A new isomorphic ℓ1
 predual not isomorphic to a complemented subspace of a C
(K
) space

Gasparis, I.

doi:10.1112/blms/bdt002

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Further Remarks and Open Problems1

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2013

2016

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Cited by 3 publications

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“…It would be interesting to add some new classes here. Reasonable candidates could be the recently constructed hereditarily indecomposable L ∞ -spaces [2,39], the preduals of ℓ 1 in [9,17]; or some Bourgain-Pisier spaces [12]. Since both G and C-spaces are Lindenstrauss spaces, one may wonder whether every L ∞ -space has an ultrapower isomorphic to a Lindenstrauss space.…”

Section: Further Remarks and Open Problemsmentioning

confidence: 99%

On ultrapowers of Banach spaces of type \mathscrL_∞

et al. 2013

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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 Gurariȋ 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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