On the Theory of L p (L q )-Banach Lattices

On the Theory of L p (L q )-Banach Lattices Given 1≤ p,q < ∞, let BL p L q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L p (L q )-Banach lattice. We show that the range of a positive contractive projection on any BL p L q -Banach lattice is itself in BL p L q . It is a consequence of this theorem and previous results that BL p L q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL p L q -Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL p L q . We also consider the class of all sublattices of L p (L q )-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L p -Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On the Theory of L p (L q )-Banach Lattices

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Publisher
Springer Journals
Copyright
Copyright © 2007 by Birkhäuser Verlag, Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-006-2023-0
Publisher site
See Article on Publisher Site

Abstract

Given 1≤ p,q < ∞, let BL p L q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L p (L q )-Banach lattice. We show that the range of a positive contractive projection on any BL p L q -Banach lattice is itself in BL p L q . It is a consequence of this theorem and previous results that BL p L q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL p L q -Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL p L q . We also consider the class of all sublattices of L p (L q )-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L p -Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.

Journal

PositivitySpringer Journals

Published: Apr 6, 2007

References

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