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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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