Characterization of projection lattices of Hilbert spaces
type: Item
instance of: scholarly article
title: Characterization of projection lattices of Hilbert spaces
zbMATH Open document ID: 0618.03033
DOI: 10.1007/BF00668822
author: Horst Szambien
published in: International Journal of Theoretical Physics
publication date: 1986-01-01T00:00:00Z
review text: This paper uses methods of topological geometry to treat the problem of lattice coordinatization which is presented in Ch. 21.3 of \textit{E. B. Beltrametti} and \textit{G. Cassinelli} [The logic of quantum mechanics (1981; Zbl 0504.03026)]. Let \({\mathcal L}\) be an irreducible, complete, orthomodular, atomic lattice of length \(\geq 4\) enjoying the covering property. The following properties (1) and (2) of \({\mathcal L}\) are shown to be equivalent: (1) \({\mathcal L}\) is isomorphic to the lattice of projections of some Hilbert space over \({\mathbb{R}}\), \({\mathbb{C}}\), or \({\mathbb{H}}\); (2) \({\mathcal L}\) carries a topology such that the set of atoms is connected, and ideals of finite height are compact paratopological sublattices. (Here a lattice with top 1 and bottom 0 carrying a topology is called paratopological iff joining of \(meet=0\) and intersecting of \(join=1\) pairs are continuous operations with open domains [cf. \textit{H. Szambien}, J. Geom. 26, 163-171 (1986; Zbl 0598.51013)]. If a lattice fulfills (2) w. r. to the topology of states then the isomorphism in (1) becomes a homeomorphism.
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 06B30
zbMATH DE Number: 4002094
zbMATH Keywords: lattice coordinatization
zbMATH Keywords: lattice of projections
zbMATH Keywords: topology of states
MaRDI profile type: Publication
cites work: Q3952094
cites work: Q3290626
cites work: Q4405003
cites work: Sur les espaces projectifs topologiques
cites work: Q3667432
cites work: Quantum Mechanics and Hilbert Space
cites work: Q5623714
cites work: Topologische projektive Räume
cites work: Q3227201
cites work: Q5516132
cites work: Q4100556
cites work: Über den Zusammenhang in topologischen projektiven Ebenen
cites work: Zum Begriff des topologischen projektiven Raumes
cites work: Q5537293
cites work: On Characterizing the Standard Quantum Logics
cites work: Axioms for non-relativistic quantum mechanics
cites work: On the lattice of closed subspaces of Hilbert spaces
description: scientific article; zbMATH DE number 4002094