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Let A be a Jordan algebra over the reals which is a Banach space with respect to a norm satisfying the requirements : c i) II a o b II ~ II a II II b ll , c i i) II a 2 11 = II a 11 2 , (iii) !la 2 11 _::: lla 2 +b 2 11 for a,b E A • It is shown that A possesses a unique norm closed Jorcian ideal J such that A/J has a faithful representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, while every "irreducible" representation of A not annihilating J is onto the exceptional Jordan algebra ~ o ' 1-a = c od = (cod) • Hence • '""' then the strong closure of cpp(A) in A is a JB-factor. '""' Proof. Let 1.' 1 =A and K be the state space of A considered as a full set of invariant state of 1.'1.

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