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The dual of the Bourgain-Delbaen space. (English) Zbl 0956.46017

A basic question in Banach space theory asks if the only \({\mathcal L}_{\infty}\)-space that is uniformly homeomorphic to \(c_0\) is \(c_0\) itself. A necessary condition that an \({\mathcal L}_{\infty}\)-spaces be uniformly homeomorphic to \(c_0\) is a consequence of the work of W.B. Johnson, J. Lindenstrauss and G. Schechtman [Geom.Funct.Anal.6, No.3, 430–470 (1996; Zbl 0864.46008)]. Using a construction developed by J. Bourgain and F. Delbaen [Acta Math. 145, 155-176 (1980; Zbl 0466.46024)], the current paper shows that the necessary condition is not sufficient to conclude that an \({\mathcal L}_{\infty}\)-space is \(c_0\). The proof is a delicate technical argument examining the \(w^*\)-topology on the dual space.

MSC:

46B25 Classical Banach spaces in the general theory
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References:

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