{
  "id": "math/0510576",
  "version": "v2",
  "published": "2005-10-26T19:58:20.000Z",
  "updated": "2005-10-31T18:15:51.000Z",
  "title": "Strongly 1-bounded von Neumann algebras",
  "authors": [
    "Kenley Jung"
  ],
  "comment": "15 pages, added references, minor corrections",
  "categories": [
    "math.OA"
  ],
  "abstract": "Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M. For $\\alpha >0$, F is $\\alpha$-bounded if the free packing $\\alpha$-entropy of F is bounded from above. We say that M is strongly 1-bounded if M has a 1-bounded finite set of selfadjoint generators F such that there exists an x in F with finite free entropy. It is shown that if M is strongly 1-bounded, then any finite set of selfadjoint generators G for M is 1-bounded and the microstates free entropy dimension of G is less than or equal to 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and the microstates free entropy dimension is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II_1-factors which have property Gamma, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of SL_n(Z), n >2. If M and N are strongly 1-bounded and their intersection is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II_1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-10-31T18:15:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54",
      "28A75"
    ],
    "keywords": [
      "microstates free entropy dimension",
      "finite set",
      "diffuse von neumann subalgebra",
      "tracial von neumann algebra",
      "selfadjoint generators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10576J"
    }
  }
}