# Positivity Webinar

# Welcome

Welcome the Positivity Webinar website. If you would like to receive email announcements on upcoming talks in the webinar, then please register your email address here.

Organizers:

Gerard Buskes

Miek Messerschmidt

Jan Harm van der Walt

# Past Talks

**Date:** 11 March 2021

**Speaker:** Tony Wickstead

**Recording & slides: **https://drive.google.com/drive/folders/1i49Qd2lZu_lOW73WplJI4UA8HLnzPP1i?usp=sharing

**Title: **The Size of the Centre of a Vector Lattice

**Abstract:**

I will survey the extreme sizes that the centre of a vector lattice can take and suggest some possible directions for related future research.

**

**Date:** 18 March 2021

**Speaker:** Semen Kutadeladze

**Recording & slides: **https://drive.google.com/drive/folders/1WG6ZfVUdQFRtBLMwGYCHq7J426quuR3h?usp=sharing

**Title: **What Is Boolean Valued Analysis?

**Abstract:**

This is a slightly informal invitation to Boolean valued analysis. We will discuss the three questions: "Why should we know anything at all about Boolean valued analysis? What need the working mathematician know this for? What do the Boolean valued models yield?"

**

**Date:** 25 March 2021

**Speaker:** Chris Schwanke

**Recording: **https://drive.google.com/drive/folders/1bTRmqhRUCkJ8rUg3UamRG8uTLLpjfMzZ?usp=sharing

**Title: **Characterizing bounded orthogonally additive polynomials

**Abstract:**

We provide several characterizations of bounded orthogonally additive polynomials on vector lattices in this talk.

**

**Date:** 1 April 2021

**Recording/slides: **https://drive.google.com/drive/folders/1gJdQ9ueQ6nzjFDmg5YOFXAHLcBE8DSLS?usp=sharing

**Speaker:** Mitchell Taylor

**Title: **The subspace structure of free Banach lattices

**Abstract:**

Given a Banach space E, one can associate a Banach lattice FBL[E], with the property that every bounded operator from E to a Banach lattice X extends uniquely to a lattice homomorphism from FBL[E] into X. We will discuss recent progress towards understanding the structure of FBL[E], and its relation to classical topics in Banach space theory.

**

**Date:** 15 April 2021

**Time:** 14h00 UTC (https://timezonewizard.com/)

**Recording/slides: **https://drive.google.com/drive/folders/1rTkqTnUWOpZO7yIPTKp5R5MCYlp3JvF7?usp=sharing

**Speaker:** Ioannis Polyrakis

**Title: **Positive bases and atomic sublattices

**Abstract:**

Suppose that the ordering in a vector lattice $E$ is defined by a countable family ${f_i|i\in\mathbf{N}\}$ of positive linear functional of $E$, i.e. $x\in E_+$ iff $f_i(x)\geq 0$, for any $i$. Based on the study of the supports of the vectors of E with respect to this family, we examine the existence of atoms in the positive cone of the lattice-subspaces $X$ of $E$ and we give sufficient conditions in order $X$ to be atomic or to have a positive basis. These results have applications in the theory of options (derivatives) in finance.

**

**Date:** 22 April 2021

**Time:** 14h00 UTC (https://timezonewizard.com/)

**Recording/Slides:** https://drive.google.com/drive/folders/1M9JWe_3JKNYj2hT8Wi47zcdObRgQjVqU?usp=sharing

**Speaker:** Niushan Gao

**Title: **Law invariance and Order

**Abstract: **Let $\mathcal{X}$ be an r.i.\ space over a non-atomic probability space. Let $\rho:\mathcal{X}\rightarrow(-\infty,\infty] $ be proper, convex, and increasing. It is known that order lower semicontinuity does not imply $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$ lower semicontinuity. However, if $\rho$ is additionally law invariant, then the implication does hold. This result indicates an interplay between law invariance and order. More surprisingly, we show that if $\rho$ is real-valued and law invariant, then both order and $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$ lower semicontinuity is automatic at every $X\in\mathcal{X}$ such that $X^-\in\mathcal{X}^a$.

The talk is based on some papers joint with S.Chen, C.Munari, D.Leung, L.Li, and F.Xanthos

**

**Date:** 29 April 2021

**Time:** 14h00 UTC (https://timezonewizard.com/)

**Recording: ****https://drive.google.com/drive/folders/10DajE2f8XPbWM557ZcJK66DEBIvs9yoo?usp=sharing**

**Speaker:**Christian Budde

**Title: **Positive Desch-Schappacher perturbations of bi-continuous semigroups on AM-spaces

**Abstract:**

We consider positive Desch–Schappacher perturbations of bi-continuous semigroups on AM-spaces with an additional property concerning the additional locally convex topology. As an examples, we discuss perturbations of the left-translation semigroup on the space of bounded continuous function on the real line and perturbations of the implemented semigroup on the space of bounded linear operators.

Dynamical processes occurring, e.g., in population models, quantum mechanics, or the financial world, are frequently expressed by a particular class of partial differential equations, the so-called evolution equations. A general operator theoretical method for dealing with those equations is the one using abstract Cauchy problems on a Banach space. In some cases, it is possible to write a given operator (A,D(A)) as a sum of simpler operators and this is where perturbation theory enters the area of evolution equations. The general question is: given a generator (A,D(A)) and another linear operator (B,D(B)), under which conditions does the operator A+B generate a semigroup?

When talking about one-parameter semigroups of linear operators on Banach spaces, mostly C0-semigroups come to mind. Nevertheless, there are operator semigroups which are not strongly continuous with respect to the norm on the Banach space but for some weaker additional locally convex topology. This is one of the reasons why people are interested in different continuity concepts of semigroups and more general solutions in order to overcome these limitations of strongly continuous semigroups. One of the auspicious approaches to this gives rise to so-called bi-continuous semigroups, which were introduced by Kühnemund.

**

**Date:** 13 May 2021

**Recording : **https://drive.google.com/drive/folders/1DRRJ6FGDBrffEFO9WkcDecriCPZU3E_p?usp=sharing

**Speaker:** Mary Angelica Tursi

**Title: **Displaying Polish group as lattice automorphism group over $C_0(X)$ spaces

**Abstract:**

A Polish group $G$ is displayable on a Banach lattice $E$ if there exists an equivalent lattice renorming $\|| \cdot \||$ on $E$ so that $G$ is isomorphic to the lattice automorphism group on $(E, \|| \cdot \|| )$. For what cases is $G$ displayable in $E$? We examine the question for proper subgroups of lattice automorphism groups over $C_0(X)$, where $X$ is a locally compact Polish space.

**

**Date:** 20 May 2021

**Speaker:** Ray Ryan

**Recording: **https://drive.google.com/drive/folders/1p30fUHQavlT3xQyGhLfF_QAEJjCnQ8ya?usp=sharing

**Title: **Polynomials on Banach Lattices

**Abstract: **Homogeneous polynomials are vital in the study of analytic functions on Banach spaces, as they are the components of the Taylor series that represent the functions locally. As most of the classical Banach spaces are Banach lattices, it is natural to work with polynomials that are coherent with the lattice structure. Thus, we study regular homogeneous polynomials, as they have a modulus that is a positive homogeneous polynomial. We demonstrate some applications, to analysis on Banach spaces with an unconditional basis, and to the radius of analyticity in a real Banach space. We also look at an application to the computation of the radius of convergence of power series on Banach lattices. We then consider the class of orthogonally additive polynomials, particularly on spaces of continuous functions, where some interesting geometric phenomena are seen.

**

**Date:** 27 May 2021

**Recording:** https://drive.google.com/drive/folders/1nbbtSI5WDMbGhMAYs5jHIW1rPivUL3mI?usp=sharing

**Speaker:** Anil Kumar Karn

**Title: **Adjoining an order unit to a normed linear space

**Abstract: **Using a technique of adjoining an order unit to a normed linear space, we have characterized strictly convex spaces among normed linear spaces and Hilbert spaces among strictly convex Banach spaces respectively. This leads to a generalization of spin factors and provides a new class of absolute order unit spaces which is denoted as tracial absolute order unit spaces. In addition, we have obtained their functional representation and studied unital absolute value preserving maps on these spaces.

**

**Date:** 3 June 2021

**Recording:** https://drive.google.com/drive/folders/1WNlw0IyfdjvtVgr6cCsmj_9bPKNwuFgH?usp=sharing

**Speaker:** Yang Deng

**Title: ** Bicommutant theorems in vector lattices and Banach lattices

**Abstract: **Von Neumann's bicommutant theorem states that the bicommutant of a unital involutive subalgebra of the bounded operators on a Hilbert space is precisely its closure in the strong operator topology. We present several analogues of this in the context of vector lattices and Banach lattices. This is joint work with Björn de Rijk and Marcel de Jeu.