{
  "id": "0710.4111",
  "version": "v2",
  "published": "2007-10-22T17:25:54.000Z",
  "updated": "2008-07-02T20:19:03.000Z",
  "title": "Lower estimates on microstates free entropy dimension",
  "authors": [
    "D. Shlyakhtenko"
  ],
  "comment": "A major revision. The previous version contained ad-hoc proofs for the case of q-semicircular variables. These have now been replaced with proofs based a more general approach involving free SDEs with analytic coefficients",
  "categories": [
    "math.OA"
  ],
  "abstract": "By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that Abstract. By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X_{1},...,X_{n}. In particular, we show that \\delta_{0}(X_{1},...,X_{n})\\geq\\dim_{M\\bar{\\otimes}M^{o}}V where M=W^{*}(X_{1},...,X_{n}) and V=\\{(\\partial(X_{1}),...,\\partial(X_{n})):\\partial\\in\\mathcal{C}\\} is the set of values of derivations A=\\mathbb{C}[X_{1},... X_{n}]\\to A\\otimes A with the property that \\partial^{*}\\partial(A)\\subset A. We show that for q sufficiently small (depending on n) and X_{1},...,X_{n} a q-semicircular family, \\delta_{0}(X_{1},...,X_{n})>1. In particular, for small q, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani (and also studied in the free case by Biane and Voiculescu).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-07-02T20:19:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54"
    ],
    "keywords": [
      "microstates free entropy dimension",
      "lower estimate",
      "free stochastic differential equations",
      "stationary solutions",
      "q-deformed free group factors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.4111S"
    }
  }
}