Hermitian matrix Jordan algebra

For $\mathbb K=\mathbb R,\mathbb C$ , or $\mathbb H$ , the Hermitian matrix Jordan algebra $\mathfrak h_n(\mathbb K)$ is the real vector space of matrices $X\in M_n(\mathbb K)$ satisfying $X^*=X$ , with product

X\circ Y=\frac12(XY+YX).

It is a Euclidean Jordan algebra for every $n\geq1$ .

Exactly when the construction is Jordan

For $\mathbb K=\mathbb R,\mathbb C,\mathbb H$ , associativity of matrix multiplication proves the Jordan identity in every size. Quaternionic matrices are associative because the quaternion algebra is associative, even though it is noncommutative.

For $\mathbb K=\mathbb O$ , the same displayed formula gives a Jordan algebra only for $n=1,2,3$ :

$\mathfrak h_1(\mathbb O)\cong\mathbb R$ ;
$\mathfrak h_2(\mathbb O)$ is a spin factor;
$\mathfrak h_3(\mathbb O)$ is the exceptional Albert algebra.

For $n\geq4$ , octonionic matrix multiplication does not satisfy the Jordan identity on all Hermitian matrices, so $\mathfrak h_n(\mathbb O)$ is not a Jordan algebra under this formula. The notation must not be presented as a uniform Jordan construction for arbitrary normed division algebras and sizes.

Dimensions and Euclidean form

If $d=\dim_{\mathbb R}\mathbb K$ , then

\dim_{\mathbb R}\mathfrak h_n(\mathbb K) =n+\frac{d\,n(n-1)}{2}.

The diagonal entries are real, while each off-diagonal pair contributes one copy of $\mathbb K$ . The standard Euclidean form is $\langle X,Y\rangle=\operatorname{Re}\operatorname{Tr}(XY)$ , equivalently the Jordan trace of $X\circ Y$ .

The algebras $\mathfrak h_2(\mathbb C)$ and $\mathfrak h_3(\mathbb C)$ are respectively the observables of a complex qubit and qutrit. Positive elements of trace one are their density matrices.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000. Publisher record.