On ultrapowers of Banach spaces of type $\mathscr{L}_{\infty} $
Volume 222 / 2013
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.
