21/01/2016, 15:00 — 16:00 — Room P3.10, Mathematics Building
Marcel de Jeu, Leiden University
Positive representations
Many spaces in analysis are ordered (real) Banach spaces, or even Banach lattices, with groups acting as positive operators on them. One can even argue that such positive representations of groups are not less natural than unitary representations in Hilbert spaces, but contrary to the latter they have hardly been studied. The same holds for representations of ordered Banach algebras such that a positive element acts as a positive operator. Whereas there is an elaborate theory of $^\ast$-representations of $C^*$-algebras, hardly anything is known about positive representations of ordered Banach algebras, even though such representations are not rare at all.
We will sketch the gradually emerging field of “positive representations”, and mention some of the main problems (of which there are many) and results (of which there are still too few), jointly obtained with Ben de Pagter, Björn de Rijk, Sjoerd Dirksen, Xingni Jiang, Miek Messerschmidt, Dusan Radicanin, Mark Roelands, Jan Rozendaal, Frejanne Ruoff, and Marten Wortel.
The talk is meant as an advertisement for the topic and, more generally, for studying groups and Banach (lattice) algebras of operators on Banach lattices. The step from single operator theory on Hilbert spaces to groups and algebras of operators was taken in the first half of the 20th century, and now the field of Positivity could be ripe for a similar development.
Note the room change!
