{
  "id": "1009.4945",
  "version": "v2",
  "published": "2010-09-24T21:00:34.000Z",
  "updated": "2013-12-05T14:51:53.000Z",
  "title": "Abelian subalgebras and the Jordan structure of a von Neumann algebra",
  "authors": [
    "Andreas Doering",
    "John Harding"
  ],
  "comment": "11 pages, no figures; v2: minor correction, improved presentation, added proof of converse of main result",
  "categories": [
    "math-ph",
    "math.MP",
    "math.OA",
    "quant-ph"
  ],
  "abstract": "For von Neumann algebras M, N not isomorphic to C^2 and without type I_2 summands, we show that for an order-isomorphism f:AbSub(M)->AbSub(N) between the posets of abelian von Neumann subalgebras of M and N, there is a unique Jordan *-isomorphism g:M->N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of a von Neumann algebra not isomorphic to C^2 and without type I_2 summands is determined by the poset of its abelian subalgebras, and has implications in recent approaches to foundational issues in quantum mechanics.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-05T14:51:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L10",
      "81P05",
      "03G12",
      "17C65",
      "18B25"
    ],
    "keywords": [
      "von neumann algebra",
      "jordan structure",
      "abelian subalgebras",
      "abelian von neumann subalgebra",
      "foundational issues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.4945D"
    }
  }
}