COTYPE OF OPERATORS FROM C(K)

For q > 2, an operator from C(K) to X is of cotype q if and only if it factors through the Lorentz space L(tq(log t)q2), 1(mu). For q = 2, if X is a rearrangement invariant space on [0, 1], the injection C([0, 1]) --> X is of cotype 2 if and only if it factors through the Lorentz space L(t2 log t), 2 ([0, 1]); but there is a cotype 2 operator C(K) --> l infinity that does not factor through L(t2 log t, 2) (mu). If a Banach lattice X satisfies the Orlicz property, any bounded lattice operator T: C(K) --> X is of cotype 2. We however construct a Banach lattice with the Orlicz property, but that fails to be of cotype 2.