{
  "id": "1804.03532",
  "version": "v2",
  "published": "2018-04-10T13:45:13.000Z",
  "updated": "2018-05-07T09:40:23.000Z",
  "title": "Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups",
  "authors": [
    "Paweł Kasprzak"
  ],
  "comment": "Section 5 added, establishing a one to one correspondence between non-degenerate, integrable, ${\\mathbb{G}}$-invariant ternary rings of operators $X\\subset L^\\infty({\\mathbb{G}})$, preserved by the scaling group and contractive idempotent functionals on ${\\mathbb{G}}$. A 1-1 correspondence between integrable coideals and group-like projections were extended beyond the compact case",
  "categories": [
    "math.OA",
    "math.QA"
  ],
  "abstract": "A one to one correspondence between shifts of group-like projections on a locally compact quantum group ${\\mathbb{G}}$ which are preserved by the scaling group and contractive idempotent functionals on the dual $\\hat{\\mathbb{G}}$ is established. This is a generalization of the Illie-Spronk's correspondence between contractive idempotents in the Fourier-Stieltjes algebra of a locally compact group $G$ and cosets of open subgroups of $G$. We also establish a one to one correspondence between non-degenerate, integrable, ${\\mathbb{G}}$-invariant ternary rings of operators $X\\subset L^\\infty({\\mathbb{G}})$, preserved by the scaling group and contractive idempotent functionals on ${\\mathbb{G}}$. Using our results we characterize coideals in $L^\\infty(\\hat{\\mathbb{G}})$ admitting an atom preserved by the scaling group in terms of idempotent states on ${\\mathbb{G}}$. We also establish a one to one correspondence between integrable coideals in $L^\\infty({\\mathbb{G}})$ and group-like projections in $L^\\infty(\\hat{\\mathbb{G}})$ satisfying an extra mild condition. Exploiting this correspondence we give examples of group like projections which are not preserved by the scaling group.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-05-07T09:40:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locally compact quantum group",
      "contractive idempotent functionals",
      "group-like projections",
      "scaling group",
      "correspondence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}