A negative solution to the complemented subspace problem for Banach spaces with unconditional bases
Antonio Acuaviva
math.FA
Sep 8, 2026 · v1
TL;DR
A Lean 4 formalisation accompanies the paper's main results on 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 $ρ>0$, we construct a projection $P_ρ$ of norm less than $1+ρ$ on a separable superreflexive space \begin{equation*} X_ρ=\left(\bigoplus_{j=1}^{\infty}\ell_{p_j}^{N_j}\right)_2, \qquad p_j\downarrow2, \end{equation*} such that $Z_ρ=P_ρ(X_ρ)$ and its dual $Z_ρ^*$ 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.
Problem
The complemented subspace problem asks whether every complemented subspace of a Banach space with an unconditional basis again has an unconditional basis, and analogously for separable Banach lattices. Prior negative solutions relied on nonseparable constructions.
Approach
For every ρ>0 a projection P_ρ of norm below 1+ρ is constructed on a separable superreflexive ℓ_2-sum of finite-dimensional ℓ_p spaces with p_j↓2. The construction separates Gordon–Lewis local unconditional structure (GL-lust) from Dubinsky–Pełczyński–Rosenthal local unconditional structure (DPR-lust). A Lean 4 formalisation of the main results accompanies the paper.
Results
The complemented subspace Z_ρ and its dual have Schauder bases but no unconditional bases, giving a negative solution over real and complex fields; in the real case they are not isomorphic to any Banach lattice, disproving a conjecture of Figiel, Johnson and Tzafriri and showing separable real Banach lattices are not primary.
