Normality of spaces of operators and quasi-lattices

Abstract

We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces X and Y with closed cones we investigate normality of B(X,Y) in terms of normality and conormality of the underlying spaces X and Y. Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples X and Y that are not Banach lattices, but for which B(X,Y) is normal. In particular, we show that a Hilbert space H endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if dimH≥3), and satisfies an identity analogous to the elementary Banach lattice identity ∥|x|∥=∥x∥ which holds for all elements x of a Ba