Let $c\_{0}\widehat{\otimes}\_{\pi} c\_{0}$ be the projective tensor product of $c\_{0}$ and $c\_{0}$. Let $(e\_{n})\_{n}$ be the unit vector basis of $c\_{0}$. For each $n$, define $z\_{n}=e\_{n}\otimes\sum\_{j=1}^{n}e\_{j}$. Then the basic sequence $(z\_{n})\_{n}$ is sub-symmetric. My question: Is each normalized block basic sequence of $(z\_{n})\_{n}$ equivalent to $(z\_{n})\_{n}$? Thank you.



On the normalized block basic sequences in $c_{0}\widehat{\otimes}_{\pi} c_{0}$
No. Consider $z\_{2n+1} - z\_{2n}$.


