Geodesic
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

Voir la notice de l'article

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 
@article{CMJ_2010_60_4_a22,
     author = {Ji, Donghai and Xue, Xiaoping and Bu, Qingying},
     title = {The {Grothendieck} property for injective tensor products of {Banach} spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1153--1159},
     year = {2010},
     volume = {60},
     number = {4},
     mrnumber = {2738976},
     zbl = {1224.46034},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_4_a22/}
}
TY  - JOUR
AU  - Ji, Donghai
AU  - Xue, Xiaoping
AU  - Bu, Qingying
TI  - The Grothendieck property for injective tensor products of Banach spaces
JO  - Czechoslovak Mathematical Journal
PY  - 2010
SP  - 1153
EP  - 1159
VL  - 60
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2010_60_4_a22/
LA  - en
ID  - CMJ_2010_60_4_a22
ER  - 
%0 Journal Article
%A Ji, Donghai
%A Xue, Xiaoping
%A Bu, Qingying
%T The Grothendieck property for injective tensor products of Banach spaces
%J Czechoslovak Mathematical Journal
%D 2010
%P 1153-1159
%V 60
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2010_60_4_a22/
%G en
%F CMJ_2010_60_4_a22
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/