{
  "id": "2011.02013",
  "version": "v1",
  "published": "2020-11-03T21:33:07.000Z",
  "updated": "2020-11-03T21:33:07.000Z",
  "title": "Geodesics of projections in von Neumann algebras",
  "authors": [
    "Esteban Andruchow"
  ],
  "categories": [
    "math.OA",
    "math.FA"
  ],
  "abstract": "Let ${\\cal A}$ be a von Neumann algebra and ${\\cal P}_{\\cal A}$ the manifold of projections in ${\\cal A}$. There is a natural linear connection in ${\\cal P}_{\\cal A}$, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of $\\mathbb{C}^n$. In this paper we show that two projections $p,q$ can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of ${\\cal A}$), if and only if $$ p\\wedge q^\\perp\\sim p^\\perp\\wedge q, $$ where $\\sim$ stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if $p\\wedge q^\\perp= p^\\perp\\wedge q=0$. If ${\\cal A}$ is a finite factor, any pair of projections in the same connected component of ${\\cal P}_{\\cal A}$ (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones' index theory for subfactors. For instance, it is shown that if ${\\cal N}\\subset{\\cal M}$ are {\\bf II}$_1$ factors with finite index $[{\\cal M}:{\\cal N}]=t^{-1}$, then the geodesic distance $d(e_{\\cal N},e_{\\cal M})$ between the induced projections $e_{\\cal N}$ and $e_{\\cal M}$ is $d(e_{\\cal N},e_{\\cal M})=\\arccos(t^{1/2})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-03T21:33:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58B20",
      "46L10",
      "53C22"
    ],
    "keywords": [
      "von neumann algebra",
      "projections",
      "finite dimensional case coincides",
      "minimal geodesic",
      "murray-von neumann equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}