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 $N$.