### Announcement

The School is for graduate and undergraduate students, PhD students and junior researchers, and aims in presenting the research area of Operator Theory, as well as the interaction with other fields in modern mathematics.

The subject areas of the School contain:
1. Group representations and C*-algebras.
2. Ergodic Theory and von Neumann algebras.
3. Dynamical systems and operator algebras.
4. Operator algebras and quantum information.

A Workshop will run in the afternoons where research talks in the wider area of Analysis will be given.

Talks will take place in the Department of Mathematics of the University of Athens, in Room Γ31 (3rd floor when entering through the main entrance of the Department of Mathematics).

Additionally, on Tuesday afternoon and Wednesday afternoon there will be a two day meeting on the "Asymptotic theory of convex bodies". See here.

For further information contact:
Mihalis Anoussis (University of the Aegean), mano@aegean.gr
Aristides Katavolos (University of Athens), akatavol@math.uoa.gr

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### Program

Monday 06 July
10:00 - 10:50: Ergodic theory and von Neumann algebras (Aristides Katavolos).
11:00 - 11:50: Introduction to C*-algebras (Dimos Drivaliaris).
12:00 - 12:30: Coffee break.
12:30 - 13:20: Group representations and operator algebras (Mihalis Anoussis).
13:30 - 14:20: Introduction to C*-algebras (Dimos Drivaliaris).

Tuesday 7 July
10:00 - 10:50: Introduction to C*-algebras (Dimos Drivaliaris).
11:00 - 11:50: Group representations and operator algebras (Mihalis Anoussis).
12:00 - 12:30: Coffee break.
12:30 - 13:20: Introduction to C*-algebras (Dimos Drivaliaris).
13:30 - 14:20: Ergodic theory and von Neumann algebras (Aristides Katavolos).

Wednesday 8 July
10:00 - 10:50: The conjugacy problem for C*-dynamics (Evgenios Kakariadis).
11:00 - 11:50: Ergodic theory and von Neumann algebras (Aristides Katavolos).
12:00 - 12:30: Coffee break.
12:30 - 13:20: Group representations and operator algebras (Mihalis Anoussis).
Workshop
13:30 - 13:55: Compact multiplication operators on nest algebras (Gabriel Andreolas).
14:05 - 14:30: Strong Morita equivalence of operator spaces (Giorgios Eleftherakis).

Thursday 9 July
10:00 - 10:50: The conjugacy problem for C*-dynamics (Evgenios Kakariadis).
11:00 - 11:50: Fourier algebras on locally compact groups (Ying-Fen Lin).
12:00 - 12:30: Coffee break.
12:30 - 13:20: Ergodic theory and von Neumann algebras (Aristides Katavolos).
13:30 - 14:20: An introduction to operator systems (Ivan Todorov).
Workshop
14:30 - 14:55: Spectral theory in hyperbolic spaces and counting in discrete subgroups of SL(2,R) (Dimitrios Chatzakos).
15.05 - 15.30: Nonselfadjoint operator algebras and reflexivity (Eleftherios Kastis).

Friday 10 July
10:00 - 10:50: An introduction to operator systems (Ivan Todorov).
11:00 - 11:50: Fourier algebras on locally compact groups (Ying-Fen Lin).
12:00 - 12:30: Coffee break.
12:30 - 13:20: The conjugacy problem for C*-dynamics (Evgenios Kakariadis).
13:30 - 14:20: An introduction to operator systems (Ivan Todorov).
Workshop
14.30 - 14.55: Ramsey theory for nets (Andreas Mitropoulos).
15.05 - 15.30: Dyadic weights on R^n and reverse Holder inequalities (Eleftherios Nikolidakis).
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### School

Speaker: Mihalis Anoussis (University of the Aegean).
Title: Group representations and operator algebras.
Abstract: We will present basic notions of the Representation Theory of Groups and its connection with Operator Algebras. We will present selfadjoint and nonselfadjoint operator algebras that are generated by representations of groups.
Slides of the talks: first talk, second talk, third talk.

Speaker: : Dimos Drivaliaris (University of the Aegean).
Title: Introduction to C*-algebras.
Abstract: We will present an elementary introduction to the theory of C*-algebras. We will give examples of C*-algebras and discuss the Gelfand theory of commutative C*-algebras (which shows that every commutative C*-algebra can be identified with the algebra of continuous functions vanishing at the infinity on a locally compact Hausdorff space X) and the Gelfand--Naimark theorem (which shows that every C*-algebra can be identified with a subalgebra of the algebra B(H) of bounded linear operators on a Hilbert space H).
Bibliography: 1. J. A. Erdos, C*-Algebras
2. D. Williams, Lecture Notes on C*-Algebras

Title: The conjugacy problem for C*-dynamics.
Abstract: In the last 50 years, noncommutative geometry focuses on the use of operator algebras for encoding geometrical and topological objects. Operator algebras may be considered as algebras of (bounded in norm) infinite matrices with complex entries. A central aspect of the program is to explore the passage from intrinsic properties of the object into properties of the associated operator algebras, and use invariants of the latter to classify the former. There are two interrelated questions that orientate the course of study:
(q.1) Which (desirable) features of the object determine the operator algebra?
(q.2) What is the (desirable) level of equivalence for classifying objects?
Examples of examined objects so far include tilings, tangles, graphs, dynamical systems, groups, semigroups, varieties, homogeneous ideals, and stochastic matrices. In these talks we will show how a class of nonsefladjoint operator algebras classifies C*-dynamical systems up to conjugacy.

Speaker: Aristides Katavolos (University of Athens).
Title: Ergodic theory and von Neumann algebras.
Abstract: We will present an elementary introduction to operator algebras on Hilbert spaces. In particular, we will be concerned with constructions of von Neumann algebras from measure (or probability) spaces acted upon by a group of transformations, and in the interaction between measure-theoretic properties of the action and algebraic properties of the corresponding algebra. In the special case when the measure space is a singleton, the corresponding algebra is the so-called von Neumann algebra of the group.
Slides from the talks.

Speaker: Ying-Fen Lin (Queen's University Belfast).
Title: Fourier algebras on locally compact groups.
Abstract: Given a locally compact group G, one can define an algebra called the Fourier algebra which consists of functions on G. The Fourier algebra is a regular commutative Banach algebra with pointwise multiplication and addition, whose spectrum is the group G; however, its structure is very different from the structure of the algebra of continuous functions. In these talks, I will introduce the notion of a Fourier algebra and its basic properties. I will also talk about some homomorphism problems between Fourier algebras if I have time.

Speaker: Ivan Todorov (Queen's University Belfast).
Title: An introduction to operator systems.
Abstract: This series of talks aims at introducing the basic properties and some applications of operator systems. I will discuss two fundamental theorems in the area: Choi-Effros' Theorem and Arveson's Extension Theorem, which provide the basis for many subsequent developments in the area. I will touch upon some constructs of categorical flavour, such as tensor products and quotients. If time permits, I will show how operator system tensor products can be used to distinguish between different classes of quantum correlations, important in Quantum Physics in general, and in Quantum Information Theory in particular.

### Workshop

Speaker: Gabriel Andreolas (University of the Aegean).
Title: Compact multiplication operators on nest algebras.
Abstract: Properties of multiplication and elementary operators have been vigorously studied over the past decades. We will present recent results about the compactness of these operators defined on nest algebras. Specifically, we obtain a characterization of the compact and weakly compact multiplication operators defined on nest algebras. We also show that there is no non-zero weakly compact multiplication operator on $Alg\mathcal{N}/(Alg\mathcal{N}\cap&space;\mathcal{K}(H))$. This is a joint work with M. Anoussis.

Speaker: Dimitrios Chatzakos (University College London).
Title: Spectral theory in hyperbolic spaces and counting in discrete subgroups of SL(2,R).
Abstract: Lattice point counting in Euclidean spaces is a classical problem in analytic number theory, where the main tools for its study come from harmonic analysis in R^n (Poisson summation formula, exponential sums). The corresponding problem in hyperbolic spaces turns out to be significantly more difficult, due to the complexity of the geometry of those spaces. The main method for its study is the spectral theory of the Laplacian, which allows us to prove results that cannot be obtained, for the moment, by other methods (dynamical systems, ergodic theory). In this talk, we study two different lattice point problems in the hyperbolic plane and we present some of our recent results.

Speaker: Giorgios Eleftherakis (University of Patras).
Title: Strong Morita equivalence of operator spaces.
Abstract: We define and examine the notion of strong Δ-equivalence and strong TRO equivalence for operator spaces. We show that these relations behave in an analogous way as strong Morita equivalence does for the category of C*-algebras. In particular, we prove that strong Δ-equivalence coincides with stable isomorphism under the expected countability hypothesis, and that strongly TRO equivalent operator spaces admit a correspondence between particular representations. As applications we show that strongly Δ-equivalent operator spaces have stably isomorphic second duals and strongly Δ-equivalent TRO envelopes. In the case of unital operator spaces, strong Δ-equivalence implies stable isomorphism of the C*-envelopes.

Speaker: Eleftherios Kastis (Lancaster University).
Title: Nonselfadjoint operator alagebras and reflexivity.
Abstract: A reflexive operator algebra is one that has sufficiently many invariant subspaces to determine it. The examination of reflexivity on non-selfadjoint algebras has its origins in 1966. Recently research has focused on reflexive algebras that arise for unitary semigroups. A characteristic example that we will indicate is the Fourier binest algebra, which derives from the Weyl commutation relations. Also we will present recent results on the triple semigroup algebra, which is generated by the one parameter semigroups for translation, dilation and multiplication. This is joint work with Professor Stephen Power.

Speaker: Andreas Mitropoulos (University of Athens).
Title: Ramsey theory for nets.
Abstract: By introducing the general new concept of the class of coideals on an infinite directed set X, a class that contains the class of ultrafilters on X consisting of cofinal subsets of X, we are able to prove partition results for the ordered finite or infinite sequences in X with respect to a given coideal on X. Our theory extends the classical infinitary Ramsey theory (including the theorems of Ramsey, Nash-Williams, Galvin-Prikry) and in addition includes as particular cases (a) the corresponding theory for coideals on the set of natural numbers proved by Louveau, Mathias, Farah and Todorcevic, (b) the Milliken-Taylor partition theorems for sequences of finite subsets of natural numbers, and (c) the partition theorems for sequences of words proved by Carlson, Bergelson-Blass-Hindman, Farmaki. This work is joint with V. Farmaki, D. Karageorgos and A. Koutsogiannis.

Speaker: Eleftherios Nikolidakis (University of Athens).
Title: Dyadic weights on R^n and reverse Holder inequalities.
Abstract: We present properties of dyadic weights on R^n, or simply on [0,1]^n, that is of functions satisfying a reverse Holder inequality upon all dyadic subcubes of [0,1]^n, for a certain value of the exponent p>1 and constant c>1. More precisely we prove that if we consider the equimeasurable decreasing rearrangement of our weight, then it also satisfies a reverse Holder inequality upon all subintervals of (0,1], of the form (0,t], with constant not more than 2^nc-2^n+1. As a consequence we conclude higher integrability properties of the initial weight under consideration. For the approach of the above problem we use elementary theory of dyadic maximal operators on R^n, as much as theory of weights on Euclidean spaces. This is a research cooperation with Antonios Melas.

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### Participants

 Ανδρεόλας Γαβριήλ, Παν. Αιγαίου Ανούσης Μιχάλης, Παν. Αιγαίου Αντωνοπούλου Ευαγγελία, Heriot Watt University, Edinburgh Βασιλόπουλος Παναγιώτης, ΕΜΠ Βέτας Κωνσταντίνος, Παν. Αιγαίου Βιδάλη Νεφέλη, ΕΚΠΑ Βλάχος Σπυρίδων, Παν. Ιωαννίνων - ΕΑΠ Βουράκης Μύρων Αριστοτέλης, ΕΚΠΑ Γατζούρας Δημήτρης, Γεωπονικό Πανεπιστήμιο Αθηνών Γεωργακόπουλος Νίκος, ΕΚΠΑ Γιανναράκης Γιώργος, ΕΚΠΑ Γκιώκας Παναγιώτης, ΕΚΠΑ Γόγλη Αθανασία, ΕΚΠΑ Γρηγοράτος Παναγιώτης, ΕΜΠ Δριβαλιάρης Δήμος, Παν. Αιγαίου Ελευθεράκης Γιώργος, Παν. Πατρών Ελευθέριος Λαμπίρης, ΕΚΠΑ Ελευθερίου Αθανάσιος, ΕΚΠΑ Ελμιρο Βετερε, Copenhagen University Ζαμπάρας Χριστόφορος, ΕΚΠΑ Ιωαννίδης Αντώνιος, Παν. Ιωαννίνων Καϊάφα Αγγελική, ΕΚΠΑ Κακαριάδης Ευγένιος, Newcastle University Καμπίτη Έλενα, ΕΚΠΑ Καραγέωργος Δημήτρης, ΕΚΠΑ Karameta Ertval, ΕΚΠΑ Καραχάλιου Ιωάννα, ΕΑΠ Καστής Ελευθέριος, Lancaster University Κατάβολος Αριστείδης, ΕΚΠΑ Κοτζαπαναγιώτου Ευγενία, Παν. Πατρών Κουλουμπού Δήμητρα, ΟΠΑ Κουσίδης Σωκράτης, ΕΑΠ Κώστα Χριστίνα, Παν. Κρήτης Λαδά Αναστασία, Παν. Πατρών Λιάπη Μυρτώ, ΕΑΠ Λιβιεράτος Ιωάννης, ΕΚΠΑ Lin Ying-Fen, Queen's University Belfast Μαγιάτης Χαράλαμπος, Παν. Αιγαίου Μάργα Κωνσταντίνα, Παν. Πατρών Μητρόπουλος Ανδρέας, ΕΚΠΑ Μπαξεβάνης Κυριάκος, Παν. Κρήτης Νικολιδάκης Ελευθέριος, ΕΚΠΑ Οικονομόπουλος Δημήτρης, ΕΚΠΑ Παλαμιώτη Νικολέττα, ΕΚΠΑ Παναγόπουλος Νικόλαος, ΕΚΠΑ Πολιτάκη Ιωάννα- Κλημεντία, Παν. Κρήτης Πολυδεύκη Γεωργία, Παν. Πατρών Σακελαρόπουλος Αλέξιος, ΕΚΠΑ Ρέντζος Στυλιανός, Παν. Αιγαίου Σοροβού Ειρήνη, ΑΠΘ Σουφλέρη Ευσταθία, ΕΚΠΑ Στάμου Μαρία-Νίκη, ΕΚΠΑ Τερεζάκης Αλέξιος, ΕΚΠΑ Todorov Ivan, Queen's University Belfast Τούρλας Παναγιώτης, Παν. Αιγαίου Τσακνάκη Ιωάννα Υβόννη, ΕΚΠΑ Τσάνκο Ιωσήφ, Παν. Κρήτης Τσουκνίδας Ιωάννης, ΕΚΠΑ Τσίγκος Δημήτριος, ΕΚΠΑ Φουκάκη Ζαχαρένια, Παν. Κρήτης Φουστέρη Φλώρα, ΑΠΘ Φρέρης Λουκάς, Παν. Αιγαίου Χατζάκος Δημήτρης, University College London Ψαρομήλιγκος Κώστας, ΕΚΠΑ

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