Let A be an unital C∗algebra where there exists a $δ > 0$ such that for any pair of distinct pure states $ϕ_1, ϕ_2$ on $A$ we have $|ϕ_1 − ϕ_2| ≥ δ$. Show that this assumption already implies that A is commutative.
The hint is to consider : $U(t)=e^{ita}$ for some hermitian element $a\in A$.
I honestly don't know where to get started with this one, and the hint made it more confusing as I am not sure why that would help.