Some model-theoretic correspondences between dimension groups and AF algebras

If A⊆B are structures for a first-order language S, A is said to be algebraically (existentially) closed in B just in case every positive existential (existential) S_A-sentence true in B_A is true in A_A. In 1976 Elliott showed that unital AF (‘approximately finite-dimensional’) algebras are classified up to isomorphism by corresponding dimension groups with order unit. This paper shows that one dimension group with order unit is algebraically (existentially) closed in another just in case the corresponding AF algebras, viewed as metric structures, fall in the same relation.