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This page is about the modular theory introduced by Tomita for von Neumann-algebras. It is important both for the structure theory of von Neumann-algebras and in the Haag-Kastler approach to AQFT, one important example is the Bisognano-Wichmann theorem. It is often called Tomita-Takesaki theory, because the first presentation beyond a preprint is due to Masamichi Takesaki.
Let $\mathcal{H}$ be a Hilbert space, $\mathcal{M}$ a von Neumann-algebra with commutant $\mathcal{M}'$ and a separating and cyclic vector $\Omega$. Then there is a modular operator $\Delta$ and a modular conjugation $J$ such that:
$\Delta$ is self-adjoint, positive and invertible (but not bounded).
$\Delta\Omega = \Omega$ and $J\Omega = \Omega$
$J$ is antilinear, $J^* = J, J^2 = \mathbb{1}$, $J$ commutes with $\Delta^{it}$. This implies
For every $A \in \mathcal{M}$ the vector $A\Omega$ is in the domain of $\Delta^{\frac{1}{2}}$ and
The unitary group $\Delta^{it}$ defines a group automorphism of $\mathcal{M}$:
$J$ maps $\mathcal{M}$ to $\mathcal{M}'$.
An introduction into Tomita-Takesaki modular theory is here:
…while a paper that puts it to serious work is this:
Many textbooks on operator algebras contain a chapter about modular theory.
MathOverflow question tomita-takesaki-versus-frobenius-where-is-the-similarity
Discussion in terms of topos theory is in