Normality of spaces of operators and quasi-lattices

TL;DR

This paper explores the properties of normality in pre-ordered Banach spaces, introduces quasi-lattices as a broader class than Banach lattices, and demonstrates their implications for operator spaces, including new examples like Lorentz cones in Hilbert spaces.

Contribution

It defines quasi-lattices, a new class of ordered Banach spaces, and shows they include spaces beyond Banach lattices, with applications to operator space normality and properties.

Findings

Quasi-lattices include all Banach lattices and more.

Hilbert spaces with Lorentz cones are quasi-lattices but not Banach lattices.

Operator spaces between quasi-lattices are absolutely monotone and have positively attained norms.

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 $\mathcal{H}$ endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if $\dim\mathcal{H}\geq3$), and satisfies an identity analogous to the elementary Banach lattice identity $\||x|\|=\|x\|$ which holds for all elements $x$ of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.