Geodesic
On ultrapowers of Banach spaces of type $\mathscr{L}_{\infty} $
Antonio Avilés1 ; Félix Cabello Sánchez2 ; Jesús M. F. Castillo2 ; Manuel González3 ; Yolanda Moreno4
1 Departamento de Matemáticas Universidad de Murcia 30100 Espinardo, Murcia, Spain
2 Departamento de Matemáticas Universidad de Extremadura Avenida de Elvas s/n 06071 Badajoz, Spain
3 Departamento de Matemáticas Universidad de Cantabria Avenida los Castros s/n 39071 Santander, Spain
4 Escuela Politécnica Universidad de Extremadura Avenida de la Universidad s/n 10071 Cáceres, Spain
Fundamenta Mathematicae, Tome 222 (2013) no. 3, pp. 195-212

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain $c_0$ 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 $c_0$, 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.
DOI : 10.4064/fm222-3-1
Mots-clés : prove ultraproduct banach spaces via countably incomplete ultrafilter contain complemented shows result widely theory ultraproducts wrong amend number results whose proofs have infected statement particular provide proofs following statements m spaces particular spaces have ultrapowers isomorphic ultrapowers their complemented subspaces isomorphic their square ultrapower gurari space complemented m space iii there exist banach spaces complemented space having ultrapowers isomorphic space

Antonio Avilés 1 ; Félix Cabello Sánchez 2 ; Jesús M. F. Castillo 2 ; Manuel González 3 ; Yolanda Moreno 4

1 Departamento de Matemáticas Universidad de Murcia 30100 Espinardo, Murcia, Spain
2 Departamento de Matemáticas Universidad de Extremadura Avenida de Elvas s/n 06071 Badajoz, Spain
3 Departamento de Matemáticas Universidad de Cantabria Avenida los Castros s/n 39071 Santander, Spain
4 Escuela Politécnica Universidad de Extremadura Avenida de la Universidad s/n 10071 Cáceres, Spain 
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Antonio Avilés; Félix Cabello Sánchez; Jesús M. F. Castillo; Manuel González; Yolanda Moreno. On ultrapowers of Banach spaces of type $\mathscr{L}_{\infty} $. Fundamenta Mathematicae, Tome 222 (2013) no. 3, pp. 195-212. doi: 10.4064/fm222-3-1

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