Square and cube?

Gowers and Maurey proved in their remarkable paper(s), that there is a Banach space $X$ such that $X$ is isomorphic to its cube $X\oplus X\oplus X$ but not to isomorphic to its square $X\oplus X$. This space seems to be not a dual space, am I correct? Is there a solution to this problem which is a dual space?

And the second quick question: what is the status of the following conjecture:

There is a Banach space $X$ such that $X$ is not isomorphic to $X^2$ but $X^2$ is isomorphic to $X^3$?

Thank you very much. S.

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Related: mathoverflow.net/questions/10128 – Martin Brandenburg Dec 12 '11 at 17:46
I think both questions are open, Sellapan. For the first one, the closest result I know is Ferenczi, V.(F-PARIS6-E) A uniformly convex hereditarily indecomposable Banach space. (English summary) Israel J. Math. 102 (1997), 199–225, which contains a proof of the theorem in the title. – Bill Johnson Dec 12 '11 at 18:28
You "know" the answer to this last question, Sellapan. For $X$ take the Gowers-Maurey space you mentioned and for $Y$ take $X^2$. – Bill Johnson Dec 13 '11 at 0:37
Sorry; I don't know whether the answer to your first question is known. The basic Gowers-Maurey space is reflexive and I would guess so are the ones constructed in their second paper. Kevin Beanland, who participates on MO, might know. It is strange that Gowers and Maurey do not mention reflexivity in their papers. I guess it just did not occur to them. – Bill Johnson Dec 17 '11 at 1:01
I must admit, I've assumed, without verifying that the spaces defined by Gowers and Maurey are reflexive. They don't seem to mention it in the papers, but if it is true it shouldn't be hard to prove. Other than these papers, I know Zsak wrote at least two papers studying these construction and Galego, Elói Medina has a series of papers studying these types of spaces. You can also check out Gowers' ICM paper at dpmms.cam.ac.uk/~wtg10/papers.html called "Recent results in the theory of infinite dimensional Banach spaces" where he gives an overview of what they prove. – Kevin Beanland Dec 20 '11 at 13:55