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ArticleOriginal scientific text

Title

Unconditional ideals in Banach spaces

Authors 1, 2, 2, 3

Affiliations

  1. Equipe d'Analyse, Université Paris VI, 4, Place Jussieu, F-75252 Paris Cedex 05, France
  2. Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211, U.S.A.
  3. Department of Mathematics, Technion, Haifa, Israel

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**.

Keywords

ENG
M-ideal, hermitian operator, unconditional convergence

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Studia Mathematica
1993/104/1
Export to PBN, Opens in new tab
Pages:
13-59
Main language of publication
English
Received
1991-09-24
Accepted
1992-09-17
Published
1993
Exact and natural sciences