Does inversion reverse order for positive elements in a unital C* algebra?

Question

Let's say that in a unital C* algebra, we have $b \geq a \geq 0$ and $a$ is invertible. Then $b$ is also invertible. Can we conclude that $a^{-1} \geq b^{-1}$? If so, why? Can any related statement be made if we only assume $a \leq b$? (And maybe that $a$, and hence $b$ are self adjoint.)

Answer 1

Yes. If $b \ge a \ge 0$ and $a$ is invertible, then $a^{-1/2} b a^{-1/2} \ge a^{-1/2} a a^{-1/2} = 1$ so $ a^{1/2} b^{-1} a^{1/2} = (a^{-1/2} b a^{-1/2})^{-1} \le 1$ and then $b^{-1} = a^{-1/2} a^{1/2} b^{-1} a^{1/2} a^{-1/2} \le a^{-1}$. Without the $\ge 0$ it's false, e.g. try $a = -1$ and $b = 1$.