On partially ordered real involutory Algebras

In this paper we study real O *-algebra, on the hermitian elements of which a partial order exists which is compatible with the algebraic structure. Such algebras occur in the axiomatic approach to the description of the space of random variables (observables) in quantum probability theory. We study the relations between these so called real O *-algebras and their complexification, and also their Jordan structure. Our main result is the theorem on the representation of abstract real O *-algebras as algebras of locally measurable (unbounded) operators affiliated with a real von Neumann algebra on a Hilbert space. © Springer Science + Business Media B.V. 2007.