  We introduce a condition for memoryless quantum channels which, when
satisfied guarantees the multiplicativity of the maximal l_p-norm with p a
fixed integer. By applying the condition to qubit channels, it can be shown
that it is not a necessary condition, although some known results for qubits
can be recovered. When applied to the Werner-Holevo channel, which is known to
violate multiplicativity when p is large relative to the dimension d, the
condition suggests that multiplicativity holds when $d \geqslant 2^{p-1}$. This
conjecture is proved explicitly for p=2, 3, 4. Finally, a new class of channels
is considered which generalizes the depolarizing channel to maps which are
combinations of the identity channel and a noisy one whose image is an
arbitrary density matrix. It is shown that these channels are multiplicative
for p = 2.
