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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.
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Hisashi Aoi (2003) studied this question.
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