Comparision of two positive invertible elements in a $C^*$-algebra

Question

Suppose $A$ is a unital $C^*$-algebra and $a,b$ are any two positive invertible elements in $A$. Must there exist $s,t>0$ such that $sa\leq b\leq ta$?

Answer 1

Put $c=a^{-1/2}ba^{-1/2}$. Then $c$ is positive and invertible. Thus there exist constants $s,t>0$ such that $s\leq c\leq t$ (namely, $t=\max\sigma(c)$ and $s=\min\sigma(c)$), and therefore $sa\leq b\leq ta$.