Jordan algebras of type I

Jordan, von Neumann, and Wigner [5] have classified all finite dimensional Jordan algebras over the reals. The present paper is an attempt to do the same in the infinite dimensional case. The following restriction will be imposed: we assume the Jordan alge- bras are weakly closed Jordan algebras of self-adjoint operators with minimal projections acting on a Hilbert space, i.e. are irreducible JW-algebras of type i.(1) The result is then quite analogous to that in [5], except we do not get hold of the Jordan algebra ~a s of that paper, as should be expected from the work of Albert [1]. We first classify all irreducible JW-algebras of type In, n>~3 (Theorem 3.9). These algebras are roughly all seif-adjoint operators on a Hilbert space over either the reals, the complexes, or the quaternions. Then all JW-factors of type

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