On separable Lindenstrauss spaces

Abstract

Isometric embeddings from l ∞ n in l ∞ n + 1 can be described by a i,n, i ⩽ n, with ∑ i = 1 n ¦ a i,n ¦ ⩽ 1 , such that e i, n = e i, n + 1 + a i, n e n + 1, n + 1 ; i = 1,…, n; holds, where e i, n and e i, n + 1 are the elements of the canonical unit vector bases of l ∞ n and l ∞ n + 1 , respectively (negative signs may occur). We study the connections between a triangular substochastic matrix A, whose nth column consists of the elements a i, n , i = 1,…, n, and the Banach space a i,n, E n ⊂ E n + 1 , E n ≅ l ∞ n , where A determines the embeddings of the E n . The class of these Banach spaces is the class of all separable Lindenstrauss spaces. Sufficient and necessary conditions are stated for a matrix A to represent c 0and c. Furthermore, we characterize the class of all extreme triangular substochastic matrices which represents C( K), where K is the Cantor set. We investigate how the special biface structure of the dual unit ball of X is reflected in the elements of a matrix A representing the separable Lindenstrauss space X. This is applicable to Gurarij spaces; we give a new proof for the maximality property of Gurarij spaces and show that they are isomorphic to A( S) where S is a Choquet simplex with dense extreme points.