Representation theory of effect algebras

Abstract

The aim of this paper is to characterize representable and weak representable effect algebras and establish a representation theory of effect algebras. An effect algebra E is said to be representable if there exists a Hilbert space H and a monomorphism π from E into the Hilbert space effect algebra e ( H ) and it is said to be weakly representable if there exists an injective morphism from E into some e ( H ). It is proved that an effect algebra E with the nonempty state space S ( E ) is representable if and only if x, y ∈ E , f ( x )+ f ( y ) ≤ 1 implies x ⊕ y is defined; it is weakly representable if and only if the state space S ( E ) separates the points of E. Some operational properties of representable effect algebras are established, and some applications of the obtained results are listed.