{
  "id": "1609.04766",
  "version": "v1",
  "published": "2016-09-15T18:28:17.000Z",
  "updated": "2016-09-15T18:28:17.000Z",
  "title": "An extension of compact operators by compact operators with no nontrivial multipliers",
  "authors": [
    "Saeed Ghasemi",
    "Piotr Koszmider"
  ],
  "categories": [
    "math.OA",
    "math.FA",
    "math.GN",
    "math.LO"
  ],
  "abstract": "We construct an essential extension of $\\mathcal K(\\ell_2({\\mathfrak{c}}))$ by $\\mathcal K(\\ell_2)$, where ${\\mathfrak{c}}$ denotes the cardinality of continuum, i.e., a $C^*$-algebra $\\mathcal A\\subseteq \\mathcal B(\\ell_2)$ satisfying the short exact sequence $$0\\rightarrow \\mathcal K(\\ell_2)\\xrightarrow{\\iota} \\mathcal A \\rightarrow\\mathcal K(\\ell_2({\\mathfrak{c}}))\\rightarrow 0,$$ where $\\iota[\\mathcal K(\\ell_2)]$ is an essential ideal of $\\mathcal A$ such that the algebra of multipliers $\\mathcal M(\\mathcal A)$ of $\\mathcal A$ is equal to the unitization of $\\mathcal A$. In particular $\\mathcal A$ is not stable which sheds light on permanence properties of the stability in the nonseparable setting. Namely, an extension of a nonseparable algebra of compact operators, even by $\\mathcal K(\\ell_2)$, does not have to be stable. This construction can be considered as a noncommutative version of Mr\\'owka's $\\Psi$-space; a space whose one point compactification equals to its Cech-Stone compactification and is induced by a special uncountable family of almost disjoint subsets of ${\\mathbb{N}}$. The role of the almost disjoint family is played by an almost orthogonal family of projections in $\\mathcal B(\\ell_2)$, but the almost matrix units corresponding to the matrix units in $\\mathcal K(\\ell_2({\\mathfrak{c}}))$ must be constructed with extra care.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-15T18:28:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact operators",
      "nontrivial multipliers",
      "matrix units",
      "short exact sequence",
      "point compactification equals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}