In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: (commutative law) (). The product of two elements x and y in a Jordan algebra is also denoted x ∘ y, particularly to avoid confusion with the product of a related associative algebra. The axioms imply that a Jordan algebra is power-associative, meaning that is independent of how we parenthesize this expression. They also imply that for all positive integers m and n. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element , the operations of multiplying by powers all commute. Jordan algebras were introduced by in an effort to formalize the notion of an algebra of observables in quantum electrodynamics. It was soon shown that the algebras were not useful in this context, however they have since found many applications in mathematics. The algebras were originally called "r-number systems", but were renamed "Jordan algebras" by , who began the systematic study of general Jordan algebras. Special Jordan algebras Given an associative algebra A (not of characteristic 2), one can construct a Jordan algebra A+ using the same underlying addition vector space. Notice first that an associative algebra is a Jordan algebra if and only if it is commutative. If it is not commutative we can define a new multiplication on A to make it commutative, and in fact make it a Jordan algebra. The new multiplication x ∘ y is the Jordan product: This defines a Jordan algebra A+, and we call these Jordan algebras, as well as any subalgebras of these Jordan algebras, special Jordan algebras. All other Jordan algebras are called exceptional Jordan algebras. The Shirshov–Cohn theorem states that any Jordan algebra with two generators is special. Related to this, Macdonald's theorem states that any polynomial in three variables, that has degree one in one of the variables, and that vanishes in every special Jordan algebra, vanishes in every Jordan algebra. Hermitian Jordan algebras If (A, σ) is an associative algebra with an involution σ, then if σ(x)=x and σ(y)=y it follows that Thus the set of all elements fixed by the involution (sometimes called the hermitian elements) form a subalgebra of A+, which is sometimes denoted H(A,σ). Examples 1. The set of self-adjoint real, complex, or quaternionic matrices with multiplication form a special Jordan algebra. 2. The