By a Rosenthal Banach space I mean one that does not contain an isomorphic copy of $\ell_1$. Is there a tensor norm $\alpha$ such that the Banach tensor product $E\otimes_\alpha F$ is Rosenthal if $E$ and $F$ are?
The maximal and minimal tensor norms $\pi,\epsilon$ do not have this property ($\ell_1$ embeds in $\ell_2\otimes_\pi\ell_2$ and in $JT\otimes_\epsilon JT$, for $JT$ the James tree space). Is anything known for the hilbertian tensor norm $w_2$, for its dual $w_2'$, or for the Chevet-Saphar norms $d_p,g_p$?
In [1] it was shown that the norms $d_p$ and $g_p$ preserve reflexive Banach spaces for $1<p\leq\infty$. Is this known for $w_2$ or $w_2'$?
I am as well interested in the same questions for symmetric tensor products. In [2] it was shown that there are tensor norms for the symmetric tensor product that preserve Asplund spaces. Is anything known with respect to reflexive and Rosenthal spaces?
[1]: R. Aharoni, P. D. Saphar, On the reflexivity of the space $\pi_p(E, F)$ of $p$-absolutely summing operators, $1\leq p \leq +\infty$. Bull. London Math. Soc. 25 (1993), 362–368.
[2]: D. Carando, D. Galicer, The symmetric Radon-Nikodým property for tensor norms. J. Math. Anal. Appl. 375 (2011), no. 2, 553–565.
