, with k\ge 4 generators, where two consecutive generators (considered cyclically) generate a Baumslag-Solitar subgroup. Higman groups have received a lot of attention and remain mysterious in many ways. We study them from the viewoint of measured group theory, and prove that many of them are superrigid for measure equivalence (a notion introduced by Gromov as a measure-theoretic analogue of quasi-isometry). I will explain the motivation and context behind this theorem, some consequences, both geometric (e.g. regarding the automorphisms of their Cayley graphs) and for associated von Neumann algebras. I will also present some of the tools arising in the proof. This is joint work with Jingyin Huang.

## Past Meetings

The speakers and abstracts from the 2020-2021 academic year can be found on Matthew Wiersma's website here.

**Time: **11:00-11:50am Pacific time (US)

**Title: **The Connes Embedding Problem, MIP*=RE, and the Completeness Theorem

**Abstract: **The Connes Embedding Problem (CEP) is arguably one of the most famous open problems in operator algebras. Roughly, it asks if every tracial von Neumann algebra can be approximated by matrix algebras. Earlier this year, a group of computer scientists proved a landmark result in complexity theory called MIP*=RE, and, as a corollary, gave a negative solution to the CEP. However, the derivation of the negative solution of the CEP from MIP*=RE involves several very complicated detours through C*-algebra theory and quantum information theory. In this talk, I will present joint work with Bradd Hart where we show how some relatively simple model-theoretic arguments can yield a direct proof of the failure of the CEP from MIP*=RE while simultaneously yielding a stronger, Gödelian-style refutation of CEP as well as the existence of “many” counterexamples to CEP. No prior background in any of these areas will be assumed.

**Time: **October 5, 2021, 11:00-11:50am

**Title: **Strong 1-boundedness and Property (T)

**Abstract: **Strong 1-boundedness is a notion introduced by Kenley Jung which captures (among Connes embeddable von Neumann algebras) the property of having "a small amount of matrix approximations". Some examples are diffuse hyperfinite von Neumann algebras, vNa's with Property Gamma, vNa's that are non prime, vNa's that have a diffuse hyperfinite regular subalgebra and so on. This is a von Neumann algebra invariant, and was used to prove in a unified approach remarkable rigidity theorems. One such is the following: A non trivial free product of von Neumann algebras can never be generated by two strongly 1-bounded von Neumann subalgebras with diffuse intersection. This result cannot be recovered by any other methods for even hyperfinite subalgebras. In this talk, I will present joint work with David Jekel and Ben Hayes, settling an open question from 2005, whether Property (T) II_1 factors are strongly 1-bounded, in the affirmative. Time permitting, I will discuss some insights and future directions.

**Time: **11:00-11:50am Pacific time (US)

**Title: **Almost commuting matrices and stability for product groups.

**Abstract: ** I will present a recent result showing that the direct product group $\Gamma=\mathbb{F}_2 \times \mathbb {F}_2$ is not Hilbert-Schmidt stable. Specifically, $\Gamma$ admits a sequence of asymptotic homomorphisms (with respect to the normalized Hilbert-Schmidt norm) which are not perturbations of genuine homomorphisms.

As we will explain, while this result concerns unitary matrices, its proof relies on techniques and ideas from the theory of von Neumann algebras. We will also explain how this result can be used to settle in the negative a natural version of an old question of Rosenthal concerning almost commuting matrices. More precisely, we derive the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$ (in the normalized Hilbert-Schmidt norm), but there are no matrices $A’,B’$ close to $A,B$ such that $A’$ commutes with $B’$ and $B’^*$.

**Time: **11:00-11:50am Pacific time (US)

**Title: **Asymptotics of polynomials via free probability

**Abstract:** Since the seminal work of Voiculescu in the early 90’s, the connection between the asymptotic behavior of random matrices and free probability has been extensively studied. More recently, in relation to the solution of the Kadison-Singer problem, Marcus, Spielman, and Srivastava discovered a deep connection between certain classical polynomial convolutions and free probability. Soon after, this connection was further understood by Marcus, who introduced the notion of finite free probability.

In this talk I will present recent results on finite free probability with applications to the asymptotic analysis of real-rooted polynomials. Our approach is based on a careful combinatorial analysis of the finite free cumulants, and allows us to study the asymptotic dynamics of the root distribution of polynomials after repeated differentiation, as well as the fluctuations of the root distributions of polynomials around their limiting measure. This is joint work with Octavio Arizmendi and Daniel Perales: arXiv:2108.08489.

**Time: **11:00-11:50am Pacific time.

**Title: **Some nice measures in infinite-dimensions

**Abstract: **Gaussian measures have long been recognized as the appropriate measures to use in infinite-dimensional analysis. Their regularity properties have allowed the development of a calculus on these measure spaces that has become an invaluable tool in the analysis of stochastic processes and their applications.

Gaussian measures arise naturally in the context of random diffusions, specifically as the end point distribution of Brownian motion, and one may see their regularity as arising from nice properties of the generator of the diffusion. More particularly, in finite dimensions, hypoellipticity of the generator is a standard assumption required for regularity of the associated measure. However, in infinite dimensions it has remained elusive to demonstrate that hypoellipticity is a sufficient condition for regularity. Using techniques first developed by Bruce Driver and Masha Gordina, there has been some recent success in proving regularity for some natural infinite-dimensional hypoelliptic models. These techniques rely on establishing uniform bounds on coefficients appearing in certain functional analytic inequalities for semi-groups on finite-dimensional approximations. We will discuss some of these successful applications, including more recent work studying models satisfying only a weak notion of hypoellipticity. This includes joint works with Fabrice Baudoin, Dan Dobbs, Bruce Driver, Nate Eldredge, and Masha Gordina.

### November 2, 2021: Felix Parraud

**Time: **11:00-11:50am Pacific time

**Title: **Free probability and random matrices: the asymptotic behaviour of polynomials in independent random matrices

**Abstract: **It has been known for a long time that as their size grow to infinity, many models of random matrices behave as free operators. This link was first explicited by Voiculescu in 1991 in a paper in which he proved that the trace of polynomials in independent GUE matrices converges towards the trace of the same polynomial evaluated in free semicircular variables. In 2005, Haagerup and Thorbjornsen proved the convergence of the norm instead of the trace. The main difficulty of their proof was to prove a sharp enough upper bound of the difference between the trace of random matrices and their free limit. They managed to do so with the help of the so-called linearization trick which allows to relate the spectrum of a polynomial of any degree with scalar coefficients with a polynomial of degree 1 with matrix coefficients. A drawback of this method is that it does not give easily good quantitative estimates. In arXiv:1912.04588, we introduced a new strategy to approach those questions which does not rely on the linearization trick and instead is based on free stochastic calculus. In this talk, I will first focus on the paper arXiv:2011.04146, in which we proved an asymptotic expansion for traces of smooth functions evaluated in independent GUE random matrices, whose coefficients are defined through free probability. And then I will talk about arXiv:2005.1383, in which we adapted the previous method to the case of Haar unitary matrices.

**Title: **Recent work on free Stein dimension

**Time: **11:00-11:50am Pacific time (US).

**Abstract: **Free information theory is largely concerned with the following question: given a tuple of non-commutative random variables, what regularity properties of the algebra they generate can be inferred from assumptions about their joint distribution? This can include von Neumann algebraic properties, such as factoriality or absence of Cartan subalgebras, and free probabilistic properties, such as a lack of non-commutative rational relations.

After giving some background, I will talk on free Stein dimension, a quantity which measures the ease of defining derivations on a tuple of non-commutative variables which turns out to be a *-algebra invariant. I will mention some recent results on its theory, including behaviour in the presence of algebraic relations as well as under direct sum and amplification of algebras. I will also mention some recent attempts to adapt its utility from polynomial algebras to W*-algebras, and time permitting, some cases where explicit estimates can be found on the Stein dimension of generating tuples of von Neumann algebras. This project is joint work with Brent Nelson.

**Time: **11:00-11:50am Pacific time (US).

**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.

**Time: **11:00-11:50am Pacific time

**Title: **Multilinear Fourier multipliers on non-commutative groups

**Abstract: **For a function m on the real line, its Fourier multiplier T_m is the operator which acts on a function f by first multiplying the Fourier transform of f by m, and then taking the inverse Fourier transform of the product. These are well-studied objects in classical harmonic analysis. Of particular interest is when the Fourier multiplier defines a bounded operator on L_p. Fourier multipliers can be generalized to arbitrary locally compact groups. If the group is non-abelian, the L_p spaces involved are now the non-commutative L_p spaces associated with the group von Neumann algebra. Fourier multipliers also have a natural extension to the multilinear setting. However, their behaviour can differ markedly from the linear case, and there is much that is unknown even about multilinear Fourier multipliers on the reals.

One question of interest is this: If m is a function on a group G which defines a bounded L_p multiplier, is the restriction of m to a subgroup H also the symbol of a bounded L_p multiplier on H? De Leeuw proved that the answer is yes, when G is R^n. This was extended to the commutative case by Saeki and to the non-commutative case (provided the group G is sufficiently nice) by Caspers, Parcet, Perrin and Ricard. In this talk, I will show how to extend these De Leeuw type theorems to multilinear Fourier multipliers on non-commutative groups. This is part of joint work with Martijn Caspers, Bas Janssens and Lukas Miaskiwskyi.

**Time: **9:00-9:50am Pacific time

**Title: **Berry-Esseen Bounds for Operator-valued Free Limit Theorems

**Abstract: **The development of free probability theory has drawn much inspiration from its deep and far reaching analogy with classical probability theory. The same holds for its operator-valued extension, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally led to operator-valued free analogues of key and fundamental limiting theorems such as the operator-valued free Central Limit Theorem due to Voiculescu and results about the asymptotic behaviour of distributions of matrices with operator-valued entries.

In this talk, we show Berry-Esseen bounds for such limit theorems. The estimates are on the level of operator-valued Cauchy transforms and the Levy distance. We address also the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the Levy distance for the operator-valued free CLT. This also yields quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile.

This is a joint work with Tobias Mai.

### January 11, 2022: Lucas Hall

**Title: **Coactions you can see

**Abstract: **We motivate the study of coactions, developing our intuition by taking a tour through topological dynamics. We reinforce this intuition by exploring the particular example of skew product topological quivers - a subject of recent study by the speaker.

### January 18, 2022: Serban Belinschi (9am!!)

**Title: **The Christoffel-Darboux kernel and noncommutative Siciak functions

**Abstract: **The Christoffel-Darboux kernel is the reproducing kernel associated to the Hilbert space containing all polynomials up to a given degree. It can be naturally written in terms of any complete set of orthonormal polynomials. In classical analysis the Christoffel-Darboux kernel is useful for studying properties of the underlying measure with respect to which the Hilbert space of polynomials is defined. In this talk, we present the version of the Christoffel-Darboux kernel for $L^2$ spaces of tracial states on noncommutative polynomials. We view this kernel as a noncommutative function, and identify its values as maxima of certain sets of non-negative matrices/operators.

In numerous cases, the classical version of the Christoffel-Darboux kernel can be used (after renormalization) to recover the measure to which it is associated as a weak derivative. This is done with the aid of the theory of plurisubharmonic functions. We use this same theory in order to introduce several noncommutative versions of the Siciak extremal function. We use the Siciak functions to prove that, in several cases of interest, the (properly normalized) limit of the evaluations of the Christoffel-Darboux kernel on matrix sets exists as a well-defined, quasi-everywhere finite plurisubharmonic function. Time permitting, we conclude with some conjectures regarding these objects. This is based on joint work with Victor Magron (LAAS) and Victor Vinnikov (Ben Gurion University).

### February 1, 2022: Jurij Volcic

**Title: **Ranks of linear pencils separate similarity orbits of matrix tuples

**Abstract: **The talk addresses the conjecture of Hadwin and Larson on joint similarity of matrix tuples, which arose in multivariate operator theory.

The main result states that the ranks of linear matrix pencils constitute a collection of separating invariants for joint similarity of matrix tuples, which affirmatively answers the two-sided version of the said conjecture. That is, m-tuples X and Y of n×n matrices are simultaneously similar if and only if rk L(X) = rk L(Y) for all linear matrix pencils L of size mn. Similar results hold for certain other group actions on matrix tuples. On the other hand, a pair of matrix tuples X and Y is given such that rk L(X) <= rk L(Y) for all L, but X does not lie in the closure of the joint similarity orbit of Y; this constitutes a counter-example to the general Hadwin-Larson conjecture.

The talk is based on joint work with Harm Derksen, Igor Klep and Visu Makam.

### February 15, 2022: Roy Araiza

**Title: **Matricial Archimedean Order Unit Spaces and Quantum Correlations

**Abstract: **During this talk I will introduce the notion of a k-AOU space, which we may think of as a matricial Archimedean order unit space. I will then describe the relationship between the category of k-AOU spaces and k-positive maps, and the category of operator systems and completely positive maps. After demonstrating the existence of injective envelopes and C*-envelopes in the category of k-AOU spaces, I will describe a connection with quantum correlations. Combined with previous work, this yields a reformulation of Tsirelson's conjecture.

### February 22, 2022: Nick Boschert

**Title: **Moment Laws in Free Probability

**Abstract: **We discuss results generalizing a result of Cordero-Erausquin and Klartag involving transport of log-concave measures to the free probabilistic setting. We also discuss open problems in extending it further.

### March 1, 2022: Rolando de Santiago

**Title:** Deformation/Rigidity and Maximal Rigid Subalgebras

**Abstract:** An important area of study in the classification of II_1 factors is to investigate the dependence of a group von Neumann algebra L(G) relative to the group G. Popa’s deformation/rigidity theory has provided novel insights into this question over the past 20 years. In this talk, we demonstrate how one can import group cohomological information into the von Neumann algebra framework to unravel the structure of a large family of von Neumann algebras.

### March 8, 2022: Therese Landry

**Title: **Developments in noncommutative fractal geometry

**Abstract: **As a noncommutative fractal geometer, I look for new expressions of the geometry of a fractal through the lens of noncommutative geometry. At the quantum scale, the wave function of a particle, but not its path in space, can be studied. Riemannian methods often rely on smooth paths to encode the geometry of a space. Noncommutative geometry generalizes analysis on manifolds by replacing this requirement with operator algebraic data. These same "point-free" techniques can also be used to study the geometry of spaces like fractals. Recently, Michel Lapidus, Fr'ed'eric Latr'emoli`ere, and I identified conditions under which differential structures defined on fractal curves can be realized as a metric limit of differential structures on their approximating finite graphs. Currently, I am using some of the same tools from that project to understand noncommutative discrete structures. Progress in noncommutative geometry has produced a rich dictionary of quantum analogues of classical spaces. The addition of noncommutative discrete structure to this dictionary would enlarge its potential to yield insights about both noncommutative sets and classically pathological sets like fractals. Time permitting, other works in progress, such as on classification of C*-algebras on fractals, may be discussed.

### March 29, 2022: Hui Tan

**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).

### April 5, 2022: Dan Ursu

**Title: **The ideal intersection property for essential groupoid C*-algebras

**Abstract: **Groupoids give a very large class of examples of C*-algebras. For example, it is known that every classifiable C*-algebra arises as the reduced C*-algebra of some twisted groupoid.

In joint work with Matthew Kennedy, Se-Jin Kim, Xin Li, and Sven Raum, we fully characterize when the essential C*-algebra of an 'etale groupoid G with locally compact unit space has the ideal intersection property. This is done in terms of the dynamics of G on the space of subgroups of the isotropy groups of G. The essential and reduced C*-algebras coincide in the case of Hausdorff groupoids, and the ideal intersection property is the same as simplicity in the case of minimal groupoids. This generalizes the case of the reduced crossed product of C(X) with G done by Kawabe, which in turn generalizes the case of the reduced C*-algebra of a discrete group done by Breuillard, Kalantar, Kennedy, and Ozawa.

No prior knowledge of groupoids will be required for this talk.

### April 19, 2022 (11am) Pieter Spaas

**Title: **Furstenberg-Zimmer structure theory for actions on von Neumann algebras

**Abstract:** In classical ergodic theory, compact and weakly mixing actions/extensions have been well-studied. The main structural result from Furstenberg and Zimmer states that every action can be written as "a weakly mixing extension of a tower of compact extensions". We will discuss some of these classical results and their motivation, and consider similar notions for actions on von Neumann algebras which have been defined throughout the years. We will then complete (part of) the picture by establishing equivalence of several such notions, followed by some consequences and open questions. This is partially based on joint work with Asgar Jamneshan.

### May 3, 2022: Changying Ding

**Title:** Properly proximal von Neumann Algebras

**Abstract:** Properly proximal groups were introduced recently by Boutonnet, Ioana, and Peterson, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will describe how the notion of proper proximality fits naturally in the realm of von Neumann algebras. I will also describe several applications, including that the group von Neumann algebra of a non-amenable inner-amenable group cannot embed into a free group factor, which solves a problem of Popa. This is joint work with Srivatsav Kunnawalkam Elayavalli and Jesse Peterson.

### May 10, 2022 Todd Kemp

**Title: **The Bifree Segal--Bargmann Transform

**Abstract: **The classical Segal--Bargmann transform (SBT) is an isomorphism between a real Gaussian Hilbert space and a reproducing kernel Hilbert space of holomorphic functions. It arises in quantum field theory, as a concrete witness of wave-particle duality. Introduced originally in the 1960s, it has been generalized and extended to many contexts: Lie Groups (Hall, Driver, late 1980s and early 1990s), free probability (Biane, early 2000s), and more recently $q$-Gaussian factors (C\'ebron, Ho, 2018).

In this talk, I will discuss current work with Charlesworth and Ho on a version of the SBT in bifree probability, a "two faced" version of free probability introduced by Voiculescu in 2014. Our work leads to some interesting new combinatorial structures ("stargazing partitions"), as well as a detailed analysis of the resultant family of reproducing kernels. In the end, the bifree SBT has a surprising connection with the $q$-Gaussian version for some $q\ne 0$.

### May 17, 2022: Krishnendu Khan

**Title:** On some structural rigidity results of group von Neumann algebras

**Abstract:**** **In this talk I will present examples of property (T) type II_{1} factors with trivial fundamental group, thus, providing progress towards the well-known open questions of Connes'94 and Popa'06. We will show that the semidirect product feature is an algebraic feature that survive passage to group von Neumann algebras for a class of inductive limit of property (T) groups arising from geometric group theory. Using Popa's deformation/rigidity in conjunction with group theoretic methods we proved that the acting group can be completely recoverable from the von Neumann algebra as well as the limit action of the acting group. In addition, the fundamental group of the group von Neumann algebras associated to these limit groups are trivial, which contrasts the McDuff case. This is based on a joint work with S. Das.

### May 31, 2022 Samantha Brooker (11:30am-12:30pm)

**Title: **Pullback diagrams of various graph C*-algebras

**Abstract:** Relative Toeplitz algebras of directed graphs were introduced by Spielberg in 2002 to describe certain subalgebras corresponding to subgraphs. They can also be used to describe *quotients* of graph algebras corresponding to subgraphs. We use the latter relationship to answer a question posed in a recent paper regarding pushout diagrams of graphs that give rise to pullback diagrams of the respective graph C*-algebras. We introduce a new category of *relative graphs* to this end, and we prove our results using graph groupoids and their C*-algebras. This is joint work with Jack Spielberg.

### October 4, 2022: Srivatsav Kunnawalkam Elayavalli

**Title: **Two full factors without isomorphic ultrapowers

**Abstract: **I will show you how to construct a full factor M such that M and L(F_2) do not have any isomorphic ultrapowers. The construction uses a combination of techniques from deformation/rigidity and free entropy theory. We also provide the first example of a II_1 factor that is full such that its ultrapower is strongly 1-bounded. This is joint work with Adrian Ioana and Ionut Chifan.

### October 18, 2022: Simon Schmidt (Zoom, 9:00am)

**Title**: Quantum symmetry vs nonlocal symmetry

**Abstract**: We will introduce the notion of nonlocal symmetry of a graph G, defined as winning quantum correlation for the G-automorphism game that cannot be produced classically. We investigate the differences and similarities between this and the quantum symmetry of the graph G, defined as non-commutativity of the algebra of functions on the quantum automorphism group of G. We show that quantum symmetry is a necessary but not sufficient condition for nonlocal symmetry. In particular, we show that the complete graph on four points does not exhibit nonlocal symmetry. We will also see that the complete graph on five or more points does have nonlocal symmetry. This talk is based on joint work with David Roberson.

### October 18, 2022: Benoit Collins (Zoom, 4:15pm)

**Title:** convergence of the spectrum of random matrices in the context of rational fractions

**Abstract:** Thanks to Voiculescu’s freeness, one knows that the normalized eigenvalue counting measure of a selfadjoint non-commutative polynomial in iid GUE’s converges in the limit of large dimension, and there exist many tools to compute its limiting distribution. On the other hand, on the limiting space (a free product algebra), lots of progress has been made in understanding non-commutative rational fractions. A question by Speicher is whether these rational fractions admit matrix models too. I will explain why the natural candidate is actually a matrix model. In other words, bearing in mind that we already understand the asymptotics of the eigenvalue counting measure of a matrix model obtained as sums, scalings products of iid random matrices, we will show that we can do the same if we allow in addition multiple uses of the matrix inverse when creating our matrix model.

This is based on arXiv/2103.05962, written in collaboration with Tobias May, Akihiro Miyagawa, Felix Parraud and Sheng Yin.

### October 25, 2022: Adam Skalski (Zoom)

**Title**: On certain operator Hecke algebras arising as deformations of group algebras of Coxeter groups.

**Abstract**: I will recall a construction of certain operator algebras arising naturally as multiparameter deformations of operator algebras of Coxeter groups, initially motivated by the study of cohomology of groups acting on buildings. We will explain that for right-angled Coxeter groups, at a certain range of multiparameters, the resulting von Neumann algebra is a factor, thus completing earlier results of Garncarek, and of Caspers, Klisse and Larsen. This result, of interest in itself, has several consequences and interpretations for the representation theory of both right-angled Coxeter groups and of certain groups acting on buildings. I will also outline further questions/results related to the classification of the related C*-algebras.

Based on joint work with Sven Raum.

### November 1, 2022: Li Gao (Zoom)

**Title:** Logarithmic Sobolev inequalities for matrices and matrix-valued functions.

**Abstract:** Logarithmic Sobolev inequalities, first introduced by Gross in 70s, have rich connections to probability, geometry, as well as information theory. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups attracted a lot of attentions for its applications in quantum information theory and quantum many-body systems. In this talk, I'll present a simple, information-theoretic approach to modified logarithmic Sobolev inequalities for both quantum Markov semigroup on matrices, and classical Markov semigroup on matrix-valued functions. In the classical setting, our results implies every sub-Laplacian of a Hörmander system admits a uniform modified logarithmic Sobolev constant for all its matrix valued functions. For quantum Markov semigroups, we improve a previous result of Gao and Rouzé by replacing the dimension constant by its logarithm. This talk is based on a joint work with Marius Junge, Nicholas, LaRacunte, and Haojian Li.

### November 8, 2022: Dolapo Oyetunbi (Zoom)

**Title: ** On $\ell$-open and $\ell$-closed $C^*$ algebras

**Abstract: **A separable $C^*$-algebra $A$ is said to be $\ell$-open ( or $\ell$-closed) when the image of Hom(A, B) is open (or closed) in Hom(A, B/I), for all separable $C^*$-algebras B and ideals I. The concept of semiprojectivity has been used many times in the classification of C*-algebras. Bruce Blackadar introduced $\ell$-open and $\ell$-closed $C^*$-algebras as a superclass of semiprojective $C^*$-algebras.

In recent work with A. Tikuisis, we characterize $\ell$-open and $\ell$-closed $C^*$-algebras and deduce that $\ell$-open $C^*$-algebras are $\ell$-closed as conjectured by Blackadar. Moreover, we show that the notion of $\ell$-open $C^*$-algebras and semiprojective $C^*$-algebras coincide for commutative unital $C^*$-algebras.

### November 15, 2022: Michael Davis (in person)

**Title: **Rigidity for von Neumann Algebras of Graph Product Groups

**Abstract:** I will discuss my ongoing joint work with Ionut Chifan and Daniel Drimbe on various rigidity aspects of von Neumann algebras arising from graph product groups whose underlying graph is a certain cycle of cliques and whose vertex groups are wreath-like product property (T) groups. In particular, I will describe all symmetries of these von Neumann algebras by establishing formulas in the spirit of Genevois and Martin’s results on automorphisms of graph product groups. In doing so, I will highlight the methods used from Popa’s deformation/rigidity theory as well as new techniques pertaining to graph product algebras.