On ultrapowers of Banach spaces of type $\mathscr L_\infty$

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.