Martingal convergence theorems on JW-algebras

The concept of almost everywhere convergence and its different variants to sequesnces in von Neumann algebra were studied by many authors. There were proved many limit and ergodic theorems with respect to almost everywhere convergence in such algebras with faithful normal state. On the other hand, in most mathematical formulations of the foundations of quantum mechanics, the bounded observables of a physical system are identified with real linear space E, of bounded selft-adjoint operators on Hilbert space H. Those bounded obervables which correspond to the projectios L from a complete orthomodular lattice P, ohterwise knowns as the lattice of the quantum logic of thephysical system. For the self adjoint operators L and y on H their Jordanprduct is difined by x a y = (xy+ yx/2 = ((x+y)2 -x2-y2)/2. So it is resonable to assume that L is JW-algebra ie. Jordan algebra of self adjoin operators on H which is closed in the weak operator topology. The main purposes of this report is to prove martingale convergence theorems in JW algebra setting. Let A be a JW-algebra with finite normal faithful trace L. LEt A be its JW-subalgebra with the unite 1. A positive unital linear mapping M(/A1): A-A1is called the conditionalexpectation wiht respect of A1 if n(xy)=r(Mx/A1)y for all x EAx y E B Let {An} be an increasing sequence of JW-subalgebras of A with conditional expectations