Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.
This is usually taken as the definition and the abstract characterization as a dual Banach space derived as a consequence.
[3] The definition of JB* algebras was suggested in 1976 by Irving Kaplansky at a lecture in Edinburgh.
JB* algebras have been used extensively as a framework for studying bounded symmetric domains in infinite dimensions.
This generalizes the theory in finite dimensions developed by Max Koecher using the complexification of a Euclidean Jordan algebra.
[11] The JBW factors not of Type I2 and I3 are all JW factors, i.e. can be realized as Jordan algebras of self-adjoint operators on a Hilbert space closed in the weak operator topology.