Intersection Properties of Balls and Subspaces in Banach Spaces

Abstract

We study intersection properties of balls in Banach spaces using a new technique. With this technique we give new and simple proofs of some results of Lindenstrauss and others, characterizing Banach spaces with L1 (u) dual spaces by intersection properties of balls, and we solve some open problems in the isometric theory of Banach spaces. We also give new proofs of some results of Alfsen and Effros characterizing M-ideals by intersection properties of balls, and we improve some of their results. In the last section we apply these results on function algebras, G-spaces and order unit spaces and we give new and simple proofs for some representation theorems for those Banach spaces with L1 (a) dual spaces whose unit ball contains extreme points. Introduction. A Banach space A is said to be a Cl-space if for every Banach space X containing A, there is a linear projection P from X onto A with IIPII A, there exists a linear extension t: Y -* A of T with II TII = I|I T1. An example of a space with this property is 1. (r) for some set r o 0. This is shown by application of the Hahn-Banach theorem coordinatewise. In 1950 Nachbin [39] and Goodner [18] characterized the real 91-spaces whose unit balls have extreme points: They are (up to isometry) the C(K)spaces for which the compact Hausdorff space K is extremally disconnected (i.e. the closure of every open set is open). In 1952 Kelley [30] showed that the assumption on the unit ball was superfluous. This was shown to hold also in the complex case by Hasumi [21] in 1958. In 1955 Grothendieck [19] showed that a real Banach space A is isometric to an LI (,a)-space for some measure t, if and only if its dual A* is a 0'1-space. This result was later proved for the complex case by Sakai [42]. Received by the editors August 20, 1975. AMS (MOS) subject classifications (1970). Primary 46B99; Secondary 46A40, 46B05, 46E15, 46E30. (1) This paper is a revised version of the author's doctoral dissertation at the University of Oslo, 1974. 0) American Mathematical Society 1977