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 factor or its fixed point subalgebra under a period 2 *-antiautomorphism of the von Neumann factor. 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 superalgebras were introduced by Kac, Kantor and Kaplansky; these are -graded algebras where is a Jordan algebra and has a "Lie-like" product with values in . Any -graded associative algebra becomes a Jordan superalgebra with respect to the graded Jordan brace Jordan simple superalgebras over an algebraically closed field of characteristic 0 were classified by . They include several families and some exceptional algebras, notably and . J-structures The concept of J-structure was introduced by 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. Quadratic Jordan algebras Quadratic Jordan algebras are a generalization of (linear) Jordan algebras introduced by . 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. See also Freudenthal algebra Jordan triple system Jordan pair Kantor–Koecher–Tits construction Scorza variety Notes References . Online HTML version. . Review Further reading External links Jordan algebra at PlanetMath Jordan-Banach and Jordan-Lie algebras at