Star Order and Topologies on von Neumann Algebras

Bohata, Martin

doi:10.1007/s00009-018-1221-x

Cited by 8 publications

(8 citation statements)

References 22 publications

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“…This follows from their close connection with usual locally convex topologies on M. For example, it was proved in [6] that the order topology τ o (M sa , ≤) is finer than the restriction of Mackey topology to M sa . Furthermore, the order topology on M with respect to the star order is finer than σ-strong* topology [5]. In this paper, we prove that τ o (M sa , ) is finer than τ o (M sa , ≤) and so it is finer than the restriction of Mackey topology to M sa .…”

Section: Introductionmentioning

confidence: 83%

Vigier's theorem for the spectral order and its applications

Bohata

2019

Journal of Mathematical Analysis and Applications

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The paper mainly deals with suprema and infima of self-adjoint operators in a von Neumann algebra M with respect to the spectral order. Let M sa be the selfadjoint part of M and let be the spectral order on M sa . We show that a decreasing net in (M sa , ) with a lower bound has the infimum equal to the strong operator limit. The similar statement is proved for an increasing net bounded above in (M sa , ). This version of Vigier's theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self-adjoint operators in terms of the strong operator limit and operator means. As an application of our results on suprema and infima, we study the order topology on M sa with respect to the spectral order. We show that it is finer than the restriction of the Mackey topology.

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Section: Introductionmentioning

confidence: 83%

Vigier's theorem for the spectral order and its applications

Bohata

2019

Journal of Mathematical Analysis and Applications

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“…However, we shall not follow this path. Our work continues on the work in [5,8,7,6,4] in which the order topology associated to a partially ordered structure arising naturally in the realm of Functional Analysis is studied and related to the functional-analytic properties of the structure.…”

Section: Introductionmentioning

confidence: 92%

On different modes of order convergence and some applications

Abela¹,

Chetcuti²,

Weber³

2020

Preprint

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Different notions for order convergence have been considered by various authors. Associated to every notion of order convergence corresponds a topology, defined by taking as the closed sets those subsets of the poset satisfying that no net in them order converges to a point that is outside of the set. We shall give a thorough overview of these different notions and provide a systematic comparison of the associated topologies. Then, in the last section we shall give an application of this study by giving a result on von Neumann algebras complementing the study started in [8]. We show that for every atomic von Neumann algebra (not necessarily σ-finite) the restriction of the order topology to bounded parts of M coincides with the restriction of the σ-strong topology s(M, M * ). We recall that the methods of [8] rest heavily on the assumption of σ-finiteness. Further to this, for a semi-finite measure space, we shall give a complete picture of the relations between the topologies on L ∞ associated with the duality L 1 , L ∞ and its order topology.

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“…Finite von Neumann algebras were also characterized by the order topological properties of their projection lattice. The study of the order topology induced by the star order on operator algebras was initiated in a very recent interesting work [3].…”

Section: Example 2 (I)mentioning

confidence: 99%

“…The pre-Hilbert space logic (S, L ) is said to be symmetric if U L = {U A : A ∈ L } ⊆ L for every U ∈ U(S) 3. We recall that an orthocomplemented poset (P, ≤, 0, 1) is atomic σ-complete if whenever (an) is a sequence of pairwise orthogonal atoms in P the supremum n an exists.…”

mentioning

confidence: 99%

Order topology on orthocomplemented posets of linear subspaces of a pre-Hilbert space

Buhagiar¹,

Chetcuti²,

Weber³

2018

Preprint

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Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair (S, L ), where S is a pre-Hilbert space and L is an orthocomplemented poset of orthogonally closed linear subspaces of S, closed w.r.t. finite dimensional perturbations, (i.e. if M ∈ L and F is a finite dimensional linear subspace of S, then M + F ∈ L ). We study the order topology τo(L ) on L and show that completeness of S can by characterized by the separation properties of the topological space (L , τo(L )). It will be seen that the remarkable lack of a proper probability-theory on pre-Hilbert space logics -for an incomplete S -comes out elementarily from this topological characterization.

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Star Order and Topologies on von Neumann Algebras

Cited by 8 publications

References 22 publications

Vigier's theorem for the spectral order and its applications

Vigier's theorem for the spectral order and its applications

On different modes of order convergence and some applications

Order topology on orthocomplemented posets of linear subspaces of a pre-Hilbert space

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