# Is there an adaptation of the theory of standard forms and Tomita-Takesaki theory to the $\mathbb{Z}_{2}$-graded case?

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 $$a \triangleright \Omega \mapsto a^{*} \triangleright \Omega.$$ It then turns out that if we consider the polar decomposition $$S = J \Delta^{1/2}$$ 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.

1. $JAJ = A'$,
2. $JaJ = a^{*}$, if $a \in A \cap A'$,
3. $Jv = v$ for all $v \in P$,
4. $aJaJP \subseteq P$ 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

1. $uau^{*} = a$ for all $a \in A$,
2. $J' = uJu^{*}$,
3. $P' = uP$.

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 $$a \triangleright \Omega \mapsto a^{\sharp} \triangleright \Omega.$$ 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.)