Small into isomorphisms on uniformly smooth spaces

Let X be a uniformly smooth infinite dimensional Banach space, and (Omega, Sigma, mu) be a sigma-finite measure space. Suppose that T : X --> L-infinity(Omega, Sigma, mu) satisfies (1 - epsilon)parallel toxparallel to less than or equal to parallel toTxparallel to less than or equal to parallel toxparallel to, For Allx is an element of X, for some positive number epsilon < 1/2 with δ(X*) (2-2ε) > 13/14. Then T is close to an isometry U:X --> L-infinity (Omega, Sigma, mu) such that parallel toT - Uparallel to less than or equal to 16(1 - 8(X*) (2 - 2epsilon)) + 1/2epsilon, where delta(X*) (t) is the modulus of convexity of the conjugate space X*. (C) 2003 Elsevier Inc. All rights reserved.