= Order preservation by the positive square root
{title2=$0\leq x\leq y\Longrightarrow x^{1/2}\leq y^{1/2}$}
For <Positive elements of a C-star algebra> $x\leq y$, <inversion reverses the order of strictly positive elements>, so $(t1+x)^{-1}\geq(t1+y)^{-1}$ for $t>0$. Subtracting their <square-root resolvent integral in a C-star algebra> formulas gives
$$
y^{1/2}-x^{1/2}=\frac1\pi\int_0^\infty t^{1/2}\bigl((t1+x)^{-1}-(t1+y)^{-1}\bigr)\,dt\geq0.
$$
The integrand is positive, and the positive cone is closed, so the norm limit of these integrals is positive. No commutativity of $x$ and $y$ is required. In particular, $b^2\geq a^2$ for positive $a,b$ implies $b\geq a$, even though squaring itself is not generally order preserving.
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