Some properties of the projective tensor product UX derived from those of U and X

Abstract

Let X be a real or complex Banach space and let U be a Banach space with an unconditional basis. We show that the projective tensor product of U and X, U<8>X, has the complete continuity property (respectively, the analytic complete continuity property) whenever U and X have the complete continuity property (respectively, the analytic complete continuity property). More general versions of these results are also obtained. Moreover, the techniques applied here to the projective tensor product, can also be used to establish some Banach space properties of the Premlin projective tensor product.

Schauder decompositions and the Fremlin projective tensor product of Banach lattices

“…Moreover, not only RNP, but other types of RNP as well, such as the analytic RNP, the near RNP, the non-containment of a copy of c 0 (see [8]), weak sequential completeness (see [26]), and the types of complete continuity properties (see [17]) are inherited from two Banach spaces to their projective tensor product if one of them has an unconditional basis.…”

Reflexivity and the Grothendieck property for positive tensor products of Banach lattices-I

“…Dowling in [4] discussed the inheritance of the complete continuity property by p⊗F X and p⊗i X . In this paper, first we use the principle of local reflexivity for Banach lattices to show that the dual of p⊗i X is isometrically lattice isomorphic to p ⊗ F X * , where 1 < p, p < ∞ and 1/ p + 1/ p = 1.…”