Functional Analysis Seminar
Spring 2022

JB- and JBW-algebras


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 at the ground floor in room 312.

The meetings are on the following Friday afternoons in the spring of 2022, starting at 14.15hr and possibly lasting until 17.00:
18 February
25 February
4 March
11 March
18 March
25 March
8 April
22 April
29 April
6 May
13 May
20 May
3 June

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: 18 February: Mark Roelands or Marten Wortel

Lecture 2: 25 February: Marten Wortel or Mark Roelands

Lecture 3: 4 March: TBA

Lecture 4: 11 March: TBA

Lecture 5: 18 March: TBA

Lecture 6: 25 March: TBA

Lecture 7: 8 April: TBA

Lecture 8: 22 April: TBA

Lecture 9: 29 April: TBA

Lecture 10: 6 May: TBA

Lecture 11: 13 May: TBA

Lecture 12: 20 May: TBA

Lecture 13: 3 June: TBA