Determinants in Jordan matrix algebras

TL;DR

The paper develops a determinant theory for matrix JB$^*$-algebras, focusing on $H_3(\mathbb{O})$ (the Cartan factor $C_6$) and $H_n(\mathbb{H}_C)$. It defines unitary determinants via spectral data and extends to general elements through an inductive Schur-type determinant formula, relating $\operatorname{dt}_n$ to $\det\widehat{\boldsymbol{x}}$. The work provides explicit classifications of minimal projections in $C_6$, characterizes automorphisms (including row/column permutations), and proves a determinant-product rule $\operatorname{dt}\boldsymbol{u}=\operatorname{dt}_{\boldsymbol{e}}\boldsymbol{u}\cdot \operatorname{dt}\boldsymbol{e}$. These results illuminate the structure of the exceptional Cartan factor $C_6$, enable reductions to biquaternion subalgebras, and supply tools for analyzing tripotents and unitary structure in JB$^*$-algebras.$

Abstract

We introduce a natural notion of determinant in matrix JB$^*$-algebras, i.e., for hermitian matrices of biquaternions and for hermitian $3\times 3$ matrices of complex octonions. We establish several properties of these determinants which are useful to understand the structure of the Cartan factor of type $6$. As a tool we provide an explicit description of minimal projections in the Cartan factor of type $6$ and a variety of its automorphisms.

Theorems & Definitions (64)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• Remark 3.2
• Proposition 3.3
• proof