Half-sided modular inclusions of von-Neumann-algebras

Hans-Werner Wiesbrock
November 26, 1993
Abstract. Let ${\cal N}\subset{\cal M}$ be von-Neumann-Algebras on a Hubert space $\cal H$, $\Omega$ a common cyclic and separating vector. Denote $\Delta_{\cal M}$, $\Delta_{\cal N}$ resp. $J_{\cal M}$, $J_{\cal N}$ the associated modular operators and conjugations. Assume $\Delta^{it}_{\cal M}{\cal N}\Delta^{-it}_{\cal M}\subset{\cal N}$ for $t\geq 0$. We call such an inclusion half-sided modular. Then we prove the existence of a one-parameter unitary group $U(a)$ on $\cal H$, $a\in\mathbb R$, with generator $\frac{1}{2\pi}(\ln\Delta_{\cal N}-ln\Delta_{\cal M})\geq 0$ and relations 1.) $\Delta^{it}_{\cal M}U(a)\Delta^{-it}_{\cal M}=\Delta^{it}_{\cal N}U(a)\Delta^{-it}_{\cal N}=U(e^{-2\pi t}a)$ for all $a,t\in\mathbb R$, 2.) $J_{\cal N}J_{\cal M}=U(2)$, 3.) $\Delta^{it}_{\cal N}=U(1)\Delta^{it}_{\cal M}U(-1)$ for all $t\in\mathbb R$, 4.) ${\cal N}=U(1){\cal M}U(-1)$. If $\cal M$ is a factor and $\Omega$ is also cyclic for ${\cal N}'\cap\cal M$, we show that $\cal M$ has to be of type $III_1$.

half-sided modular inclusions, Borchers triples