Facial characterizations of complex Lindenstrauss spaces

Abstract

We characterize complex Banach spaces A whose Banach dual spaces are '(/*) spaces in terms of L-ideals generated by certain extremal subsets of the closed unit ball K of A*.Our treatment covers the case of spaces A containing constant functions and also spaces not containing constants.Separable spaces are characterized in terms of w*-compact sets of extreme points of K, whereas the nonseparable spaces necessitate usage of the w'-closed faces of K. Our results represent natural extensions of known characterizations of Choquet simplexes.We obtain also a characterization of complex Lindenstrauss spaces in terms of boundary annihilating measures, and this leads to a characterization of the closed subalgebras of C^X) which are complex Lindenstrauss spaces.