  We find an analytic formulation of the notion of Hopf image, in terms of the
associated idempotent state. More precisely, if $\pi:A\to M_n(\mathbb C)$ is a
finite dimensional representation of a Hopf $C^*$-algebra, we prove that the
idempotent state associated to its Hopf image $A'$ must be the convolution
Ces\`aro limit of the linear functional $\phi=tr\circ\pi$. We discuss then some
consequences of this result, notably to inner linearity questions.
