Star order on JBW algebras

The goal of the paper is to extend the star order from associative algebras to non-associative Jordan Banach structures. Let A be a JBW algebra. We define a relation on A as the set of all pairs ( a , b ) ∈ A × A such that the range projections of a and b − a are orthogonal. We show that this relation defines a partial order on A which, in the case of the self-adjoint part of a von Neumann algebra, gives the star order. After showing basic properties of this order we shall prove the following preserver theorem: Let A be a JBW algebra without Type I 2 direct summand and let φ be a continuous map from A to B preserving the star order in both directions. If for each scalar λ one has φ ( λ 1 ) = f ( λ ) z , where f is a (continuous) function and z is a central invertible element, then there is a unique Jordan isomorphism ψ : A → B such that φ ( a ) = ψ ( f ( a ) ) z . Moreover, we show that if A is a Type I n factor, where n ≠ 2 , then the equation above holds for all continuous maps preserving the star order in both directions.