Related to [Jon's question](https://mathoverflow.net/questions/103306/strong-monotone-limits-and-dense-subalgebras-of-von-neumann-algebras-again), I have two questions. Let $\mathcal{A}$ be a concrete $C^\*$-algebra on a Hilbert space $\mathcal{H}$. For any selfadjoint subset $S$ of $\mathbb{B}(\mathcal{H})$, let $S^m$ denote the set of elements of $\mathbb{B}(\mathcal{H})\_{sa}$ that can be obtained as the strong limits of monotone increasing nets from $S$.



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> **Question 1.** Is $((\mathcal{A}\_{sa})^m)^m=(\mathcal{A}\_{sa})^m$? (Maybe this is very basic.)  
> **Question 2.** Does the $C^\*$-algebra $C^\*((\mathcal{A}\_{sa})^m)$ generated by $(\mathcal{A}\_{sa})^m$ in $\mathbb{B}(\mathcal{H})$ coincide with the strong closure of $\mathcal{A}$ in $\mathbb{B}(\mathcal{H})$?
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For Question 2, I have been thinking that Pedersen's up-down-up theorem [Theorem 2 in *American Journal of Mathematics* **94** (1972), 955-962] might be useful, but I couldn't figure out.



The monotone closure of a $C^*$-algebra
Edited. As Masayoshi points out, my reading of Hamana's paper was incorrect. I'm quite sure question 1 is false in general but I don't have a reference. (Masayoshi, did you look in Pedersen's book? I feel the answer may be there but I don't have access to it right now.)


I guess I'd better be more explicit about question 2. For example, take $A = C[0,1]$ acting by multiplication on $l^2[0,1]$. Then the strong closure of $A$ equals $l^\infty[0,1]$, but $A\_{sa}^m$ is contained in e.g. the set of bounded Borel measurable functions, so the C\*-algebra it generates is also contained in that class.


