Editing Jordan algebra (section)

===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>