When an invertible element in a $C^{*}$-algebra is unitary

Question

I am trying to show that if $a$ is an invertible element of a unital $C^{*}$-algebra, and $||a||=||a^{-1}||=1$, then $a$ is unitary. I can do this if I think of $a$ as a Hilbert space operator using Gelfand-Naimark. I think it can be done without using Gelfand-Naimark but I haven't been able to do it so this is what I would like to ask.

Answer 1

Is there a result in $C^*$-algebras that if $b$ satisfies $b = b^*$ and $\sigma(b) \subset [\alpha,\beta]$ where $\alpha,\beta > 0$, then $\alpha I \le b \le \beta I$?

Then you could prove it as follows. Consider $b = a^* a$. Then a short argument shows that $\|b\| = \|b^{-1}\| = 1$. But $b$ is positive, and $\inf\sigma(b) = \sup\sigma(b) = 1$. Hence $I \le b \le I$. Therefore $b=I$.