On ultrapowers of Banach spaces of type ${ℒ}_{\infty }$

Antonio Avilés, et al. "On ultrapowers of Banach spaces of type $ℒ_{∞}$." Fundamenta Mathematicae 222.3 (2013): 195-212. <http://eudml.org/doc/282646>.

@article{AntonioAvilés2013,
abstract = {We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain c₀ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose proofs have been infected by that statement. In particular we provide proofs for the following statements: (i) All M-spaces, in particular all C(K)-spaces, have ultrapowers isomorphic to ultrapowers of c₀, as also do all their complemented subspaces isomorphic to their square. (ii) No ultrapower of the Gurariĭ space can be complemented in any M-space. (iii) There exist Banach spaces not complemented in any C(K)-space having ultrapowers isomorphic to a C(K)-space.},
author = {Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno},
journal = {Fundamenta Mathematicae},
keywords = {ultraproducts of Banach spaces; spaces},
language = {eng},
number = {3},
pages = {195-212},
title = {On ultrapowers of Banach spaces of type $ℒ_\{∞\}$},
url = {http://eudml.org/doc/282646},
volume = {222},
year = {2013},
}

TY - JOUR
AU - Antonio Avilés
AU - Félix Cabello Sánchez
AU - Jesús M. F. Castillo
AU - Manuel González
AU - Yolanda Moreno
TI - On ultrapowers of Banach spaces of type $ℒ_{∞}$
JO - Fundamenta Mathematicae
PY - 2013
VL - 222
IS - 3
SP - 195
EP - 212
AB - We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain c₀ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose proofs have been infected by that statement. In particular we provide proofs for the following statements: (i) All M-spaces, in particular all C(K)-spaces, have ultrapowers isomorphic to ultrapowers of c₀, as also do all their complemented subspaces isomorphic to their square. (ii) No ultrapower of the Gurariĭ space can be complemented in any M-space. (iii) There exist Banach spaces not complemented in any C(K)-space having ultrapowers isomorphic to a C(K)-space.
LA - eng
KW - ultraproducts of Banach spaces; spaces
UR - http://eudml.org/doc/282646
ER -