{
  "id": "2001.10941",
  "version": "v1",
  "published": "2020-01-29T16:31:53.000Z",
  "updated": "2020-01-29T16:31:53.000Z",
  "title": "On disjointness, bands and projections in partially ordered vector spaces",
  "authors": [
    "Jochen Glück"
  ],
  "comment": "22 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Disjointness, bands, and band projections are a classical and essential part of the structure theory of vector lattices. If $X$ is such a lattice, those notions seem -- at first glance -- intimately related to the lattice operations on $X$. The last fifteen year, though, have seen an extension of all those concepts to a much larger class of ordered vector spaces. In fact if $X$ is an Archimedean ordered vector space with generating cone, or a member of the slightly larger class of pre-Riesz spaces, then the notions of disjointness, bands and band projections can be given proper meaning and give rise to a non-trivial structure theory. The purpose of this note is twofold: (i) We show that, on any pre-Riesz space, the structure of the space of all band projections is remarkably close to what we have in the case of vector lattices. In particular, this space is a Boolean algebra. (ii) We give several criteria for a pre-Riesz space to already be a vector lattice. These criteria are coined in terms of disjointness and closely related concepts, and they mark how lattice-like the order structure of pre-Riesz spaces can get before the theory collapses to the vector lattice case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-29T16:31:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partially ordered vector spaces",
      "pre-riesz space",
      "disjointness",
      "band projections",
      "larger class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}