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

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

2005-05-20

arxiv.org/abs/math/0505426v2

Journal of Algebra 301, no. 2 (2006) 728--747

Mathematics

General Mathematics

To appear in Journal of Algebra

Scientific paper

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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