Abstract

Let A be a Jordan algebra over the reals which is a Banach space with respect to a norm satisfying the requirements: (i) ∥ a ° b ∥ ≤ ∥ a ∥ ∥ b ∥, (ii) ∥ a 2 ∥ = ∥ a ∥ 2, (iii) ∥ a 2 ∥ ≤ ∥ a 2 + b 2 ∥ for a, b ϵ A. It is shown that A possesses a unique norm closed Jordan 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 M 3 8.

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