Takesaki II Lemma 1.13: stuck in proof

Question

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:

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

2. 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 $$\pi_r(\mathcal{D}^b)\mathcal{D}^b \subseteq \mathcal{D}^b?$$

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

Answer 1

1. 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 $g\in C_0([0,\infty))$, we have $g(\lvert x\rvert)=u^\ast g(\lvert x^\ast\rvert)u+(1-u^\ast u)g(0)$. In particular, if $g(0)=0$, then $g(\lvert x\rvert)=u^\ast g(\lvert x^\ast\rvert)u$.
   To see this, first let $g_\lambda(t)=(t+\lambda)^{-1}$ for $\lambda \notin \mathbb R$. We have \begin{align*} (u^\ast g_\lambda(|x^\ast|)u+\lambda^{-1}(1-u^\ast u))(|x|+\lambda)&=u^\ast(|x^\ast|+\lambda)^{-1}u(u^\ast|x^\ast|u+\lambda)+1-u^\ast u\\ &=u^\ast(|x^\ast|+\lambda)^{-1}(|x^\ast|+\lambda)u+1-u^\ast u\\ &=1. \end{align*} Here we used that $u^\ast u|x|=|x|$ and $u u^\ast|x^\ast|=|x^\ast|$ by the definition of the polar decomposition. Thus $u^\ast g_\lambda(|x^\ast|)u+(1-u^\ast u)g_\lambda(0)=(|x|+\lambda)^{-1}=g_\lambda(|x|)$.
   By Stone-Weierstraß, the functions $t\mapsto (t+\lambda)^{-1}$ with $\lambda\notin \mathbb R$ generate a dense subspace of $C_0([0,\infty))$, and the claim for arbitary $g\in C_0([0,\infty))$ follows by approximation.

2. 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.

3. 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.