(1+)-complemented, (1+)-isomorphic copies of L1 in dual Banach spaces

0562913 - MÚ 2023 RIV CH eng J - Journal Article
Chen, D. - Kania, Tomasz - Ruan, Y.
(1+)-complemented, (1+)-isomorphic copies of L1 in dual Banach spaces.
Archiv der Mathematik. Roč. 119, č. 5 (2022), s. 495-505. ISSN 0003-889X. E-ISSN 1420-8938
Institutional support: RVO:67985840
Keywords : Banach spaces * complemented subspaces * isomorphic copies of L1 * quotient maps
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Open access
https://doi.org/10.1007/s00013-022-01778-2

The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on the Hagler-Stegall characterisation of dual spaces containing complemented copies of L-1. As a corollary, we obtain the following quantitative version of the Hagler-Stegall theorem asserting that for a Banach space X, the following statements are equivalent:

X contains almost isometric contains almost isometric copies of (circle plus(infinity)(n=1) l(infinity)(n))(l1),

for all epsilon > 0, X * contains a (1 + epsilon)-complemented, (1 + epsilon)-isomorphic copy of L-1,

for all epsilon > 0, X * contains a (1 + epsilon)-complemented, (1 + epsilon)-isomorphic copy of C[0, 1]*. Moreover, if X is separable, one may add the following assertion:

for all epsilon > 0, there exists a (1 + epsilon)-quotient map T : X -> C(Delta) so that T*[C(Delta)*] is (1 + epsilon)-complemented in X*, where Delta is the Cantor set
Permanent Link: https://hdl.handle.net/11104/0335079