modular theory



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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 Ω\Omega. Then there is a modular operator Δ\Delta and a modular conjugation JJ such that:

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

  2. ΔΩ=Ω\Delta\Omega = \Omega and JΩ=Ω J\Omega = \Omega

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

    Ad(J)Δ=Δ 1 Ad(J) \Delta = \Delta^{-1}
  4. For every AA \in \mathcal{M} the vector AΩA\Omega is in the domain of Δ 12\Delta^{\frac{1}{2}} and

    JΔ 12AΩ=A *Ω=:SAΩ J \Delta^{\frac{1}{2}} A \Omega = A^* \Omega =: SA \Omega
  5. The unitary group Δ it\Delta^{it} defines a group automorphism of \mathcal{M}:

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


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)

Revised on September 11, 2017 11:10:42 by Urs Schreiber (