Tomita graph and standard modular data

Assume the Axiom of Choice. Let $H$ be a complex Hilbert space with inner product linear in its first argument, let $M\subseteq \mathcal{B}(H)$ be a concrete von Neumann algebra, and let $\mathrm{\Omega }$ be a unit vector cyclic and separating for $M$. Define $D(S_0)=M\mathrm{\Omega }$ and $S_0(x\mathrm{\Omega })=x^{\ast }\mathrm{\Omega }$ for $x\in M$. Let $S=\overline{S_0}$ and write its polar decomposition as $S=J{\mathrm{\Delta }}^{1\mathrm{/}2}$. The preceding Tomita-operator lemma ensures this closure and polar data are well defined, with $\mathrm{\Delta }$ positive self-adjoint and injective. For $x\in \mathcal{B}(H)$ and dimensionless $t\in \mathbb{R}$, define ${\sigma }_t(x):={\mathrm{\Delta }}^{it}x{\mathrm{\Delta }}^{-it}\mathrm{.}$ Set $\omega (x)=⟨x\mathrm{\Omega },\mathrm{\Omega }⟩$. The powers ${\mathrm{\Delta }}^{it}$ are the bounded unitary imaginary powers from the unbounded Borel functional calculus. At this definition stage ${\sigma }_t$ is defined on $\mathcal{B}(H)$; preservation of $M$ is established by a separate theorem.

Concrete von Neumann algebra and faithful vector

Involution and polar identities for the Tomita operator

Unbounded Borel functional calculus: domains, products, spectral mapping

The Axiom of Choice