no. 6
Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem
Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2413-2435.

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).

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).

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2991
Classification: 46B10
Keywords: Banach spaces, duality, descriptive set theory, Zippin’s theorem 
@article{AIF_2015__65_6_2413_0,
     author = {Braga, Bruno de Mendon\c{c}a},
     title = {Duality on {Banach} spaces and a {Borel} parametrized version of {Zippin{\textquoteright}s} theorem},
     journal = {Annales de l'Institut Fourier},
     pages = {2413--2435},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2991},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2991/}
}
TY  - JOUR
TI  - Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem
JO  - Annales de l'Institut Fourier
PY  - 2015
DA  - 2015///
SP  - 2413
EP  - 2435
VL  - 65
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2991/
UR  - https://doi.org/10.5802/aif.2991
DO  - 10.5802/aif.2991
LA  - en
ID  - AIF_2015__65_6_2413_0
ER  - 
%0 Journal Article
%T Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem
%J Annales de l'Institut Fourier
%D 2015
%P 2413-2435
%V 65
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2991
%R 10.5802/aif.2991
%G en
%F AIF_2015__65_6_2413_0
Braga, Bruno de Mendonça. Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem. Annales de l'Institut Fourier, Volume 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, Tome 233, Springer, New York, 2006, xii+373 pages | MR: 2192298 | Zbl: 1094.46002

[2] Argyros, Spiros A.; Dodos, Pandelis Genericity and amalgamation of classes of Banach spaces, Adv. Math., Tome 209 (2007) no. 2, pp. 666-748 | Article | MR: 2296312 | Zbl: 1109.03047

[3] Bossard, Benoît An ordinal version of some applications of the classical interpolation theorem, Fund. Math., Tome 152 (1997) no. 1, pp. 55-74 | EuDML: 212199 | MR: 1434377 | Zbl: 0901.46011

[4] Davis, W. J.; Figiel, T.; Johnson, W. B.; Pełczyński, A. Factoring weakly compact operators, J. Functional Analysis, Tome 17 (1974), pp. 311-327 | MR: 355536 | Zbl: 0306.46020

[5] Dodos, Pandelis Banach spaces and descriptive set theory: selected topics, Lecture Notes in Mathematics, Tome 1993, Springer-Verlag, Berlin, 2010, xii+161 pages | Article | MR: 2598479 | Zbl: 1215.46002

[6] Dodos, Pandelis Definability under duality, Houston J. Math., Tome 36 (2010) no. 3, pp. 781-792 | MR: 2727002 | Zbl: 1226.03053

[7] Dodos, Pandelis; Ferenczi, Valentin Some strongly bounded classes of Banach spaces, Fund. Math., Tome 193 (2007) no. 2, pp. 171-179 | Article | MR: 2282714 | Zbl: 1115.03061

[8] Kechris, Alexander S. Classical descriptive set theory, Graduate Texts in Mathematics, Tome 156, Springer-Verlag, New York, 1995, xviii+402 pages | Article | MR: 1321597 | Zbl: 0819.04002

[9] Schechtman, Gideon On Pełczyński’s paper “Universal bases” (Studia Math. 32 (1969), 247–268), Israel J. Math., Tome 22 (1975) no. 3-4, pp. 181-184 | MR: 390730 | Zbl: 0316.46014

[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., Tome 310 (1988) no. 1, pp. 371-379 | Article | MR: 965758 | Zbl: 0706.46015

Cited by Sources: