  In this paper we study two semigroups of completely positive unital
self-adjoint maps on the von Neumann algebras of the free orthogonal quantum
group $O_N^+$ and the free permutation quantum group $S_N^+$. We show that
these semigroups satisfy ultracontractivity and hypercontractivity estimates.
We also give results regarding spectral gap and logarithmic Sobolev
inequalities.
