Complex Lindenstrauss spaces with extreme points

Abstract

We prove that a complex Lindenstrauss space whose unit ball has at least one extreme point is isometric to the space of complex valued continuous affine functions on a Choquet simplex. If X is a compact Hausdorff space and A C Cc(X) is a function space then A is a Lindenstrauss space iff A is selfadjoint and Re A is a real Lindenstrauss space. 1. In [3] and [4] E. Effros proposed and investigated the complex analogue of the preduals of real L '-spaces, also called Lindenstrauss spaces. We aim to discuss those complex Lindenstrauss spaces whose unit balls have at least one extreme point. First, we prove a result which is well known in the case of real scalars (cf. [8]): Theorem 1. Let E be a Lindenstrauss space and u an extreme point of the closed unit ball of E. Let S = {x* E E* |X|*|| = x*(u) = 1}. For every x E E let x E Cc(S) be defined by x(x*) = x*(x). Then S is a w*-compact subset of E* and the map x -a x of E into Cc(S) is an isometry such that u =ls. M4ore information about Lindenstrauss spaces as function spaces is given by Theorem 2. Let X be a compact Hausdorff space and A C Cc(X) a closed linear subspace, separating the points of X and containing the constant functions. Let S denote the state space of A. Then the following statements are equivalent: (i) A is a Lindenstrauss space; (ii) y e A1n M(aAX) = 0; (iii) Z = conv (SU iS) is a Choquet simplex; (iv) A is selfadjoint and Re A is a real Lindenstrauss space. From Theorems 1 and 2 we shall get a characterization of cC(X) spaces identical to that given for real scalars in [8, p. 76] and we shall see that Theorem 2 implies that no uniform algebra is a Lindenstrauss space unless it is Cc(X). We shall follow the notations of [1]. By I' we shall denote the linear space of all sequences a = (al, a2, .., an) of n complex numbers with the norm jaj = max 1<i<n I aj. If E is a Banach space we shall denote its closed unit ball by B(E). Received by the editors November 28, 1972 and, in revised form, January 13, 1973. AMS (MOS) subject classifications (1970). Primary 46E15.