{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08921",
  "author_total": 1,
  "authors": [
    {
      "name": "Antonio Acuaviva"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cgfc5egj5x25.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08921",
  "fulltext_url": null,
  "id": "1cgfc5egj5x25",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08921",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7212082939",
  "pdf_url": "https://arxiv.org/pdf/2609.08921",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cgfc5egj5x25.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08921",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08921"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A negative solution to the complemented subspace problem for Banach spaces with unconditional bases",
  "title_text": "A negative solution to the complemented subspace problem for Banach spaces with unconditional bases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/1cgfc5egj5x25",
  "venue": "arXiv",
  "versions": []
}