Asymptotic Unconditionality

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 ${\lim }_{n\to \infty }\Vert x^{\ast }+x_n^{\ast }\Vert ={\lim }_{n\to \infty }\Vert x^{\ast }-x_n^{\ast }\Vert$ whenever $x^{\ast }\in X^{\ast }$, $(x_n^{\ast })$ is a weak${}^{\ast }$-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.

26 pages. Submitted for publication. This is a replacement submission. The paper is unchanged but the "Comments" field has been edited