The axioms imply[1] that a Jordan algebra is power-associative, meaning that
It was soon shown that the algebras were not useful in this context, however they have since found many applications in mathematics.
The Shirshov–Cohn theorem states that any Jordan algebra with two generators is special.
[3] Related to this, Macdonald's theorem states that any polynomial in three variables, having degree one in one of the variables, and which vanishes in every special Jordan algebra, vanishes in every Jordan algebra.
The set of self-adjoint real, complex, or quaternionic matrices with multiplication form a special Jordan algebra.
Over the real numbers there are three isomorphism classes of simple exceptional Jordan algebras.
The Jordan identity implies that if x and y are elements of A, then the endomorphism sending z to x(yz)−y(xz) is a derivation.
A simple example is provided by the Hermitian Jordan algebras H(A,σ).
In many important examples, the structure algebra of H(A,σ) is A. Derivation and structure algebras also form part of Tits' construction of the Freudenthal magic square.
A (possibly nonassociative) algebra over the real numbers is said to be formally real if it satisfies the property that a sum of n squares can only vanish if each one vanishes individually.
In 1932, Jordan attempted to axiomatize quantum theory by saying that the algebra of observables of any quantum system should be a formally real algebra that is commutative (xy = yx) and power-associative (the associative law holds for products involving only x, so that powers of any element x are unambiguously defined).
In finite dimensions, the simple formally real Jordan algebras come in four infinite families, together with one exceptional case: Of these possibilities, so far it appears that nature makes use only of the n×n complex matrices as algebras of observables.
However, the spin factors play a role in special relativity, and all the formally real Jordan algebras are related to projective geometry.
If e is an idempotent in a Jordan algebra A (e2 = e) and R is the operation of multiplication by e, then so the only eigenvalues of R are 0, 1/2, 1.
If the Jordan algebra A is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces A = A0(e) ⊕ A1/2(e) ⊕ A1(e) of the three eigenspaces.
It was later studied in full generality by Albert (1947) and called the Peirce decomposition of A relative to the idempotent e.[6] In 1979, Efim Zelmanov classified infinite-dimensional simple (and prime non-degenerate) Jordan algebras.
The norm on the real Jordan algebra must be complete and satisfy the axioms: These axioms guarantee that the Jordan algebra is formally real, so that, if a sum of squares of terms is zero, those terms must be zero.
They have been used extensively in complex geometry to extend Koecher's Jordan algebraic treatment of bounded symmetric domains to infinite dimensions.
In particular the JBW factors—those with center reduced to R—are completely understood in terms of von Neumann algebras.
Apart from the exceptional Albert algebra, all JWB factors can be realised as Jordan algebras of self-adjoint operators on a Hilbert space closed in the weak operator topology.
Of these the spin factors can be constructed very simply from real Hilbert spaces.
Jordan superalgebras were introduced by Kac, Kantor and Kaplansky; these are
The concept of J-structure was introduced by Springer (1998) to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation.
In characteristic not equal to 2 the theory of J-structures is essentially the same as that of Jordan algebras.
The fundamental identities of the quadratic representation of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic.