The Takesaki equivalence relation for maximal abelian subalgebras

For a maximal abelian subalgebra $A\subset M$ in a finite von Neumann algebra, we consider an invariant due to Takesaki which is an equivalence relation on a standard probability space. We give several characterization of this invariant and show that it can be reconstructed from the A-bimodule structure of the GNS Hilbert space $L^2(M)$. In particular, we show that this invariant is induced by the action of the normalizer on A. Hence, this gives a new proof to a question of Takesaki.

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