{
  "id": "1910.02335",
  "version": "v1",
  "published": "2019-10-05T22:26:28.000Z",
  "updated": "2019-10-05T22:26:28.000Z",
  "title": "Non-asymptotic $\\ell_1$ spaces with unique $\\ell_1$ asymptotic model",
  "authors": [
    "Spiros A. Argyros",
    "Alexandros Georgiou",
    "Pavlos Motakis"
  ],
  "comment": "20 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "A recent result of D. Freeman, E. Odell, B. Sari, and B. Zheng states that whenever a separable Banach space not containing $\\ell_1$ has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis of $c_0$ then the space is Asymptotic $c_0$. We show that if we replace $c_0$ with $\\ell_1$ then this result is no longer true. Moreover, a stronger result of B. Maurey - H. P. Rosenthal type is presented, namely, there exists a reflexive Banach space with an unconditional basis admitting $\\ell_1$ as a unique asymptotic model whereas any subsequence of the basis generates a non-Asymptotic $\\ell_1$ subspace.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-05T22:26:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B03",
      "46B06",
      "46B25",
      "46B45"
    ],
    "keywords": [
      "non-asymptotic",
      "unit vector basis",
      "unique asymptotic model",
      "reflexive banach space",
      "rosenthal type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}