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A $K\_0$-avoiding dimension group with an order-unit of index two

Friedrich Wehrung
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Abstract

We prove that there exists a dimension group $G$ whose positive cone is not isomorphic to the dimension monoid Dim$L$ of any lattice $L$. The dimension group $G$ has an order-unit, and can be taken of any cardinality greater than or equal to $\aleph\_2$. As to determining the positive cones of dimension groups in the range of the Dim functor, the $\aleph\_2$ bound is optimal. This solves negatively the problem, raised by the author in 1998, whether any conical refinement monoid is isomorphic to the dimension monoid of some lattice. Since $G$ has an order-unit of index two, this also solves negatively a problem raised in 1994 by K.R. Goodearl about representability, with respect to $K\_0$, of dimension groups with order-unit of index 2 by unit-regular rings.

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Friedrich Wehrung. 2005-06-01. A $K\_0$-avoiding dimension group with an order-unit of index two. https://arxiv.org/abs/math/0505426

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