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
HTML articles powered by AMS MathViewer

by Gilles Godefroy PDF

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

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.

References

Éric Amar, Sur un théorème de Mooney relatif aux fonctions analytiques bornées, Pacific J. Math. 49 (1973), 311–314 (French, with English summary). MR 344862, DOI 10.2140/pjm.1973.49.311
T. Ando, On the predual of $H^{\infty }$, Comment. Math. Special Issue 1 (1978), 33–40. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. MR 504151
Sheldon Axler, I. David Berg, Nicholas Jewell, and Allen Shields, Approximation by compact operators and the space $H^{\infty }+C$, Ann. of Math. (2) 109 (1979), no. 3, 601–612. MR 534765, DOI 10.2307/1971228
Ehrhard Behrends, $M$-structure and the Banach-Stone theorem, Lecture Notes in Mathematics, vol. 736, Springer, Berlin, 1979. MR 547509, DOI 10.1007/BFb0063153
J. Bourgain, On weak completeness of the dual of spaces of analytic and smooth functions, Bull. Soc. Math. Belg. Sér. B 35 (1983), no. 1, 111–118. MR 712065
Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125

On sets closed in measure

2

En Russe

34

F. Delbaen, The Pełczyński property for some uniform algebras, Studia Math. 64 (1979), no. 2, 117–125. MR 537115, DOI 10.4064/sm-64-2-117-125
G. A. Edgar, An ordering for the Banach spaces, Pacific J. Math. 108 (1983), no. 1, 83–98. MR 709700, DOI 10.2140/pjm.1983.108.83
Hicham Fakhoury, Existence d’une projection continue de meilleure approximation dans certains espaces de Banach, J. Math. Pures Appl. (9) 53 (1974), 1–16 (French). MR 358183
Tadeusz Figiel, Stanisław Kwapień, and Aleksander Pełczyński, Sharp estimates for the constants of local unconditional structure of Minkowski spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 12, 1221–1226 (English, with Russian summary). MR 487397

Nouvelles classes d’espaces de Banach à prédual unique

Gilles Godefroy, Points de Namioka. Espaces normants. Applications à la théorie isométrique de la dualité, Israel J. Math. 38 (1981), no. 3, 209–220 (French, with English summary). MR 605379, DOI 10.1007/BF02760806
Gilles Godefroy, Parties admissibles d’un espace de Banach. Applications, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 1, 109–122 (French). MR 719765, DOI 10.24033/asens.1442

Propriétés de la classe de espaces de Banach qui sont I’ unique prédual de leur dual

35

Projections becontractantes du bidual

d’un espace

sur l’espace

Banach spaces which are

idèals in their bidual

Henry Helson and David Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165–202. MR 97688, DOI 10.1007/BF02392425
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Daniel H. Luecking, The compact Hankel operators form an $M$-ideal in the space of Hankel operators, Proc. Amer. Math. Soc. 79 (1980), no. 2, 222–224. MR 565343, DOI 10.1090/S0002-9939-1980-0565343-7
Jean-Pierre Kahane, Another theorem on bounded analytic functions, Proc. Amer. Math. Soc. 18 (1967), 827–831. MR 215068, DOI 10.1090/S0002-9939-1967-0215068-6
H. Milne, Banach space properties of uniform algebras, Bull. London Math. Soc. 4 (1972), 323–326. MR 322485, DOI 10.1112/blms/4.3.323
D. J. Newman, Pseudo-uniform convexity in $H^{1}$, Proc. Amer. Math. Soc. 14 (1963), 676–679. MR 151834, DOI 10.1090/S0002-9939-1963-0151834-X
Nils Øvrelid, Integral representation formulas and $L^{p}$-estimates for the $\bar \partial$-equation, Math. Scand. 29 (1971), 137–160. MR 324073, DOI 10.7146/math.scand.a-11041

Banach spaces of analytic functions and absolutely summing operators

Joel H. Shapiro, Subspaces of$L^{p}(G)$ spanned by characters: $0<p<1$, Israel J. Math. 29 (1978), no. 2-3, 248–264. MR 477605, DOI 10.1007/BF02762013
Gilles Pisier, Une nouvelle classe d’espaces de Banach vérifiant le théorème de Grothendieck, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, x, 69–90 (French, with English summary). MR 487403
Roman Pol, On a question of H. H. Corson and some related problems, Fund. Math. 109 (1980), no. 2, 143–154. MR 597061, DOI 10.4064/fm-109-2-143-154
Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
P. Wojtaszczyk, On weakly compact operators from some uniform algebras, Studia Math. 64 (1979), no. 2, 105–116. MR 537114, DOI 10.4064/sm-64-2-105-116
Laurence D. Hoffmann, Pseudo-uniform convexity of $H^{1}$ in several variables, Proc. Amer. Math. Soc. 26 (1970), 609–614. MR 268656, DOI 10.1090/S0002-9939-1970-0268656-5

Espaces

sur des domaines généraux