# Unconditional ideals in Banach spaces

G. Godefroy; N. Kalton; P. Saphar

Studia Mathematica (1993)

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

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topGodefroy, 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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