In most mathematical formulations of the foundations of quantum mechanics, the bounded observables of a physical system are identified with a real linear space, L, of bounded self-adjoint operators on a Hilbert space, H. Those bounded observables which correspond to the projections in L form a complete orthomodular lattice, P, otherwise known as the lattice of ”questions” or the quantum logic of the physical system. For x,y in \(B(H)_{sa}\), the physicist P. Jordan defined the Jordan product of x and y by \[ x\circ y=(xy+yx)=(x+y)^ 2+x^ 2-y^ 2. \] So it is reasonable to assume that L is a Jordan algebra of self- adjoint operators on H which is closed in the weak operator topology. Hence L is a JBW-algebra. Mackey’s Axiom VII, makes the much stronger assumption that \(L=B(H)_{sa}\). Mackey states that, unlike his other axioms, Axiom VII has no physical basis but is made for technical convenience. He goes on to say ”It would be interesting to have a thorough study of the consequences of modifying Axiom VII... ” One of the technical advantages of Axiom VII is that it allows the use of Gleason’s theorem to identify the completely additive probability measures on P with the normal states.
Recently, the results of Christensen together with Yeadon’s work on Type \(II_ 1\) algebras, has shown that Gleason’s theorem can be extended to any von Neumann algebra with no Type \(I_ 2\) direct summand.
The main purposes of this paper is to extend the Gleason-Christensen- Yeadon theorem from von Neumann algebras to JBW-algebras. This removes one of the mathematical difficulties arising from weakening Axiom VII to the physically plausible assumption that L is a JBW-algebra.
The main results of this paper may be summarized as follows:
Theorem. Let M be a JBW-algebra with P(M) its lattice of projections. Let M be either type \(I_ n\) for \(3\leq n<\infty\), or type \(I_{\infty}\), or without type \(I_ 2\) direct summand. Then any measure on P(M) is the restriction of a positive linear functional on M.
Corollary. Let M be a JBW-algebra with no type \(I_ 2\) direct summand. Then any countably additive measure on P(M) is the restriction of a positive linear functional on M.
We make essential use of the methods of Christensen, Yeadon, Gunson, and Aarnes. It turns out that for the type \(II_ 1\) case the methods of Yeadon can be generalized fairly easily. On the other hand, we need to surmount a number of technical obstacles before we can cope with the properly infinite JBW-algebras. Indeed, type \(I_ n\), for \(n\geq 3\) gives rise to some non-trivial difficulties.