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

Published in: Quaestiones Mathematicae
Volume 32 , issue 3 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 339–350

DOI: 10.2989/QM.2009.32.3.5.906

Author(s): Qingying Bu* Department of Mathematics, USA, Michelle Craddock Department of Mathematics, USA, Donghai Ji Department of Mathematics, P.R. China,

Keywords: Banach lattice, reflexivity, Grothendieck property, positive tensor product,

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Abstract

Let ϕ be an Orlicz function such that ϕ and its complementary function ϕ* satisfy the Δ₂-condition, let ℓ_ϕ be an Orlicz sequence space associated to ϕ, and let X be a Banach lattice. Then ℓ_ϕ ⊗_F X (respectively, ℓ_ϕ ∼⊗_i X), the Fremlin projective (respectively, the Wittstock injective) tensor product of ℓ_ϕ and X, has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from ℓ_ϕ (respectively, from ℓ_ϕ*) to X* (respectively, to X**) is compact.

*The author is partially supported by the NSF of China, Grant No. 10871213.

^† The author is supported by the NSF of China, Grant No. 10671048.

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