Jordan operator algebra

Builds on Hilbert space Self-adjoint operator Jordan algebra Special Jordan algebra

Definition

Let $H$ be a complex Hilbert space. Let $B(H)_{\mathrm{sa}}$ be the real vector space of bounded self-adjoint operators on $H$ , with the operator norm $\norm{\cdot}$ . For $a, b \in B(H)_{\mathrm{sa}}$ put $a \circ b = \frac{1}{2}(ab + ba)$ . A Jordan operator algebra, or JC-algebra, on $H$ is a real linear subspace $A \subseteq B(H)_{\mathrm{sa}}$ such that:

$a \circ b \in A$ for all $a, b \in A$ ;
$A$ is closed in the norm topology.

Remarks

Each JC-algebra is a special Jordan algebra, as a Jordan subalgebra of $B(H)^+$ . Clause 1 is equivalent to $a^2 \in A$ for all $a \in A$ , since $a \circ b = \frac{1}{2}((a + b)^2 - a^2 - b^2)$ .

Example: $H = \C^n$ and $A$ the real symmetric $n \times n$ matrices. For symmetric $a$ and $b$ , $(ab + ba)^T = ba + ab$ , so clause 1 holds, and $A$ is closed because it is finite-dimensional.

Non-example: for $H = \C^2$ , the real span of the Pauli matrices $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ fails clause 1. Indeed $\sigma_x \circ \sigma_x = I$ , and $I$ is not a combination of $\sigma_x$ and $\sigma_z$ .

A JC-algebra that is closed in the weak operator topology is called a JW-algebra.

Reading path

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Tag	`QZH32DH8SX`
URI	`m2://QZH32DH8SX`
Kind	definition
Writing	W0 stub
MSC	46L70, 17C65
Defines	Jordan operator algebra
Uses	Hilbert space, Self-adjoint operator, Jordan algebra, Special Jordan algebra
Changed	3 November 2026

Jordan operator algebra

Definition

Remarks

Reading path

Further reading

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