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→∞ ∥x* + xn* = lim _n→∞ ∥x*-xn*∥ whenever x* ∈ X*, (x*_n)^∞_n=1 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.