On realization of generalized effect algebras

TL;DR

This paper establishes conditions under which generalized effect algebras can be represented within operator generalized effect algebras of a Hilbert space, extending previous results and confirming the existence of order determining sets of states.

Contribution

It extends the representation theory of effect algebras to generalized effect algebras, showing their representability in operator algebras of Hilbert spaces under certain conditions.

Findings

Generalized effect algebra is representable iff it has an order determining set of generalized states.

Operator generalized effect algebras always have an order determining set of generalized states.

Extends previous results for effect algebras to generalized effect algebras.

Abstract

A well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice $L({\mathcal H})$ of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra ${\mathcal G}_D({\mathcal H})$ of effects of a complex Hilbert space ${\mathcal H}$ iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Rie\v{c}anov\'a and Zajac. Further, any operator generalized effect algebra ${\mathcal G}_D({\mathcal H})$ possesses an order determining set of generalized states.