The Takesaki equivalence relation for maximal abelian subalgebras

TL;DR

This paper studies Takesaki's invariant for maximal abelian subalgebras in finite von Neumann algebras, characterizing it via bimodule structures and normalizer actions, providing new insights and proofs.

Contribution

It offers new characterizations of Takesaki's invariant and shows it can be reconstructed from the A-bimodule structure and normalizer action.

Findings

Invariant is induced by the normalizer action on A

Provides new characterization of Takesaki's invariant

Reconstructs the invariant from bimodule structures

Abstract

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.