Abstract
It is known that every dimension group with order-unit of size at most ℵ1 is isomorphic toK 0(R) for some locally matricial ringR (in particular,R is von Neumann regular); similarly, every conical refinement monoid with orderunit 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-unitE of cardinality ℵ2 (or whatever larger) with strong non-measurability properties. These properties yield in particular thatE + is not measurable in Dobbertin’s sense, or thatE is not isomorphic to theK 0 of any von Neumann regular ring, or that the maximal semilattice quotient ofE + 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.
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