It is known that there are characterizations of weak compactness in most of classical nonreflexive spaces (e.g. $L_{1}$spaces and $C(K)$spaces). I wonder whether there are characterizations of weak compactness in James space $J$ or its dual $J^{*}$. Can we establish a criterion for it if there is no? Thank you!

1$\begingroup$ I am not an expert on Banach spaces, but I am not sure it is accurate to say weak compactness has been characterized in "most classical nonreflexive spaces". $L_1$ and $C(K)$ are pretty special cases, it seems to me $\endgroup$– Yemon ChoiMay 5, 2016 at 16:21
3 Answers
James space is a commutative Banach algebra with pointwise operations. $J^{\ast\ast} = J\oplus\mathbb{C}$ is just $J$ with a unit attached [https://doi.org/10.4153/CJM19800837].
Second, since $J$ contains no copy of $\ell^1$, every bounded sequence $(x_n)$ in $J$ has a weakly Cauchy subsequence, say $(u_n)$, that converges weak* in $J^{**}$ to an element $x+\lambda 1\in J^{**}$. If $\lambda\neq 0$, then let $e_n = \lambda^{1}(u_nx)$ so that $(e_n)$ is a bounded approximate identity for $J$. If a subset $C\subseteq J$ is not weakly compact, then it contains a weakly Cauchy sequence $(u_n)$ with $\lambda\neq 0$.
Consequently, a subset $C\subseteq J$ is weakly compact if and only if $y+\lambda C$ contains no (sequential) bounded approximate identity for all $y\in J$ and all $\lambda\in\mathbb{C}\backslash\{0\}$.
About the weak$^\ast$ convergence in $J^{\ast\ast}$: The spectrum $\sigma(J)=\{\delta_n:n\in\mathbb{N}\}$ consists of coordinate functionals, defined by $\delta_n(x_m)_{m\in\mathbb{N}} = x_n$ [Proposition 2.7 in https://doi.org/10.4153/CJM19800837]. Furthermore, $J^{\ast}$ is the closed linear span of $\sigma(J)$ since $J$ is semisimple, $J$ is an ideal (in fact, maximal) in $J^{\ast\ast}$, and $J$ has a bounded approximate identity. Consequently, weak* convergence in $J^{\ast\ast}$ is just the pointwise convergence.
Thus, $C\subseteq J$ is weakly compact if and only if $C$ is closed under the pointwise convergence of sequences, i.e., whenever $(x_n)\subseteq C$ is a sequence such that $x_n\to x$ pointwise, then $x\in C$.
I'm not sure this deserves to be posted as an "answer," but it is far too long for a comment. But since there is already an answer posted, I do not believe this will detract from your question getting attention. Anyway...
First, note that $J$ has a unique (up to equivalence) spreading basis, which we shall call "the" spreading basis for $J$. As every seminormalized weakly null sequence in $J$ contains a subsequence equivalent to the $\ell_2$ basis, the spreading basis cannot be weakly null. This was all established here: http://www.mscand.dk/article/viewFile/11904/9920
From the same paper comes the following (as part (b) from Theorem 2.1).
Theorem. Let $(z_n)_{n=1}^\infty$ be a seminormalized sequence in $J$ having no weak cluster point. Then there is a subsequence $(z_{n_k})_{k=1}^\infty$ equivalent to the spreading basis of $J$, such that its closed linear span $[z_{n_k}]_{k=1}^\infty$ is complemented in $J$.
Corollary. Let $C\subset J$ be a subset which is weakly closed. Then the following are equivalent.
(i) $C$ is not weakly compact.
(ii) $C$ contains a sequence $(c_n)_{n=1}^\infty$ equivalent to the spreading basis of $J$ such that its closed linear span $[c_n]_{n=1}^\infty$ is complemented in $J$.
(iii) $C$ contains a basic sequence $(c_n)_{n=1}^\infty$ which is not weakly null.
Proof. (i) $\Rightarrow$ (ii): By EberleinSmulian we can find $(c_n)_{n=1}^\infty\subset C$ with no weak cluster point in $C$, and hence (by weak closure of $C$) no weak cluster point in $J$. Now apply the above theorem to obtain a subsequence equivalent to $J$ and complemented in $J$.
(ii) $\Rightarrow$ (iii): This follows from the fact that the spreading basis is not weakly null.
(iii) $\Rightarrow$ (i): By passing to a subsequence if necessary we may assume that the weak closure of $(c_n)_{n=1}^\infty$ (as a set) fails to contain zero. Hence, by a wellknown criterion of KadetsPelczynski (e.g. Theorem 1.5.6 in the Albiac/Kalton book here), $(c_n)_{n=1}^\infty$ is not relatively weakly compact, and by EberleinSmulian, has no weakly convergent subsequence. Thus $C$ is not weakly compact. $\square$
Note that the equivalence (i) $\Leftrightarrow$ (iii) holds for arbitrary Banach space in place of $J$. So, the only special thing here is (ii).

$\begingroup$ You are very welcome. I am also interested in the properties of the James space, so please feel free to post anything else you learn. the same goes for the generalized $p$James spaces. $\endgroup$– Ben WMay 7, 2016 at 15:11
One of the things which you can do is: to combine the following theorem of James [James, Robert C. Weakly compact sets. Trans. Amer. Math. Soc. 113 1964 129–140]: A weakly closed subset $C$ of a Banach space $B$ is weakly compact if and only if each member of $B^*$ attains a maximum on $C$ with the description of the dual space of $J$ and $J^*$.
Another thing: analyze when, say, the weak$^*$limit of a sequence in $J$, considered as a sequence in $J^{**}$, is outside $J$. Do the same for the pair $J^*$ and $J^{***}$
I think that one can definitely get something on these lines. Possibly this has already been done somewhere, but it could be difficult to find.

$\begingroup$ I search characterizations of weak compactness in $J$ by google, but fail. $\endgroup$ May 7, 2016 at 1:15

$\begingroup$ Such results are difficult to find either because nobody ever needed such results or because they were published as auxiliary results in a paper devoted to something else. You can look for large papers dealing with the James space (e.g. James, Robert C. Banach spaces quasireflexive of order one. Studia Math. 60 (1977), no. 2, 157–177), you can also check the book Fetter, Helga; Gamboa de Buen, Berta, The James forest. Cambridge University Press, Cambridge, 1997 (if you have not yet done this). $\endgroup$ May 7, 2016 at 3:16

$\begingroup$ Thank you for your suggestions, Mikhail. I'll look for it. $\endgroup$ May 7, 2016 at 4:20