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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 \(\mathcal {H}\) 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

  1. 1.

    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].

  2. 2.

    The term ‘regularity’ is due to Davies [9]. Our naming convention is to attach the names of the persons who (to the author’s knowledge) first proved the relevant normality-conormality duality results of the defining properties (cf. Theorem 2).

  3. 3.

    There is a small error in the statement of (1) in [5, Proposition 1.7.8.]. We give its correct statement and proof.

References

  1. 1.

    Aliprantis, C.D., Border, K.C.: Infinite dimensional analysis, 3rd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. 2.

    Aliprantis, C.D., Tourky, R.: Cones and duality. American Mathematical Society, Providence (2007)

    MATH  Book  Google Scholar 

  3. 3.

    Andô, T.: On fundamental properties of a Banach space with a cone. Pacific J. Math. 12, 1163–1169 (1962)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Barbu, V., Precupanu, Th: Convexity and optimization in Banach spaces. In: Reidel, D. (ed.) Dordrecht (1986)

  5. 5.

    Batty, C.J.K., Robinson, D.W.: Positive one-parameter semigroups on ordered Banach spaces. Acta Appl. Math. 2(3–4), 221–296 (1984)

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    Bonsall, F.F.: The decomposition of continuous linear functionals into non-negative components. Proc. Univ. Durham Philos. Soc. Ser. A 13, 6–11 (1957)

    MathSciNet  Google Scholar 

  7. 7.

    Bonsall, F.F.: Linear operators in complete positive cones. Proc. Lond. Math. Soc. 3(8), 53–75 (1958)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Clément, Ph, Heijmans, H.J.A.M., Angenent, S., van Duijn, C.J., de Pagter, B.: One-parameter semigroups. North-Holland Publishing Co., Amsterdam (1987)

    MATH  Google Scholar 

  9. 9.

    Davies, E.B.: The structure and ideal theory of the predual of a Banach lattice. Trans. Am. Math. Soc. 131, 544–555 (1968)

    MATH  Article  Google Scholar 

  10. 10.

    de Jeu, M., Messerschmidt, M.: Crossed products of Banach algebras III. arXiv:1306.6290 (2013)

  11. 11.

    de Jeu, M., Messerschmidt, M.: A strong open mapping theorem for surjections from cones onto Banach spaces. Adv. Math. 259, 43–66 (2014)

  12. 12.

    Ellis, A.J.: The duality of partially ordered normed linear spaces. J. Lond. Math. Soc. 39, 730–744 (1964)

    MATH  Article  Google Scholar 

  13. 13.

    Grosberg, J., Krein, M.: Sur la décomposition des fonctionnelles en composantes positives. C. R. (Doklady) Acad. Sci. URSS (N.S.) 25, 723–726 (1939)

    MathSciNet  Google Scholar 

  14. 14.

    Krein, M.: Propriétés fondamentales des ensembles coniques normaux dans l’espace de Banach. C. R. (Doklady) Acad. Sci. URSS (N.S.) 28, 13–17 (1940)

    MathSciNet  Google Scholar 

  15. 15.

    Megginson, R.E.: An introduction to Banach space theory. Springer, New York (1998)

    MATH  Book  Google Scholar 

  16. 16.

    Ng, K.F.: The duality of partially ordered Banach spaces. Proc. Lond. Math. Soc. 3(19), 269–288 (1969)

    Article  Google Scholar 

  17. 17.

    Ng, K.F.: On order and topological completeness. Math. Ann. 196, 171–176 (1972)

    MATH  MathSciNet  Article  Google Scholar 

  18. 18.

    Ng, K.F., Law, C.K.: Monotonic norms in ordered Banach spaces. J. Aust. Math. Soc. Ser. A 45(2), 217–219 (1988)

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Robinson, D.W., Yamamuro, S.: The canonical half-norm, dual half-norms, and monotonic norms. Tohoku Math. J. (2) 35(3), 375–386 (1983)

    MATH  MathSciNet  Article  Google Scholar 

  20. 20.

    Walsh, B.: Ordered vector sequence spaces and related classes of linear operators. Math. Ann. 206, 89–138 (1973)

    MATH  MathSciNet  Article  Google Scholar 

  21. 21.

    Wickstead, A.W.: Spaces of linear operators between partially ordered Banach spaces. Proc. Lond. Math. Soc. 28(3), 141–158 (1974)

    MATH  MathSciNet  Article  Google Scholar 

  22. 22.

    Wong, Y.C., Ng, K.F.: Partially ordered topological vector spaces. Clarendon Press, Oxford (1973)

    MATH  Google Scholar 

  23. 23.

    Yamamuro, S.: On linear operators on ordered Banach spaces. Bull. Aust. Math. Soc. 27(2), 285–305 (1983)

    MATH  MathSciNet  Article  Google Scholar 

  24. 24.

    Zaanen, A.C.: Introduction to operator theory in Riesz spaces. Springer, Berlin (1997)

    MATH  Book  Google Scholar 

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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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Correspondence to Miek Messerschmidt.

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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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Keywords

  • (Pre)-ordered Banach space
  • Operator norm
  • Quasi-lattice
  • Normality
  • Conormality
  • Lorentz cone

Mathematics Subject Classification

  • Primary 46B40
  • Secondary 47B60
  • 47H07
  • 46B42
  • 46A40