Unconditional banach space ideal property

Abstract

A Banach X is an M-ideal in its bidual if the relation \\y^t\\ = \\y\\ + 11^1 1 holds for every y € X* and every t e X1 £ j^***. The spaces Co(J) — / any set-equipped with their canonical norm belong to this class, which also contains e.g. certain spaces K(E,F) of compact operators between reflexive spaces (see [11]) and certain spaces of the form C(G)/CA (G) where G is an abelian compact group and A is a subset of the discrete dual group (see [5]). This class has been carefully investigated, in particular by A. Lima and by the «West-Berlin school», since the notion of Af-ideal was introduced by Alfsen and Effros in 1972 [1]. We will show in this paper that these spaces somehow «look like » C o; more precisely, that they share the property (u) with this latter space. This solves affirmatively a question that was pending for several years, and provides improvements of some results of [6] and [10]. Our proof uses non-linear arguments. The key lemma is actually a special case of a fundamental lemma ([I], lemma 1.4.) of the original article of Alfsen and Effros. Notation. — The closed unit ball of a Banach space Z is denoted by Zi, and its dual by Z*. The topology defined on Z* by the pointwise convergence on Z is denoted by co*. The canonical injection from a Banach space X into its bidual X** is denoted by i. A sequence (Xm) in X is said to be a weakly unconditionally convergent series —