TITLE: Banach spaces whose second conjugates are separable
QUESTION [8 upvotes]: It was known that the James space $J$ has separable second conjugate, is non-reflexive and isometric to its second conjugate. I want to know whether there are Banach spaces $X$ with separable second conjugates $X^{**}$, but $X$ is not a dual space (the James space $J$ is a dual space). Furthermore, are there any references about Banach spaces with separable second conjuates ?
Thank you !

REPLY [3 votes]: This post is intended not as an answer, but rather to list several Banach space properties of $X$ given above (that is $X^{**}$ is separable, $X$ is not a dual space).

Clearly $X$ is not reflexive.
Clearly $X$ and $X^{*}$ are also separable.
Being separable dual spaces, $X^{*}$ and $X^{**}$ have RNP (Radon-Nikodym property). $X$ also has RNP, being a closed subspace of a space with RNP.
$X$, $X^{*}$ do not contain copies of $\ell^1$.
$X$, $X^{*}$, $X^{**}$ do not contain copies of $c_0$.
$X$, $X^{*}$ do not possess an unconditional basis.
$X$, $X^{*}$ are not w.s.c. (weakly sequentially complete) since any Banach space which is w.s.c. and contains no copy of $\ell^1$ is reflexive.
$X$, $X^{*}$ do not have Schur property, for any Banach space that has Schur property and contains no copy of $\ell^1$ is finite dimensional.
$X$ does not have DPP (Dunford-Pettis property), since otherwise $X^{*}$ would have Schur property. Consequently, $X^{*}$ and $X^{**}$ do not have DPP (and thus not have Schur property)
$X$, $X^{*}$, $X^{**}$ do not have Pełczyński property (V), for any Banach space that has property (V) and contains no copy of $c_0$ is reflexive.
$X$, $X^{*}$ have Dieudonne property, for any Banach space that contains no copy of $\ell^1$ has Dieudonne property.
$X$, $X^{*}$, $X^{**}$ are not Grothendieck spaces, since every separable Grothendieck space is reflexive.

The properties above are also shared by the James space, since we merely used that "$X$ is not a dual space" to deduce that $X$ is not reflexive.