Let $A$ be a $C^\*$-algebra. Endow $A$ with the strict topology for which a net $\{a\_i\}\_{i \in I}$ converges to $a \in A$ if $$\|a\_i b-ab\| + \|ba\_i-ba\| \to 0$$
for all $b \in A$. Is it true that if $\{a\_i\}\_{i \in I}$ is a bounded net of positive elements that converges strictly to $a \in A$, then $a$ is also positive?


Remarks: (1) If $A$ is unital, the strict topology coincides with the norm-topology and the result is well-known then.


(2) If $A= B\_0(H)$, then the strict topology on bounded subsets of $A$ is the strong$^\*$- topology and then the statement is obviously true.



Convergent bounded net of positive operators converges to a positive operator
Yes, $a$ is positive, too.


*Proof.*
For every $b \in A$ we have $b^\* a\_i b \to b^\* a b$, so $b^\*a b$ is positive.
Now, let $(e\_j)$ be an approximate identity in $A$. Then it follows that $(e\_j a e\_j)$ is a net of positive elements that converges in norm to $a$; hence, $a$ is positive.


