The monotone closure of a $C^*$-algebra Related to Jon's question, 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 $S$.

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 $\mathbb{B}(\mathcal{H})$?

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.
 A: 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.
A: I can give you a partial answer to question 1, which you may already be aware of if you're familiar with Pedersen's "C*-algebras and their automorphism groups".  Specifically, if $A\subseteq A^{**}$, i.e. if we are considering the universal representation of $A$, and if $A$ is either unital or separable then you do indeed have $(A_\mathrm{sa}^m)^m=A_\mathrm{sa}^m=\overline{A_\mathrm{sa}^m}(=$ norm closure of $A_\mathrm{sa}^m$).  For if $A$ is unital then, by Pedersen 3.11.7, $A_\mathrm{sa}^m=\overline{A_\mathrm{sa}^m}$, while if $A$ is separable then we again have $A_\mathrm{sa}^m=\overline{A_\mathrm{sa}^m}$, by Corollary 3.25(a) of Lawrence G. Brown's "Semicontinuity and Multipliers of C*-algebras".  By Pedersen 3.11.5, $\overline{A_\mathrm{sa}^m}$ consists precisely of those $x\in A^{**}_\mathrm{sa}$ that are lower semicontinuous on $Q=A^{*1}_+$, which are immediately seen to be closed under taking supremums of bounded increasing nets.
If we consider $A$ with its atomic representation instead, then the above remarks still apply, as there is then a normal morphism $\pi$ from $A^{**}$ onto $A''$ which is faithful on $A_\mathrm{sa}^m$, by Pedersen 4.3.13 and 4.3.15.  Arbitrary representations can be faithful on $A$ but not on $A^m_\mathrm{sa}$, although in general you might perhaps try to use the predual $A''_*$ of $A''$, in the topology induced by $A$, as a replacement for $A^*$ with the weak* topology.  The only problem is that $A''^1_{*+}$ may not be compact in this topology so you would have to somehow adjust the proof of Pedersen 3.11.2, on which 3.11.5 relies.
Incidentally, it is a problem of Akemann and Pedersen from 1973 (still open as of 2014 according to Brown) whether $A_\mathrm{sa}^m=\overline{A_\mathrm{sa}^m}$ for arbitrary $A$ in its universal representation, so if you or Nik have a counterexample it would be quite important.
