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 spaceL and II is also embedded in the dual order-unit Banach spaceL *. 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 ofL *. 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.
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Cook, T.A. The geometry of generalized quantum logics. Int J Theor Phys 17, 941–955 (1978). https://doi.org/10.1007/BF00678422
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DOI: https://doi.org/10.1007/BF00678422