Non-measurability properties of interpolation vector spaces

International audienceIt is known that every dimension group with order-unit of size at most $\aleph_1$ is isomorphic to $K_0(R)$ for some locally matricial ring $R$ (in particular, $R$ is von Neumann regular); similarly, every conical refinement monoid with order-unit of size at most $\aleph_1$ is the image of a V-measure in Dobbertin's sense, the corresponding problems for larger cardinalities being open. We settle these problems here, by showing a general functorial procedure to construct ordered vector spaces with interpolation and order-unit $E$ of cardinality $\aleph_2$ (or whatever larger) with strong non-measurability properties. These properties yield in particular that $E^+$ is not measurable in Dobbertin's sense, or that $E$ is not isomorphic to the $K_0$ of any von Neumann regular ring, or that the maximal semilattice quotient of $E^+$ is not the range of any weak distributive homomorphism (in E.T. Schmidt's sense) on any distributive lattice, thus respectively solving problems of Dobbertin, Goodearl and Schmidt