An indefinite measure in J-space (Q2639303)

Let H be a space with an indefinite metric [.,.] and a canonical symmetry J, i.e. H is a J-space. Then H is a Hilbert space with respect to the inner product \((x,y)=[Jx,y]\). And let \({\mathcal P}\) be the quantum logic of all J-projections in a J-selfadjoint weakly closed unital operator algebra \({\mathfrak A}\) in H. If \(J\in {\mathfrak A}\) then \({\mathfrak A}\) is called \(W^*J\)-algebra. The main result of the paper is an analogue of Gleason's theorem for measures on the logic of J-projections in an approximately finite dimensional \(W^*J\)-factor \({\mathfrak A}\).