Fixed points and limits of convolution powers of contractive quantum measures

We study fixed points of contractive convolution operators associated to quantum measures on locally compact groups. characterise the existence non-zero respectively $L^\infty(\mathbb{G})$ and $C_0(\mathbb{G})$, exploit these results obtain for example structure non-commutative $L_p$-spaces. Some consequences classical Herz-Schur multipliers are also indicated.