Editing Jordan algebra (section)

==Special kinds and generalizations==

===Infinite-dimensional Jordan algebras===
In 1979, [[Efim Zelmanov]] classified infinite-dimensional simple (and prime non-degenerate) Jordan algebras. They are either of Hermitian or Clifford type. In particular, the only exceptional simple Jordan algebras are finite-dimensional [[Albert algebra]]s, which have dimension 27.

===Jordan operator algebras===
{{Main|Jordan operator algebra}}
The theory of [[operator algebras]] has been extended to cover [[Jordan operator algebra]]s.

The counterparts of [[C*-algebra]]s are JB algebras, which in finite dimensions are called [[Euclidean Jordan algebra]]s. The norm on the real Jordan algebra must be [[Complete metric space|complete]] and satisfy the axioms:

:<math>\displaystyle{\|a\circ b\|\le \|a\|\cdot \|b\|,\,\,\, \|a^2\|=\|a\|^2,\,\,\, \|a^2\|\le \|a^2 +b^2\|.}</math>

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. The complexifications of JB algebras are called Jordan C*-algebras or JB*-algebras. They have been used extensively in [[complex geometry]] to extend [[Max Koecher|Koecher's]] Jordan algebraic treatment of [[bounded symmetric domain]]s to infinite dimensions. Not all JB algebras can be realized as Jordan algebras of self-adjoint operators on a Hilbert space, exactly as in finite dimensions. The exceptional [[Albert algebra]] is the common obstruction.

The Jordan algebra analogue of [[von Neumann algebra]]s is played by JBW algebras. These turn out to be JB algebras which, as Banach spaces, are the dual spaces of Banach spaces. Much of the structure theory of von Neumann algebras can be carried over to JBW algebras. 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. All other JWB factors are either the self-adjoint part of a [[Von Neumann algebra#Factors|von Neumann factor]] or its fixed point subalgebra under a period 2 *-antiautomorphism of the von Neumann factor.<ref>See:
*{{harvnb|Hanche-Olsen|Størmer|1984}}
*{{harvnb|Upmeier|1985}}
*{{harvnb|Upmeier|1987}}
*{{harvnb|Faraut|Koranyi|1994}}</ref>

===Jordan rings===
A Jordan ring is a generalization of Jordan algebras, requiring only that the Jordan ring be over a general ring rather than a field. Alternatively one can define a Jordan ring as a commutative [[nonassociative ring]] that respects the Jordan identity.

===Jordan superalgebras===

Jordan [[superalgebra]]s were introduced by Kac, Kantor and Kaplansky; these are <math>\mathbb{Z}/2</math>-graded algebras <math>J_0 \oplus J_1</math> where <math>J_0</math> is a Jordan algebra and <math>J_1</math> has a "Lie-like" product with values in <math>J_0</math>.<ref>{{harvnb|McCrimmon|2004|pp=9–10}}</ref>

Any <math>\mathbb{Z}/2</math>-graded associative algebra <math>A_0 \oplus A_1</math> becomes a Jordan superalgebra with respect to the graded Jordan brace

:<math>\{x_i,y_j\} = x_i y_j + (-1)^{ij} y_j x_i \ . </math>

Jordan simple superalgebras over an algebraically closed field of characteristic 0 were classified by {{harvtxt|Kac|1977}}.  They include several families and some exceptional algebras, notably <math>K_3</math> and <math>K_{10}</math>.

===J-structures===
{{Main|J-structure}}
The concept of [[J-structure]] was introduced by {{harvtxt|Springer|1998}} to develop a theory of Jordan algebras using [[linear algebraic group]]s and axioms taking the Jordan inversion as basic operation and [[Hua's identity]] as a basic relation.  In [[characteristic of a field|characteristic]] not equal to 2 the theory of J-structures is essentially the same as that of Jordan algebras.

===Quadratic Jordan algebras===
{{Main|Quadratic Jordan algebra}}
Quadratic Jordan algebras are a generalization of (linear) Jordan algebras introduced by {{harvs|txt|first=Kevin|last=McCrimmon|year=1966}}. 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. There is a uniform description of finite-dimensional simple quadratic Jordan algebras, independent of characteristic: in characteristic not equal to 2 the theory of quadratic Jordan algebras reduces to that of linear Jordan algebras.