Recall that a Banach space $X$ is said to have the Schur property if any weakly null sequence in $X$ is norm null, equivalently, every weakly Cauchy sequence in $X$ is norm Cauchy. It follows from Rosenthal's $l_{1}$-theorem that if a Banach space $X$ has the Schur property and contains no isomorphic copy of $l_{1}$, then $X$ is finite dimensional.

Question. Is there an infinite-dimensional Banach space $X$ such that $X^{*}$ has the Schur property and $X^{**}$ contains no isomorphic copies of $l_{1}$  ?

Thank you  !

Recall that a Banach space $X$ is said to have the Schur property if any weakly null sequence in $X$ is norm null, equivalently, every weakly Cauchy sequence in $X$ is norm Cauchy. It follows from Rosenthal's $l_{1}$-theorem that if a Banach space $X$ has the Schur property and contains no isomorphic copy of $l_{1}$, then $X$ is finite dimensional.

Question. Is there an infinite-dimensional Banach space $X$ such that $X^{*}$ has the Schur property and $X^{**}$ contains no isomorphic copies of $l_{1}$?

Thank you!