Past Functional Analysis Seminar

Let $X$ be a compact Hausdorff space and $C(X)$ be the space of continuous real-valued functions on $X$ endowed with the topology of uniform convergence. Assume we are given a finite number of closed subalgebras $A_1, \dots A_k$ of $C(X)$. Our talk is devoted to the following problem. What conditions imposed on $A_1, \dots, A_k$ are necessary and/or sufficient for the representation $C(X) = A_1 +\dots + A_k$? For the case $k = 1$, the history of this problem goes back to 1937 and 1948 papers by M. Stone. A version of the corresponding famous result, known as the Stone-Weierstrass theorem, states that a closed subalgebra $A \subset C(X)$, which contains a nonzero constant function, coincides with the whole space $C(X)$ if and only if $A$ separates points of $X$.

A self-similar groupoid action (G,E) consists of a faithful action of a groupoid G on the path space of a graph which displays a notion of self-similarity. In this talk I will explain this concept and consider KMS states on associated Cuntz-Pimsner C*-algebras. This talk is based on joint work with Marcelo Laca, Iain Raeburn, and Jacqui Ramagge

The classical Banach-Stone theorem describes the structure of onto linear isometries of the Banach space $C(K)$ of all continuous functions on a compact Hausdorff space $K$. Namely, such an isometry is always a product of a composition operator with a homeomorphism symbol and a multiplication operator with a continuous symbol which has modulus 1.

Recently, similar results have been obtained in the setting of certain class of probability measures. In my talk first, I will give an overview of these results, and then I will present the main ideas of a recent work. Namely, I will provide a characterisation of all surjective isometries of the (non-linear) space of all Borel probability measures on an arbitrary separable Banach space with respect to the famous Levy-Prokhorov distance (which metrises the weak convergence). This is a recent joint work with Tamas Titkos (MTA Alfred Renyi Institute of Mathematics, Budapest, Hungary).

We study a spectral problem which is related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. Under certain geometrical assumptions, the underlying Maxwell operator reduces to an elliptic operator and we study the effect of the perturbation by the waveguide on its spectrum. We show that the perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem and use variational arguments to prove that guided mode spectrum can be created by arbitrarily small perturbations.

I will report on the results of my recent work with Dmitri Yafaev (Univeristy of Rennes-1). We consider functions $\omega$ on the unit circle with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions in the BMO norm. We find explicitly the leading term of the asymptotics of the distance in the BMO norm between $\omega$ and the set of rational functions of degree $n$ as $n$ goes to infinity. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.
 

The talk will be a survey of recent work on the connections between group operator systems and non-signalling correlations. I will show how tensor products in the operator system category can be interpreted as physical theories, and will exhibit a refinement of non-signalling corrections giving rise to quantum attributes of graphs, including quantum versions of the chromatic and the fractional chromatic number. Relations between the parameters will be examined and further directions will be discussed.

Although much of the theory of Fourier multipliers has focused on the $(L^{p},L^{p})$-boundedness

 of such operators, for many applications it suffices that a Fourier multiplier operator is bounded

 from $L^{p}$ to $L^{q}$ with p and q not necessarily equal. Moreover, one can derive 

(L^{p},L^{q})-boundedness results for $p\neq q$ under different, and often weaker, assumptions

 than in the case $p=q$. In this talk I will explain some recent results on the

 $(L^{p},L^{q})$-boundedness of operator-valued Fourier multipliers. Also, I will sketch some

 applications to the stability theory for $C_{0}$-semigroups and functional calculus theory. 

This talk will be transmitted from Warsaw to us and Dresden, provided that Warsaw get things set up.  We will not be using the TCC facility,

so the location will be C1.

We consider a system of coupled second order integro-differential evolution equations in a Hilbert space, which is partially damped through memory effects. A global existence theorem regarding the solutions to its Cauchy problem is given, only under basic conditions that the memory kernels possess positive definite primitives but without nonnegative/decreasing assumptions. Following this, we find an approach to successfully obtain the stability of the system energy and various decay rates. Moreover, the abstract results are applied to several concrete systems in the real world, including the Timoshenko type. This is a joint work with Professor Ti-Jun Xiao (Fudan University) and Professor Jin Liang (Shanghai Jiaotong University)

A subshift is characterized by a set of allowable words on $d$ symbols. In a sense it encodes the allowable operations an automaton performs. In the late 1990's Matsumoto constructed a C*-algebra associated to a subshift, deriving initially his motivation from the work of Cuntz-Krieger. These C*-algebras were then studied in depth in a series of papers. In 2009 Shalit-Solel discovered a relation of the subshift algebras with their variants of operator algebras related to homogeneous ideals. In particular a subshift corresponds to a monomial ideal under this prism.

In a recent work with Shalit we take a closer look at these cases and study them in terms of classification programmes on nonselfadjoint operator algebras and Arveson's Programme on the C*-envelope. We investigate two nonselfadjoint operator algebras from one SFT and show that they completely classify the SFT: (a) up to the same allowable words, and (b) up to local conjugacy of the quantized dynamics. In addition we discover that the C*-algebra fitting Arveson's Programme is the quotient by the generalized compacts, rather than taking unconditionally all compacts as Matsumoto does. Actually there is a nice dichotomy that depends on the structure of the monomial ideal.

Nevertheless in the process we accomplish more in different directions. This happens as our case study is carried in the intersection of C*-correspondences, subproduct systems, dynamical systems and subshifts. In this talk we will give the basic steps of our results with some comments on their proofs.

The Bergman space $A_2(\mathbb D)$ is the closed subspace of $L^2(\mathbb D)$ consisting of analytic functions, where $\mathbb D$ denotes the unit
disk. One considers the projection from $L^2(\mathbb D)$ into $A_2(\mathbb D)$, such a projection can be written as an integral operator
with a singular kernel. In this talk, we will present the recent advances on the one weight and two weight theory for the
Bergman projection, in particular we will discuss the Sarason Conjecture for the Bergman space, sharp weighted estimates for the Bergman projection and a description of a $B_{\infty}$ class that has been until now absent. This is joint work with A. Aleman and S. Pott from Lund University (Sweden).

 An old result of Dixmier, Day and others states that every continuous bounded representation of an amenable group on Hilbert space is similar to a unitary representation. In similar vein, one can ask if amenable subalgebras of $B(H)$ are always similar to self-adjoint subalgebras. This problem was open for many years, but it was recently shown by Farah and Ozawa that in general the answer is negative; their approach goes via showing that the Dixmier--Day result is false when $B(H)$ is replaced by the Calkin algebra.



In this talk, I will give some of the background, and then outline a simplified and more explicit version of their construction; this is taken from joint work with Farah and Ozawa (2014) . It turns out that the key mechanism behind these negative results is the large supply of projections in $\ell_\infty / c_0$, rather than the complicated structure of $B(H)$.

This talk will be by videolink from Warsaw.  The starting-time will be a little after 17:00 due to a TCC lecture and time needed to establish video connections.

Abstract: The Haagerup approximation property for finite von Neumann algebras  (i.e.von Neumann algebras with a tracial faithful normal state) has been studied for more than 30 years. The original motivation to study this property came from the case of group von Neumann algebras of discrete groups, where it corresponds to the geometric Haagerup property of the underlying group. Last few years brought a lot of interest in the Haagerup property for discrete and general locally compact quantum groups. If the discrete quantum group in question is not unimodular, the associated (quantum) group von Neumann algebra cannot be finite, so we need a broader framework for the operator algebraic property. In this talk, I will present recent developments regarding the Haagerup approximation property for arbitrary von Neumann algebras and will also discuss some questions relating it to the issues related to the classical Schoenberg correspondence. (Mainly based on joint work with Martijn Caspers.)

We present several quantified versions of Ingham’s Tauberian theorem in
which the rate of decay is determined by the behaviour of a certain boundary
function near its singularities. The proofs of these results are modified
versions of Ingham’s own proof and, in particular, involve no estimates of
contour integrals. The general results are then applied in the setting of C_0-
semigroups, giving both new proofs of previously known results and, in one
important case, a sharper result than was previously available.

Stochastic homogenization shows that solutions to an elliptic problem 

with rapidly oscillating, ergodic random coefficients can be effectively 

described by an elliptic problem with homogeneous, deterministic 

coefficients. The definition of the latter is based on the construction 

of a "corrector" and invokes an elliptic operator that acts on the 

probability space of admissible coefficient fields. While qualitative 

homogenization is well understood and classical, quantitative results 

(e.g. estimates on the homogenization error and approximations to the 

homogenized coefficients) have only been obtained recently.  In the talk 

we discuss an optimal estimate on the associated semigroup (usually 

called the "random walk in the random environment") and show that it 

decays with an algebraic rate. The result relies on a link between a 

Spectral Gap of a Glauber dynamics on the space of coefficient fields (a 

notion that we borrow from statistical mechanics) and heat kernel 

estimates. As applications we obtain moment bounds on the corrector and 

an optimal convergence rate for the approximation of the homogenized 

coefficients via periodic representative volume elements.