On ultrapowers of Banach spaces of type \mathscr L_∞

2013/07/16 by Avilés, Antonio, Sánchez, Félix Cabello, Castillo, Jesús M. F. +2
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1307.4387

Abstract

We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain c₀ 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 c₀, as well as all their complemented subspaces isomorphic to their square. (ii) No ultrapower of the Gurari\u ı 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.

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