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Non-measurability properties of interpolation vector spaces

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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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  1. Département de Mathématiques, Université de Caen, 14032, Caen Cedex, France

    Friedrich Wehrung

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  1. Friedrich Wehrung
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Wehrung, F. Non-measurability properties of interpolation vector spaces. Isr. J. Math. 103, 177–206 (1998). https://doi.org/10.1007/BF02762273

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  • DOI: https://doi.org/10.1007/BF02762273

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