Spring 2022

* Organisers*

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

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

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

* Mathematical guides*

Mark Roelands (m.roelands@math.leidenuniv.nl)

Marten Wortel (marten.wortel@up.ac.za)

*Contact*

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

*Covid: go/no go decision on 1 February at the latest*

This type of seminar is not suitable for an online or hybrid version. On 1 February 2022 at the latest, the organisers will decide whether or not to start up this on campus seminar, depending on the perspectives at that moment.

*Topic*

In quantum mechanics, observables are self-adjoint operators on a complex Hilbert space. The set of bounded observables forms a real partially ordered Banach space and a commutative but non-associative algebra under the Jordan product (AB+BA)/2. Such a structure is called a JB-algebra, and the typical example of a JB-algebra is the set of self-adjoint elements of a C*-algebra. In this seminar we will investigate the general theory of JB-algebras. After the general theory of JB-algebras, we will focus on JBW-algebras, which are the Jordan analogue of von Neumann algebras: the typical example of a JBW-algebra is the set of self-adjoint elements of a von Neumann algebra. It is possible to derive some of the results from the theory of C*-algebras and von Neumann algebras, but the literature that we will use builds up the theory solely using the order and Jordan algebra structure.

*Intended for*

Advanced MSc students, PhD students, postdoscs, 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*

Erik M. Alfsen and Frederic W. Shultz: `Geometry of State Spaces of Operator Algebras'. Birkhäuser Boston, Inc., Boston, MA, 2003. ISBN: 0-8176-4319-2.

We intend to cover most of the Chapters 1, 2, and 3, and then select additional material from this book.

Please note: do not confuse this with `State Spaces of Operator Algebras. Basic Theory, Orientations, and C*-products' by the same authors.

*Background literature*

For those who want to read more on the subject: the book `Jordan Operator Algebras' by Hanche-Olsen and Størmer, which is another standard reference in the field, can be dowloaded freely. Chapter 0 in McCrimmon's `A Taste of Jordan Algebras' gives a nicely written overview of the links between Jordan algebras and various parts of mathematics.

*Venue*

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

The lectures are all at the ground floor in room 312, with one exception: the meeting on 3 June is in room 412, one floor up.

*The meetings are from 14.15-16.00 on the following Friday afternoons:*

25 February

4 March

11 March

18 March

25 March

22 April

29 April

13 May

20 May

3 June

10 June

6 EC for participation and delivering a 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:* 25 February: Marten Wortel (University of Pretoria)

Introduction to JB-algebras; inverses; p.3-14.

Pictures of the blackboards are here.

*Lecture 2:* 4 March: Marten Wortel and Mark Roelands (Leiden University)

Orthogonality; quotients; p.15-22.

Pictures of the blackboards are here.

*Lecture 3:* 11 March: Sven van Dijk (Leiden University)

Projections; compressions; operator commutativity; p.22-30.

Pictures of the blackboards are here.

*Lecture 4:* 18 March: Qingchong Zhu (Leiden University)

Introduction to JBW-algebras; orthogonality; range projections; spectral resolution; p.37-48.

Pictures of the blackboards are here.

*Lecture 5:* 25 March: Cyriel van Velzen (University of Amsterdam)

Lattice of projections; compressions; hereditary subalgebras; center; p.48-58.

Pictures of the blackboards are here.

*Lecture 6:* 22 April: Jesse Reimann (Delft University of Technology)

Bidual; predual; p.58-69.

Background notes are here. Pictures of the blackboards are here.

*Lecture 7:* 29 April: Onno van Gaans (Leiden University)

JW-algebras; equivalence of projections; comparison theory; p.70-72 and p.79-84.

Pictures of the blackboards are here.

*Lecture 8:* 13 May: Marcel de Jeu (Leiden University)

Type I JBW-algebras; p.85-95.

Pictures of the blackboards are here.

*Lecture 9:* 20 May: Onno van Gaans (Leiden University)

Atomic JBW-algebras (p.96-100) and own research.

Pictures of the blackboards are here.

*Lecture 10:* 3 June: Marten Wortel (University of Pretoria) and/or Mark Roelands (Leiden University)

JB- and JBW-algebras in research.

Pictures of the blackboards are here.

*Lecture 11:* 10 June: Mark Roelands (Leiden University) and/or Marten Wortel (University of Pretoria)

JB- and JBW-algebras in research.