On blocks in quantum logics
type: Item
instance of: scholarly article
title: On blocks in quantum logics
zbMATH Open document ID: 0607.03018
DOI: 10.1016/0034-4877(85)90010-2
published in: Reports on Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: Let L be a quantum logic, \(\Omega\) (L) the convex set of states on L and M a property, i.e. a convex subset of \(\Omega\) (L). For any \(P\subseteq L\) we define \(A_ M(P)=\{p\in L| \mu,\nu \in M\) and \(\mu /P=\nu /P\Rightarrow \mu (p)=\nu (p)\}\). The subset \(A_ M(P)\subseteq L\) is orthomodular and \(A_ M\) is a closure operator on the subsets of L. We call \(P\subseteq L\) M-dense, provided \(A_ M(P)=L.\) We show that a non-classical quantum logic satisfying the chain condition and having a full and unital property M has no block which is M-dense. We also prove that a quantum logic with a property M for which every counter is expectational and no block is M-dense necessarily has uncountably many blocks. In this setting we then discuss projection lattices of von Neumann algebras.
Mathematics Subject Classification ID: 03G12
zbMATH DE Number: 3981165
zbMATH Keywords: chain condition
zbMATH Keywords: projection lattices of von Neumann algebras
author: David Cohen
author: Gottfried T. Rüttimann
MaRDI profile type: Publication
full work available at URL: 0034-4877(85)90010-2
OpenAlex ID: W2066124510
cites work: Q3952094
cites work: Q5565478
cites work: Operational Statistics. I. Basic Concepts
cites work: On the structure of orthomodular lattices satisfying the chain condition
cites work: Q4091259
cites work: Q5569793
cites work: Jordan-Hahn decomposition of signed weights on finite orthogonality spaces
cites work: Q3929738
cites work: Expectation functionals of observables and counters
cites work: Q5624220
cites work: Q5620935
description: scientific article; zbMATH DE number 3981165