Let $\mathcal A$ be a unital C* algebra. Consider the Gelfand transform on the unital, commutative C* algebra $\mathcal A_A$ generated by $A$, i.e. $\Gamma: \mathcal A_A \rightarrow C(\mathcal M(\mathcal A_A))$, where $C(\mathcal M(\mathcal A_A))$ are the continous functions on the linear, multiplicative functionals on $\mathcal A_A$.
Now, let $A\in \mathcal A$ be self-adjoint.
What does $\vert\vert \Gamma(A) - \Gamma(\mathbb I)\vert\vert$ have to fulfill for $\Gamma(A)$ to be a positive functional on $\mathcal M(\mathcal A_A)$? For example, is it sufficient that $\vert\vert \Gamma(A) - \Gamma(\mathbb I)\vert\vert\leq 1$ ?