Let $M$ be a von Neumann algebra acting standardly on a [Hilbert space](/page/Hilbert%20Space) $H$, and let $\Omega\in H$ be cyclic and separating for $M$. Let $S$, $J$, and $\Delta$ be the Tomita operator, modular conjugation, and modular operator associated to $(M,\Omega)$. Then $JMJ=M'$. Also, \begin{align*} \Delta^{it}M\Delta^{-it}=M \quad \text{for all } t\in\mathbb R. \end{align*} The maps $\sigma_t(x)=\Delta^{it}x\Delta^{-it}$ define a one-parameter group of $*$-automorphisms of $M$.