Spring 2020

* Organisers*

Marcel de Jeu (mdejeu@math.leidenuniv.nl)

Bram Mesland (b.mesland@math.leidenuniv.nl)

Onno van Gaans (vangaans@math.leidenuniv.nl)

*Contact*

If you are interested, please contact Marcel de Jeu, Bram Mesland, or Onno van Gaans.

*Topic*

The following is a quotation from Christian le Merdy's review in MathSciNet of the book that we shall use:

``Operator space theory grew from papers by Effros and Ruan and by Blecher and Paulsen published at the beginning of the nineties. It has now become a well-established and deep theory, with various and remarkable applications resulting from the interplay between operator spaces and other topics such as C^{*}-algebras and von Neumann algebras, Banach spaces, nonselfadjoint operator algebras, noncommutative harmonic analysis, noncommutative integration theory, free probability, etc. The monograph under review is the first one dealing with this recently developed theory. Parts I and II, which are about 200 pages, contain a comprehensive and detailed exposition of all the basic results concerning operator spaces. They should definitely be recommended to anyone who aims at being an operator space theorist. Then parts III to V contain a smart selection of some of the most remarkable advances of the theory. Despite the depth of some of them, the book is essentially self-contained and accessible to a large audience.''

The book is a standard reference in the field and has a very good reputation. During this seminar, we intend to cover part I and as much from part II as is feasible. This will prepare the participants for a futher study of operator spaces and their applications.

*Intended for*

Students, PhD students, and faculty.

*Prerequisites*

Proficiency in the functional analytic language at the level corresponding to a solid `pass' for the national functional analysis course in the Mastermath programme. An introductory course in functional analysis is *not* sufficient.

*Literature*

Edward Effros and Zhong-Jin Ruan: *Operator spaces*.

London Mathematical Society Monographs. New Series, 23. The Clarendon Press, Oxford University Press, New York, 2000.

ISBN: 0-19-853482-5.

The organisers have been informed that paper versions of this book appear to be harder to obtain than electronic ones.

*Venue*

Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building), Leiden. See here for directions.

The lectures are in room 174, with the exception of 21 February, when we shall convene in room 401.

*Dates (these may still change a little on basis of the participants' preferences) and time *

Friday afternoons, 14.00-17.00hr (at the latest), on:

14 February

21 February

28 February

6 March

13 March

27 March

3 April

17 April

24 April

1 May

8 May

15 May

29 May

*EC*

6 EC for participation and delivering an afternoon filling lecture.

*Grade*

It is enevitable that some topics are more suitable for an attractive presentation than others, so, as in previous years, there will be no grades but simply a `pass'.

*Please note*

If you are a student, but not from Leiden, contact your study advisor or exam committee beforehand about the eligibility of this seminar for your own programme, in order to prevent unwanted surprises. If your institute should require this, then, although this is not the preferred method, a grade could be supplied instead of a `pass'.

**Programme**

*Lecture 1:* 14 February: YeongChyuan Chung

This lecture was of a preparatory nature, to make sure that everyone has seen the basic material in Chapter 1 that will be used in subsequent chapters. Some of the material (polar decomposition, Schatten classes) was covered for general Hilbert spaces, rather than for the finite-dimensional ones that are predominant in Chapter 1 of the book. References for this are, e.g., J.R. Ringrose, *Compact non-self-adjoint operators*, (Van Nostrand Reinhold Co., London, 1971), and J. Diestel, H., Jarchow, and A. Tonge, * Absolutely summing operators*, (Cambridge University Press, Cambridge, 1995). Attention was also paid to tensor products. These are basic for the remainder of the book but, in spite of their elementary nature, they are not in the standard undergraduate curriculum for functional analysts.

Pictures of the blackboards are here.

*Lecture 2:* 21 February: Bram Mesland

Chapter 2.

*Lecture 3:* 28 February: Francesca Arici

First part of Chapter 3.

*Lecture 4:* 6 March: Onno van Gaans

Second part of Chapter 3.

*Lecture 5:* 13 March: Esmée Teeuwis

Chapter 4.

*Lecture 6:* 27 March: speaker TBA

First part of Chapter 5.

*Lecture 7:* 3 April: speaker TBA.

Second part of Chapter 5.

*Lecture 8:* 17 April: speaker TBA

Chapter 6.

*Lecture 9:* 24 April: speaker TBA

Chapter 7.

*Lecture 10:* 1 May: speaker TBA

Chapter 8.

*Lecture 11:* 8 May: speaker TBA

First part of Chapter 9.

*Lecture 12:* 15 May: speaker TBA

Second part of Chapter 9.

*Lecture 13:* 29 May: speaker TBA

Topic TBA (Chapter 10? Examples of applications?)