{
  "abstract": "In this paper, we study Kubo-Ando means of finite order over the real division algebras $\\mathbb{R}$, $\\mathbb{C}$, and $\\mathbb{H}$. We establish an explicit bijection between Kubo-Ando means of order $n$ and normalized positive functions $f$ such that both $f$ and its transpose $f^{\\top}(t)=tf(t^{-1})$ are matrix monotone of order $n$. We also prove that matrix monotonicity of a fixed order is independent of the underlying division algebra. We then investigate finite-order Kubo-Ando means under the canonical embeddings relating quaternionic, complex, and real positive definite matrices. These embeddings preserve functional calculus and the Loewner order, inducing natural correspondences between means on the original and embedded cones. We characterize their extensions to the ambient cones and study the geometry of the embeddings with respect to the Log-Euclidean metric. As an application, we obtain explicit affine formulas for Kubo-Ando means of order $2$ on $\\mathscr{P}_{2}(\\mathbb{D})$ and transfer them to the associated embedded cones of $4\\times4$ positive definite matrices. In particular, we derive trace-determinant formulas for the geometric mean and show that they do not, in general, extend to the full ambient cones.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.27707",
  "author_total": 2,
  "authors": [
    {
      "name": "Jose Franco"
    },
    {
      "name": "Allan Merino"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vtqtmczh4xcf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.27707",
  "fulltext_url": null,
  "id": "5vtqtmczh4xcf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.27707",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7162664499",
  "pdf_url": "https://arxiv.org/pdf/2605.27707",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vtqtmczh4xcf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2605.27707",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.27707"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Kubo-Ando Means of Finite Order over Real Division Algebras",
  "title_text": "Kubo-Ando Means of Finite Order over Real Division Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/5vtqtmczh4xcf",
  "venue": "arXiv",
  "versions": []
}