Non-measurability properties of interpolation vector spaces

It is known that every dimension group with order-unit of size at most א1 is isomorphic to K0(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 א1 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 א2 (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 K0 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.