nLab
modular theory

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.

Definition

Let $\mathscr{H}$ be a Hilbert space, $ℳ$ a von Neumann-algebra with commutant $ℳ\prime$ 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, $J^{*}=J,J^2=𝟙$, $J$ commutes with ${\Delta }^{\mathrm{it}}$. This implies

   $$\mathrm{Ad}(J)\Delta ={\Delta }^{-1}$$Ad(J) \Delta = \Delta^{-1}

4. For every $A\in ℳ$ the vector $A\Omega$ is in the domain of ${\Delta }^{\frac{1}{2}}$ and

   $$J{\Delta }^{\frac{1}{2}}A\Omega =A^{*}\Omega =:\mathrm{SA}\Omega$$J \Delta^{\frac{1}{2}} A \Omega = A^* \Omega =: SA \Omega

5. The unitary group ${\Delta }^{\mathrm{it}}$ defines a group automorphism of $ℳ$:

   $$\mathrm{Ad}({\Delta }^{\mathrm{it}})ℳ=ℳ\text{for all}t\in ℝ$$Ad(\Delta^{it}) \mathcal{M} = \mathcal{M} \; \; \text{for all} \; t \in \mathbb{R}

6. $J$ maps $ℳ$ to $ℳ\prime$.

References

Many textbooks on operator algebras contain a chapter about modular theory.

MathOverflow question tomita-takesaki-versus-frobenuis-where-is-the-similarity

• Alain Connes, Carlo Rovelli, Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories, arXiv:gr-qc/9406019, pdf