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Ordered Banach Algebras
Scientific background and motivation
The fields of Banach algebras and Positivity have developed largely independently of each other. Still, there are many instances where these two disciplines in Functional Analysis meet in the common object of an ordered Banach algebra, often an algebra of operators on an ordered Banach space. One could even argue that such ordered Banach algebras of operators occur at least as naturally in mathematics and its applications as the much better studied algebras of operators on a Hilbert space. The notion of "positivity" is a quite common one in the sciences! Yet a systematic study of such algebras of operators, and of ordered Banach algebras in general, is essentially in its infancy.
The aim of this workshop was to further this study by bringing together experts from both the Banach algebra community and the Positivity community, and a number of researchers who have already pioneered this common ground.
The workshop had only six lectures on Monday through Wednesday, with the explicit request to the audience to interrupt the lecturers for question and discussion whenever they felt this was necessary. This worked very well: the moderators for the sessions, with the task of starting the discussions, were not necessary. The lecturers had ample time to present their material, which was of a non-technical nature. The emphasis was on the overview, the ideas and -- of course -- on the open questions, of which there was no shortage.
On Thursday and Friday there was time for what had already started naturally on the previous days, namely working on themes in groups. Three themes were selected, and for each of these a mixed group (both in discipline and in seniority) had the task to identify the leading questions for that theme and, if possible, sketch an approach that might yield answers. On Friday afternoon these groups gave a plenary presentation, and a short written account of each of these presentations was sent to all participants after the workshop.
Prior to the workshop, a more or less complete bibliography for ordered Banach algebras had been prepared by the organisers. The abstracts of the papers in the bibliography were (and still are) available for each participant in a reader.
All participants were enthusiastic about the topic of the workshop and its format. The idea of bird's-eye-view lectures and a loose programme in the informal environment of the Snellius venue was a definite success: it really led to a hands-on workshop rather than to a mini-conference as many workshops are. Ideas for future collaboration have already grown and the written accounts of the plenary presentations contain enough questions for years of research. As organisers we therefore think that the workshop, with participants from 15 countries on 4 continents, has met its goal. If all goes well with the applications, there will be a sequel in two years' time: one of the participants has already agreed to organise this. This can then be held using the ordered Banach algebra logo, designed by Tony Wickstead for the field on the occasion of this workshop and already used on its poster.
Apart from the funding by the Lorentz
Center itself and the availability of the facilities, this workshop was made
possible by the support of (in alphabetical order) the Delft Institute of
Applied Mathematics (DIAM), Foundation Compositio Mathematica, the Mathematical Institute of
Leiden University, the national Dutch research cluster Nonlinear Dynamics of
Natural Sciences (NDNS+), and the Netherlands Organisation for Scientific
Research (NWO). Their generous support is gratefully acknowledged.
Garth Dales (Lancaster, United Kingdom)
Marcel de Jeu (Leiden, Netherlands)
Sonja Mouton (Stellenbosch, South Africa)
Ben de Pagter (Delft, Netherlands)
Tony Wickstead (Belfast, United Kingdom)