Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $\Omega$ be a cyclic and separating vector in $H$. Let $J$ and $\Delta$ be the corresponding modular conjugation and modular operator from Tomita–Takesaki theory, i.e. $JMJ=M'$ and $\Delta^{it}M\Delta^{-it} = M$, for all $t \in \mathbb{R}$.
What is known about the unital inclusions of von Neumann algebras $N \subset M$ such that $\Delta^{it}N\Delta^{-it} = N$?
What if $N \subset M$ is a unital inclusion of $\mathrm{III}_1$ factors? What if it is (moreover) of finite index?
Ziyun Xu just pointed out to me the following theorem by M. Takesaki (in this book, Chapter 9) which should be useful here.
In fact, I would like to know whether a unital inclusion of $\mathrm{III}_1$ factors can always be reduced (i.e. keeping the quantum symmetries) into a unital inclusion of $\mathrm{II}_{\infty}$ factors just using the Tomita–Takesaki theory (and then into a unital inclusion of $\mathrm{II}_{1}$ factors using $\mathrm{II}_{1} \otimes \mathrm{I}_{\infty} = \mathrm{II}_{\infty}$), with a finite index assumption if required.
There exists such reduction by Sorin Popa using Jones' tower, see the following extract from the introduction of his book.
But this post asks about using the Tomita-Takesaki theory only.
