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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 Carlenand Lieb, as a byproduct of their analysis of certain convex functions onmatrix spaces. The second norm was defined by Pisier and others using resultsfrom the theory of operator spaces. It is shown that the second norm is upperbounded by a constant multiple of the first for all 1 <= p <= 2, q >= 1. In onecase (2 = p < q) it is also shown that there is no such lower bound, and hencethat the norms are inequivalent. It is conjectured that the norms areinequivalent in all cases.

Comparison of matrix norms on bipartite spaces