Summary: Let \(\mathcal{A}\) and \(\mathcal{B}\) be two unital \(C^\ast\)-algebras. Denote by \(W(a)\) the numerical range of an element \(a\in\mathcal{A}\). We show that the condition \(W(ax)=W(bx)\) for all \(x\in\mathcal{A}\) implies that \(a=b\). Using this, among other results, it is proved that, if \(\phi:\mathcal{A}\to\mathcal{B}\) is a surjective map such that \(W(\phi(a)\phi(b)\phi(c)) = W(abc)\) for all \(a, b\) and \(c\in\mathcal{A}\), then \(\phi(1)\in Z(B)\) and the map \(\psi=\phi(1)^2\phi\) is multiplicative.