Let $A$ be a von Neumann algebra acting on a Hilbert space $H$, and suppose that $\Omega \in H$ is a cyclic and separating vector for $A$. Then in Tomita-Takesaki theory one defines an unbounded operator $S$ to be the closure of the operator \begin{equation} a \triangleright \Omega \mapsto a^{*} \triangleright \Omega. \end{equation} It then turns out that if we consider the polar decomposition \begin{equation} S = J \Delta^{1/2} \end{equation} into an anti-linear unitary involution $J$ and an unbounded positive operator $\Delta^{1/2}$ it follows that the map $A \ni a \mapsto Ja^{*}J \in (A')^{\operatorname{opp}}$ is an isomorphism. (Here the superscript opp stands for opposite algebra.)
Now, let $A$ be any von Neumann algebra, then a standard form for $A$ is a quadruple $(A, H, J, P)$, where $H$ is a Hilbert space in which $A$ acts, $J$ is an antilinear operator in $H$, and $P$ is a self-dual cone, which satisfy the following properties.
- $JAJ = A'$,
- $JaJ = a^{*}$, if $a \in A \cap A'$,
- $Jv = v$ for all $v \in P$,
- $aJaJP \subseteq P$ for all $a \in A$.
Theorem: [Haagerup] If $(A,H',J',P')$ is a second standard form for $A$, then there exists a unique unitary $u :H \rightarrow H'$ such that
- $uau^{*} = a$ for all $a \in A$,
- $J' = uJu^{*}$,
- $P' = uP$.
It turns out that if $\Omega \in H$ is cyclic and separating as above, then the quadruple $(A,H,J, P_{\Omega})$, with $P_{\Omega} = \{ aJaJ\Omega \mid a \in A \}$ is a standard form for $A$.
Does there exist an adaptation for the $\mathbb{Z}_{2}$-graded case?
What I mean by this is as follows. Suppose that $A$ is a $\mathbb{Z}_{2}$-graded von Neumann algebra acting on a $\mathbb{Z}_{2}$ graded Hilbert space $H$, which is equipped with a cyclic and separating vector $\Omega \in H$. (Suppose that $\Omega$ is even.) Furthermore, assume that the representation of $A$ on $H$ respects the grading. Define an involution on $A$ by $a^{\sharp} = a^{*}$ if $a$ is even and $a^{\sharp} = -ia^{*}$ if $a$ is odd. We then consider the unbounded operator $S^{\sharp}$ defined to be the closure of the operator \begin{equation} a \triangleright \Omega \mapsto a^{\sharp} \triangleright \Omega. \end{equation} Then consider the polar decomposition $S^{\sharp} = J^{\sharp} \Delta^{1/2\sharp}$.
Is it true that the map $A \ni a \mapsto J^{\sharp} a^{\sharp} J^{\sharp} \in (A^{\backprime})^{\operatorname{opp}}$ is an isomorphism? (Here $A^{\backprime}$ is the graded commutant of $A$.)
Similarly, one can define a graded standard form of $A$, by replacing $A'$ by $A^{\backprime}$ and $*$ by $\sharp$.
Does Haagerup's theorem still hold? If so, what circumstances guarantee that $u$ is even?
(This question was motivated by the analysis done on pages 57 and 58 of these lecture notes on conformal nets.)
