Sous-espaces bien disposés de 𝐿¹-applications

Godefroy, Gilles

doi:10.1090/S0002-9947-1984-0756037-1

Sous-espaces bien disposés de $L^{1}$-applications
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by Gilles Godefroy

Trans. Amer. Math. Soc. 286 (1984), 227-249

DOI: https://doi.org/10.1090/S0002-9947-1984-0756037-1

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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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