Symmetries and half-sided modular inclusions of von Neumann algebras
Let \({\mathcal N},{\mathcal M}\) be von Neumann algebras on a Hilbert space H. Let \(\Omega\) be a cyclic and separating vector for all \({\mathcal N},{\mathcal M}\), and \({\mathcal N}'\cap{\mathcal M}\). Denote by \(\Delta_{\mathcal M}\), \(\Delta_{\mathcal N},\) \(\Delta_{{\mathcal N}'\cap{\mathcal M}}\) the modular operators associated with \(({\mathcal M},\Omega)\), \(({\mathcal N},\Omega)\), resp. \(({\mathcal N}'\cap{\mathcal M},\Omega)\). Assume that \(\Delta_{\mathcal M}^{-it}{\mathcal N}\Delta_{\mathcal M}^{it}\subset{\mathcal N}\) for all \(t\geq 0\). Then \(\Delta_{\mathcal M}^{- it}\), \(\Delta_{{\mathcal M}_ 1}^{-ir}\), \(\Delta_{{\mathcal M}_ 1'\cap{\mathcal M}}^{-is}\), where \(t,r,s\in\mathbb{R}\), generate a representation of the group \(\text{Sl}(2,\mathbb{R})/Z_ 2\).
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