Unconditional ideals in Banach spaces

G. Godefroy; N. Kalton; P. Saphar

Studia Mathematica (1993)

  • Volume: 104, Issue: 1, page 13-59
  • ISSN: 0039-3223

Abstract

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We show that a Banach space with separable dual can be renormed to satisfy hereditarily an “almost” optimal uniform smoothness condition. The optimal condition occurs when the canonical decomposition X * * * = X ⊥ ⊕ X * is unconditional. Motivated by this result, we define a subspace X of a Banach space Y to be an h-ideal (resp. a u-ideal) if there is an hermitian projection P (resp. a projection P with ∥I-2P∥ = 1) on Y* with kernel X ⊥ . We undertake a general study of h-ideals and u-ideals. For example we show that if a separable Banach space X is an h-ideal in X** then X has the complex form of Pełczyński’s property (u) with constant one and the Baire-one functions Ba(X) in X** are complemented by an hermitian projection; the converse holds under a compatibility condition which is shown to be necessary. We relate these ideas to the more familiar notion of an M-ideal, and to Banach lattices. We further investigate when, for a separable Banach space X, the ideal of compact operators K(X) is a u-ideal or an h-ideal in ℒ(X) or K(X)**. For example, we show that K(X) is an h-ideal in K(X)** if and only if X has the “unconditional compact approximation property” and X is an M-ideal in X**.

How to cite

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Godefroy, G., Kalton, N., and Saphar, P.. "Unconditional ideals in Banach spaces." Studia Mathematica 104.1 (1993): 13-59. <http://eudml.org/doc/215957>.

@article{Godefroy1993,
abstract = {We show that a Banach space with separable dual can be renormed to satisfy hereditarily an “almost” optimal uniform smoothness condition. The optimal condition occurs when the canonical decomposition $X*** = X^\{⊥\} ⊕ X*$ is unconditional. Motivated by this result, we define a subspace X of a Banach space Y to be an h-ideal (resp. a u-ideal) if there is an hermitian projection P (resp. a projection P with ∥I-2P∥ = 1) on Y* with kernel $X^\{⊥\}$. We undertake a general study of h-ideals and u-ideals. For example we show that if a separable Banach space X is an h-ideal in X** then X has the complex form of Pełczyński’s property (u) with constant one and the Baire-one functions Ba(X) in X** are complemented by an hermitian projection; the converse holds under a compatibility condition which is shown to be necessary. We relate these ideas to the more familiar notion of an M-ideal, and to Banach lattices. We further investigate when, for a separable Banach space X, the ideal of compact operators K(X) is a u-ideal or an h-ideal in ℒ(X) or K(X)**. For example, we show that K(X) is an h-ideal in K(X)** if and only if X has the “unconditional compact approximation property” and X is an M-ideal in X**.},
author = {Godefroy, G., Kalton, N., Saphar, P.},
journal = {Studia Mathematica},
keywords = {M-ideal; hermitian operator; unconditional convergence; unconditional ideals; summand; principle of local reflexivity; Hermitian operator; unconditional compact approximation property; Banach space with separable dual; renormed; optimal uniform smoothness condition; canonical decomposition; h-ideal; u-ideal; Pełczyński’s property (u); Baire-one functions; Banach lattices; ideal of compact operators},
language = {eng},
number = {1},
pages = {13-59},
title = {Unconditional ideals in Banach spaces},
url = {http://eudml.org/doc/215957},
volume = {104},
year = {1993},
}

TY - JOUR
AU - Godefroy, G.
AU - Kalton, N.
AU - Saphar, P.
TI - Unconditional ideals in Banach spaces
JO - Studia Mathematica
PY - 1993
VL - 104
IS - 1
SP - 13
EP - 59
AB - We show that a Banach space with separable dual can be renormed to satisfy hereditarily an “almost” optimal uniform smoothness condition. The optimal condition occurs when the canonical decomposition $X*** = X^{⊥} ⊕ X*$ is unconditional. Motivated by this result, we define a subspace X of a Banach space Y to be an h-ideal (resp. a u-ideal) if there is an hermitian projection P (resp. a projection P with ∥I-2P∥ = 1) on Y* with kernel $X^{⊥}$. We undertake a general study of h-ideals and u-ideals. For example we show that if a separable Banach space X is an h-ideal in X** then X has the complex form of Pełczyński’s property (u) with constant one and the Baire-one functions Ba(X) in X** are complemented by an hermitian projection; the converse holds under a compatibility condition which is shown to be necessary. We relate these ideas to the more familiar notion of an M-ideal, and to Banach lattices. We further investigate when, for a separable Banach space X, the ideal of compact operators K(X) is a u-ideal or an h-ideal in ℒ(X) or K(X)**. For example, we show that K(X) is an h-ideal in K(X)** if and only if X has the “unconditional compact approximation property” and X is an M-ideal in X**.
LA - eng
KW - M-ideal; hermitian operator; unconditional convergence; unconditional ideals; summand; principle of local reflexivity; Hermitian operator; unconditional compact approximation property; Banach space with separable dual; renormed; optimal uniform smoothness condition; canonical decomposition; h-ideal; u-ideal; Pełczyński’s property (u); Baire-one functions; Banach lattices; ideal of compact operators
UR - http://eudml.org/doc/215957
ER -

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Citations in EuDML Documents

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  1. Ăsvald Lima, Property (wM*) and the unconditional metric compact approximation property
  2. J. Cabello, E. Nieto, An ideal characterization of when a subspace of certain Banach spaces has the metric compact approximation property
  3. Kamil John, On a result of J. Johnson
  4. Eve Oja, Märt Põldvere, On subspaces of Banach spaces where every functional has a unique norm-preserving extension
  5. Kamil John, Dirk Werner, M -ideals of compact operators into ℓ p
  6. Sophie Grivaux, Jan Rychtář, Invariant subspaces of X * * under the action of biconjugates
  7. Trond A. Abrahamsen, Asvald Lima, Vegard Lima, Unconditional ideals of finite rank operators
  8. Kamil John, U-ideals of factorable operators
  9. Åsvald Lima, Eve Oja, Ideals of finite rank operators, intersection properties of balls, and the approximation property
  10. Giovanni Emmanuele, Kamil John, Uncomplementability of spaces of compact operators in larger spaces of operators

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