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\(Q\)-reflexive Banach spaces. (English) Zbl 0916.46011
The authors consider Banach spaces \(E\) such that every (continuous) polynomial on \(E\) is weakly continuous on bounded sets. In this case, every \(n\)-homogeneous polynomial \(P\in{\mathcal P}(^nE)\) has a unique extension to \(E^{**}\) which is weak\(^*\) continuous on bounded sets. So using the adjoint of the extension map, it is possible to define in a unique way a continuous linear map from \({\mathcal P}(^nE)^{**}\) into \({\mathcal P}(^nE^{**})\). The space \(E\) is said to be \(Q\)-reflexive if this map is an isomorphism.
The authors show that if no spreading model of a Banach space \(E\) admits a lower \(q\)-estimate for any \(q<\infty\) and \(E^{**}\) has the Radon-Nikodým property and the approximation property, then \(E\) is \(Q\)-reflexive. The quasi-reflexive James space modeled on the original Tsirelson space \(T^*_J\) satisfies these properties. So it is an example of a nonreflexive \(Q\)-reflexive space.

MSC:
46B28 Spaces of operators; tensor products; approximation properties
46B10 Duality and reflexivity in normed linear and Banach spaces
46G20 Infinite-dimensional holomorphy
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