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Jordan operator algebra
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Jordan operator algebras are real or complex <a href="/facts/Jordan_algebra/4BBGsQd9">Jordan algebras</a> with the compatible structure of a Banach space. When the coefficients are <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a>, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algebras were devised by Alfsen, Shultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the <a href="/facts/Operator_norm/a5l0TkFW">operator norm</a> are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by <a href="/facts/Irving_Kaplansky/nRLyp7K9">Irving Kaplansky</a> in 1976, require an involution and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of <a href="/facts/Von_Neumann_algebra/8NyWUIRH">von Neumann algebras</a> as <a href="/facts/C*_algebra/B3tnGHIo">C* algebras</a> for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras. Those that can be realised using <a href="/facts/Weak_operator_topology/upQgUu90">ultraweakly closed</a> Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional <a href="/facts/Albert_algebra/YThFsbVl">Albert algebra</a> and the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> and in <a href="/facts/Complex_geometry/ekCsxAXb">complex geometry</a>, where <a href="/facts/Max_Koecher/MJHkXltW">Koecher's</a> description of <a href="/facts/Bounded_symmetric_domain/fLBcoPbU">bounded symmetric domains</a> using <a href="/facts/Jordan_algebra/4BBGsQd9">Jordan algebras</a> has been extended to infinite dimensions. </p>