# Banach space with an unconditional basis but not a quasi-greedy one?

A few years ago, Schechtman showed that $\ell_p(\ell_q)$ fails to admit a greedy basis whenever $1\leq p\neq q<\infty$. This furnishes an example of a Banach space with an unconditional basis but not a greedy one. However, I am also interested in the following:

Question 1. Does there exist a Banach space $X$ admitting an unconditional basis but not a quasi-greedy basis?

It was shown in Dilworth/Kalton/Kutzarova ("On the existence of almost greedy bases") that $c_0$ is the unique (up to isomorphism) $\mathcal{L}_\infty$-space admitting a quasi-greedy basis. Thus, to find a positive answer to question 1, it suffices to find a negative answer to the following:

Question 2. Is $c_0$ the unique $\mathcal{L}_\infty$-space admitting an unconditional basis?

It is proved in Albiac/Kalton that if $K$ is metrizable then $C(K)$ admits an unconditional basis if and only if $C(K)\approx c_0$. The fact that they used metrizability in their proof makes me think perhaps there is a lurking counter-example when $K$ is not metrizable. Hence, we could ask:

Question 3. Is $c_0$ the unique $C(K)$ space (where $K$ is compact Hausdorff) admitting an unconditional basis?

As every $C(K)$ space is an $\mathcal{L}_\infty$-space, a negative answer to question 3 would give a negative answer to question 2, and hence a positive answer to question 1.

Recently, Argyros/Gasparis/Motakis showed that $X$ is a separable $\mathcal{L}_\infty$-space if and only if it is isomorphic to a so-called Bourgain Delbaen space (whose definition is complicated but can be found in their paper). Hence, question 2 is equivalent to the following:

Question 4. Is $c_0$ the unique Bourgain Delbaen space admitting an unconditional basis?

Again, a negative answer here would mean a positive answer to question 1, which is what I hope to find.

In their Absolutely Summing Operators paper, Lindenstrauss and Pelczynski gave positive answers to questions 2, 3, and 4. You can find a proof on p. 29 of the Basic Concepts article Joram and I wrote for the Handbook of the Geometry of Banach Spaces. (This proof uses Khintchine's inequality rather than Grothendieck's inequality.) To see that you get 2 and 4, observe that the proof gives that if $P$ is a projection on $\ell_\infty^n$ and the range of $P$ has a normalized $\lambda$-unconditional basis, then the basis is $f(\lambda, \|P\|)$- equivalent to the unit vector basis of $\ell_\infty^k$, where $k$ is the rank of $P$.