G.L. l. u. st. for subspaces of Banach spaces with an unconditional basis

Question

A Banach space $X$ has Gordon-Lewis local unconditional structure (G.L. l. u. st.) if for every finite dimensional subspace $E$ of $X$, the inclusion operator $i:E\to X$ factors through a finite dimensional space $U$ with an unconditional basis in a uniform manner, i.e., there exists a constant $C$ and operators $A:U\to X$ and $B:E\to U$ such that $AB=i$ and $\|A\|\cdot\|B\|\chi(U)\leq C$, where $\chi(U)$ is the unconditional constant of $U$.

I am interested in conditions on a Banach space $Y$ with an unconditional basis implying that every closed subspace of $Y$ has G.L. l. u. st.

What happens if $Y$ is super-reflexive?

Answer 1

The condition implies superreflexivity and for every $\epsilon >0$ that $Y$ has cotype $2-\epsilon$ and type $2+\epsilon$. The main open problem is whether the condition implies that $Y$ is isomorphic to a Hilbert space. This is open even if you strengthen the condition to "every subspace has an unconditional basis", which is Problem 1.10 in Casazza's article in the Handbook of the Geometry of Banach Spaces.