# Contents

## Idea

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.

## nPOV definition

The modern approach to defining the modular automorphism group is through the theory of noncommutative L_p-spaces?. This was pioneered by Haagerup in 1979 Haagerup and Yamagami in 1992 Yamagami.

In this approach, given a von Neumann algebra $M$, a faithful semifinite normal weight $\mu$ on $M$, and an imaginary number $t$, the modular automorphism associated to $M$, $\mu$, and $t$ is

$\sigma_\mu^t\colon M\to M,\qquad m\mapsto \mu^t m \mu^{-t}.$

This approach makes it easy to deduce various properties of the modular automorphism group.

For more details, see a MathOverflow answer.

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:

1. $\Delta$ is self-adjoint, positive and invertible (but not bounded).

2. $\Delta\Omega = \Omega$ and $J\Omega = \Omega$

3. $J$ is antilinear, $J^* = J, J^2 = \mathbb{1}$, $J$ commutes with $\Delta^{it}$. This implies

$Ad(J) \Delta = \Delta^{-1}$
4. For every $A \in \mathcal{M}$ the vector $A\Omega$ is in the domain of $\Delta^{\frac{1}{2}}$ and

$J \Delta^{\frac{1}{2}} A \Omega = A^* \Omega =: SA \Omega$
5. The unitary group $\Delta^{it}$ defines a group automorphism of $\mathcal{M}$:

$Ad(\Delta^{it}) \mathcal{M} = \mathcal{M} \; \; \text{for all} \; t \in \mathbb{R}$
6. $J$ maps $\mathcal{M}$ to $\mathcal{M}'$.

## References

### General

An introduction into Tomita-Takesaki modular theory is here:

• Stephen J. Summers: “Tomita-Takesaki Modular Theory” (arXiv)

…while a paper that puts it to serious work is this:

• H.J. Borchers: “On Revolutionizing of Quantum Field Theory with Tomita’s Modular Theory”, (ESI preprint page)

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

• Simon Henry, From toposes to non-commutative geometry through the study of internal Hilbert spaces, 2014 (pdf)