Let $\mathcal{A}$ be a unital C*-algebra (not necessarily commutative) and let $A^*=A\in \mathcal{A}$ be a self-adjoint element with $\vert\vert A \vert\vert \leq 2$?
I want to show that $\vert\vert \mathbb{1}-A \vert\vert \leq 1 \Leftrightarrow \sigma(A)\subset [0,\infty)$ (i.e. $A$ is positive), but don't really see the connection.
How can approach this problem?
Let $\mathcal B\subseteq \mathcal A$ be the sub-$C^*$-algebra generated by $1$ and $A.$ Then $\mathcal B$ is commutative, so $\mathcal B\simeq C(X).$ Let $f_A\in C(X)$ be the function corresponding to $A.$ The following implications hold: \begin{gather*} ||1-A||\leq 1 \Longleftrightarrow ||1-f_A||_\infty\leq 1 \Longleftrightarrow |1-f_A|\leq 1\ \mbox{on}\ X\\ \Longleftrightarrow 0\leq f_A\leq 2\ \mbox{on}\ X\Longleftrightarrow 0\leq A\leq 2\ (\mbox{in } \mathcal B)\Longleftrightarrow 0\leq A\leq 2\ (\mbox{in } \mathcal A) \end{gather*} (Here $||\cdot||_\infty$ is the supremum norm on $X,$ i.e. norm in $C(X)$)
It solves your problem.