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Preprints, Working Papers, ... Year : 2005

A $K_0$-avoiding dimension group with an order-unit of index two

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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Dates and versions

hal-00004942 , version 1 (20-05-2005)
hal-00004942 , version 2 (01-06-2005)

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Friedrich Wehrung. A $K_0$-avoiding dimension group with an order-unit of index two. 2005. ⟨hal-00004942v1⟩
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