no. 6
Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem
[Dualité sur les espaces de Banach et une version borélienne du théorème de Zippin]
Braga, Bruno de Mendonça1
1 Department of Mathematics, Statistics, and Computer Science (M/C 249) University of Illinois at Chicago 851 S. Morgan St. Chicago, IL 60607-7045 (USA)
Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2413-2435.

Soit SB le codage standard des espaces de Banach séparables comme sous-espaces de C(Δ). Dans ce papier, on montre que si 𝔹⊂SB est un sous-ensemble borélien d’espaces à dual séparable, alors l’application X↦X * peut être réalisée par une fonction borélienne de 𝔹 à SB. En outre, cette application peut être construite de manière que l’évaluation fonctionnelle est toujours bien définie (Théorème 1). Par ailleurs, on démontre une version borélienne du théorème de Zippin. Plus précisément, on démontre qu’il existe Z∈SB et une fonction borélienne qui à chaque X associe une copie isomorphe à X à l’intérieur de Z (Théorème 5).

Let SB be the standard coding for separable Banach spaces as subspaces of C(Δ). In these notes, we show that if 𝔹⊂SB is a Borel subset of spaces with separable dual, then the assignment X↦X * can be realized by a Borel function 𝔹→SB. Moreover, this assignment can be done in such a way that the functional evaluation is still well defined (Theorem 1). Also, we prove a Borel parametrized version of Zippin’s theorem, i.e., we prove that there exists Z∈SB and a Borel function that assigns for each X∈𝔹 an isomorphic copy of X inside of Z (Theorem 5).

DOI : 10.5802/aif.2991
Classification : 46B10
Keywords: Banach spaces, duality, descriptive set theory, Zippin’s theorem
Mot clés : espaces de Banach, dualité, théorie descriptive des ensembles, théorème de Zippin.

Braga, Bruno de Mendonça 1

1 Department of Mathematics, Statistics, and Computer Science (M/C 249) University of Illinois at Chicago 851 S. Morgan St. Chicago, IL 60607-7045 (USA) 
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Braga, Bruno de Mendonça. Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2413-2435. doi : 10.5802/aif.2991. https://aif.centre-mersenne.org/articles/10.5802/aif.2991/

[1] Albiac, Fernando; Kalton, Nigel J. Topics in Banach space theory, Graduate Texts in Mathematics, 233, Springer, New York, 2006, pp. xii+373 | MR | Zbl

[2] Argyros, Spiros A.; Dodos, Pandelis Genericity and amalgamation of classes of Banach spaces, Adv. Math., Volume 209 (2007) no. 2, pp. 666-748 | DOI | MR | Zbl

[3] Bossard, Benoît An ordinal version of some applications of the classical interpolation theorem, Fund. Math., Volume 152 (1997) no. 1, pp. 55-74 | EuDML | MR | Zbl

[4] Davis, W. J.; Figiel, T.; Johnson, W. B.; Pełczyński, A. Factoring weakly compact operators, J. Functional Analysis, Volume 17 (1974), pp. 311-327 | MR | Zbl

[5] Dodos, Pandelis Banach spaces and descriptive set theory: selected topics, Lecture Notes in Mathematics, 1993, Springer-Verlag, Berlin, 2010, pp. xii+161 | DOI | MR | Zbl

[6] Dodos, Pandelis Definability under duality, Houston J. Math., Volume 36 (2010) no. 3, pp. 781-792 | MR | Zbl

[7] Dodos, Pandelis; Ferenczi, Valentin Some strongly bounded classes of Banach spaces, Fund. Math., Volume 193 (2007) no. 2, pp. 171-179 | DOI | MR | Zbl

[8] Kechris, Alexander S. Classical descriptive set theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995, pp. xviii+402 | DOI | MR | Zbl

[9] Schechtman, Gideon On Pełczyński’s paper “Universal bases” (Studia Math. 32 (1969), 247–268), Israel J. Math., Volume 22 (1975) no. 3-4, pp. 181-184 | MR | Zbl

[10] Sclumprecht, Th. Notes on Descriptive Set Theory, and Applications to Banach Spaces (Class notes for Reading Course in Spring/Summer 2008)

[11] Zippin, M. Banach spaces with separable duals, Trans. Amer. Math. Soc., Volume 310 (1988) no. 1, pp. 371-379 | DOI | MR | Zbl

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