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Positivity

, Volume 22, Issue 1, pp 301–339 | Cite as

New Axiomatizable classes of Banach spaces via disjointness-preserving isometries

  • Yves Raynaud
Article

Abstract

Let \(\mathcal {C}\) be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of \(\mathcal {C}\) in their ultrapowers preserve disjointness, the class \(\mathcal {C}^\mathcal {B}\) of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if \(\mathcal {C}\) coincides with its “script class” \(\mathcal {SC}\), so does \(\mathcal {C}^\mathcal {B}\) with \(\mathcal {SC}^\mathcal {B}\). This allows us to give new examples of axiomatizable classes of Banach spaces, namely certain Musielak–Orlicz spaces, Nakano spaces, and mixed norm spaces.

Keywords

Banach space Banach lattice Axiomatizable class Ultraproducts and ultraroots Disjointness preserving isometries 

Mathematics Subject Classification

Primary 46B42 46E30 Secondary 03C65 46M07 

Notes

Acknowledgements

The author thanks C. W. Henson for his critical reading of a previous version of the manuscript and his suggestions for improving it.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut Math. Jussieu-Paris-Rive GaucheCNRS, UPMC (Univ. Paris 06) and Univ. Paris-DiderotParis Cedex 05France

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