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This post is closely related to this one. (In fact if I copied some of its content.)

Let $H$ be an infinite dimensional separable complex Hilbert space. All $C^{\star}$-subalgebras of $B(H)$ are assumed to be non-degenerate. The spectral projections of a self-adjoint element $T$ of $B(H)$ lie in the weakly closed algebra generated by $T$. In the early 1970s Pedersen proved that if a $C^{\star}$-subalgebra $A$ of $B(H)$ contains all of the spectral projections of each of its self-adjoint elements, then $A$ is weakly closed, i.e. $A=A''$. Now, concerning the consequences of this result, Pedersen says in his book:

"For any $C^\star$-subalgebra $A$ of $B(H)$ define $a(A)$ as the smallest $C^{\star}$-subalgebra of $B(H)$ containing all spectral projections of each self-adjoint element in $A$. [...] Note that [...] a transfinite (but countable) application of the operation $a$ will produce $A''$."

Question 1: What is the reasoning here?

I see that $\omega_1$ applications of the operation $a$ produce $A''$ and also that each element of $A''$ appears at the $\alpha$-th application for some $\alpha < \omega_1$ (This is due to the fact that the closure in the norm-topology is a sequential closure and hence, only countably many elements play a role.); but why is $a^{\alpha}(A)=A''$ for some $\alpha < \omega_1$?

Maybe there is something deeper behind:

Question 2: Is there an $\omega_1$-chain of unital subalgebras of $B(H)$?

Here $\omega_1$-chain means an $\omega_1$-index family $(A_{\alpha})_{\alpha< \omega_1}$ of subalgebras of $B(H)$ such that for all $\beta < \omega_1$ we have $$\overline{\cup_{\alpha< \beta}A_{\alpha}}^{\|.\|} \subsetneq A_{\beta}.$$
show/hide this revision's text 2 deleted 2 characters in body

This post is closely related to this one. (In fact if copied some of its content.)

Let $H$ be an infinite dimensional separable complex Hilbert space. All $C^{\star}$-subalgebras of $B(H)$ are assumed to be non-degenerate. The spectral projections of a self-adjoint element $T$ of $B(H)$ lie in the weakly closed algebra generated by $T$. In the early 1970s Pedersen proved that if a $C^{\star}$-subalgebra $A$ of $B(H)$ contains all of the spectral projections of each of its self-adjoint elements, then $A$ is weakly closed, i.e. $A=A''$. Now, concerning the consequences of this result, Pedersen says in his book:

"For any $C^\star$-subalgebra $A$ of $B(H)$ define $a(A)$ as the smallest $C^{\star}$-subalgebra of $B(H)$ containing all spectral projections of each self-adjoint element in $A$. [...] Note that [...] a transfinite (but countable) application of the operation $a$ will produce $A''$."

Question 1: What is the reasoning here?

I see that $\omega_1$ applications of the operation $a$ produce $A''$ and also that each element of $A''$ appears at the $\alpha$-th application for some $\alpha < \omega_1$; but why is $a^{\alpha}(A)=A''$ for some $\alpha < \omega_1$? omega_1$ (This is due to the fact that the closure in the norm-topology is a sequential closure and hence, only countably many elements play a role.)role.); but why is $a^{\alpha}(A)=A''$ for some $\alpha < \omega_1$?

Maybe there is something deeper behind:

Question 2: Is there an $\omega_1$-chain of unital subalgebras of $B(H)$?

Here $\omega_1$-chain means an $\omega_1$-index family $(A_{\alpha})_{\alpha< \omega_1}$ of subalgebras of $B(H)$ such that for all $\beta < \omega_1$ we have $$\overline{\cup_{\alpha< \beta}A_{\alpha}}^{\|.\|} \subsetneq A_{\beta}.$$
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Transfinite induction, a theorem of Pedersen, and chains of subalgebras of $B(H)$

This post is closely related to this one. (In fact if copied some of its content.)

Let $H$ be an infinite dimensional separable complex Hilbert space. All $C^{\star}$-subalgebras of $B(H)$ are assumed to be non-degenerate. The spectral projections of a self-adjoint element $T$ of $B(H)$ lie in the weakly closed algebra generated by $T$. In the early 1970s Pedersen proved that if a $C^{\star}$-subalgebra $A$ of $B(H)$ contains all of the spectral projections of each of its self-adjoint elements, then $A$ is weakly closed, i.e. $A=A''$. Now, concerning the consequences of this result, Pedersen says in his book:

"For any $C^\star$-subalgebra $A$ of $B(H)$ define $a(A)$ as the smallest $C^{\star}$-subalgebra of $B(H)$ containing all spectral projections of each self-adjoint element in $A$. [...] Note that [...] a transfinite (but countable) application of the operation $a$ will produce $A''$."

Question 1: What is the reasoning here?

I see that $\omega_1$ applications of the operation $a$ produce $A''$ and also that each element of $A''$ appears at the $\alpha$-th application for some $\alpha < \omega_1$; but why is $a^{\alpha}(A)=A''$ for some $\alpha < \omega_1$? (This is due to the fact that the closure in the norm-topology is a sequential closure and hence, only countably many elements play a role.)

Maybe there is something deeper behind:

Question 2: Is there an $\omega_1$-chain of unital subalgebras of $B(H)$?

Here $\omega_1$-chain means an $\omega_1$-index family $(A_{\alpha})_{\alpha< \omega_1}$ of subalgebras of $B(H)$ such that for all $\beta < \omega_1$ we have $$\overline{\cup_{\alpha< \beta}A_{\alpha}}^{\|.\|} \subsetneq A_{\beta}.$$