ASYMPTOTIC UNCONDITIONALITY

We show that a separable real Banach space embeds almost isometrically in a space Y with a shrinking 1-unconditional basis if and only if lim(n ->infinity) parallel to x* + x(n)*parallel to = lim(n ->infinity) parallel to x* - x(n)*parallel to whenever x* is an element of X*, (x(n)*)(n=1)(infinity) is a weak*-null sequence and both limits exist. If X is reflexive then Y can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.