  We provide a complete picture of contractivity of trace preserving positive
maps with respect to $p$-norms. We show that for $p>1$ contractivity holds in
general if and only if the map is unital. When the domain is restricted to the
traceless subspace of Hermitian matrices, then contractivity is shown to hold
in the case of qubits for arbitrary $p\geq 1$ and in the case of qutrits if and
only if $p=1,\infty$. In all non-contractive cases best possible bounds on the
$p$-norms are derived.
