On blocks in quantum logics
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.
projection lattices of von Neumann algebras
scientific article; zbMATH DE number 3981165