Hilbert Space Effect-Representations of Effect Algebras

In answer to open questions (posed in [12]) we prove that an effect algebra has a Hilbert space effect-representation iff E   possesses an ordering set of states. These are, up to isomorphism, all intervals and all their sub-effect algebras in the set of all positive linear operators on any Hilbert space HH. Nevertheless, there are effect algebras E, elements of which are linear operators in a Hilbert space, but E does not have such a representation.