Let $A$ be a $C^\ast$-algebra and $f\colon A\to\mathbb{C}$ a positive$^[2]$ linear functional. Suppose $a\leq b^{[1]}$, is it true that $f(a)\leq f(b)$? This is used in a proof, but for approximate identities, maybe they are needed.
$^{[1]}$ Note that $a\leq b$ means $b-a$ is a positive element, i.e. $\sigma(b-a)\subset[0,\infty)$.
$^{[2]}$ Note that positive linear functional means $f(a^\ast a)\geq0$ for every $a\in A$.