The geometry of generalized quantum logics

Abstract

Let II be a quantum logic; by this we mean an orthocomplemented, orthomodular, partially ordered set. We assume that II carries a sufficiently large collection Δ of states (probability measures). Then, Δ is embedded as a base for the cone of a partially ordered normed space L and II is also embedded in the dual order-unit Banach space L*. We consider conditions on the pairs (Δ, II) and (L, L*) that guarantee that II is a dense subset of the extreme points of the positive part of the unit ball of L*. We demonstrate a connection of these conditions in noncommutative measure theory. The assumptions made here are far weaker than the assumptions of the traditional quantum mechanical formalisms and also apply to situations quite different from quantum mechanics. Finally, we show the connections of this theory to the well-known models of quantum mechanics and classical measure theory. © 1979 Plenum Publishing Corporation.