On ultrapowers of Banach spaces of type $ℒ_{∞}$

• Autorzy: Antonio Avilés , Félix Cabello Sánchez , Jesús M. F. Castillo , Manuel González , Yolanda Moreno
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46M07: Ultraproducts
• 46B26: Nonseparable Banach spaces
• 46B08: Ultraproduct techniques in Banach space theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2013
• Tom: 222
• Numer: 3
• Strony: 195-212

Daty

wydano

2013

Twórcy

autor

Antonio Avilés

• Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain

autor

Félix Cabello Sánchez

• Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas s/n, 06071 Badajoz, Spain

autor

Jesús M. F. Castillo

• Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas s/n, 06071 Badajoz, Spain

autor

Manuel González

• Departamento de Matemáticas, Universidad de Cantabria, Avenida los Castros s/n, 39071 Santander, Spain

autor

Yolanda Moreno

• Escuela Politécnica, Universidad de Extremadura, Avenida de la Universidad s/n, 10071 Cáceres, Spain

Identyfikatory

DOI

10.4064/fm222-3-1