Type III factors (Araki-Woods 1963): The von Neumann algebra of a free Bose field in the Fock vacuum is a Type III₁ factor — no trace, no minimal projections, spectrum of modular operator Δ is ℝ₊.
Tomita-Takesaki (1970): Every cyclic separating vector Ω defines a modular operator Δ and antilinear involution J. Modular flow: σ_t(A) = Δ^{it}AΔ^{-it}. The algebra is its own commutant up to J.
KMS condition: ω(Aσ_{iβ}(B)) = ω(BA) — the state ω is thermal w.r.t. modular flow with period β.
Unruh effect (1976): An accelerating observer with a = const sees the Minkowski vacuum as a thermal KMS state with β = 2π/a (in natural units). The Rindler wedge algebra IS a Type III₁ factor.