Complex Lindenstrauss spaces with extreme points

Abstract

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 CCçJLX) is a function space then A is a Lindenstrauss space iff A is selfadjoint and Re A is a real Lindenstrauss space.

Intersection properties of balls in spaces of compact operators

“…(See also [13] and [14]). Hence Theorem 1 of Hirsberg and Lazar [7] is an easy consequence of Theorem 2.1 and Theorem 2.2. Theorem 1 of [7] says that if A is predual of a complex L^-space and e € 3^A^ , then the map $ of Theorem 2.1 is an isometry.…”

Intersection Properties of Balls and Subspaces in Banach Spaces

“…First we give a new proof of a representation theorem for Banach spaces whose dual spaces are isometric to Lj (ji)-space and whose unit balls have extreme points. The real version of this theorem was proved by Nachbin [39], Kadison [27] and Lindenstrauss [35], and the complex version was proved by Hirsberg and Lazar [23]. For the complex case our proof is considerably simpler than the original one.…”

Facial characterizations of complex Lindenstrauss spaces

“…In §4 we extend a measure-theoretic characterization of complex Lindenstrauss spaces, due to Hirsberg and Lazar [14], to the case without constants. This enables us to characterize the closed subalgebras of C^X) which are complex Lindenstrauss spaces.…”

“…The space A is selfadjoint with state-space S, and since 5 is not a simplex it follows that A is not a complex Lindenstrauss space (see Hirsberg and Lazar [14]). The map (p ~»<p/(p(\) is a w*-continuous mapping from 3A" onto 35", and hence if E is a compact subset of 3A' then line E = 7 is finite dimensional, say 7 = linc {x" .…”

“…To prove that A is a complex Lindenstrauss space it will be sufficient, using the result of Hirsberg and Lazar [14,Theorem 2], to show that the set Z = co(5' u -iS) is a simplex. By Briem's characterization of simplexes it will be sufficient to show that each A(Z)-peak face of Z is parallel.…”