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The Grothendieck property for injective tensor products of Banach spaces
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 4, pp. 1153-1159 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $X$ be a Banach space with the Grothendieck property, $Y$ a reflexive Banach space, and let $X\check{\otimes}_{\varepsilon} Y$ be the injective tensor product of $X$ and $Y$. \item {(a)} If either $X^{\ast \ast }$ or $Y$ has the approximation property and each continuous linear operator from $X^\ast $ to $Y$ is compact, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property. \item {(b)} In addition, if $Y$ has an unconditional finite dimensional decomposition, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property if and only if each continuous linear operator from $X^\ast $ to $Y$ is compact.
Let $X$ be a Banach space with the Grothendieck property, $Y$ a reflexive Banach space, and let $X\check{\otimes}_{\varepsilon} Y$ be the injective tensor product of $X$ and $Y$. \item {(a)} If either $X^{\ast \ast }$ or $Y$ has the approximation property and each continuous linear operator from $X^\ast $ to $Y$ is compact, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property. \item {(b)} In addition, if $Y$ has an unconditional finite dimensional decomposition, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property if and only if each continuous linear operator from $X^\ast $ to $Y$ is compact.
Classification : 46B28, 46M05
Keywords: Banach space; Grothendieck property; tensor product 
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Ji, Donghai; Xue, Xiaoping; Bu, Qingying. The Grothendieck property for injective tensor products of Banach spaces. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 4, pp. 1153-1159. http://geodesic.mathdoc.fr/item/CMJ_2010_60_4_a22/

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