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.
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asked Nov 11, 2015 at 17:07
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$\begingroup$ Out of curiosity, are you trying to prove subprojectivity of this space? $\endgroup$– Tomasz KaniaCommented Nov 11, 2015 at 17:28
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$\begingroup$ No. I am trying to prove the space $\overline{span}\{z_{n}:n=1,2,...\}$ contains no uniformly copies of $l^{n}_{\infty}$ for all $n$. $\endgroup$– Dongyang ChenCommented Nov 11, 2015 at 18:14
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No. Consider $z_{2n+1} - z_{2n}$.
answered Nov 12, 2015 at 20:01
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$\begingroup$ You are right,Bill. The purpose of my question is to prove that the closed subspace spanned by $(z_{n})_{n}$ has nontrivial cotype. $\endgroup$ Commented Nov 12, 2015 at 23:11