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!