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.