{
  "id": "1211.6037",
  "version": "v3",
  "published": "2012-11-26T17:49:28.000Z",
  "updated": "2013-09-07T15:43:26.000Z",
  "title": "Liberation of Projections",
  "authors": [
    "Benoit Collins",
    "Todd Kemp"
  ],
  "comment": "53 pages",
  "categories": [
    "math.FA",
    "math.AP",
    "math.CV",
    "math.OA",
    "math.PR"
  ],
  "abstract": "We study the liberation process for projections: $(p,q)\\mapsto (p_t,q)= (u_tpu_t^\\ast,q)$ where $u_t$ is a free unitary Brownian motion freely independent from $\\{p,q\\}$. Its action on the operator-valued angle $qp_tq$ between the projections induces a flow on the corresponding spectral measures $\\mu_t$; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure $\\mu_t$ possesses a piecewise analytic density for any $t>0$ and any initial projections of trace $\\frac12$. We us this to prove the Unification Conjecture for free entropy and information in this trace $\\frac12$ setting.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-07T15:43:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54",
      "42B37",
      "60H30"
    ],
    "keywords": [
      "projections",
      "liberation",
      "unitary brownian motion freely independent",
      "free unitary brownian motion",
      "corresponding spectral measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.6037C"
    }
  }
}