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 \(\dim \mathcal {H}\ge 3\)), and satisfies an identity analogous to the elementary Banach lattice identity \(\Vert |x|\Vert =\Vert x\Vert \) 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.
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Notes
A note on terminology: The terms ‘normality’ (due to Krein [14]) and ‘monotonicity’ are fairly standard terms throughout the literature. Our consistent use of the adjective ‘absolute’ is inspired by [22] and mimics its use in the term ‘absolute value’.
The concept that we will call ‘conormality’ has seen numerous equivalent definitions and the nomenclature is rather varied in the existing literature. The term ‘conormality’ is due to Walsh [20], who studied the property in the context of locally convex spaces. What we will call ‘\(1\)-max-conormality’ occurs under the name ‘strict bounded decomposition property’ in [7]. The properties that we will call ‘approximate \(1\)-absolute conormality’ and ‘approximate \(1\)-conormality’, were first defined (but not named) respectively by Davies [9] and Ng [16]. Batty and Robinson give equivalent definitions for our conormality properties which they call ‘dominating’ and ‘generating’ [5].
There is a small error in the statement of (1) in [5, Proposition 1.7.8.]. We give its correct statement and proof.
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Acknowledgments
The author would like to thank Marcel de Jeu, Ben de Pagter, Onno van Gaans, Anthony Wickstead and Marten Wortel for many fruitful discussions on this topic. Thanks should also be given to the MathOverflow community, in particular to Anton Petrunin for ideas contained in Theorem 9. The author’s research was supported by a Vrije Competitie Grant of the Netherlands Organisation for Scientific Research (NWO).
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Messerschmidt, M. Normality of spaces of operators and quasi-lattices. Positivity 19, 695–724 (2015). https://doi.org/10.1007/s11117-015-0323-y
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DOI: https://doi.org/10.1007/s11117-015-0323-y