A construction of equivalence subrelations for intermediate subalgebras

Abstract

If M is a (separable) von Neumann algebra and A is a Cartan subalgebra of M, then M is determined by an equivalence relation and a 2-cocycle. By constructing an equivalence subrelation, we show that for any intermediate von Neumann subalgebra N between M and A, there exists a faithful normal conditional expectation from M onto N.