This is an announcement for the paper "Asymptotic unconditionality" by S. R. Cowell and N. J. Kalton.
Abstract: 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 $\lim_{n \to \infty} |x^* + x_n^*| = \lim_{n \to \infty} |x^* - x_n^*|$ whenever $x^* \in X^*$, $(x_n^*)$ 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.
Archive classification: math.FA
Mathematics Subject Classification: 46B03; 46B20
Remarks: 26 pages. Submitted for publication. This is a replacement
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http://front.math.ucdavis.edu/0809.2294
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http://arXiv.org/abs/0809.2294
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