  Two non-commutative versions of the classical L^q(L^p) norm on the algebra of
(mn)x(mn) matrices are compared. The first norm was defined recently by Carlen
and Lieb, as a byproduct of their analysis of certain convex functions on
matrix spaces. The second norm was defined by Pisier and others using results
from the theory of operator spaces. It is shown that the second norm is upper
bounded by a constant multiple of the first for all 1 <= p <= 2, q >= 1. In one
case (2 = p < q) it is also shown that there is no such lower bound, and hence
that the norms are inequivalent. It is conjectured that the norms are
inequivalent in all cases.
