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

摘要: We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain $c_0$ complemented. This shows "result" widely used in the theory ultraproducts is wrong. then amend number results whose proofs had been infected by statement. In particular we provide for following statements: (i) All $M$-spaces, all $C(K)$-spaces, have ultrapowers isomorphic to $c_0$, as well their complemented subspaces square. (ii) No ultrapower Gurari\u \i\ space be any $M$-space. (iii) There exist not $C(K)$-space having $C(K)$-space.