Half-Sided modular inclusions of von-Neumann-Algebras

Abstract

LetN ⊂M be von-Neumann-Algebras on a Hilbert spaceH, Ω a common cyclic and separating vector. Denote Δ _M ,Δ _N resp. J _M ,J _N the associated modular operators and conjugations. Assume Δ ^-it _M ,Δ ^+it _N ⊂N fort≧0. We call such an inclusion half-sided modular. Then we prove the existence of a oneparameter unitary groupU(a) onH,a ∈R, with generator$\frac{1}{{2\pi }}(\ln \Delta _{\cal N} - \ln \Delta _M ) \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 0$ and relations

• 1 Δ ^it_ℳ U(a)Δ ^-it_ℳ =Δ ^it_ℳ = Δ ^it _N U(a) Δ ^it _N = U(e^-2πta) for alla, t ∈R,

• 2 J _M J _N =U(2),

• 3 Δ ^it _N = U (1) Δ ^it_ℳ U (- 1) for allt ∈R

• 4 N = U (1) ℳ U (- 1)

If ℳ is a factor and Ω is also cyclic forN ℳ, we show that ℳ has to be of typeIII ₁.