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Lattice Structures on Banach Spaces
Nigel J. Kalton
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Memoirs of the American Mathematical Society
1993; 92 pp; softcover
Volume: 103
ISBN-10: 0-8218-2557-7
ISBN-13: 978-0-8218-2557-0
List Price: US$34
Individual Members: US$20.40
Institutional Members: US$27.20
Order Code: MEMO/103/493
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The general problem addressed in this work is to characterize the possible Banach lattice structures that a separable Banach space may have. The basic questions of uniqueness of lattice structure for function spaces have been studied before, but here the approach uses random measure representations for operators in a new way to obtain more powerful conclusions. A typical result is the following: If \(X\) is a rearrangement-invariant space on \([0,1]\) not equal to \(L_2\), and if \(Y\) is an order-continuous Banach lattice which has a complemented subspace isomorphic as a Banach space to \(X\), then \(Y\) has a complemented sublattice which is isomorphic to \(X\) (with one of two possible lattice structures). New examples are also given of spaces with a unique lattice structure.

Readership

Research mathematicians specializing in Banach space theory.

Table of Contents

  • Introduction
  • Banach lattices and Köthe function spaces
  • Positive operators
  • The basic construction
  • Lower estimates on \(P\)
  • Reduction to the case of an atomic kernel
  • Complemented subspaces of Banach lattices
  • Strictly \(2\)-concave and strictly \(2\)-convex structures
  • Uniqueness of lattice structure
  • Isomorphic embeddings
  • References
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