A COMPLETE CLASSIFICATION OF THE SPACES OF COMPACT OPERATORS ON C([1, α], lp) SPACES, 1 < p < 8

We complete the classification, upto isomorphism, of the spaces of compact operators on C([ 1, gamma], lp) spaces, 1 < p < infinity. In order to do this, we classify, upto isomorphism, the spaces of compact operators K(E, F), where E = C([1, lambda], l(p)) and F = C([1, xi], l(q)) for arbitrary ordinals lambda and xi and 1 < p <= q < infinity. More precisely, we prove that it is relatively consistent with ZFC that for any infinite ordinals lambda, mu, xi and. the following statements are equivalent: (a) K(C([1, lambda], l(p)), C([1, xi], l(q))) is isomorphic to K(C([1, mu], l(p)), C([1, eta], l(q))). (b) lambda and mu have the same cardinality and C([1, xi]) is isomorphic to C([1, eta]) or there exists an uncountable regular ordinal a and 1 <= m, n < omega such that C([1, xi]) is isomorphic to C([1, alpha m]) and C([1, eta]) is isomorphic to C([1, alpha n]). Moreover, in ZFC, if lambda and mu are finite ordinals and xi and eta are infinite ordinals, then the statements (a) and (b') are equivalent. (b') C([1, xi]) is isomorphic to C([1, eta]) or there exists an uncountable regular ordinal a and 1 <= m, n <= omega such that C([1, xi]) is isomorphic to C([1, alpha m]) and C([1, eta]) is isomorphic to C([1, alpha n]).