The order topology for a von Neumann algebra

Emmanuel Chetcuti ¹   ; Jan Hamhalter ²   ; Hans Weber ³

¹ Department of Mathematics Faculty of Science University of Malta Msida, Malta
² Faculty of Electrical Engineering Czech Technical University in Prague Technicka 2 166 27, Praha 6, Czech Republic
³ Dipartimento di matematica e informatica Università degli Studi di Udine 1-33100 Udine, Italy

Studia Mathematica, Tome 230 (2015) no. 2, pp. 95-120

Résumé

The order topology $\tau _o(P)$ (resp. the sequential order topology $\tau _{os}(P)$) on a poset $P$ is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra $M$ we consider the following three posets: the self-adjoint part $M_{sa}$, the self-adjoint part of the unit ball $M_{sa}^1$, and the projection lattice $P(M)$. We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology to the other standard locally convex topologies on $M$, and relate the properties of the order topology to the underlying operator-algebraic structure of $M$.

Détail

DOI : 10.4064/sm8041-1-2016

Mots-clés : order topology tau resp sequential order topology tau poset topology has its closed sets those contain order limits their order convergent nets resp sequences von neumann algebra consider following three posets self adjoint part self adjoint part unit ball projection lattice study order topology corresponding sequential variant these posets compare order topology other standard locally convex topologies relate properties order topology underlying operator algebraic structure