×

On positive injective tensor products being Grothendieck spaces. (English) Zbl 1462.46076

Let \(\lambda\) be a Banach sequence lattice and \(X\) be a Banach lattice. In [Y.-J. Li and Q.-Y. Bu, Houston J. Math. 43, No. 2, 569–575 (2017; Zbl 1410.46001)], a characterization of the positive projective tensor products \(\lambda \widehat{\otimes}_{|\pi|} X\) that are Grothendieck spaces was obtained. In the paper under review, the authors show that if \(\lambda\) is also reflexive, then the positive injective tensor product \(\lambda \widehat{\otimes}_{|\varepsilon|} X\) is a Grothendieck space if and only if \(X\) is a Grothendieck space and every positive linear operator from \(\lambda^{*}\) to \(X^{**}\) is compact.

MSC:

46M05 Tensor products in functional analysis
46B42 Banach lattices
46B28 Spaces of operators; tensor products; approximation properties

Citations:

Zbl 1410.46001
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bu, Q.; Buskes, G., Schauder decompositions and the Fremlin projective tensor product of Banach lattices, J. Math. Anal. Appl., 355, 335-351 (2009) · Zbl 1180.46014 · doi:10.1016/j.jmaa.2009.01.061
[2] Bu, Q.; Wong, N. C., Some geometric properties inherited by the positive tensor products of atomic Banach lattices, Indag. Math. (N.S.), 23, 199-213 (2012) · Zbl 1262.46050 · doi:10.1016/j.indag.2011.11.004
[3] Diestel, J., Grothendieck spaces and vector measures, Vector and operator valued measures and applications, 97-108 (1973), New York: Academic Press, New York · Zbl 0316.46009
[4] Fremlin, D. H., Tensor products of Archimedean vector lattices, Amer. J. Math., 94, 778-798 (1972) · Zbl 0252.46094 · doi:10.2307/2373758
[5] Fremlin, D. H., Tensor products of Banach lattices, Math. Ann., 211, 87-106 (1974) · Zbl 0272.46050 · doi:10.1007/BF01344164
[6] Grothendieck, A., Sur les applications lineaires faiblement compactes d’espaces du type C(K), Canad. J. Math., 5, 129-173 (1953) · Zbl 0050.10902 · doi:10.4153/CJM-1953-017-4
[7] Li, Y.; Bu, Q., New examples of non-reflexive Grothendieck spaces, Houston J. Math., 43, 569-575 (2017) · Zbl 1410.46001
[8] P. Meyer-Nieberg, Banach lattices, Springer-Verlag, 1991. · Zbl 0743.46015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.