Extremal structure in operator spaces

Abstract

a) A characterization of extreme operators (in the unit ball of operators) between L 1'-spaces is given, together with other related proper- ties. (b) A general theorem of Kreln-Milman type for the unit ball of operator spaces is proved, and is applied to operators between Ll-spaces and to oper- ators into C-spaces. 1. Introduction. In this paper we study in detail extreme points in the unit ball of operator spaces (which we shall briefly call extreme operators). Although we treat mostly operators in L '-spaces and C-spaces, some of the results are general (?3). Before discussing the results, we go briefly over the notations. They are mostly standard and we refer to (14) for some of the notations, though we remark that if E is a Banach space, the action of x* E E* on x E E will be denoted either by x*(x) or by (x*, x). The canonical imbedding of E into E** will be denoted by JE. The scalars are either real or complex; only in Theorem 2.10 are they assumed to be real. Generalizing a notation of Morris and Phelps (121, we call an operator T: E -. F, for two Banach spaces E and F, a nice operator, if T*(ext S(F*)) C ext S(E*). It is easy to verify that every nice oper- ator is extreme, and to construct examples where there exist no nice operators