Hermitian Jordan triple system

Definition

A Hermitian Jordan triple system is a complex vector space $V$ with a sesquilinear triple product

\{x,y,z\}:V\times V\times V\longrightarrow V

that is complex-linear in $x,z$ , conjugate-linear in $y$ , symmetric in the outer variables, and satisfies the Jordan triple identity

\{x,y,\{u,v,w\}\} =\{\{x,y,u\},v,w\} -\{u,\{y,x,v\},w\} +\{u,v,\{x,y,w\}\}.

In the finite-dimensional positive case, the Hermitian trace form

(x\mid y)=\operatorname{tr}D(x,y), \qquad D(x,y)z=\{x,y,z\},

is positive definite.

Operator-algebra example

If $V$ is an associative complex $*$ -algebra, or a complex vector subspace of one that is closed under the following operation, then

\{x,y,z\}=\frac12(xy^*z+zy^*x)

defines a Hermitian Jordan triple product. In particular, rectangular complex matrices form a triple system even though ordinary matrix multiplication does not close on a rectangular matrix space. A C*-algebra with this triple product is a basic analytic example.

Relation to Jordan algebras

A unital Hermitian Jordan algebra gives a triple system, but the triple-system language does not require a distinguished unit or even a binary product. Conversely, choosing a suitable tripotent can produce Peirce spaces and binary Jordan products on appropriate components. Thus a Hermitian Jordan triple system should not be identified with a Jordan algebra without extra data.

Geometry

Finite-dimensional positive Hermitian Jordan triple systems classify bounded symmetric domains: the open unit ball for the associated spectral norm is a bounded symmetric domain, and every bounded symmetric domain arises this way up to biholomorphism. Simple triple systems correspond to irreducible bounded symmetric domains.

Convention warning

Some authors reverse which outer variable is linear or rescale the triple product. The Jordan triple identity must be read with the same convention. The adjective Hermitian records the conjugate-linearity; an algebraic Jordan triple system over a general field is linear in all variables and is a different notion.

References

Ottmar Loos, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977, Parts I–II.
Harald Upmeier, Symmetric Banach Manifolds and Jordan C-Algebras*, North-Holland, 1985, Chapters 2–4.
John C. Baez, Eric H. Bokor, and Latham Boyle, “Jordan Pair Quantum Mechanics and the Standard Model,” 2026. arXiv:2607.10833.