Let $x$ be an element of a $C^*$-algebra. If $0 \leq x \leq 1$, then $\|x\| \leq 1$?
Here, $0 \leq x \leq 1$ means that both $x$ and $1-x$ are positive elements. I think this statement should be true (and seems to be used in a proof, regarded as a trivial statement). This is true when the $C^*$-algebra is the space of bounded operators on a Hilbert space.
I want an elementary proof, not using something like that every $C^*$-algebra is isomorphic to the space of bounded operators on a Hilbert space.