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A superreflexive Banach space which does not admit complex structure. (English) Zbl 0604.46019
An infinite-dimensional superreflexive real Banach space which does not admit complex structure and consequently is not isomorphic to the Cartesian square of any Banach space is constructed; before only nonreflexive such examples were known. Also, a variant of Bourgain’s example of a complex Banach space with nonunique complex structure is presented. Other related results were since obtained by the author, see ”A Banach space without a basis, which has the bounded approximation property”, Acta. Math., in print.

MSC:
46B20 Geometry and structure of normed linear spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
47L05 Linear spaces of operators
52A22 Random convex sets and integral geometry (aspects of convex geometry)
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