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The order topology for a von Neumann algebra

Volume 230 / 2015

Emmanuel Chetcuti, Jan Hamhalter, Hans Weber Studia Mathematica 230 (2015), 95-120 MSC: Primary 46L10; Secondary 06F30. DOI: 10.4064/sm8041-1-2016 Published online: 19 January 2016


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 $M$.


  • Emmanuel ChetcutiDepartment of Mathematics
    Faculty of Science
    University of Malta
    Msida, Malta
  • Jan HamhalterFaculty of Electrical Engineering
    Czech Technical University in Prague
    Technicka 2
    166 27, Praha 6, Czech Republic
  • Hans WeberDipartimento di matematica e informatica
    Università degli Studi di Udine
    1-33100 Udine, Italy

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