# Lattice Structures on Banach Spaces

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*Nigel J. Kalton*

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.

#### Table of Contents

# Table of Contents

## Lattice Structures on Banach Spaces

- Table of Contents v6 free
- 1. Introduction 18 free
- 2. Banach lattices and Kothe function spaces 714 free
- 3. Positive operators 2027
- 4. The basic construction 2532
- 5. Lower estimates on P 3037
- 6. Reduction to the case of an atomic kernel 4148
- 7. Complemented subspaces of Banach lattices 5158
- 8. Strictly 2-concave and strictly 2-convex structures 5764
- 9. Uniqueness of lattice structure 7077
- 10. Isomorphic embeddings 7986
- References 8996