The paper

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

Antonio Acuaviva · 5 September 2026

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

X_\rho=\left(\bigoplus_{j=1}^{\infty}\ell_{p_j}^{N_j}\right)_2,\qquad p_j\downarrow2,

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–Pełczyński–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.

The abstract describes further results beyond the five Lean theorems presented on this site.

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