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 I2 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(λ1)=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 In factor, where n≠2, then the equation above holds for all continuous maps preserving the star order in both directions.