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 $X$.

# Past Functional Analysis Seminar

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.