This is an announcement for the paper "On ultrapowers of Banach spaces of type $\mathscr L_\infty$" by Antonio Aviles, Felix Cabello Sanchez, Jesus M. F. Castillo, Manuel Gonzalez and Yolanda Moreno.

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 had 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 well as all their complemented subspaces isomorphic to their square. (ii) No ultrapower of the Gurari\u \i\ 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.

Archive classification: math.FA

Remarks: This paper is to appear in Fundamenta Mathematica

Submitted from: castillo@unex.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1307.4387

or

http://arXiv.org/abs/1307.4387