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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-1,
author = {Antonio Avil\'es and F\'elix Cabello S\'anchez and Jes\'us M. F. Castillo and Manuel Gonz\'alez and Yolanda Moreno},
title = {On ultrapowers of Banach spaces of type $L\_{[?]}$
},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {195-212},
zbl = {1300.46010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-1}
}
Antonio Avilés; Félix Cabello Sánchez; Jesús M. F. Castillo; Manuel González; Yolanda Moreno. On ultrapowers of Banach spaces of type $ℒ_{∞}$
. Fundamenta Mathematicae, Tome 220 (2013) pp. 195-212. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-3-1/