Topological properties in tensor products of Banach spaces

Abstract

Given two Banach spaces X and Y, we analyze when the projective tensor product X⊗ˆπY has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) X⊗ˆπY 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⊗ˆπY 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) ℓp(Γ)⊗ˆπℓq(Γ) is subspace of a Hilbert generated space for any 1<p,q<∞ such that 1/p+1/q<1 and for any infinite set Γ. We also pay attention to the injective tensor product X⊗ˆεY. In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that all regular Borel probability measures on the dual ball have countable Maharam type. Along this way, we generalize a result of Plebanek and Sobota that if K is a compact space such that C(K×K) has property (C), then all regular Borel probability measures on K have countable Maharam type. This generalization provides a consistent negative answer to a question of Ruess and Werner about the preservation of the w⁎-angelicity of the dual unit ball under injective tensor products.