Let $1<p<\infty$. The James $p$-space $J\_{p}$ is the Banach space of all sequences of real numbers $(a\_{i})\_{i}\in c\_{0}$ such that 
$$\|(a\_{i})\_{i}\|=\sup\{(\sum\_{j=1}^{n}|a\_{p\_{j-1}}-a\_{p\_{j}}|^{p})^{\frac{1}{p}}:p\_{0}<p\_{1}<\cdots<p\_{n}, n\in\mathbb{N}\}<\infty.$$
Let $(e\_{n})\_{n}$ be the unit vector basis of $J\_{p}$ and $(e^{\*}\_{n})\_{n}$ be the sequence of biorthogonal functionals. It is known that $(e\_{n})\_{n}$ is a shrinking monotone basis for $J\_{p}$. I have the following two questions:


Question 1: Let $z^{\*}\_{n}=\sum\_{i=k\_{n-1}+1}^{k\_{n}}c\_{i}e^{\*}\_{i}$ be a semi-normalized block basic sequence in $J^{\*}\_{p}$ and suppose that $\sum\_{i=k\_{n-1}+1}^{k\_{n}}c\_{i}=0$ for each $n\in \mathbb{N}$. Is the sequence $(z^{\*}\_{n})\_{n}$ equivalent to the unit vector basis of $l\_{p}$?


Question 1 is true for $p=2$ as proved by Alfred Andrew in Israel J. Math.(1981).


Question 2: Is the dual $J\_{p}^{\*}$ of $J\_{p}$ weakly sequentially complete?


Thank you!



Two questions on the James $p$-space $J_{p}(1<p<\infty)$
I think that this question should be considered as answered, following the comments of Bill Johnson as:


(1) One can follow the argument of Andrew and show that $z\_n^\*$ is equivalent to the unit vector basis of $\ell \_q$, where $q$ is conjugate to $p$.


(2) The well-known result of Rosenthal implies that each bounded sequence either contains a weakly Cauchy subsequence or a subsequence equivalent to the unit vector basis of $\ell\_ 1$ Hence, by the Eberlein-Smulian theorem, if the space is weakly complete, it should either contain $\ell\_1$ or be reflexive. (See Lindenstrauss and Tzafriri, vol. 1, page 99 for Rosenthal's theorem.) 


