On complemented subspaces of $L_1[0,1]$

Abstract

We construct two complemented subspaces of $L_1[0,1]$. The first has the Schur property but fails the Radon--Nikodým property. The second contains a copy of ${\ell }_2$ but no copy of $L_1[0,1]$. Neither space is isomorphic to a Banach lattice. This gives a negative answer to the complemented-subspace question of Lindenstrauss and Rosenthal. A Lean 4 formalisation of the main results accompanies the paper.