Let $$ \ell_1 = \{ (x_n)_{n \in \mathbb{N}} \subset \mathbb{C} : \sum_{n=1}^\infty |x_n| < \infty \}.$$ I would like to know if there is any result that describes the subspaces of $\ell_1$ that are isomorphic to $\ell_1$, and some references. The closest result I have found says that $\ell_1$ is isomorphically embeddable into each of its infinite dimensional subspaces, but I'm looking for subspaces that are actually isomorphic, not for those that contain an isomorphic copy of $\ell_1$.
Thank you very much.
It is easy to deduce that every complemented subspace of $\ell_1$ is isomorphic to $\ell_1$. (The same holds true for $\ell_p$, see here)
It goes deeper if the complementability is removed. First, it follows from Pitt's Theorem that subspaces of $\ell_1$ cannot contain $c_0$ and $\ell_2$. Then a paper by Linderstrauss and Zippin shows that a subspace $X$ of $\ell_1$ is isomorphic to $\ell_1$ if and only if $X$ has a unique (up to equivalence) unconditional basis.