Let $A$ be a $C^*$-algebra. It is known that the order relation on the set of self-adjoint elements of $A$ is well-behaved with respect to addition, i.e. if $a\leq b$ and $c\leq d$ then $a+c\leq b+d$.
Question: What are some circumstances under which it is also well-behaved under multiplication?
Thoughts: For example, if $a\leq b$ and $ab=ba$, then provided $a+b$ is positive, we have $a^2\leq b^2$, since we would then have the factorisation $b^2-a^2=(b-a)(b+a)$ into positive elements. So one concrete question is: if $a,b,c,d$ are commuting positive elements with $a\leq b$ and $c\leq d$, then can we say that $ac\leq bd$?
A product of two positive commuting elements $x$ and $y$ is positive as $$xy=x^{1/2}yx^{1/2}=(y^{1/2}x^{1/2})^*y^{1/2}x^{1/2}\ge 0$$ Then we obtain $$bd-ac=(b-a)d+a(d-c)\ge 0$$