Asymptotic unconditionality in Banach spaces

Abstract

We show that a separable real Banach space embeds almost isometrically in a space [Upsilon] with a shrinking 1-unconditional basis if and only if lim [subscript n] [subscript arrow] [subscript infinity] [norms] [chi] [group of units] [plus][chi] [group of units] [subscript n] [norms] [equals] lim [subscript n] [subscript arrow] [subscript infinity][nearest integer function] [norms] [chi][group of units] [minus] [chi] [group of units] [subscript n] [norms] whenever [chi] [group of units] [element of][Chi] [group of units] ([chi][group of units][subscript n]) [superscript infinity] [subscript n[equals]₁ is a weak [group of units] - null sequence and both limits exist. If [Chi] is reflexive then [Upsilon] can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.