| dc.description.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 — | en_US |
