* Organisers*

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

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

*Contact*

If you are interested please contact Marcel de Jeu.

*Intended for*

Students, PhD students and staff.

*Topic*

Many spaces in functional analysis are not only Banach spaces, but they also have a natural partial ordering. For example, one can say for continuous functions that f is greater than or equal to g if this holds pointwise. For L_p-spaces one can require this to hold almost everywhere. With this ordering, every pair has a least upper bound and greatest lower bound in the space: these spaces are (vector) lattices. This combination of norm and ordering is characteristic for so-called Banach lattices. Many concrete real Banach spaces in functional analysis are, in fact, Banach lattices, but their ordered structure often gets less attention than it deserves. In this seminar we will cover some of the general theory for vector lattices and Banach lattices, but primarily we will look at concrete examples of Banach lattices and their (regular) operators. Literature will be supplied by the organisers.

*Prerequisites*

Some maturity in the functional analytic language. Prior knowledge of ordered structures is not assumed: we will build this up from the start. The national functional analysis course is more than enough preparation, and those with a liking for functional analysis could participate after having taken a first course in that direction.

*Venue*

Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building), Leiden. All meetings are in room 407 in the corridor on the first floor (Dutch count) of the lecture room wing of the building.

*Date and time*

Friday afternoons, 14.00-17.00hr at the latest.

*EC*

6 for attendance and delivering an afternoon filling lecture with break(s). These lectures are not public exams before an audience: the atmosphere in this seminar has been friendly in the past years and we will keep it that way. Think of it as a group of people who like functional analysis and who are learning a new subject together.

*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 about the eligibility of this seminar for your own programme beforehand, in order to prevent unwanted surprises. In past years, other institutes have always validated participation. If your institute should require this, then, although this is not the preferred method, a grade could be supplied instead of a "pass".

**Schedule**

*Lecture 1:* February 20, 2015: Onno van Gaans

General introduction to vector lattices and Banach lattices. See his list of topics and references.

*Lecture 2:* March 6, 2015: Dusan Radicanin

Description of lattice homomorphisms between C(K)-spaces (Thm 4.25 in Invitation to Operator Theory; see Liset Sloof's bachelor thesis for a somewhat more precise statement and proof) and related material. Also covered: Theorems 7.2 and 7.3 from B.A. Davey and H.A. Priestley "Introduction to lattices and order" (2ed.) on closure operators on general (ordered) sets.

*Lecture 3:* March 13, 2015: Xingni Jiang

C(K)-spaces: general theory on p.77-88 in De Jonge & Van Rooij, with an emphasis on Theorem 12.9 (description of the lattice of bands) and Theorem 12.16 (Dedekind completeness).

*Lecture 4:* March 20, 2015: Erwin van der Meer

L^p-spaces and super Dedekind completeness: Riesz Spaces I p.126(iv).

For a Banach lattice the norm and order dual coincide: Introduction to Operator Theory in Riesz Spaces Thm 25.8 (i) and (iii). Hence L^q is the order dual of L^p for finite p>1.

*Lecture 5:* March 27, 2015: Wouter Hetebrij

Banach function spaces and Orlicz spaces: p.167-179 in De Jonge & Van Rooij.

*Lecture 6:* April 10, 2015: Liset Sloof

Measures, the Radon-Nikodym theorem and bands: Chapter 14 in Introduction to Operator Theory in Riesz Spaces.

*Lecture 7:* May 1, 2015: Willem Schouten

The Yosida-Kakutani representation theorem, showing that Archimedean Riesz spaces with order unit are lattice-isomorphic with an order dense sublattice of a C(K)-space: p.89-97 in De Jonge & Van Rooij.

*Lecture 8:* May 8, 2015: David Kok

Abstract M- en L-spaces: p.138-145 (zonder 16.10) in De Jonge & Van Rooij; review of the Maeda-Ogasaware representation theorem as far as necessary; Proposition 1.2.13 in Meyer-Nieberg.

*Lecture 9:* May 22, 2015: Feng Zhang

Finite dimensional Perron-Frobenius theory: Theorem 8.26 in Invitation to Operator Theory, and what is necessary for this from Sections 8.1-8.5; mention of the infinite dimensional version (Theorem 44.9 in Introduction to operator theory on Riesz spaces); complefixication of a Banach lattice as on p.103-105 in Invitation to operator theory.

*Lecture 10:* May 29, 2015: Hent van Imhoff

Krein-Rutman theorem (Theorem 41.2 in Introduction to operator theory on Riesz spaces) and De Pagter's theorem on positive ideal irreducible compact operators (Theorem 43.4 in the same book).

*Lecture 11:* June 5, 2015: Josse van Dobben de Bruyn

Ordering in operator algebras