Facial characterizations of complex lindenstrauss spaces

Abstract

We characterize complex Banach spaces A whose Banach dual spaces are L¹(μ) 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_c(X) which are complex Lindenstrauss spaces.