Spring 2017

* Organizers*

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

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

*Contact*

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

*Topic*

Many vector spaces in functional analysis have a natural partial order. Spaces of real valued functions, for example, always carry the natural pointwise order. The space of all real valued functions on a set is a vector lattice, i.e. any two elements have a least upper bound. This need not be the case, however, for an arbitrary space of real valued functions, as the example of the differentiable functions on [0,1] shows. The self-adjoint operators on a Hilbert space provide another example of a partially ordered vector space that -- degenerate cases aside -- is not a vector lattice. The theory of vector lattices and their operators is a world in itself, but there is also a rich theory for general partially ordered spaces and the operators between them. This theory, which -- in view of the ubiquity of such spaces -- is less well known than it deserves, is the topic of the upcoming issue of the seminar.

It is the objective of the seminar to give a detailed overview of the main results of the theory. We will naturally include the basic theory of partially ordered vector spaces and their operators `as such', including e.g. extension theorems, but we will also consider the context where there is an additional vector space topology present that is related to the partial order. In that context, the interaction between topology and order is worth understanding, but in addition there is also a well-developed theory of these ordered topological vector spaces and operators between them. For instance, we will see that for suitable topologies every positive operator (defined in the obvious way) is continuous and every continuous operator is the difference of two positive ones. In a similar vein, it is, for instance, possible to give characterizations of the locally convex topologies (or of the seminorms generating them) for which results such as the squeeze theorem from elementary analysis hold true.

Towards the end of the semester, some more specialized topics may be discussed as well, some of them of a recent nature, such as Dedekind completions and the (im)possibility of embedding a partially ordered space as a subspace of a vector lattice.

*Intended for*

Students, PhD students, and faculty.

*Prerequisites*

Proficiency in the functional analytic language beyond an introductory course; basic knowledge of topological vector spaces. The national functional analysis course in the Mastermath programme is an amply sufficient preparation.

*Literature*

Will provided by the organizers.

Here is a list of books, notes, and theses, meant to be reasonably complete at the time of writing, that contain material on general partially ordered vector spaces and their operators (there is much more on Riesz spaces and their operators), and on topologies as related to ordering (also for Riesz spaces):

Alfsen and Shultz, *State spaces of operator algebras. Basic theory, orientations and C*-products*, Birkhäuser, 2001

Aliprantis and Burkinshaw, *Locally solid Riesz spaces with applications to economics (2ed.)*, American Mathematical Society, 2003.

Aliprantis and Tourky, *Cones and duality*, American Mathematical Society, 2007.

Becker, *Ordered Banach spaces*, Hermann, 2008.

Cristescu, *Ordered vector spaces and linear operators*, Abacus Press, 1976.

Fremlin, *Topological Riesz spaces and measure theory*, Cambridge University Press, 1974.

van Gaans, *Seminorms on ordered vector spaces*, Thesis, Radboud University Nijmegen, 1999.

van Gaans and Kalauch, *Book in preparation*.

van Haandel, *Completions in Riesz space theory*, Thesis, Radboud University Nijmegen, 1993.

Husain and Khaleelulla, *Barrelledness in topological and ordered vector spaces*, Springer, 1978.

Jameson, *Ordered linear spaces*, Springer, 1970.

Kadelburg and Radenovic, *Subspaces and quotients of topological and ordered vector spaces*, Novi Sad, 1997.

Kantorovich and Akilov, *Functional analysis*, Pergamon Press, 1982.

Kelley and Namioka, *Linear topological spaces*., Springer, 1963.

Krasnoselskii, *Positive solutions of operator equations* (translated from the Russian), Noordhoff, 1964.

Nakano, *Modulared semi-ordered linear spaces*, Maruzen, Tokyo, 1950.

Namioka, *Partially ordered linear topological spaces*, American Mathematical Society, 1974.

Peressini, *Ordered topological vectors spaces*, Harper and Row, 1967.

Schaefer and Wolff, *Topological vector spaces (2ed.)*, Springer, 1999.

Vulikh, *Introduction to the theoy of partially ordered vector spaces*, Wolters-Noordhoff, 1967.

Wickstead, *Linear operators between partially ordered Banach spaces and some related topics*, Thesis, University of London, 1973.

Wong and Ng, *Partially ordered topological vector spaces*, Oxford University Press, 1973.

Some of the titles in the above list are certainly too specialized to be used for the seminar. Those who want to get an impression of the flavour of the theory are advised to browse through the books by Jameson, Namioka, or Peressini.

*Venue*

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

For the (varying) lecture room, see the detailed schedule below.

*Dates and time*

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

17 February

24 February

10 March

24 March

31 March

7 April

28 April

12 May

19 May

9 June

16 June

23 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. In past years, there have always been non-Leiden participants. 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:* 17 February 2017 (room 402): Hent van Imhoff

Basic theory of partially ordered vector spaces. I

Pictures of the blackboards are here.

*Lecture 2:* 24 February 2017 (room 405): Onno van Gaans

Basic theory of partially ordered vector spaces. II

Pictures of the blackboards are here.

*Lecture 3:* 10 March (room 402): Dusan Radicanin

Order and topology. I. Locally full topologies

Pictures of the blackboards are here.

*Lecture 4:* 24 March 2017 (room 405): Emiel Lorist

Order and topology. II. Locally solid topologies

Pictures of the blackboards are here.

*Lecture 5:* 31 March 2017 (room 402): Geerten Koers

Positive operators. I

Pictures of the blackboards are here.

*Lecture 6:* 7 April 2017 (room 408): Loek Veenendaal

Positive operators. II

Pictures of the blackboards are here.

*Lecture 7:* 28 April 2017 (room 402): Ronalda Benjamin

Geometric duality of cones

Pictures of the blackboards are here.

*Lecture 8:* 12 May 2017 (room 402): Sibel Kalkan

The Dedekind completion of an Archimedean Riesz space

Pictures of the blackboards are here.

*Lecture 9:* 19 May 2017 (room 402): Feng Zhang

The Riesz completion of an Archimedean partially ordered vector space

Pictures of the blackboards are here.

*Lecture 10:* 9 June 2017 (room 402): Xingni Jiang

Riesz homomorphisms and Riesz *-homomorphisms

Pictures of the blackboards are here.

*Lecture 11:* 16 June 2017 (room 402): Yang Deng

Ideals in pre-Riesz spaces

Pictures of the blackboards are here.

*Lecture 12:* 23 June 2017 (room 402): Nick Lindemulder

Geometry of cones

Pictures of the blackboards are here.