{
  "id": "2308.10955",
  "version": "v1",
  "published": "2023-08-21T18:01:15.000Z",
  "updated": "2023-08-21T18:01:15.000Z",
  "title": "The Space of Traces of the Free Group and Free Products of Matrix Algebras",
  "authors": [
    "Joav Orovitz",
    "Raz Slutsky",
    "Itamar Vigdorovich"
  ],
  "categories": [
    "math.GR",
    "math.OA"
  ],
  "abstract": "We show that the space of traces of the free group is a Poulsen simplex, i.e., every trace is a pointwise limit of extremal traces. We prove that this fails for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $\\mathbf{M}_n(\\mathbb{C}) * \\mathbf{M}_n(\\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R{\\o}rdam for $n \\geq 4$. Similar results are shown for certain faces of the simplices above, most notably, for the face of finite-dimensional traces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-21T18:01:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "46L05",
      "20E06",
      "46L09",
      "46L10",
      "81P45",
      "47L25"
    ],
    "keywords": [
      "free product",
      "matrix algebras",
      "poulsen simplex",
      "similar results",
      "finite-dimensional traces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}