Symmetries and half-sided modular inclusions of von Neumann algebras

Abstract

LetN, ℳ be a von Neumann algebras on a Hilbert space ℋ, Ω a common cyclic and separating vector. Assume Ω to be cyclic and also separating forN′ ⋂ ℳ. Denote by Δ_ℳ, Δ _N , Δ _N′⋂ℳ the modular operators to (ℳ, Ω), (N, Ω), resp (N′ ⋂ ℳ, Ω). Assume now Δ ^-it_ℳ N Δ ^it_ℳ ⊂N for allt ⩾ 0. (Such type of inclusions ((N ⊂U, Ω) ⊂ ℳ, Ω) are called half-sided modular.) Then the modular groups Δ ^it_ℳ , Δ ^ir _N , Δ ^is _N′ ⋂ℳ,t, r, s ∈ ℝ generate a unitary representation of the group S1(2, ℝ)/Z ₂ of positive energy.

Another result is related to two half-sided modular inclusions (ℳ₁ ⊂ ℳ, Ω) and (ℳ₂ ⊂ ℳ, Ω). Under proper conditions the three modular groups Δ ^it_ℳ , Δℳ ^ir₁ , Δℳ ^is₂ ,t, r, s ∈ ℝ generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.