Takesaki II "Connes cocycle derivative"

Question

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" (chapter VIII $\oint 3$, Modular Automorphism groups, p107-108:

Why are the second and third line in $(12)$ true? For example, I am not able to convince myself that we have the inclusion $$S(\mathfrak{D}^\sharp \cap \mathfrak{H}_1)\subseteq \mathfrak{H}_1.$$ Probably, some general fact about unbounded operators and their closures is used here but I am unable to identify what exactly I need.

Thanks in advance for any help/comments/insights!

Answer 1

The operator $S$ is defined as the closure of an operator $S_0$ that is described by (10) and (6*). So, by definition, the graph $K$ of $S_0$ (viewed as a linear subspace of $\mathfrak{H}_\rho \oplus \mathfrak{H}_\rho$) is of the form $$K = K_1 + K_2 + K_3 + K_4$$ with $$K_1 \subset \mathfrak{H}_1 \oplus \mathfrak{H}_1 \;\; , \;\; K_2 \subset \mathfrak{H}_2 \oplus \mathfrak{H}_3 \;\; , \;\; K_3 \subset \mathfrak{H}_3 \oplus \mathfrak{H}_2 \;\; , \;\; K_4 \subset \mathfrak{H}_4 \oplus \mathfrak{H}_4 \;\; .$$ Since the subspaces $\mathfrak{H}_1 \oplus \mathfrak{H}_1 \;$, $\; \mathfrak{H}_2 \oplus \mathfrak{H}_3 \;$, $\; \mathfrak{H}_3 \oplus \mathfrak{H}_2 \;$ and $\; \mathfrak{H}_4 \oplus \mathfrak{H}_4 \;$ of $\mathfrak{H}_\rho \oplus \mathfrak{H}_\rho$ are closed and orthogonal to each other, the closure of $K$ is the sum of the closures of the $K_i$, $i=1,2,3,4$. Since this closure of $K$ is the domain of $S$, the formulas in (12) hold.

Answer 2

Here is concrete way to see this. Let us for example show that $$S(\mathfrak{D}^\sharp\cap \mathfrak{H}_2)= \mathfrak{D}^\sharp \cap \mathfrak{H}_3.$$ It suffices to show that the inclusion $\subseteq $ holds since $S$ is an involution.

Let $\xi \in \mathfrak{D}^\sharp\cap \mathfrak{H}_2$. There exists a sequence $\{\xi_n\}_n\subseteq \mathfrak{A}_\rho$ such that $\xi_n\to \xi$ and $\{\xi_n^\sharp\}_n$ is convergent. We can write $$\xi_n=\eta_\rho\begin{pmatrix}x_{11}^n & x_{12}^n\\ x_{21}^n& x_{22}^n\end{pmatrix}.$$ Then with $p_i: \mathfrak{H}_\rho\to \mathfrak{H}_\rho$ the projection onto $\mathfrak{H}_i$, we have $$(p_2\xi_n)^\sharp= \eta_\rho\begin{pmatrix}0 & (x_{21}^n)^*\\0 & 0\end{pmatrix}= p_3 \xi_n^\sharp \to p_3\xi^\sharp$$ and $$p_2\xi_n\to p_2\xi = \xi.$$ It follows that $\xi^\sharp = p_3\xi^\sharp\in \mathfrak{H}_3\cap \mathfrak{D}^\sharp$.