# Jordan Algebra

In abstract algebra, a Jordan algebra is an (not necessarily associative) algebra over a field whose multiplication satisfies the following axioms:

1. (commutative law)
2. (Jordan identity).

The product of two elements x and y in a Jordan algebra is also denoted xy, particularly to avoid confusion with the product of a related associative algebra. The axioms imply that a Jordan algebra is power-associative and satisfies the following generalization of the Jordan identity: for all positive integers m and n.

Jordan algebras were first introduced by Pascual Jordan (1933) to formalize the notion of an algebra of observables in quantum mechanics. They were originally called "r-number systems", but were renamed "Jordan algebras" by Albert (1946), who began the systematic study of general Jordan algebras.

### Other articles related to "jordan algebra":

Jordan Algebra - Peirce Decomposition
... If e is an idempotent in a Jordan algebra A (e2=e) and R is the operation of multiplication by e, then R(2R−1)(R−1) = 0 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 ...

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