This seminar has come to its conclusion for the academic year and will startup again in Fall 2021 under the organization of David Jekel.
June 1, 2021
Title: On Connes' rigidity conjecture
Abstract: A bold conjecture due to Connes (1980) predicts that the group von Neumann algebra of an i.c.c. property (T) group completely remembers the group. The strong form of Connes' conjecture, due to Popa (2005) predicts that the factors arising from property (T) groups have trivial fundamental group. In this talk I shall discuss recent progress towards these conjectures, and present the first examples of property (T) factors with trivial fundamental group. This talk is based on a joint work with Ionut Chifan, Cyril Houdayer, and Krishnendu Khan.
May 25, 2021
Title: Amenability, proximality and higher order syndeticity
Abstract: I will discuss new descriptions of some universal flows associated to a discrete group, obtained using what we view as a kind of “topological Furstenberg correspondence.” The descriptions are algebraic and relatively concrete, involving subsets of the group satisfying a higher order notion of syndeticity. We utilize them to establish new necessary and sufficient conditions for strong amenability and amenability. Throughout, I will discuss connections to operator algebras. This is joint work with Sven Raum and Guy Salomon.
May 18, 2021
Title: Central sequence algebras via nilpotent elements
Abstract: A central sequence in a C*-algebra is a sequence (x_n) of elements such that [x_n, a] converges to zero, for any element a of the C*-algebra. In von Neumann algebra setting one typically means the convergence with respect to tracial norms, while in C*-theory it is with respect to the C*-norm. In this talk we will consider the C*-theory version of central sequences. We will discuss properties of central sequence algebras and in particular address a question of J. Phillips and of Ando and Kirchberg of which separable C*-algebras have abelian central sequence algebras.
Joint work with Dominic Enders.
May 4, 2021
Title: Lower bounds on the radius of comparison of the crossed product by a minimal homeomorphism.
Abstract: Let X be a compact metric space, and let h be a homeomorphism of X. The mean dimension of h is an invariant invented by people in topological dynamics, with no consideration of C*-algebras. The shift on the product of copies of [0, 1]^d indexed by Z has mean dimension d.
The radius of comparison of a C*-algebra A is an invariant introduced with no consideration of dynamics, and originally applied to C*-algebras which are not given as crossed products. It is a numerical measure of bad behavior in the Cuntz semigroup of A, and its original use was to distinguish counterexamples to the original formulation of the Elliott conjecture.
It is conjectured that if h is a minimal homeomorphism of a compact metric space, then the radius of comparison of C* (Z, X, h) is equal to half the mean dimension of h. There is a generalization to countable amenable groups. Considerable progress has been made on proving that the radius of comparison of C* (Z, X, h) is at most half the mean dimension; in particular, this is known in full generality for minimal homeomorphisms. We give the first systematic results for the opposite inequality. We do mot get the exact expected lower bound, but, for many known examples of actions of amenable groups with large mean dimension, we come close.
The methods depend on "mean cohomological independence dimension", Cech cohomology, and the Chern character.
This is joint work with Ilan Hirshberg.
April 27, 2021 at 4 pm (not the usual time)
Title: The eigenvalues of principal submatrices in rotationally invariant hermitian random matrices and the Markov-Krein Correspondence
Abstract: This talk establishes a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy. More precisely, if the EED of the whole matrix converges to some deterministic probability measure 𝔪, then its difference from the EED of its principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with 𝔪 by the Markov-Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of 𝔪. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.
This is a joint work with Katsunori Fujie.
April 20, 2021
Title: Stable decompositions and rigidity for product equivalence relations
Abstract: After discussing the motivation behind the talk and some necessary preliminaries, we will consider the ''stabilization'' of a countable ergodic p.m.p. equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group. Using a new local characterization of the Schmidt property, we show that this always gives rise to a so-called stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. We will also discuss some new structural results for product equivalence relations, which we will obtain using von Neumann algebraic techniques.
April 13, 2021
Title: Local lifting and approximation properties for operator modules
Abstract: We introduce notions of finite presentation which serve as analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we then introduce analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules, we show that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where, for a locally compact group G, we show that A(G)-nuclearity of the inclusion C*_r(G) --> C^*_r(G)** and A(G)-semi-discreteness of VN(G) are both equivalent to amenability of G. We also present the equivalence between A(G)-injectivity of the crossed product G\bar{\ltimes}M, A(G)-semi-discreteness of G\bar{\ltimes} M, and amenability of W*-dynamical systems (M,G,\alpha) with M injective.
April 6, 2021
Title: On Complete Logarithmic Sobolev Inequalities.
Abstract: Logarithmic Sobolev inequalities (LSI) were first introduced by Gross in 70s, and later found rich connections to geometry, probability, graph theory, optimal transport as well as information theory. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups have attracted a lot of attentions and found applications in quantum information theory and quantum many-body system. For classical Markov semigroup on a probability space, an important advantage of log-Sobolev inequalities is the tensorization property that if two semigroups satisfies LSI, so does their tensor product semigroup. Nevertheless, tensoraization property fails for LSI in the quantum cases. In this talk, I'll present some recent progresses on tensor stable log-Sobelev inequalities for quantum Markov semigroups. This talk is based on joint works with Michael Brannan, Marius Junge, Nicholas LaRacuente, Haojian Li and Cambyse Rouze.
March 30, 2021
Title: Self-similar k-graph C*-algebras
Abstract: A self-similar k-graph is a pair consisting of a discrete group and a k-graph, such that the group acts on the k-graph self-similarly. For such a pair, one can associate it a universal C*-algebra, called the self-similar k-graph C*-algebra. This class of C*-algebras embraces many important and interesting C*-algebras, such as the higher rank graph C*-algebras of Kumjian-Pask, the Katsura algebra, the Nekrashevych algebra, and the Exel-Pardo algebra. In this talk, I will present some results about those C*-algebras, which are based on joint work with Hui Li.
March 2, 2021
Title: Right angled Hecke operator algebras and representation theory
Abstract: With every Coxeter system one can associate a family of algebras considered as deformation of its group algebra. These are so-called Hecke algebras, which are classical objects of study in combinatorics and representation theory. Complex Hecke algebras admit a natural *-structure and a *-representation on Hilbert space. Taking the norm- and SOT-closure in such representation, one obtains Hecke operator algebras, which have recently seen increased attention.
In this talk, I will introduce Hecke operator algebras from scratch, focusing on the case of right-angled Coxeter groups. This case is is particularly interesting from an operator algebraic perspective, thanks to its description by iterated amalgamated free products. I will survey known results on the structure of Hecke operator algebras, before I describe recent work that clarified the factor decomposition of Hecke von Neumann algebras. Two applications to representation theory will be presented. I will finish with some results on the scope and limits of K-theoretic classification of right-angled Hecke C*-algebras.
This is joint work with Adam Skalski.
February 23, 2021
Title: Noncommutative boundary maps and C*-algebras of quasi-regular representations
Abstract: We investigate some structural properties of C*-algebras generated by quasi-regular representations of stabilizers of boundary actions of discrete groups G. Our main tool is the notion of (noncommutative) boundary maps, namely G-equivariant unital positive maps from G-C*algebras to C(B), where B is the Furstenberg boundary of G. We completely describe the tracial structure and characterize the simplicity of these C*-algebras. As an application, we show that the C*-algebra generated by the quasi-regular representation associated to Thompson's groups F < T does not admit traces and is simple.
This is joint work with Eduardo Scarparo.
February 16, 2021
Title: A random matrix approach to absorption in free products
Abstract: I'll discuss joint with Jekel-Nelson-Sinclair. We give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using 1-bounded entropy. The main techniques are concentration of measure on unitary groups as well as Voiculescu's asymptotic freeness theorem.
February 9, 2021
Title: Gelfand pairs, spherical functions and (exotic) group $C^*$-algebras
Abstract: For a non-amenable group $G$, there may be many (exotic) group $C^*$-algebras that lie naturally between the universal and the reduced $C^*$-algebra of $G$. Let $G$ be a simple Lie group or an appropriate locally compact group acting on a tree. I will explain how the $L^p$-integrability properties of different spherical functions on $G$ (relative to a maximal compact subgroup) can be used to distinguish between different (exotic) group $C^*$-algebras. This recovers results of Samei and Wiersma. Additionally, I will explain that under certain natural assumptions, the aforementioned exotic group $C^*$-algebras are the only ones coming from $G$-invariant ideals in the Fourier-Stieltjes algebra of $G$. This is based on joint work with Dennis Heinig and Timo Siebenand.
February 2, 2021
Title: The Invariant Subspace Problem for rank-one perturbations
Abstract: The Invariant Subspace Problem is one of the most famous problem in Operator Theory, and is concerned with the search of non-trivial, closed, invariant subspaces for bounded operators acting on a separable Banach space. Considerable success has been achieved over the years both for the existence of such subspaces for many classes of operators, as well as for non-existence of invariant subspaces for particular examples of operators. However, for the most important case of a separable Hilbert space, the problem is still open.
A natural, related question deals with the existence of invariant subspaces for perturbations of bounded operators. These type of problems have been studied for a long time, mostly in the Hilbert space setting. In this talk I will present a new approach to these “perturbation” questions, in the more general setting of a separable Banach space. I will focus on the recent history, presenting several new results that were obtained along the way with this new approach, and examining their connection and relevance to the Invariant Subspace Problem.
January 26, 2021
Title: Noncommutative $C^k$ Functions, Multiple Operator Integrals, and Derivatives of Operator Functions
Abstract: Let $A$ be a $C^*$-algebra, $f \colon \mathbb{R} \to \mathbb{C}$ be a continuous function, and $\tilde{f} \colon A_{\text{sa}} \to A$ be the functional calculus map $A_{\text{sa}} \ni a \mapsto f(a) \in A$. It is elementary to show that $\tilde{f}$ is continuous, so it is natural to wonder how the differentiability properties of $f$ relate/transfer to those of $\tilde{f}$. This turns out to be a delicate, complicated problem. In this talk, I introduce a rich class $NC^k(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of noncommutative $C^k$ functions $f$ such that $\tilde{f}$ is $k$-times differentiable. I shall also discuss the interesting objects, called multiple operator integrals, used to express the derivatives of $\tilde{f}$.
January 12, 2021
Title: On decomposability and spectrality of operators in finite von Neumann algebras.
Abstract: Linear operators on finite dimensional spaces can be analyzed in terms of their spectra using Schur's upper triangular form and Jordan's canonical form. Regarding spectral analysis for operators on infinite dimensional Banach spaces, Dunford (in the 1950's) introduced the notions of (a) spectral operators and (b) operators of scalar type, while Apostol and Foias (in the 1960's) introduced the notion of (c) decomposable operators. About a decade ago, Haagerup and Schultz, studying elements of finite von Neumann algebras, proved existence of invariant projections that behave well with respect to Brown measure, which is a sort of spectral distribution. We use these invariant projections to construct analogues of Schur's upper triangular forms for such operators, and explore connections to decomposability and specrality of elements of finite von Neumann algebras. We cover results that are the result of joint work with (in various combinations) Ian Charlesworth, Amudhan Krishnaswamy-Usha, Joseph Noles, Fedor Sukochev and Dmitriy Zanin.
January 5, 2021
Title: W* and C*-superrigid groups
Abstract: Due to the work of Murray and von Neumann, there is a natural way to associate a von Neumann algebra $L(\Gamma)$ to every countable discrete group $\Gamma$. The problem of classifying $L(\Gamma)$ in terms of $\Gamma$ is often challenging since these algebras tend to have only a “faded memory” of the underlying group. A good illustration is a remarkable theorem of Connes which asserts that that all icc amenable groups give rise to isomorphic von Neumann algebras. In the non-amenable case the situation is much more complex and revealed a striking rigidity phenomenon; many examples where the von Neumann algebraic structure is sensitive to various algebraic group properties have been discovered via Popa’s deformation/rigidity theory. The most extreme situation is when $\Gamma$ is W$^*$-superrigid, meaning that $L(\Gamma)$ completely remembers the underlying group $\Gamma$. There have been discovered only two types of group theoretic constructions that lead to W$^*$-superrigid groups: some classes of generalized wreath products groups with abelian base (Ioana-Popa-Vaes '10, Berbec-Vaes '12) and amalgamated free products (Chifan-Ioana '17).
In this talk I will introduce several new constructions of W$^*$-superrigid groups which include direct product groups, semidirect products with non-amenable core and HNN-extensions. I will also present some applications of these results to C$^*$-algebras by presenting new examples of groups that are completely remembered by their reduced C$^*$-algebra. This is based on a joint work with Ionut Chifan and Alec Diaz-Arias.
December 8, 2020
Title: Synchronous Games and the Connes Embedding Problem
Abstract: In MIP*=RE the authors settle the CEP in the negative by showing that two computational complexity classes are equal. But the heart of their argument is the construction of a synchronous game with certain properties. In this talk we will describe the theory of synchronous games, and show how our construction of the algebra of a game leads more directly to the CEP. This approach to the CEP still depends on their construction of a synchronous game with particular properties, but stays within the context of operator algebras and games.
Held jointly with UCSD Group Actions Seminar
December 1, 2020
Title: Stationary actions of higher rank lattices on non-commutative spaces
Abstract: I will present new results about stationary actions of higher rank semi-simple lattices on compact spaces, in the spirit of Nevo and Zimmer's work. Then I will explain how these results generalize to stationary actions on C*-algebras (i.e. "non-commutative" spaces) and give consequences about unitary representations of these lattices and their characters. All these results can be seen as generalizations of Margulis normal subgroup theorem at different levels. This is based on joint works with Cyril Houdayer, Uri Bader and Jesse Peterson.
November 24, 2020
Title: Amenable actions of locally compact groups on C*-algebras
Abstract: I will talk about joint work with Siegfried Echterhoff and Rufus Willett in which we introduce and study amenable actions of locally compact groups on C*-algebras, building on previous similar notions by Anantharaman-Delaroche for actions of discrete groups. Among the new results we prove an extension of Matsumura’s theorem giving a characterisation of the weak containment property (coincidence of full and reduced crossed products) for actions on commutative C*-algebras and give examples showing that this result does not extend to general noncommutative C*-algebras.
November 17, 2020
Title: Non-commutative smooth functions and non-commutative probability distributions
Abstract: In free probability theory, there is no direct analog of density for probability distributions, but there is something like a notion of "log-density" in the study of free Gibbs laws and free score functions. Non-commutative notions of smoothness are important for studying both these log-densities and the functions used for changes of variables (or transport of measure) in free probability. In the single-variable setting, we have a good understanding of the smoothness properties of a function $f: \mathbb{R} \to \mathbb{R}$ applied to self-adjoint operators thanks to the work of Peller, Aleksandrov, and Nazarov; see the recent paper of Evangelos Nikitopoulos. However, in the multivariable setting, much of the literature has used classes of functions that are either too restrictive (such as analytic functions) or very technical to define (such as Dabrowski, Guionnet, and Shlyakhtenko's smooth functions where Haagerup tensor norms were used for the derivatives). We will discuss a notion of tracial non-commutative smooth functions that is modeled on trace polynomials. These smooth functions have many desirable properties, such as a chain rule, good behavior under conditional expectations, and a natural way to incorporate the one-variable functional calculus. We will sketch current work about the smooth transport of measure for free Gibbs laws as well as future directions in relating these smooth functions to free SDE.
November 3, 2020
Title: Malnormal matrices
Abstract: In this talk, I will revisit an old result of von Neumann on the genericity of matrices which commute poorly with trace zero, self-adjoint matrices. Some partial improvements and several conjectures will be discussed. This is joint work with Garrett Mulcahy.
October 27, 2020
Title: Versions of Bi-Free Entropy
Abstract: In a series of papers, Voiculescu generalized the notions of entropy and Fisher information to the free probability setting. In particular, the notions of free entropy have several applications in the theory of von Neumann algebras and free probability such as demonstrating certain von Neumann algebras do not have property Gamma, demonstrating the absence of atoms in the distributions of polynomials of random matrices, and the construction of free monotone transport. With the recent bi-free extension of free probability being sufficiently developed, it is natural to ask whether there are bi-free extensions of Voiculescu's notions of free entropy. In this talk, we will provide an introduction to a few notions of bi-free entropy and discuss the difficulties and peculiarities that occur. This is joint work with Ian Charlesworth.
October 13, 2020
Title: New examples of W* and C*-superrigid groups
Abstract: In the mid thirties F. J. Murray and J. von Neumann found a natural way to associate a von Neumann algebra L(G) to every countable discrete group G. Classifying L(G) in terms of G emerged as a natural yet quite challenging problem as these algebras tend to have very limited “memory” of the underlying group. This is perhaps best illustrated by Connes’ famous result asserting that all icc amenable groups give rise to isomorphic von Neumann algebras; therefore, in this case, besides amenability, the algebra has no recollection of the usual group invariants like torsion, rank, or generators and relations. In the non-amenable case the situation is radically different; many examples where the von Neumann algebraic structure is sensitive to various algebraic group properties have been discovered via Popa’s deformation/rigidity theory.
In my talk I will focus on an extreme situation, namely when L(G) completely remembers the underlying group G; such groups G are called W*-superrigid. Currently there have been identified only two types of group theoretic constructions that lead to W*-superrigid groups: some classes of generalized wreath products groups with abelian base (Ioana-Popa-Vaes '10, Berbec-Vaes ‘13) and amalgamated free products (Chifan-Ioana'16). After briefly surveying these results I will introduce several new constructions of W*-superrigid groups which include direct product groups, semidirect products with non-amenable core, and tree groups (iterations of amalgams and HNN-extensions). In addition, I will present several applications of these results to the study of rigidity in the C*-setting. This is based on a very recent joint work with Alec Diaz-Arias and Daniel Drimbe.
October 6, 2020
Title: Traces on locally compact groups
Abstract: Let $G$ be a locally compact group. There is a natural correspondence between continuous positive definite functions $u$ on $G$ with $u(e)=1$, and states on the universal C*-algebra $C^*(G)$. With this correspondence traces are those continuous positive definite functions $u$ for which $u(st)=u(ts)$ and $u(e)=1$. Then $N_u = \{ s \in G : u(s)=1 \}$ is a closed normal subgroup of $G$ and the intersection of such ``trace kernels” is the trace kernel $N_{Tr}$. I wish to say as much as I can about the structure of $G/N_{Tr}$, sometimes for certain classes of groups. I also wish to speak about reduced traces, those which define traces on the reduced C*-algebra $C^*_r(G)$; and add non-discrete examples of groups with unique reduced trace. I will also discuss amenable traces and factorization property for some classes of locally compact groups. This is joint work with Brian Forrest (Waterloo) and Matthew Wiersma (UC San Diego).
Students taking MATH 243 for credit will receive a grade of either S or U based on their attendance.