It is known that there are characterizations of weak compactness in most of classical non-reflexive 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!



Weak compactness in the James space and its dual
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/CJM-1980-083-7].


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 $e\_n = \lambda^{-1}(u\_n-x)$ 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/CJM-1980-083-7]. 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 $x\in C$.


