{
  "id": "1112.3571",
  "version": "v2",
  "published": "2011-12-15T17:18:46.000Z",
  "updated": "2012-11-15T02:23:11.000Z",
  "title": "Trivial automorphisms",
  "authors": [
    "Ilijas Farah",
    "Saharon Shelah"
  ],
  "comment": "Thoroughly revised version",
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove that the statement `For all Borel ideals I and J on $\\omega$, every isomorphism between Boolean algebras $P(\\omega)/I$ and $P(\\omega)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(\\omega)/I$ and any other quotient $P(\\omega)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $\\omega$. We can also assure that the dominating number is equal to $\\aleph_1$ and that $2^{\\aleph_1}>2^{\\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(\\omega)/Fin$ are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of suslin proper forcings.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-11-15T02:23:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "trivial automorphisms",
      "borel ideal",
      "suslin proper forcings",
      "boolean algebras",
      "outer automorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.3571F"
    }
  }
}