Extreme and nice operators (Q1077697)

If E is a complex (or real) Banach space, we denote its closed unit ball by S(E) and the set of extreme points of S(E) by Ext S(E). If E and F are complex (or real) Banach spaces, S(E,F) is the closed unit ball of the Banach space L(E,F) of all bounded linear operators from E into F (with the usual sup-norm). An extreme point of S(E,F) is called extreme operator for E into F. We say that T in L(E,F) is a nice operator if: \(T^*(Ext S(F^*))\subset Ext S(E^*)\), where \(E^*\) and \(F^*\) are the topological duals of E and F. We can easily see that every nice operator is extreme. But, the converse is not true: for example, the embedding operator from \(\ell^ 1\) into \(\ell^ 2\) is an extreme, non-nice operator. Several authors proved that, under various additional assumptions on E and F, every extreme operator is also nice; among them R. R. Phelps, J. Lindenstrauss, M. Sharir. Also, every extreme operator having some additional property is nice. In this paper, the main published results concerning the relationship between extreme and nice operators are listed, and some new proofs and ideas are also given. In particular, if X and Y are compact Hausdorff spaces, we give the following new characterization of a nice operator of L(C(X),C(Y)), where C(X) is the Banach space of all complex valued continuous functions on X (with the usual sup-norm): T in S(C(X),C(Y)) is nice if, and only if, T is essentially multiplicative i.e. \(T(fg)=T(1)\) T(f) T(g) for all f and g in C(X) and \(| T(1)| =1\); T is also nice if, and only if, T is extreme and verifies \(| T(1)| =1.\) Let \(C_ R(X)\) be the Banach space of all real valued continuous functions on X. By using the algebraic and order properties of \(C_ R(X)\) and \(C_ R(Y)\) and the last previous characterization of a nice operator from C(X) into C(Y), we give a short proof of the following result: if \({\mathcal A}\) is a Banach sublattice of \(C_ R(X)\) possessing an order unit and if Y is stonian, every extreme operator of L(\({\mathcal A},C_ R(Y))\) is nice. In this special context, we write explicitely the expression of the operators \(T^+\) and \(T^-\), when T is an operator of \(S(C_ R(X),C_ R(Y))\) such that \(| T(1)| =1\), and we characterize the extreme operators of \(L(C_ R(X),C_ R(Y))\) in terms of \(T^+\) and \(T^-.\) In this paper, the weaker notion of an almost-nice operator is also introduced: T in L(E,C(Y)) is almost-nice if the set \(J=\{y\in Y|\) \(T^*(\delta_ y)\in Ext S(E^*)\}\) is dense in Y. The two definitions of a nice operator and of an almost-nice operator do not differ when Ext S(E\({}^*)\) is \(w^*\)-closed. This new notion might lead to better results than those obtained for nice operators.