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NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here.

Background. Let $T$ denote the Figiel-Johnson construction of the Tsirelson space, that is, the completion of $c_{00}$ under the implicitly-defined norm

\begin{equation*}\|x\|_T=\max\left\{\|x\|_{\ell_\infty},\frac{1}{2}\sup\sum_i\|E_ix\|_T\right\}\end{equation*}

where the "sup" is taken over all $j\in\mathbb{N}$ and all sequences $E_1<\cdots<E_j$ of finite subsets of $\mathbb{N}$ satisfying $j\leq\min E_1$.

It is well-known that $T$ and its dual $T^*$ are reflexive spaces which each fail to contain any copy of $c_0$ or $\ell_p$, $1\leq p\leq\infty$. Furthermore, $T^*$ can be represented as the completion of $c_{00}$ under some other norm $\|\cdot\|_{T^*}$, such that elements of $T^*$ act on elements of $T$ in the natural way, i.e. if $f=(\beta_n)\in T^*$ and $x=(\alpha_n)\in T$ then $f(x)=\sum_{n=1}^\infty\alpha_n\beta_n$.

Problem. Given an arbitrary $C>0$ and an arbitrary subsequence $(e_{n_k})$ of the canonical basis for $T^*$, I would like to find the coordinates for a sequence $(\gamma_k)\in c_{00}$ such that

\begin{equation*}C\|(\gamma_k)\|_{\ell_\infty}\leq\|\sum_{k=1}^\infty\gamma_ke_{n_k}\|_{T^*}.\end{equation*}

Discussion. The existence of such a sequence is easy to show given the above facts about $T$ and $T^*$. Notice that $\|f\|_{\ell_\infty}\leq\|f\|_{T^*}$ for all $f\in c_{00}$. This is because, due to $c_{00}$ being dense in $T$, for any $\epsilon>0$ we can always find $x\in c_{00}$ with

\begin{equation*}\|f\|_{T^*}\|x\|_T\geq|f(x)|\geq(\|f\|_{\ell_\infty}-\epsilon)\|x\|_{\ell_1}\geq(\|f\|_{\ell_\infty}-\epsilon)\|x\|_T.\end{equation*}

Thus, since $T^*$ fails to contain a copy of $c_0$, we get the desired sequence $(\gamma_k)$.

However, I am interested in a more constructive method of obtaining $(\gamma_k)$. This is because I will be looking at variations on Tsirelson's space for which the deep result about $T^*$ containing no copy of $c_0$ is not always established. Is there a way to actually write down the coordinates of $(\gamma_k)$? For instance, what about something similar to Argyros's construction of repeated averages (special convex combinations)?

Thanks!

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