Kubo-Ando Means of Finite Order over Real Division Algebras
Abstract
In this paper, we study Kubo-Ando means of finite order over the real division algebras ℝ, ℂ, and 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⊤(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 𝒫2(𝔻) and transfer them to the associated embedded cones of 4×4 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.
Related
- Graded identities for matrix algebras of order two over a finite field
- Division Algebras, Triality, and Exceptional Magic
- Factorization of central polynomials over division algebras
- Uniqueness of twisted linear periods over Archimedean division algebras
- Simple Torsion-Free Linear Groups from Global Division Algebras
BibTeX
@misc{franco2026kuboando,
title = {{Kubo-Ando Means of Finite Order over Real Division Algebras}},
author = {Jose Franco and Allan Merino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.27707},
eprint = {2605.27707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.27707},
}
RIS
TY - UNPB
TI - Kubo-Ando Means of Finite Order over Real Division Algebras
AU - Jose Franco
AU - Allan Merino
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2605.27707
UR - https://arxiv.org/abs/2605.27707
AB - In this paper, we study Kubo-Ando means of finite order over the real division algebras ℝ, ℂ, and 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^⊤(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 𝒫_2(𝔻) and transfer them to the associated embedded cones of 4×4 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.
ER -