Topological properties in tensor products of Banach spaces

Given two Banach spaces X and Y, we analyze when the projective tensor product X circle times �pi Y has Corson's property (C) or is weakly Lindelof determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) X circle times ⠂pi Y is WLD if and only if both X and Y are WLD and all operators from X to Y * and from Y to X* have separable range; (ii) X circle times-pi Y is subspace of a WCG space if the same holds for both X and Y under the assumption that every operator from X to Y * is compact; (iii) tp(Gamma)circle times �pi 4(Gamma) is subspace of a Hilbert generated space for any 1 < p, q < infinity such that 1/p + 1/q < 1 and for any infinite set Gamma. We also pay attention to the injective tensor product X circle times �epsilon Y . In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that

Automated summary

This paper analyzes different properties of Banach spaces such as Corson's property, weakly Lindelof determined, weakly compactly generated, and Hilbert generated spaces, and provides some conclusions and properties.