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Lattice Structures on Banach Spaces
 
Lattice Structures on Banach Spaces
eBook ISBN:  978-1-4704-0070-5
Product Code:  MEMO/103/493.E
List Price: $36.00
MAA Member Price: $32.40
AMS Member Price: $21.60
Lattice Structures on Banach Spaces
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Lattice Structures on Banach Spaces
eBook ISBN:  978-1-4704-0070-5
Product Code:  MEMO/103/493.E
List Price: $36.00
MAA Member Price: $32.40
AMS Member Price: $21.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1031993; 92 pp
    MSC: Primary 46;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Banach lattices and Köthe function spaces
    • 3. Positive operators
    • 4. The basic construction
    • 5. Lower estimates on $P$
    • 6. Reduction to the case of an atomic kernel
    • 7. Complemented subspaces of Banach lattices
    • 8. Strictly 2-concave and strictly 2-convex structures
    • 9. Uniqueness of lattice structure
    • 10. Isomorphic embeddings
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1031993; 92 pp
MSC: Primary 46;

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.

  • Chapters
  • 1. Introduction
  • 2. Banach lattices and Köthe function spaces
  • 3. Positive operators
  • 4. The basic construction
  • 5. Lower estimates on $P$
  • 6. Reduction to the case of an atomic kernel
  • 7. Complemented subspaces of Banach lattices
  • 8. Strictly 2-concave and strictly 2-convex structures
  • 9. Uniqueness of lattice structure
  • 10. Isomorphic embeddings
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.