Stochastic coupling for von Neumann algebras

We consider even and odd stochastic transitions of von Neumann algebras when dual mappings intertwine (couple) modular groups of the corresponding states (with the occurrence of a sign exchange for the odd case). We show that one can define modular objects and cones associated to linear combinations of von Neumann algebras, which generalize objects and cones in the standard modular theory. In the odd case, we find sufficient conditions for the intertwining property and consider several applications to noncommutative Markov processes. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 760–774, May, 1999.