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Sous-espaces bien disposés de $ L\sp{1}$-applications


Author: Gilles Godefroy
Journal: Trans. Amer. Math. Soc. 286 (1984), 227-249
MSC: Primary 46B25; Secondary 32A35, 46J15
DOI: https://doi.org/10.1090/S0002-9947-1984-0756037-1
MathSciNet review: 756037
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Abstract: RÉsumÉ. On montre que le quotient d'un espace $ {L^1}$ par un sous-espace fermé dont la boule unité est fermée dans $ {L^0}$ est faiblement séquentiellement complet; cette situation se présente dans de nombreux cas concrets, tels que le quotient $ {L^1}/{H^1}$. On applique le résultat général dans diverses situations: duaux de certaines algères uniformes, analyse harmonique, fonctions de plusieurs variables complexes. On montre ensuite comment peuvent s'appliquer les métheodes de $ M$-structure; on considère aussi de nouvelles classes d'uniques préduaux. A titre d'exemples, on montre:

(1) Le caractère f.s.c. d'espaces $ {\mathcal{C}_E}{(G)^\ast}$, pour de "gros" sous-ensembles $ E$ du groupe dual $ \Gamma = \hat G$.

(2) Le caractère f.s.c. d'espaces $ {L^1}/{H^1}$ mutli-dimensionnels, tels que $ {L^1}/{H^1}({D^n})$ et $ {L^1}/{H^1}({B^n})$.

(3) L'unicité du prédual pour certaines sous-algèbres ultrafaiblement fermées non-autoadjointes de $ \mathcal{L}(H)$.

One shows that the quotient of an $ {L^1}$-space by a closed subspace, whose unit ball is closed in $ {L^0}$, is weakly sequentially complete. This situation occurs in many natural cases, like $ {L^1}/{H^1}$. This result is applied in several situations: uniform algebras, harmonic analysis, functions of several complex variables. One shows how to apply $ M$-structure theory; several new classes of unique preduals are also obtained. As an example, one shows:

(1) If $ E$ is a "big" subset of the dual group $ \Gamma = \hat G$, then $ {\mathcal{C}_E}{(G)^\ast}$ is w.s.c.

(2) The spaces $ {L^1}/{H^1}({D^n})$ and $ {L^1}/{H^1}({B^n})$ are w.s.c.

(3) Several classes of $ {\omega ^\ast}$-closed non-self-adjoint subalgebras of $ \mathcal{L}(H)$ have unique preduals.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0756037-1
Keywords: Measure convergence, small subsets of discrete groups, Dirichlet algebras, Hardy spaces, unicity of preduals
Article copyright: © Copyright 1984 American Mathematical Society

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