Take the tour ×  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.  share|improve this question
add comment  1 Answer  up vote 7 down vote accepted  Edited. As
Masayoshi points out, my reading of Hamana's paper was
incorrect. I'm quite sure question