{
  "id": "1309.7856",
  "version": "v2",
  "published": "2013-09-30T14:23:28.000Z",
  "updated": "2019-12-21T23:14:21.000Z",
  "title": "Algebraic tensor products and internal homs of noncommutative L^p-spaces",
  "authors": [
    "Dmitri Pavlov"
  ],
  "comment": "22 pages. Comments and questions are very welcome. v2: Contains the journal version together with additional expository material",
  "journal": "Journal of Mathematical Analysis and Applications 456:1 (2017), 229-244",
  "doi": "10.1016/j.jmaa.2016.11.060",
  "categories": [
    "math.OA",
    "math.FA"
  ],
  "abstract": "We prove that the multiplication map L^a(M)\\otimes_M L^b(M)\\to L^{a+b}(M) is an isometric isomorphism of (quasi)Banach M-M-bimodules. Here L^a(M)=L_{1/a}(M) is the noncommutative L_p-space of an arbitrary von Neumann algebra M and \\otimes_M denotes the algebraic tensor product over M equipped with the (quasi)projective tensor norm, but without any kind of completion. Similarly, the left multiplication map L^a(M)\\to Hom_M(L^b(M),L^{a+b}(M)) is an isometric isomorphism of (quasi)Banach M-M-bimodules, where Hom_M denotes the algebraic internal hom. In particular, we establish an automatic continuity result for such maps. Applications of these results include establishing explicit algebraic equivalences between the categories of L_p(M)-modules of Junge and Sherman for all p\\ge0, as well as identifying subspaces of the space of bilinear forms on L^p-spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-30T14:23:28.000Z",
      "title": "Algebraic tensor products and internal homs of noncommutative L_p-spaces",
      "abstract": "We prove that the multiplication map L_a(M)\\otimes_M L_b(M)\\to L_{a+b}(M) is an isometric isomorphism of (quasi)Banach M-M-bimodules. Here L_a(M)=L^{1/a}(M) is the noncommutative L_p-space of an arbitrary von Neumann algebra M and \\otimes_M denotes the algebraic tensor product over M equipped with the (quasi)projective tensor norm, but without any kind of completion. Similarly, the left multiplication map L_a(M)\\to Hom_M(L_b(M),L_{a+b}(M)) is an isometric isomorphism of (quasi)Banach M-M-bimodules, where Hom_M denotes the algebraic internal hom without any continuity assumptions. In a forthcoming paper these results will be applied to L_p-modules introduced by Junge and Sherman, establishing explicit algebraic equivalences between the categories of right L_p(M)-modules for all p\\ge0.",
      "comment": "20 pages. Comments and questions are very welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2019-12-21T23:14:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L51",
      "46L52",
      "46L08",
      "46L10"
    ],
    "keywords": [
      "algebraic tensor product",
      "banach m-m-bimodules",
      "arbitrary von neumann algebra",
      "isometric isomorphism",
      "noncommutative"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.7856P"
    }
  }
}