Forthcoming events in this series


Thu, 16 Apr 2020

16:45 - 17:30

Introduction to non-commutative L_p-space

Runlian Xia
(University of Glasgow)
Further Information

This is a meeting of the UK virtual operator algebras seminar: see https://sites.google.com/view/uk-operator-algebras-seminar/home

Abstract

This talk will give an easy introduction to non-commutative L_p spaces associated with a tracial von Neumann algebra. Then I will focus on non-commutative L_p spaces associated to locally compact groups and talk about some interesting completely bounded multipliers on them.

Thu, 16 Apr 2020

16:00 - 16:30

Introduction to C_0 semigroups

David Seifert
(University of Newcastl)
Further Information

UK Virtual Operator Algebras seminar held by zoom.  See https://sites.google.com/view/uk-operator-algebras-seminar/home

Abstract

This talk will introduce some of the basic notions and results in the theory of C_0-semigroups, including generation theorems, growth and spectral bounds. If time permits, I will also try to discuss one or two classical results in the asymptotic theory of C_0-semigroups.

Thu, 02 Apr 2020
16:00

What is the Jiang Su algebra (Virtual Seminar)

Sam Evington
(University of Oxford)
Further Information

This is the first meeting of the virtual operator algebra seminar in collaboration with colleagues in Glasgow and UCLan.  The seminar will take place by zoom, and the meeting details will be available here.

Tue, 10 Mar 2020
16:00
C1

Pick's theorem and the Kadison-Singer problem

Michael Hartz
(University of Saarbrucken)
Abstract

Pick's theorem is a century-old theorem in complex analysis about interpolation with bounded analytic functions. The Kadison-Singer problem was a question about states on $C^*$-algebras originating in the work of Dirac on the mathematical description of quantum mechanics. It was solved by Marcus, Spielman and Srivastava a few years ago.

I will talk about Pick's theorem, the Kadison-Singer problem and how the two can be brought together to solve interpolation problems with infinitely many nodes. This talk is based on joint work with Alexandru Aleman, John McCarthy and Stefan Richter.

Tue, 03 Mar 2020
16:00
C1

Amenability, paradoxicality and uniform Roe algebras.

Fernando Lledo
(Madrid)
Abstract

There is a classical mathematical theorem (due to Banach and Tarski) that implies the following shocking statement: An orange can be divided into finitely many pieces, these pieces can be rotated and rearranged in such a way to yield two oranges of the same size as the original one. In 1929 J.~von Neumann recognizes that one of the reasons underlying the Banach-Tarski paradox is the fact that on the unit ball there is an action of a discrete subgroup of isometries that fails to have the property of amenability ("Mittelbarkeit" in German).

In this talk we will address more recent developments in relation to the dichotomy amenability vs. existence of paradoxical decompositions in different mathematical situations like, e.g., for metric spaces, for algebras and operator algebras. We will present a result unifying all these approaches in terms of a class of C*-algebras, the so-called uniform Roe algebras.

P. Ara, K. Li, F. Lledó and J. Wu, Amenability of coarse spaces and K-algebras , Bulletin of Mathematical Sciences 8 (2018) 257-306;
P. Ara, K. Li, F. Lledó and J. Wu, Amenability and uniform Roe algebras, Journal of Mathematical Analysis and Applications 459 (2018) 686-716;

Tue, 25 Feb 2020

16:00 - 17:00
C1

Functional calculus for analytic Besov functions

Charles Batty
(Oxford)
Abstract

There is a class $\mathcal{B}$ of analytic Besov functions on a half-plane, with a very simple description.   This talk will describe a bounded functional calculus $f \in \mathcal{B} \mapsto f(A)$ where $-A$ is the generator of either a bounded $C_0$-semigroup on Hilbert space or a bounded analytic semigroup on a Banach space.    This calculus captures many known results for such operators in a unified way, and sometimes improves them.   A discrete version of the functional calculus was shown by Peller in 1983.

Tue, 18 Feb 2020
16:00
C1

Quasi-locality and asymptotic expanders

Jan Spakula
(University of Southampton)
Abstract

Let $X$ be a countable discrete metric space, and think of operators on $\ell^2(X)$ in terms of their $X$-by-$X$ matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of $X$. This algebra "encodes" the large-scale (a.k.a. coarse) structure of $X$. Quasi-locality, coined by John Roe in '88, is a property of an operator on $\ell^2(X)$, designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work, and an expander-ish condition on graphs that came out of trying to find a counterexample. (Joint with: A. Tikuisis, J. Zhang, K. Li and P. Nowak.)
 

Tue, 11 Feb 2020
16:00
C1

Fredholm theory and localisation on Banach spaces

Raffel Hagger
(University of Reading)
Abstract

Let $\mathcal{B}$ be a (unital) commutative Banach algebra and $\Omega$ the set of non-trivial multiplicative linear functionals $\omega : \mathcal{B} \to \mathbb{C}$. Gelfand theory tells us that the kernels of these functionals are exactly the maximal ideals of $\mathcal{B}$ and, as a consequence, an element $b \in \mathcal{B}$ is invertible if and only if $\omega(b) \neq 0$ for all $\omega \in \Omega$. A generalisation to non-commutative Banach algebras is the local principle of Allan and Douglas, also known as central localisation: Let $\mathcal{B}$ be a Banach algebra, $Z$ a closed subalgebra of the center of $\mathcal{B}$ and $\Omega$ the set of maximal ideals of $Z$. For every $\omega \in \Omega$ let $\mathcal{I}_{\omega}$ be the smallest ideal of $\mathcal{B}$ which contains $\omega$. Then $b \in \mathcal{B}$ is invertible if and only if $b + \mathcal{I}_{\omega}$ is invertible in $\mathcal{B} / \mathcal{I}_{\omega}$ for every $\omega \in \Omega$.

 

From an operator theory point of view, one of the most important features of the local principle is the application to Calkin algebras. In that case the invertible elements are called Fredholm operators and the corresponding spectrum is called the essential spectrum. Therefore, by taking suitable subalgebras, we can obtain a characterisation of Fredholm operators. Many beautiful results in spectral theory, e.g.~formulas for the essential spectrum of Toeplitz operators, can be obtained in this way. However, the central localisation is often not sufficient to provide a satisfactory characterisation for more general operators. In this talk we therefore consider a generalisation where the ideals $\mathcal{I}_{\omega}$ do not originate from the center of the algebra. More precisely, we will start with general $L^p$-spaces and apply limit operator methods to obtain a Fredholm theory that is applicable to many different settings. In particular, we will obtain characterisations of Fredholmness and compactness in many new cases and also rediscover some classical results.

 

This talk is based on joint work with Christian Seifert.

Tue, 04 Feb 2020
16:00
C1

Lipschitz spaces from the semigroup language point of view

Marta de Leon Contreaas
(University of Reading)
Abstract

 Lipschitz (or H\"older) spaces $C^\delta, \, k< \delta <k+1$, $k\in\mathbb{N}_0$, are the set of functions that are more regular than the $\mathcal{C}^k$ functions and less regular than the $\mathcal{C}^{k+1}$ functions. The classical definitions of H\"older classes involve  pointwise conditions for the functions and their derivatives.  This implies that to prove   regularity results for an operator among these spaces  we need its pointwise expression.  In many cases this can be a rather involved formula, see for example the expression of $(-\Delta)^\sigma$  in (Stinga, Torrea, Regularity Theory for the fractional harmonic oscilator, J. Funct. Anal., 2011.)

In  the 60's of last century, Stein and Taibleson, characterized bounded H\"older functions via some integral estimates of the Poisson semigroup, $e^{-y\sqrt{-\Delta}},$ and of  the Gauss semigroup, $e^{\tau{\Delta}}$. These kind of semigroup descriptions allow to obtain regularity results for fractional operators in these spaces in a more direct way.

 In this talk we shall see that we can characterize H\"older spaces adapted to other differential operators $\mathcal{L}$ by means of semigroups and that these characterizations will allow us to prove the boundedness of some fractional operators, such as $\mathcal{L}^{\pm \beta}$, Riesz transforms or Bessel potentials, avoiding the long, tedious and cumbersome computations that are needed when the pointwise expressions are handled.

Tue, 28 Jan 2020

16:00 - 17:00
C1

A Fourier-analytic approach to the transport AKT theorem.

Sergey Bobkov
(University of Minnesota)
Abstract

We will be discussing a Fourier-analytic approach
to optimal matching between independent samples, with
an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.

Tue, 14 Jan 2020
16:00
C3

Structure theory for groupoid C*-algebras

Christian Bonicke
(University of Glasgow)
Abstract

C*-algebras constructed from topological groupoids allow us to study many interesting and a priori very different constructions
of C*-algebras in a common framework. Moreover, they are general enough to appear intrinsically in the theory. In particular, it was recently shown
by Xin Li that all C*-algebras falling within the scope of the classification program admit (twisted) groupoid models.
In this talk I will give a gentle introduction to this class of C*-algebras and discuss some of their structural properties, which appear in connection
with the classification program.
 

Tue, 26 Nov 2019
17:00
C1

Semigroup C*-algebras

Xin Li
(Queen Mary London)
Abstract

I will give an overview of semigroup C*-algebras, which are C*-algebras generated by left regular representations of semigroups. The main focus will be on examples from number theory and group theory.

Tue, 19 Nov 2019

17:00 - 18:00
C1

Semigroup C*-algebras associated with arithmetic progressions

Chris Bruce
(University of Victoria)
Abstract

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.

Tue, 05 Nov 2019
17:00
C1

Schroedinger operator with non-zero accumulation points of complex eigenvalues

Sabine Boegli
(Durham)
Abstract

We consider Schroedinger operators on the whole Euclidean space or on the half-space, subject to real Robin boundary conditions. I will present the construction of a non-real potential that decays at infinity so that the corresponding Schroedinger operator has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum. This proves that the Lieb-Thirring inequalities, crucial in quantum mechanics for the proof of stability of matter, do no longer hold in the non-selfadjoint case.

Tue, 29 Oct 2019

17:00 - 18:00
C1

Functional and Geometric Inequalities via Optimal Transport

Andrea Mondino
(University of Oxford)
Abstract

I will give an overview of the localization technique: a powerful dimension-reduction tool for proving geometric and functional inequalities.  Having its roots in a  pioneering work of Payne-Weinberger in the 60ies about sharp Poincare’-Wirtinger inequality on Convex Bodies in Rn, recently such a technique found new applications for a range of sharp geometric and functional inequalities in spaces with Ricci curvature bounded below.

Tue, 22 Oct 2019
17:00
C1

Asymptotics of semigroups: quantified Katznelson--Tzafriri theorems

Abraham Ng
(Oxford)
Abstract

The well known Katznelson-Tzafriri theorem states that a power-bounded operator $T$ on a Banach space $X$ satisfies $\|T^n(I-T)\| \to 0$ as $n \to \infty$ if and only if the spectrum of $T$ touches the complex unit circle nowhere except possibly at the point $\{1\}$. As it turns out, the rate at which $\|T^n(I-T)\|$ goes to zero is largely determined by estimates on the resolvent of $T$ on the unit circle minus $\{1\}$ and not only is this interesting from a purely spectral and operator theoretic perspective, the applications of such quantified decay rates are myriad, ranging from the mean ergodic theorem to so-called alternating projections, from probability theory to continuous-in-time evolution equations. In this talk, we will trace the story of these so-called quantified Katznelson-Tzafriri theorems through previously known results up to the present, ending with a new result proved just a few weeks ago that largely completes the adventure.

Thu, 16 May 2019
17:00
C2

The least doubling constant of a metric measure space

Pedro Tradacete
(Madrid)
Further Information

ABSTRACT: Given a metric measure space $(X,d,\mu)$, its doubling constant is given by
$$
C_\mu=\sup_{x\in X, r>0} \frac{\mu(B(x,2r))}{\mu(B(x,r))},
$$
where $B(x,r)$ denotes the open ball of radius $r$ centered at $x$. Clearly, $C_\mu\geq1$, and in the case $X$ reduces to a singleton $C_\mu=1$. One might think that for a metric space with more than one point, the constant $C_\mu$ could be very close to one. However, we will show that in general $C_\mu\geq2$. The talk is based on a joint work with Javier Soria (Barcelona).

Tue, 30 Apr 2019
17:00
C2

Extensions of Uniform Algebras

Sam Morley
(East Anglia)
Further Information

The theory of algebraic extensions of commutative Banach algebras is well established and have been used to solve many problems. In his thesis, Cole constructed algebraic extensions of a certain uniform algebra to give a counterexample to the peak point conjecture. Cole’s method for extending uniform algebras ensures that certain properties of the original algebra are preserved by the extension. In this talk, we discuss the general theory of uniform algebra extensions and a certain class of uniform algebra extensions which generalise Cole’s construction.
 

Tue, 26 Feb 2019
14:00
N3.12

The behaviour of the $(p, q)$-version of Fourier's series

Houry Melkonian
(Exeter University)
Abstract

Consider a periodic function $f$, such that its restriction to the unit segment lies in the Banach space $L^2=L^2(0,1)$. Denote by $S$ the family of dilations $f(nx)$ for all $n$ positive integer.    The purpose of this talk is to discuss the following question: When does $S$ form a Riesz basis of $L^2$?

In this talk, we will present a new \textit{mutli-term} criteria for determining Riesz basis properties of $S$ in $L^2$. This method was established in [L. Boulton, H. Melkonian; arXiv: 1708.08545 J. (2017), to appear at the Journal of Analysis and its Applications (ZAA)] and it relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients.

Tue, 12 Feb 2019
17:00
C1

Admissibility problem of some classes of state-delayed systems

Radoslaw Zawiski
(Silesian University of Technology)
Further Information


Beginning with a short introduction and a review of Hilbert space 
techniques used in the admissibility analysis of dynamical systems, 
we will focus on state-delayed systems. 
Using the "lifting" method to reformulate the problem, we will firstly 
analyse a retarded delay system assuming only contraction property 
of the undelayed semigroup. Next, we will turn our attention to problems 
where more can be said about the underlying semigroup. 


In particular, we will investigate diagonal systems.
This talk will present results of a joint work with Jonathan Partington.