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Separable lifting property and extensions of local reflexivity
Johnson, William B. ; Oikhberg, Timur
Illinois J. Math., Tome 45 (2001) no. 4, p. 123-137 / Harvested from Project Euclid
A Banach space X is said to have the {\it separable lifting property} if for every subspace Y of X^{**} containing X and such that Y/X is separable there exists a bounded linear lifting from Y/X to Y. We show that if a sequence of Banach spaces E_1, E_2, \ldots has the joint uniform approximation property and E_n is c-complemented in E_n^{**} for every n (with c fixed), then \eco has the separable lifting property. In particular, if E_n is a {\mathcal{L}}_{p_n, \lambda}-space for every n (1 < p_n < \infty, \lambda independent of n), an L_\infty or an L_1 space, then \eco has the separable lifting property. We also show that there exists a Banach space X which is not extendably locally reflexive; moreover, for every n there exists an n-dimensional subspace E \hra X^{**} such that if u : X^{**} \raw X^{**} is an operator (= bounded linear operator) such that u (E) \subset X, then ||(u|_E)^{-1}|| \cdot ||u|| \geq c \sqrt{n}, where c is a numerical constant.
Publié le : 2001-01-15
Classification:  46B10,  46B03,  46B28
@article{1258138258,
     author = {Johnson, William B. and Oikhberg, Timur},
     title = {Separable lifting property and extensions of local reflexivity},
     journal = {Illinois J. Math.},
     volume = {45},
     number = {4},
     year = {2001},
     pages = { 123-137},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1258138258}
}
Johnson, William B.; Oikhberg, Timur. Separable lifting property and extensions of local reflexivity. Illinois J. Math., Tome 45 (2001) no. 4, pp.  123-137. http://gdmltest.u-ga.fr/item/1258138258/