Pairs of simple dimension groups

Given a pair of simple dimension groups with isomorphic state spaces we try to express it as the pair K-0(A), K-0(A x(alpha) R) of K-0 groups for a C* -algebra A with an action alpha of R, where both A and the crossed product A x(alpha) R are supposed to be simple AT algebras of real rank zero. We so lve this when the state spaces are finite-dimensional.