Characterization of projection lattices of Hilbert spaces
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.
scientific article; zbMATH DE number 4002094
scientific article