When do the Banach lattices C([0,α],X) determine the ordinal intervals [0,α]?

Suppose that X   is a Banach lattice containing no Banach sublattice isomorphic to c0c0 and consider the Banach lattices C([0,α],X)C([0,α],X) of X-valued continuous functions defined on the ordinal intervals [0,α][0,α], provided with the supremum norm. We prove that if the finite sums of X satisfy a certain geometric condition, then for all ordinals α and β the following assertions are equivalent:(1)The Banach lattices C([0,α],X)C([0,α],X) and C([0,β],X)C([0,β],X) are isomorphic.(2)The intervals of ordinals [0,α][0,α] and [0,β][0,β] are homeomorphic. As application of this cancellation law we obtain the complete classification, up to Banach lattices isomorphism, of certain C(K⊕[0,α],X)C(K⊕[0,α],X) spaces, where K is an arbitrary perfect compact Hausdorff space.