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 ρ>0, we construct a projection Pρ of norm less than 1+ρ on a separable superreflexive space Xρ=(⨁j=1∞ℓpjNj)2, pj↓2, such that Zρ=Pρ(Xρ) and its dual Zρ* 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.
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BibTeX
@misc{acuaviva2026negative,
title = {{A negative solution to the complemented subspace problem for Banach spaces with unconditional bases}},
author = {Antonio Acuaviva},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08921},
eprint = {2609.08921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08921},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A negative solution to the complemented subspace problem for Banach spaces with unconditional bases
AU - Antonio Acuaviva
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08921
UR - https://arxiv.org/abs/2609.08921
AB - 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 X_ρ=(⨁_j=1^∞ℓ_p_j^N_j)_2, p_j↓2, 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.
ER -