A Coordinatization Theorem for Jordan Algebras

Abstract

Throughout this note, the term “algebra” is used for algebra over a field Φ of characteristic ≠2, not necessarily associative or of finite dimensionality. Let D be such an algebra with an identity 1 and an involution d → $\bar d$. We can form the matrix algebra D _n of n × n matrices with entries in D and the usual matrix compositions of addition, multiplication by elements of Φ, and matrix multiplication. If γ = diag {γ₁, γ₂,…, γ _n } is a diagonal matrix with entries γ _i in the nucleus of D such that the γ _i are self-adjoint ($\left( {{{\bar \gamma }_i} = {\gamma_i}} \right)$ = γ _i ) and have inverses, then γ determines an involution

$$ X \to X* \equiv \gamma ^{ - 1} \bar X'\gamma $$

(1)

in D _n . Here $\bar X'$ is the matrix whose (i,j) entry is the conjugate ${\bar x_{ji}}$ of the (j,i) entry of X. No parentheses are needed in the last term in (1) since γ is in the nucleus of D _n . An involution of the type just described is called a canonical involution in D _n and the special case obtained by taking γ = 1 is called a standard involution. Let ℌ(D _n ,γ) be the set of γ-hermitian matrices, that is, the matrices AÎD _n such that A* = γ⁻¹ $\bar A'$ γ = A. This is a subspace of D _n over Φ, and it is closed under the Jordan product

$$ A \cdot B = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( {AB + BA} \right) $$

((2))

This research was supported in part by the U.S. Air Force under the grant SAR-R-AFOSR 61–29. Reproduction in whole or in part is permitted for any purpose of the United States Government.