CHAIN RULES FOR CANONICAL STATE EXTENSIONS ON VON NEUMANN ALGEBRAS BY

In previous papers we introduced and studied the extension of a state defined on a von Neumann subalgebra to the whole of the von Neumann algebra with respect to a given state. This was done by using the standard form of von Neumann algebras. In the case of the existence of a norm one projection from the algebra to the subalgebra preserving the given state our construction is simply equivalent to taking the composition with the norm one projection. In this paper we study couples of von Neumann subalgebras in connection with the state extension. We establish some results on the ω-conditional expectation and give a necessary and sufficient condition for the chain rule of our state extension to be true. We extensively use the language of spatial derivatives developed by Connes [5] and summarized also in [2], [3]. Given two normal states ϕ and ω of a von Neumann algebra M ⊂ B(H) we set (1) [ϕ/ω] = ∆(ϕ, ω′)1/2∆(ω, ω′)−1/