ON C0(S, X)-DISTORTION OF THE CLASS OF ALL SEPARABLE BANACH SPACES

Suppose that K is a compact Hausdorff space, S is a locally compact Hausdorff space and X is a Banach space with Schaffer constant S(X). In this paper, we prove that if there is a map T from C(K) to C-0(S, X) satisfying 1/M parallel to f - g parallel to <= parallel to T(f) - T(g)parallel to <= M parallel to f - g parallel to, (sic)f, g is an element of C(K), with 1 <= M-2 <= S(X), then there exists a compact subset S-0 of S and a continuous function phi from S-0 onto K. This theorem on Lipschitz embeddings of C(K) into C-0(S, X) is the first nonlinear vector extension of the classical 1966 Holszty ' nski Theorem. Our result is optimal for many Banach spaces X including the spaces l(p)(n), l(p) and L-p([0, 1]), 1 < p < infinity, n >= 2, even when T is linear. The motivation to prove this result comes from the fact that it immediately yields a nontrivial lower bound for the C-0(S, X)-distortion of the class of all separable Banach spaces whenever S is a scattered space and S(X) > 1, namely S(X) itself.