Tomita–Takesaki theory

Summary

Tomita–Takesaki theory starts from a von Neumann algebra with a cyclic and separating vector. It attaches to it a positive self-adjoint modular operator $\Delta$ and an antiunitary modular conjugation $J$. Its main theorem says that $J$ turns the algebra into its commutant. It also says that the unitaries $\Delta^{it}$ give a one-parameter group of automorphisms, the modular automorphism group. So every faithful normal state comes with a natural time evolution. The theory is the basis of Connes' classification of type $\mathrm{III}$ factors and of the use of KMS states in quantum statistical mechanics.