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?



Unconditional Convergence of Positive Terms in a $C*$-algebra
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 $x = y$.


