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?

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