# Analysis Seminar

Department of Mathematics

University of Houston

Fall 2024

(Fridays 1pm-2pm CST at PGH 646)

August 30

Speaker: Jesse Peterson (Vanderbilt)

Title: Biexact von Neumann algebras

Abstract: The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying indecomposability properties of von Neumann algebras. We will introduce the notion of a biexact von Neumann algebra, which allows us to place many previous indecomposability results in a more systematic context, and naturally leads to extensions of these results. The techniques also give a new characterization of weakly exact von Neumann algebras, answering a problem of Brown and Ozawa. This is based on joint work with Changying Ding.

September 20

Speaker: David Sherman (University of Virginia)

Title: Universality for contractive Hilbert space operators

Abstract: Coming to Texas, I’m bringing the biggest operators I can find. A contractive Hilbert space operator T is “universal” if, for any noncommutative *-polynomial q, the norm of q(T) is as large as it can be. I’ll compare this notion of universality with others, and I’ll investigate properties that a universal contraction can or must have. I’ll also discuss the unique unital C*-algebra generated by any universal contraction, which has interesting properties and uses.

September 27

Speaker: Daniel Perales (Texas A&M)

Title: S-transform in finite free probability

Abstract: The finite free additive and multiplicative convolutions are binary operations of polynomials that behave well with respect to the roots and can be understood as a finite analogue of free probability that involves only discrete measures.

We show a simple way to obtain the limiting spectral distribution of a sequence of polynomials (with increasing degree) directly using their coefficients. Specifically, we relate the asymptotic behavior of the ratio of consecutive coefficients to Voiculescu's S-transform of the limiting measure.

The proof uses some new developements that are of independent interest such as a partial order in the set of polynomials, and a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials.

Finaly we define a new notion of finite S-transform, which satisfies several analogous properties to those of the S-transform in free probability, including multiplicativity and monotonicity. And we will mention some applications of our results.

Joint work with Octavio Arizmendi, Katsunori Fujie and Yuki Ueda (arXiv:2408.09337).

October 18

Speaker: Ryo Toyota (Texas A&M)

Title: Operator space structures of hyperbolic group C*-algebras

Abstract: The Haagerup inequality is a Sovolev-type inequality (a comparison between operator norm and \ell^2-norm) for the reduced group C*-algebra C*(F_n) on the free group. More precisely, if f is a function on the free group F_n which is supported on words of length k, then the operator norm of the convolution operator by f is dominated by k+1 times \ell^2-norm of f. To study the structure of subspaces of C*(F_n) as operator spaces, U Haagerup and G. Pisier extended the Haagerup inequality for the case where a function f takes operator values and the most complete form was given by A. Buchholz, where \ell^2-norm is replaced by sum of k+1 different matrix norms associated to word decompositions. We explain how to generalize this inequality for Gromov-Hyperbolic groups. This talk is based on a joint work with Z. Yang.

October 25

Speaker: Akihiro Miyagawa (UCSD)

Title: Strong Haagerup inequality for q-circular systems

Abstract: U. Haagerup proved the inequality for operator norms of homogeneous polynomials in generators of free groups with respect to the left regular representation, which was improved by T. Kemp and R. Speicher for the polynomials which do not contain the adjoint operators. In this talk, I will explain another approach to their inequality in the case of the q-circular system. This talk is based on a joint project with T. Kemp.

November 1

Speaker: Tattwamasi Amrutam (Institute of Mathematics of the Polish Academy of Sciences)

Title: Boomerang subalgebras of the group von Neumann algebra

Abstract: Consider a countable discrete group $\Gamma$ and its subgroup space-$\text{Sub}(\Gamma)$, the collection of all subgroups of $\Gamma$. $\text{Sub}(\Gamma)$ is a compact metrizable space with respect to the Chabauty topology (the topology induced from the product topology on $\{0, 1\}^{\Gamma}$). The normal subgroups of $\Gamma$ are the fixed points of $(\text{Sub}(\Gamma), \Gamma)$. Furthermore, the $\Gamma$-invariant probability measures of this dynamical system are known as invariant random subgroups (IRSs).

Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. They generalize the notion of normal subgroups. Among many other remarkable results, they strengthen the well-known Margulis's normal subgroup Theorem.

To a countable discrete group $\Gamma$, we can also associate an algebraic object $L(\Gamma)$, called the group von Neumann algebra. More recently, in a joint work with Hartman and Oppelmayer, we introduced the notion of Invariant Random Algebra (IRA), an invariant probability measure on the collection of sub algebras of $L(\Gamma)$.

Motivated by the works of Glasner and Lederle, in ongoing joint work with Yair Glasner, Yair Hartman, and Yongle Jiang, we introduce the notion of Boomerang subalgebras in the context of $L(\Gamma)$. In this talk, we shall show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group (an example of this is a Free group generated by two elements $a$ and $b$) comes from a Boomerang subgroup. We shall also discuss its connection to understanding IRAs in such groups.

November 8

Speaker: Ping Zhong (University of Houston)

Title: Some snapshots of free probability theory and its applications

Abstract: Free probability theory was originally developed to address longstanding questions about von Neumann algebras of free groups, where usual independence is replaced with a concept known as free independence. Abstract free random variables can be modeled by concrete random matrix models, making free probability theory an essential framework for studying universality in random matrix theory, following the pioneering work of Voiculescu. The fruitful interactions between abstract operator algebras and random matrix theory led to applications in a wide range of areas including operator algebras, combinatorics, mathematical physics, quantum information theory, and high-dimensional statistics.

In this talk, I will review some foundational results in free probability and highlight applications in operator algebras and random matrix theory. Toward the end, I will present recent works on the spectral measure of non-normal free random variables (known as the Brown measure) and applications in non-Hermitian random matrix theory. In addition to discussing some PDE techniques, I will report new applications of a unified approach based on Hermitian reduction and (noncommutative) complex analysis. This approach resolves several open problems related to Brown measure formulas and has applications to operator algebras, the study of full-rank perturbations in random matrix models, and outliers in large random matrices.

This talk aims to provide an introduction to free probability theory and my recent research, requiring no prior knowledge of operator algebras or random matrix theory.

November 16-17

Brazos Analysis Seminar at UH

Spring 2024

(Fridays 1pm-2pm, CST)

February 2

Speaker: David Blecher and Mehrdad Kalantar (Parts I and II)

Title: Operator space complexification, and complexification transfigured

Abstract: This is a two part series, with the second part in March. Real structure occurs naturally and crucially in very many areas of mathematics. With collaborators we have recently developed the theory of real operator spaces and (possibly nonselfadjoint) real operator algebras to a somewhat mature level. We begin by describing this theory and how standard constructions interact with the complexification. We characterize real structure in complex operator spaces, and characterize some of the most important objects in the subject. Generalizing further, joint work of the two speakers gives a novel framework that contains the operator space complexification, as well as the less-studied quaternification, as special cases. It also may be viewed as the appropriate variant of Frobenius and Mackey's induced representations for the category of operator spaces.

February 9

Speaker: Pawel Sarkowicz (University of Waterloo)

Title: Embeddings of unitary groups

Abstract: We discuss unitary groups of C*-algebras with an interest in group homomorphisms between them, and how they give relationships between the K-theory and traces. With this information, one can use the state-of-the-art K-theoretic classification of embeddings to conclude that there are certain embeddings between C*-algebras if and only if there are appropriate embeddings between their unitary groups. Despite the use of some heavy machinery (in the background), these topics should be accessible to anyone with a basic understanding of operator algebras.

February 16

Speaker: Ellen Weld (Sam Houston State University)

Title: A Brief Introduction to L^p-Operator Algebras and an Interesting Example

Abstract: The study of C*-algebras is an extremely active area in Functional Analysis. Every C*-algebra may be realized concretely as a norm-closed self-adjoint subalgebra of B(H), the set of bounded operators on a Hilbert space H. As we know, every Hilbert space is isometrically isomorphic to an L2-space and so we see that every C*-algebra may be realized on $B(L^2(\mu))$ under suitable closure conditions. A natural question is what spaces may be realized on $B(L^p(\mu))$ for p in [1,\infty)? We call such spaces $L^p$-operator algebras.

In this talk, we introduce Lp-operator algebras, discuss some properties which distinguish them from Lp-algebras, and then investigate an intriguing example of an L1-operator algebra.

This is joint work with Alonso Delf\'{i}n.

February 23

Speaker: Matthew Neal (Denison University)

Title: Metric-linear characterizations of algebraic structures

Abstract: For an operator algebra X, a metric-linear condition is a condition involving only the vector space structure and operator norms on M_n(X). Since the dawn of operator algebras, there has been great interest in the relationship between an operator algebra’s algebraic properties and it's metric-linear structure. For example, L is a unital linear complete isometry between two C*-algebras if and only if it is an algebraic isomorphism. In this talk, we will investigate which purely algebraic properties of C*-algebras, TROs, JB*-triples, and non-selfadjoint operator algebras can be characterized by purely metric-linear conditions. We will also consider metric-linear conditions that determine if an operator space X is completely isometric to one of these types of operator algebras. At the end we will give some additional results about the characterization of operator algebras among operator spaces X using the holomorphic structure of X.

Wednesday February 28

Speaker: Alain Valette (University of Neuchâtel, Switzerland)

Title: Maximal Haagerup subgroups in $\Z^n\rtimes SL_2(\Z)$

Abstract: The Haagerup property is a strong negation of Kazhdan's property (T). In a countable group, every Haagerup subgroup is contained in a maximal one. We propose to classify maximal Haagerup subgroups in the semi-direct product $G_n=\Z^n\rtimes SL_2(\Z)$, where the action of $SL_2(\Z)$ on $\Z^n$ is induced by the unique irreducible representation of $SL_2(\R)$ on $\R^n$ (with $n>1$). We prove that there is a dichotomy for maximal Haagerup subgroups in $G_n$: either (amenable case) they are of the form $\Z^n\rtimes K$, with $K$ maximal amenable in $SL_2(\Z)$; or (non-amenable case) they are transverse to $\Z^n$. This extends work by Jiang and Skalski for $n=2$.

In joint work with P. Jolissaint, for n even, we prove the stronger result that the von Neumann algebra of $$\Z^n\rtimes K$ (K as above) is maximal Haagerup in the von Neumann algebra of $G_n$. This involves looking at the orbit equivalence relation induced by $SL_2(\Z)$ on the n-torus, and proving that it satisfies a dichotomy: every ergodic sub-equivalence relation is either rigid or hyperfinite. This extends a result by Ioana for $n=2$.

March 2-3

March 8

Speaker: Mehrdad Kalantar (joint with David Blecher)

Title: Operator space complexification, and complexification transfigured

Abstract: Given a finite group G, a central subgroup H of G, and an operator space X equipped with an action of by complete isometries, we construct an operator space G(X) equipped with an action of that is unique under a “reasonable” condition. This generalizes the operator space complexification of X. As a linear space G(X) is the space obtained from inducing the representation of H to G (in the sense of Frobenius). Indeed a main achievement of our paper is the induced representation construction in the category of operator spaces. This has been hitherto elusive even for Banach spaces since it is not clear how to norm the induced space. We show that for a large class of group actions the induced space has a unique operator space norm.

March 15

(Spring Break)

March 22

Speaker: Vern Paulsen (IQC Waterloo and UH)

Title: Positive Maps and Entanglement in Real Hilbert Spaces

Abstract's: We look at various results concerning separability and entanglement to compare and contrast the real and complex theories.

March 29

Speaker: Travis Russell (Texas Christian University)

Title: Products of unital operator spaces

Abstract: In this talk, I will summarize some preliminary results from ongoing joint work with Adam Dor-On. We will consider two notions of "product" for unital operator spaces: tensor products and operator products. We will define a notion of tensor product for unital operator spaces and describe several examples, including minimal, maximal, and Haagerup tensor products. We will discuss some properties associated to the Haagerup tensor product and consider some canonical C*-covers. Finally, we will introduce the notion of operator products for unital operator spaces. These are unital operator spaces densely spanned by the product of two unital subspaces of a C*-algebra. We will provide an abstract characterization for operator products of unital operator spaces and conclude by discussing some connections between tensor products and operator products of unital operator spaces.

April 5

Speaker: Irina Holmes (Texas A&M University)

Title: Sharp Restricted Weak-Type Estimates for Sparse Operations

Abstract: We discuss a recent result, obtained in collaboration with Guillermo Rey (University of Madrid) and Kristina Skreb (University of Zagreb), in which we find the exact Bellman function associated with level sets of sparse operators acting on characteristic functions. We will start with a lighter introduction to modern dyadic harmonic analysis and the role these operators play in the so-called “sparse revolution” in harmonic analysis. Then, we will try to explain the key ideas of our Bellman function proof.

April 12

Speaker: Operator space seminar!

Title: Duality

Abstract: TBA

April 19

Speaker: Roger Smith (TAMU)

Title: Crossed products by compact actions of abelian groups

Abstract: Let $G$ be a discrete group acting ergodically and freely on an abelian von Neumann algebra $A$ by automorphisms $\{\alpha_g:g\in G\}$.In the crossed product $A\rtimes_\alpha G$, there are two distinguished subalgebras $A$ and $L(G)$, and the problem I will address is how to characterize intermediate von Neumann algebras containing $A$ or $L(G)$. The first of these is well understood, but the second is more complicated, and is probably intractable without further assumptions on $G$ and the action. Recently Chifan and Das made progress in the case that $G$ is I.C.C. and the action is compact. In this talk I will discuss the case when $G$ is abelian (the opposite end of the spectrum from I.C.C.), and here complete descriptions of $L(G)$-bimodules and intermediate algebras can be obtained.

All terminology will be explained in the talk, and little background will be assumed. This work is taken from an ongoing project with Jan Cameron and Alan Wiggins.

April 26

Speaker: Florent Baudier (TAMU)

Title: Metric Invariants and bi-Lipschitz Distortion of Graphs

Abstract: A central theme in the 40-year-old Ribe program is the quest for metric invariants that characterize local properties of Banach spaces. These invariants are usually closely related to the geometry of certain sequences of finite graphs (Hamming cubes, binary trees, diamond graphs...) and provide quantitative bounds on the bi-Lipschitz distortion of those graphs. A more recent program, deeply influenced by the late Nigel Kalton, has a similar goal but for asymptotic properties instead, The metric invariants there tell us a lot about the geometry of sequences of (infinite) graphs with countable degree. In this talk, we will focus on metric invariants capturing the geometry of trees and if time permits of diamond graphs. In particular, we will motivate the (asymptotic) notion of umbel convexity, inspired by the (local) notion of Markov convexity, and discuss the values of these invariants for Banach or metric spaces, and Heisenberg groups.

Much of this talk is based on joint work with Chris Gartland (UC San Diego).

Fall 2023

(Fridays 1pm-2pm, CST)

All talks are in PGH 646, unless otherwise noted

September 15

Speaker: Jitendra Prakash (University of New Orlean)

Title: Constant-sized robust self-tests for states and measurements of unbounded dimension

Abstract: We consider correlations, $p_{n,x}$, arising from measuring a maximally entangled state using $n$ measurements with two outcomes each, constructed from $n$ projections that add up to $xI$. We show that the correlations $p_{n,x}$ robustly self-test the underlying states and measurements. To achieve this, we lift the group-theoretic Gowers-Hatami based approach for proving robust self-tests to a more natural algebraic framework. A key step is to obtain an analogue of the Gowers-Hatami theorem allowing to perturb an "approximate" representation of the relevant algebra to an exact one. For $n = 4$ , the correlations $p_{n,x}$ self-test the maximally entangled state of every odd dimension as well as 2-outcome projective measurements of arbitrarily high rank.

September 22

Speaker: Louis E. Labuschagne (North-West University)

Title: Quantum Fokker-Planck dynamics

Abstract: The Fokker-Planck equation is a partial differential equation which is a key ingredient in many models in physics. Given that relevant models relate to the description of large systems, quantization of the Fokker-Planck equation should be done in a manner that respects this fact. With this in mind we develop a quantum counterpart of Fokker-Planck dynamics within the context of non-commutative analysis based on general von Neumann algebras. We achieve this by presenting a quantization of the generalized Laplace operator, and by also proposing a potential term conditioned to noncommutative analysis. We next obtain conditions under which the composite term generates Markov dynamics, before in closing examining the asymptotic behaviour of the Markov semigroup thus obtained. We also present a noncommutative Csiszar-Kullback inequality formulated in terms of a notion of relative entropy, and show that for more general systems, good behaviour with respect to this notion of entropy similarly ensures good asymptotic behaviour of the dynamics.

Speaker: Hao-chung Cheng (National Taiwan University )

Title: One-Shot Analysis for Classical Communication over Quantum Channels

Abstract: One of the fundamental tasks in quantum information theory is to design a good coding strategy for communication against quantum noise. In this talk, I will demonstrate a simple coding method based on the so-called pretty-good measurement for achieving the state-of-the-art one-shot channel capacity of sending classical information over quantum channels with or without entanglement assistance. I will also discuss some open problems along this line of research. This talk is based on arXiv:2208.02132.

September 29

Speaker: Zhen-Chuan Liu (Baylor)

Title: An unconditional decomposition of the Schatten-$p$ classes.

Abstract: My talk will be about the (complete) boundedness of the Schur multipliers on the Schatten $p$-classes. In 1980’s, J. Bourgain proved a Marcinkiewicz-type testing condition for Toeplitz type Schur multipliers. In a recent joint work, we show that an analogue of J. Bourgain’s theory holds for non-Toeplitz type Schur multipliers as well. As an application, we obtain an unconditional decomposition for the Schatten-$p$ class with $1<p<\infty$.

This talk is based on joint work with Chian Yeong Chuah and Tao Mei.

October 13

Speaker: Zhiyuan Yang (TAMU)

Title: Factoriality of Yang-Baxter deformed Gaussian von Neumann algebras via conjugate variables

Abstract: The Yang-Baxter deformed Gaussian von Neumann algebras were introduced in 1994 by Marek Bozejko and Roland Speicher using Fock representation of operators with general commutation relations. Among those algebras, the most studied ones are the q-Gaussian algebras and the mixed q-Gaussian algebras which are known to always be II_1 factors due to the generator masas. The same argument, however, becomes difficult when dealing a general Yang-Baxter deformation or a nontracial deformation (for example, q-Araki-Woods algebras). In this talk, we will show the factoriality of a general Yang-Baxter deformed Gaussian (as well as Araki-Woods algebras) when the Yang-Baxter is strictly contracting by computing the conjugate variables and applying the abtract results on nontracial conjugate variables by Brent Nelson. This generalizes the previous results by A. Miyagawa and R. Speicher for q-Gaussian algebras. We will also explain how to use the free monotone transport to show the isomorphism with free group factors when the deformation is small. (based on arXiv:2304.13856)

October 27

Speaker: Adam Skalski (Mathematical Institute of the Polish Academy of Sciences)

Title: Separation properties for quantum positive-definite functions and associated von Neumann algebras

Abstract: It is well known that amenability of a discrete (quantum) group is equivalent to the existence of a net of finitely supported (quantum) positive-definite functions converging pointwise to 1. We will show that using the (quantum) Godement mean one can weaken the latter condition to the existence of a net of finitely supported normalised (quantum) positive-definite functions which is `pointwise strictly positive in the limit'. This further implies that von Neumann algebras of unimodular discrete quantum groups enjoy a strong form of non-w*-CPAP, which we call the matrix epsilon-separation property. Based on the joint work with Jacek Krajczok.

November 3

Speaker: Changying Ding (UCLA)

Title: Biexact von Neumann algebras

Abstract: The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.

November 17 (Cancelled)

Speaker: Irina Holmes (Texas A&M University)

Title: A Bellman function with no Bellman

Abstract: We discuss a work in progress, joint with Guillermo Rey (Universidad Madrid) and Kristina Skreb (Zagreb University), about some Bellman functions for sparse operators — with the long term goal of obtaining a weighted bound. I will discuss a possible new avenue to obtain such Bellman functions without any Bellman methods.

Spring 2023

(Fridays 1pm-2pm, CST)

February 17

Speaker: Bang Xu (University of Houston)

Title: Lp multipliers on quantum tori

Abstract: In this seminar, I will introduce the Fourier multipliers, which are the most important operators in analysis. In the first lecture, we study the Fourier multipliers on quantum tori. The arguments based on transference technique and operator-valued harmonic analysis.

February 24

Speaker: Bang Xu (UH)

Title: Fourier multipliers and Calderon-Zygmund theory

Abstract: In the first part, we continue to study the transference principle on quantum tori. In the second part, I will introduce the Calderon-Zygmund theory, which is an effective way to deal with the boundedness theory of Fourier multipliers.

March 3

Speaker: Chris Gartland (Texas A&M University)

Title: Embeddability of Wasserstein and Transportation Cost Spaces into $L^1$

Abstract: The Wasserstein-1 metric $W_1(X)$ over a metric space $X$ is a canonical distance on the space of probability measures over $X$. A natural problem to study is how the geometry of $X$ influences the geometry of $W_1(X)$, and in particular, identifying the properties of $X$ that determine the biLipschitz embeddability of $W_1(X)$ into Banach spaces such as $L^1$. In this talk, we will survey some recent results of the speaker on embeddability and nonembeddability. Based on joint works with Florent Baudier, David Freeman, and Thomas Schlumprecht.

March 7 (Tuesday 4pm--5pm)

Speaker: Srivatsav Kunnawalkam Elayavalli (IPAM, UCLA)

Title: Proper proximality for groups

Abstract: Recently in 2018 Boutonnet, Ioana and Peterson introduced the property of proper proximality for groups as a generalization of bi exactness of Ozawa. This property has several applications for rigidity associated to the group von Neumann algebras. Proper proximality is a very robust class and is stable under natural constructions such as direct products. I will discuss some results obtained by myself and my collaborator Ding establishing new examples and techniques.

March 10

Speaker: Sherry Gong (Texas A&M University)

Title: The Novikov conjecture, operator K theory, and diffeomorphism groups

Abstract: In this talk, I will discuss some recent work on a version of the Novikov conjecture for certain subgroups of diffeomorphism groups. This talk will be about joint work with Jianchao Wu, Zhizhang Xie, and Guoliang Yu.

March 17 (Spring Break)

March 24 (2pm--3pm on zoom)

Speaker: Trung Hoa Dinh (Troy University, AL)

Title: TBA

Abstract: TBA

March 31 (Reserved for Brazos Analysis Seminar)

April 7

Speaker: Irina Holmes (Texas A&M University)

Title: Two-weight bounds for paraproducts and sparse operators

Abstract: We discuss a sparse operator approach to Bloom-type two-weight bounds for paraproducts, and introduce a new type of weighted sparse operator.

April 14 (Virtual)

Speaker: Yifan Jia (Technical University of Munich)

Title: Hay from the haystack: explicit examples of exponential quantum circuit complexity

Abstract: The vast majority of quantum states and unitaries have circuit complexity exponential in the number of qubits. In a similar vein, most of them also have exponential minimum description length, which makes it difficult to pinpoint examples of exponential complexity. In this work, we construct examples of constant description length but exponential circuit complexity. We provide infinite families such that each element requires an exponential number of two-qubit gates to be generated exactly from a product and where the same is true for the approximate generation of the vast majority of elements in the family. The results are based on sets of large transcendence degree and discussed for tensor networks, diagonal unitaries, and maximally coherent states. (based on arXiv:2205.06977)

April 21 (Two talks)

Speaker (1-2pm, virtual): Ivan Todorov (University of Delaware)

Title: Quantum non-local game values and operator space structures

Abstract: The values of a quantum non-local game of different types (local, quantum, quantum commuting and no-signalling) will be placed in a unified context, using different operator space structures on the trace class. The quantum commuting value of a quantum game will be expressed as the norm of a suitable tensor, associated with the game, in the maximal tensor products of canonically arising TRO's. Restricting to the local value case, this will lead to a metric characterisation of state convertibility via local operations with shared randomness.

Speaker (2-3pm, in person): Tim Möbus (Technical of University Munich)

Title: Perturbation and generation theory of quantum Markov semigroups on continuous variable systems and their invariant subsets

Abstract: The exponential convergence to invariant subsets of quantum Markov semigroups plays a crucial role in quantum information theory. One such example is the utilization of bosonic continuous error correction schemes, where a dissipation is utilized to contract the space exponentially to the code space --- an invariant subspace of the Markov semigroup which is protected against certain errors. In this paper, we investigate perturbed quantum dynamical semigroups that operate on continuous variable (CV) systems and admit an invariant subspace without perturbation. First, we prove a generation theorem for quantum Markov semigroups on CV systems under the physical assumptions that the (possibly unbounded) generator increases all moments only uniformly bounded in the input state and that the generator has GKSL form with Lindbladians defined by polynomials of annihilation and creation operators. Additional, we show that the level sets of operators with bounded k-moment are admissible subsets of the evolution, which makes the perturbation theory work. Furthermore, we extend our results to quantum time-dependent semigroups. Finally, we prove that all gates in the bosonic CAT-code are well-defined and are exponentially dampening outside the code space. This exponential dampening is also demonstrated for the quantum Ornstein-Uhlenbeck semigroup.

April 28

Speaker: Sarah Chehade (Oak Ridge National Laboratory)

Title: How many unitaries does it take to reach a good solution state?

Abstract: Variational quantum algorithms (VQA), which use classical optimization techniques to train a parameterized quantum circuit, are a great tool to help solve linear and non-linear systems, factoring, combinatorial optimization, etc. Once a problem is encoded in a VQA, the question of how many layers in the circuit are required to guarantee existence of a good approximation to a solution has been partially answered through a concept of overparameterization. Overparameterization occurs whenever a quantum circuit has surpassed a critical number of layers that allows it to explore all relevant directions in state space. In many instances, the number of layers needed is bounded by the size of the Lie algebra which circuit unitaries are generated from. The deeper the circuit, i.e., the more layers in the algorithm, the more of state space can be explored. Deep circuits sometimes scale exponentially and due to limiting computing power, such circuits cannot always be evaluated.

In this talk, we address the expressibility of shallow/intermediate-depth circuits that are used when computing power is limited. We define a good notion of dimension for sets of generated unitaries and states that increases until overparameterization is achieved. As an interesting application, we highlight that only a finite number of layers are needed to reach an optimal solution for a particular VQA called the quantum approximate optimization algorithm (QAOA). This result improves on previous proofs that show the optimal solution can be reached in the infinite-layer case. This is based on joint work with Phil Lotshaw and Ryan Bennink.

Fall 2022

(Fridays 1pm-2pm, CST)

September 2 (Time: 5pm, please note the different time)

Speaker (Vitual): Bang Xu (National Seoul University)

Title: Matrix-valued maximal singular integral operators and applications

Abstract: In this talk, I shall start with the original problems in harmonic analysis that concerns the convergence properties of Fourier series on torus. The square and maximal functions will be the central reseach objects. Then I will present our recent work on singular integral theory in the framework of noncommutative setting. And we will see that the martingale theory and the variaous transference techniques that play important roles. This is joint work with Guixiang Hong, Xudong Lai and Samya Ray.

September 9

Speaker: Nikolaos Panagopoulos (UH)

Title: On invariant subalgebras of group C* and von Neumann algebras

Abstract: Given an irreducible lattice $\Gamma$ in the product of higher rank simple Lie groups we prove that: (i) every $\Gamma$-invariant von Neumann subalgebra of $L(\Gamma)$ is generated by a normal subgroup and (ii) given a non-amenable unitary representation $\pi$ of $\Gamma$, every $\Gamma$-equivariant conditional expectation on $C*_\pi(\Gamma)$ is the canonical conditional expectation onto the C*-subalgebra generated by a normal subgroup.

September 16

Speaker: Sheng Yin (Baylor University)

Title: Rank inequality done by free probability and random matrices

Abstract:

October 1-2

October 7

Speaker: Antonio Ismael Cano Mármol (ICMAT)

Title: Xp inequalities in von Neumann algebras

Abstract: Naor and Schechtman recently introduced the so-called \emph{metric Xp

inequalities}, an obstruction for embeddings of $Lq$ into $Lp$

whenever $2 < q < p < \infty$. This invariant was refined by Naor via

a fundamental inequality in the Hamming cube which strongly relies on

Fourier analysis. In this talk, we will show that this latter result

can be understood within the frame of noncommutative harmonic

analysis, providing a general realization in the context of von

Neumann algebras associated to discrete groups. Moreover, we will

briefly discuss some metric consequences.

October 14

Speaker: Alexander Kliesch (Technische Universität München)

Title: Quantum Approximation Algorithms for Combinatorial Optimization

Abstract: Many important real-world combinatorial optimization problems are NP-complete, meaning that it is not possible to solve them classically, exactly, and efficiently unless the complexity classes P and NP coincide. Approximation Algorithms are efficient algorithms that output approximate solutions to such problems with the hope that the quality of the output (as measured by an objective function) is close to the quality of the exact solution.

In this seminar, I will talk about my work concerning a popular quantum approximation algorithm called QAOA (Farhi, Goldstone, Gutmann 2014). In particular, I will focus on mathematically provable limitations of this algorithm and modified versions of QAOA aimed at overcoming these limitations. This talk is based on joint works with Sergey Bravyi (IBM), Libor Caha (TUM), Robert Koenig (TUM), and Eugene Tang (MIT).

October 21

Speaker: Arianna Cecco (UH)

Title: When is one injective thing the same as another injective thing?

Abstract: In this talk, I will discuss injectivity and injective envelopes of objects in different categories. I will present our recent work which attempts to answer the question ``What happens to injective objects under some particular functors?" This is based on joint work with David Blecher and Mehrdad Kalantar.

October 28

Speaker: Zhenhua Wang (University of Georgia)

Title: Relative operator entropies in Jordan algebras

Abstract: The relative operator entropy is a generalized operator version of entropy, which is a fundamental notion in quantum systems. In the setting of Hilbert space operators, the relative operator entropies recently are utilized as a tool to study the quantum coherence. In this talk, I will introduce the notion of relative operator entropies in the more general setting of Jordan algebras and present some recent progress on their lower and upper bounds.

November 4

Speaker: Raphael Clouatre (University of Manitoba)

Title: Minimal boundaries for operator algebras

Abstract: One spectacular achievement of modern dilation theory is the construction, for any operator algebra, of sufficiently many boundary representations. These representations are the driving force behind an ambitious program aiming to clarify the structure of non self-adjoint operator algebras through the use of non-commutative function theoretic methods. Indeed, the collection $B$ of boundary representations can be meaningfully interpreted as a non-commutative analogue of the Choquet boundary of a function algebra. Guided by classical intuition, one can thus think of the closure of $B$ as the minimal closed boundary of an operator algebra. In this talk, we will explore the general question of minimality for boundaries of operator algebras. We will relate this question to non-commutative notions of peak points and to what we call the Bishop property.

This is based on joint work with Ian Thompson.

November 11

Speaker: TBA

Title: TBA

Abstract: TBA

November 18

Speaker: TBA

Title: TBA

Abstract: TBA

Spring 2022

(Fridays 1pm-2pm)

April 29

Speaker: Haojian Li (Baylor University)

Title: Matrix-valued Beckner inequalities

Abstract: Beckner inequalities were introduced by Beckner in 1989 for the canonical Gaussian measures on $\mathbb{R}^{n}$. Beckner inequalities can be viewed an interpolation between logarithmic Sobolev inequalities and Poincare inequalities. I will briefly review classical Beckner inequalities and introduce the matrix-valued Beckner inequalities. The talk is based on the joint work with Li Gao and Cambyse Rouze.

April 1

Speaker: Tim Möbus (Technical University of Munich)

Title: Optimal convergence rate in the quantum Zeno effect for open quantum systems in infinite dimensions

Abstract: In open quantum systems, the quantum Zeno effect consists in frequent applications of a given quantum operation, e.g. a measurement, used to restrict the time evolution (due e.g. to decoherence) to states that are invariant under the quantum operation. In an abstract setting, the Zeno sequence is an alternating concatenation of a contraction operator (quantum operation) and a strongly continuous contraction semigroup (time evolution) on a Banach space. In this paper, we prove the optimal convergence rate of order 1/n of the Zeno sequence by proving explicit error bounds. For that, we derive a new Chernoff-type lemma, which we believe to be of independent interest. Moreover, we generalize the Zeno effect in two directions: We weaken the assumptions on the generator, which induce a Zeno dynamics generated by an unbounded generator and we improve the convergence of one result to the uniform topology. Finally, we provide a large class of examples arising from our assumptions.

March 25

Speaker: Alexander Frei (University of Copenhagen)

Title: The role of operator algebras for non-local games in QIT

Abstract: We start with a swift introduction to non-local games within quantum information theory (QIT) and a quick example to get everybody on board. After this swift introduction, we will then describe the main classes of strategies studied in QIT given by local, quantum-spatial, quantum-approximate and quantum-commuting strategies, and mention their relation to the Connes embedding problem. We then move on to linear BCS-games as a special class of XOR-games, and take a closer look on the famous Mermin--Peres magic square game. We discuss here how groups and their operator algebras come into stage. Finally we will talk about the CHSH game and uniqueness of optimal strategies and how this relates to computing the operator norm in "C*(F_n)\otimes C*(F_n)". All will be handled from an operator algebraic point of view and the relation to the Connes embedding problem.

March 11

Speaker: Alex Bearden (UT Tyler)

Title: Amenability of C*-dynamical systems

Abstract: The notion of amenability of a locally compact group was generalized to certain operator algebraic settings by Anantharaman-Delaroche in the late 70s. We will review the history and triumphs of amenability in W*- and C*-dynamical systems, and then describe recent progress in the understanding of the various competing notions of amenability from the work of Buss/Echterhoff/Willett, Ozawa/Suzuki, and our joint work with Jason Crann.

February 25

Speaker: Yair Hartman (Ben-Gurion University)

Title: Stationary Random Subgroups and their absence

Abstract: In this talk, we will discuss the notion of a stationary random subgroup. This is a natural extension of the notion of Invariant Random Subgroups, where one considers stationary action instead of measure-preserving actions. After setting the definitions we will wonder if such objects exist, and will provide some answers.

February 18

Speaker: Haonan Zhang (Institute of Science and Technology Austria)

Title: A variational method and its applications

Abstract: In a celebrated paper in 1973, Lieb proved what we now call Lieb's Concavity Theorem and resolved a conjecture of Wigner-Yanase-Dyson. In this talk, I will present a very simple variational method to study the concavity/convexity of trace functionals. This allows us to reduce many concavity/convexity problems to certain fundamental results: mainly this 1973 concavity theorem of Lieb and its complementary convexity result of Ando in 1979. Along the way, we settle a conjecture of Carlen-Frank-Lieb concerning "double convexity". A weaker form of this conjecture was made earlier by Audenaert-Datta when studying the data processing inequalities for alpha-z Rényi relative entropies. We also prove "triple convexity" theorems, which extend some results of Hiai-Petz and Carlen-Frank-Lieb. Such "triple convexity" results are related to a conjecture by Al-Rashed and Zegarliński, which we will prove using complex interpolation. Moreover, this variational method is also helpful in proving non-concavity/non-convexity results. One can use this to disprove a strategy to study equality conditions of data processing inequalities. The talk is based on arXiv:1811.01205, arXiv:2007.06644 and arXiv:2108.05785.

February 11

Speaker: Haonan Zhang (Institute of Science and Technology Austria)

Title: Curvature-dimension conditions for symmetric quantum Markov semigroups

Abstract: The curvature-dimension condition consists of the lower Ricci curvature bound and upper dimension bound of the Riemannian manifold, which has a number of geometric consequences and is very helpful in proving many functional inequalities. The Bakry--Émery theory and Lott--Sturm--Villani theory allow to extend this notion beyond the Riemannian manifold setting and have seen great progress in the past decades. In this talk, I will first review several notions around lower Ricci curvature bounds in the noncommutative setting and present our work on gradient estimates. Then I will speak about two noncommutative versions of curvature-dimension conditions for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet--Myers theorem, and concavity of entropy power in the noncommutative setting. I will also give some examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz-Schur multipliers over group algebras, and depolarizing semigroups. Our work also gives new proofs and new results in the discrete setting. This is based on joint work with Melchior Wirth (IST Austria).

Fall 2021

September 24

Speaker: Nikolaos Panagopoulos (UH)

Title: Noncommutative just-infinite groups

Abstract: A group G is just-infinite, if every non-trivial normal subgroup of G has finite index in G. This notion has been studied extensively in various contexts such as combinatorial group theory, ergodic theory, dynamics, etc. Recent breakthroughs in the field of Operator Algebras and their deep relation with Ergodic Theory have already provided us with important connections between operator algebraic properties of a group and properties of the space of its subgroups. In this talk, we will present a C*-algebraic analogue of just-infiniteness for countable discrete groups, which relates the space of normal subgroups of the given group with the space of the unital C*-subalgebras of its (reduced) group C*-algebra. Groups with this property will be called nc-just-infinite and as we will prove prominent examples of such groups are the irreducible lattices in certain Lie groups of higher rank.

October 8

Speaker: Tao Mei (Baylor University)

Title: Lacunary Fourier Series—Old and new stories.

Abstract: The Lacunary Fourier Series is a well-studied subject with several interesting properties in the classical analysis. In this talk, I will review these properties, especially the unconditionality of Lacunary series in the Lp spaces of the torus. I will then explain an open question on the unconditionality of length-lacunary Fourier series on free group C*-algebras and discuss the related problems and results. This talk is based on joint work with C. Chuah, Y. Han and Z. Liu.

October 22

Speaker: Anna Vershynina (UH)

Title: Trace functions in quantum information theory

Abstract: We will discuss the convexity of several operator trace functions motivated by problems in quantum information theory. In particular, we will focus on the data processing inequality (or monotonicity) for several relative entropies. The inequality effectively states that quantum states become harder to distinguish after they pass through a noisy quantum channel. It has been shown that this inequality for some relative entropies is equivalent to convexity of certain trace functions. Additionally, the convexity of these trace functions give rise to the conditions on states that ensure equality in the data processing inequality. I will report on the newest result by my collaborators and I on the convexity of a recently-introduced operator trace function relevant to the data processing inequality for alpha-z Renyi relative entropy. I will present the full classification of its convexity and concavity region. Moreover, I will show that the convexity of two other well-known trace functions is not stable with the change of one parameter.

November 5

Speaker: Li Gao (UH)

Title: Complete Logarithmic Sobolev inequalities

Abstract: Logarithmic Sobolev inequalities (LSI) were first introduced by L. Gross in 70s, and later found rich connections to probability, geometry, optimal transport as well as information theory. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups attracted a lot of attentions in quantum information theory and quantum many-body system. For classical Markov semigroups 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 in the quantum setting. In this talk, I'll present some recent progress on tensor stable log-Sobelev inequalities for finite dimensional quantum Markov semigroups. This talk is based on a joint work with Cambyse Rouze.

November 19 (Different time: 12:00pm-1:00pm)

Speaker: Arthur Parzygnat (IHES)

Title: Quantum Bayesian Updating

Abstract: I will begin by reviewing and describing Bayesian updating from a new perspective, along with several examples. This new perspective allows us to define quantum Bayesian updating and Jeffrey conditioning. However, this procedure cannot always be implemented as a quantum channel (i.e., a completely positive map). A quantum Bayes’ theorem will provide the necessary and sufficient conditions needed to perform this procedure. Special cases of quantum Bayesian updating include conditional expectations, error-correcting codes, the Einstein-Podolsky-Rosen experiment, and much more.