Some geometric properties inherited by the positive tensor products of atomic Banach lattices
containment of copies of \(c_0, \ell_1, \ell_\infty\)
Reflexivity of and containment of copies of \(c_0, \ell_1, \ell_\infty\) in tensor products of Banach spaces and related spaces of operators or of vector valued sequences were studied in many papers already from the 1970s through the 1990s.NEWLINENEWLINEThe positive projective (or Fremlin) tensor product and the positive injective (or Wittstock) tensor product of Banach lattices are, respectively, the order analogues of the projective and the injective tensor products of Banach spaces.NEWLINENEWLINELet \(X\) be a Banach lattice and let \(E\) be an atomic Banach lattice. In the paper under review, the authors study these properties in the positive projective tensor product \(E\hat{\otimes}_{|\pi |}X\) and in the positive injective tensor product \(E\check{\otimes}_{|\varepsilon |}X\). They prove order analogues of characterizations which are known for the tensor products of Banach spaces. The main results are as follows.NEWLINENEWLINEThe tensor product \(E\hat{\otimes}_{|\pi |}X\) (respectively, \(E\hat{\otimes}_{|\pi |}X^*\)) contains no sublattice isomorphic to \(c_0\) (respectively, to \(\ell_\infty\)) if neither \(E\) nor \(X\) contains a sublattice isomorphic to \(c_0\) (respectively, to \(\ell_\infty\)). Moreover, \(E\check{\otimes}_{|\varepsilon |}X\) contains no sublattice isomorphic to \(\ell_1\) if neither \(E\) nor \(X\) contains a sublattice isomorphic to \(\ell_1\). If \(E\) and \(X\) are reflexive, then \(E\hat{\otimes}_{|\pi |}X\) (respectively, \(E\check{\otimes}_{|\varepsilon |}X\)) is reflexive whenever every positive linear operator from \(E\) to \(X^*\) (respectively, from \(E^*\) to \(X\)) is compact.NEWLINENEWLINEThe main general idea of the proof consists in identifying a separable \(E\) with a sequence Banach lattice and the corresponding positive tensor products with Banach lattice valued sequence spaces. To make this idea work, the authors give a sequential representation of the positive injective tensor product \(\lambda\check{\otimes}_{|\varepsilon |}X\), where \(\lambda\) is a \(\sigma\)-order continuous solid sequence Banach lattice. The proof relies on [\textit{Q.-Y. Bu} and \textit{G. Buskes}, J. Math. Anal. Appl. 355, No. 1, 335--351 (2009; Zbl 1180.46014)], where a sequential representation of the corresponding positive projective tensor product was obtained.
scientific article; zbMATH DE number 6070477
scientific article