A negative solution to the complemented subspace problem for Banach spaces with unconditional bases

Abstract

We give a negative solution to the complemented subspace problem for Banach spaces with unconditional bases over both the real and complex fields. For every $\rho>0$, we construct a projection $P_\rho$ of norm less than $1+\rho$ on a separable superreflexive space \begin{equation*} X_\rho=\left(\bigoplus_{j=1}^{\infty}\ell_{p_j}^{N_j}\right)_2, \qquad p_j\downarrow2, \end{equation*} such that $Z_\rho=P_\rho(X_\rho)$ and its dual $Z_\rho^*$ have Schauder bases but admit no unconditional bases. Over the real field, both spaces have Gordon--Lewis local unconditional structure (GL-lust) but fail Dubinsky--Pe\l czy\'nski--Rosenthal local unconditional structure (DPR-lust), disproving a conjecture of Figiel, Johnson and Tzafriri. In particular, neither is isomorphic to a Banach lattice, giving a negative solution to the separable Banach-lattice complemented subspace problem. A modification of the construction also shows that the class of separable real Banach lattices is not primary. A Lean 4 formalisation of the main results accompanies the paper.