Idempotent states and the inner linearity property

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 $C^{\ast }$-algebra, we prove that the idempotent state associated to its Hopf image $A^{\mathrm{\prime }}$ must be the convolution Ces`aro limit of the linear functional $\varphi =tr\circ \pi$. We discuss then some consequences of this result, notably to inner linearity questions.