Takesaki II Lemma 1.13: stuck in proof Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"): 

Here, we associate with an operator $\eta \in \mathcal{D}^{b}$ a closed (unbounded) operator $\pi_r(\eta)$ with polar decompositions $\pi_r(\eta) = uh = ku$ as in the following two lemmas:


For further notations and conventions, I refer to the previous pages in the book.
Two questions about the proof of this lemma:

*

*Why are the marked equalities true? It probably suffices to show the first one, the second one will be similar.


*Why do we have $f(k)\eta\in \mathfrak{A}'?$ Maybe we have
$$\pi_r(\mathfrak{A}')\mathcal{D}^b \subseteq \mathcal{D}^b?$$
This would be sufficient to conclude. Or maybe we even have
$$\pi_r(\mathcal{D}^b)\mathcal{D}^b \subseteq \mathcal{D}^b?$$


*Why does there exist a net $\{a_i\}\subseteq \mathfrak{A}$ such that $\pi_l(a_i)\to 1$ strongly?
 A: *

*The marked inequalities follow from properties of the polar decomposition of closed densely defined operators. More precisely, if $x=u\lvert x\rvert$ is the polar decomposition of a closed densely defined operator $x$ on $H$ and $g$ is a continuous function on $[0,\infty)$ with $g(0)=0$, we have $g(\lvert x\rvert)=u^\ast g(\lvert x^\ast\rvert)u$.

To see this, let $\xi\in 1_{[0,R]}(\lvert x\rvert)H$ and first assume that $g$ is a polynomial, say $g(\lambda)=\sum_{k=1}^m \alpha_k\lambda^k$. We have
$$
g(\lvert x\rvert)\xi=\sum_{k=1}^m \alpha_k \lvert x\rvert^k\xi=\sum_{k=1}^m \alpha_k(u^\ast\lvert x^\ast\rvert u)^k\xi=\sum_{k=1}^m\alpha_ku^\ast\lvert x^\ast\rvert^ku\xi=u^\ast g(\lvert x^\ast\rvert)u\xi.
$$
If $g\in C_0((0,\infty))$ is arbitrary, we can approximate it uniformly on $[0,R]$ by polynomials vanishing at $0$ to see that this identity continues to hold. Finally, since $\bigcup_{R>0}1_{[0,R]}(\lvert x\rvert)H$ is dense in $H$, we obtain $g(\lvert x\rvert)=u^\ast g(\lvert x^\ast\rvert)u$.



*Write $f= \overline{f_1} f_2$ with $f_1, f_2 \in \mathcal{K}(0, \infty)$. Then
$$f(k)\eta= f_1(k)^*f_2(k)\eta \in \pi_r(\mathfrak{A}')^*\mathfrak{B}'\subseteq \pi_r(\mathfrak{B'})^*\mathfrak{B}'\subseteq \mathfrak{A}'$$
by lemma 1.9(1) in the same chapter of Takesaki's book.


*The algebra $\pi_\ell(\mathfrak A)$ is non-degenerate by definition of a left Hilbert algebra, hence the identity is contained in its strong closure.
