On the Radon-Nikodym property and the Lewis-Radon-Nikodym property for a tensor norm

Abstract

The Radon-Nikodym property and the Lewis-Radon-Nikodym property for tensor norms are introduced and discussed. In particular, it is shown that the Hilbertian tensor norm h introduced in [6, Section 3] has the Lewis-Radon-Nikodym property but does not have the Radon-Nikodym property. Instead, if α is a tensor norm that has the Lewis-Radon-Nikodym property, then it holds that α/ has the Radon-Nikodym property. However, we single out another of Grothendieck's natural tensor norms, namely the projective tensor norm ∧, that does have the Radon-Nikodym property. Furthermore, it is shown that if α is a tensor norm with the Radon-Nikodym property, then \α and /α have the property as well, but in general α\ need not have the property. However, both tensor norms γ_p and γ_p \ are shown to have the Lewis-Radon-Nikodym property. Furthermore, it is deduced that the tensor norms γ_p /, /h and \h have the Radon-Nikodym property. We also bring on board the least of things: we show that the injective tensor norm ∨ does not have the Lewis-Radon-Nikodym property.