TITLE: Hereditarily primary Banach spaces
QUESTION [6 upvotes]: A Banach space $X$ is said to be prime if every infinite dimensional complemented subspace is isomorphic to the space $X$. The space $X$ is primary if it has an infinite dimensional subspace $Y$ such that every complemented subspace is either isomorphic to $X$ or to the subspace $Y$. The space $X$ is quasiprime if it is primary and the only decomposition of $X$, as a direct sum, into two infinite dimensional subspaces is $X +Y$. Prime spaces are quasiprime which are primary.
Also there are examples that separate the above classes.
A Banach space $X$ is hereditarily prime (primary, quasiprime) if every infinite dimensional subspace is prime (primary, quasiprime). Hilbert spaces are hereditarily prime. Also there exists a non-Hilbertian space $X$ which is hereditarily quasiprime.
Question I Is every hereditarily prime space isomorphic to a Hilbert space?
Question II Do there exist subspaces of $\ell_p$, which are not primary?

REPLY [2 votes]: Actually the above three conditions yielding a counterexample are satisfied by any counterexample and this is easy to be checked. However such a space has extremely peculiar structure. In particular it does not contain any subspace of the form Y + Z with Y Hilbertian and Z indecomposable. This follows from the previous positive partial answer. That means that for every indecomposable subspace Z and every Hilbertian subspace Y dist( $S_Z$ , $S_Y$ )= 0. Further the same holds for all Y subspaces of X. The last follows from the fact that X is $l_2$ saturated. It is hard to think how such a space is defined.
However if the following has a positive answer then the original question I
has also a positive one.
Question III Let X be a separable  non Hilbertian and $l_2$ saturated Banach space.Does the space X contain Y +Z with Y Hilbertian and Z non Hilbertian?