Denting and strongly extreme points in the unit ball of spaces of operators
The following are the main results of this interesting paper: (1) There are no denting points in the unit ball of \(\mathcal L(\ell^p,Y)\) whenever there is a non-compact operator and the space of compact operators \(\mathcal K(\ell^p,Y)\) is an M-ideal in \(\mathcal L(\ell^p,Y)\). (2) The unit ball of the space \(\mathcal L(L^1(\mu),X)\) has a point of weak-norm continuity iff \(L^1(\mu)\) is finite dimensional and the unit ball of X has a point of weak-norm continuity. (3) There are no points of weak-norm continuity in the unit ball of \(\mathcal L(X,C(K))\), where \(K\) is an infinite, totally disconnected compact Hausdorff space. (4) Suppose \(X^*\) has the extreme point intersection property (i.e., \(|x*(x)|\) = 1 for all \(x \in \delta_e X_1\) and all \(x* \in \delta_eX*_1\)). Then any `nice' operator \(T\) in the unit ball of \(\mathcal L(X,Y)\) (i.e., \(T*(\delta_eB(Y*)) \subset \delta_eB(X*)\), where B denotes the unit ball of the space indicated) is a strongly extreme point.
scientific article; zbMATH DE number 1340037