On Positive Injective Tensor Products Being Grothendieck Spaces

Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. In this paper, we show that the positive injective tensor product λ⊗⌣|ε|X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over \otimes } _{\left| \varepsilon \right|}}X$$\end{document} is a Grothendieck space if and only if X is a Grothendieck space and every positive linear operator from λ* to X** is compact.