A+ CATEGORY SCIENTIFIC UNIT

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

Volume 222 / 2013

Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno Fundamenta Mathematicae 222 (2013), 195-212 MSC: 46B08, 46M07, 46B26. DOI: 10.4064/fm222-3-1

Abstract

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.

Authors

  • Antonio AvilésDepartamento de Matemáticas
    Universidad de Murcia
    30100 Espinardo, Murcia, Spain
    e-mail
  • Félix Cabello SánchezDepartamento de Matemáticas
    Universidad de Extremadura
    Avenida de Elvas s/n
    06071 Badajoz, Spain
    e-mail
  • Jesús M. F. CastilloDepartamento de Matemáticas
    Universidad de Extremadura
    Avenida de Elvas s/n
    06071 Badajoz, Spain
    e-mail
  • Manuel GonzálezDepartamento de Matemáticas
    Universidad de Cantabria
    Avenida los Castros s/n
    39071 Santander, Spain
    e-mail
  • Yolanda MorenoEscuela Politécnica
    Universidad de Extremadura
    Avenida de la Universidad s/n
    10071 Cáceres, Spain
    e-mail

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