Unconditional Convergence of Positive Terms in a $C*$-algebra I am reading the paper Frames and Outer Frames for Hilbert $C^*$-modules by L.J. Arambasic and D. Bakic. They have mentioned in passing, the following:

"...Since in each $C^*$-algebra, a convergent series of positive elements necessarily
converges unconditionally..."

Unfortunately they did not give a direct reference for this claim, and I was not successful in looking for a paper that could verify this. Perhaps this is a basic result that I am not aware of, and I can't seem to prove it myself, could someone verify this for me?
 A: Let $x = \sum x_n$ be a convergent sum of positive elements of a C${}^\ast$-algebra $A$. Then for any state $\phi$ on $A$ we have $\sum \phi(x_n) = \phi(x)$, converging unconditionally since it is a series of positive terms. So if some rearrangement of the series sums to $y$, we would have $\phi(x) = \sum \phi(x_n) = \phi(y)$. That is,
$\phi(x - y) = 0$ for every state $\phi$. This implies that $x = y$.
A: My preferred definition of unconditional convergence for  series
is as follows:   a family $\{x_i\}_{i\in I}$ of vectors in a normed space is unconditionally summable, with sum $y$, provided for
every $\varepsilon >0$, there is a finite subset $F_0\subseteq I$, such that for every finite subset $F\subseteq I$, with $F_0\subseteq F$, one has that
$$
   \|s_F-y\|<\varepsilon ,
  $$
where
$$
  s_F=\sum_{i\in F}x_i.
  $$
This is precisely the same as saying that the net $\{s_F\}_{F\in \mathscr F}$ converges to $y$, where $\mathscr F$ is the
directed set of all finite subsets of $I$, ordered by inclusion.
This totally ignores any ordering of the index set $I$ and it makes sense even if $I$ is uncountable.
Moreover,
it is very easy to see that, when $I={\mathbb N}$,  the above implies convergence in any reordering of the
summands, with the same sum $y$, hence unconditional convergence in the usual sense.
If $\{x_n\}_{n\in {\mathbb N}}$ are positive elements in a C*-algebra,  and $\sum_{n=1}^\infty x_n$ is summable,  then the series converges
unconditionally in the above strong sense because, given $\varepsilon >0$, we can take some $n_0$ such that
$$
  \Big \|\sum_{i=1}^nx_i-y\Big \|<\varepsilon , \quad \forall n\geq n_0.
  $$
Setting $F_0=\{1, 2, \ldots , n_0\}$, and taking any finite set $F$ of indices, containing $F_0$, we may choose $n_1 >n_0$ such that
$F\subseteq \{1, 2, \ldots , n_1\}$.  Therefore
$$
  \Vert s_F-y\Vert  \leq  \Big \Vert s_F-\sum_{i=1}^{n_0}x_i\Big \Vert  + \Big \Vert \sum_{i=1}^{n_0}x_i-y\Big \Vert  < \Big \Vert \sum_{i=n_0+1}^{n_1}x_i\Big \Vert  + \varepsilon ,
  $$
because the $x_i$ are positive.   The rest of the argument is now easy to fill in.
