Firstly, for a unital C*-algebra $A$, and an hermitian element $x \in A$, the following are equivalent: 1) The spectrum $Sp_A x \geq 0$; 2) $x$ is of the form $yy^*$ for some $y \in A$; 3) $x$ is of the form $h^2$ for some hermitian $h \in A$.
Let $P$ be the set of $x \in A$ satisfying any of these conditions.
On page 16 of Dixmier's C*-algebra, there is a lemma:
Let $A$ be a unital C*-algebra. If $x \in A$ is hermitian and $||1-x|| \leq 1$, then $x \in P$. If $x \in P$ and $||x|| \leq 1$ then $||1-x|| \leq 1$.
The author reduced the proof to the case of $A$ being the C*-algebra of continuous complex-valued functions on a compact space. Then he said that this was clear. But I don't know why is this clear in this case.
I'll be very grateful for the explanation.