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

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: Lattice; monoid; dimension monoid; dimension group; index; V-homomorphism; modular lattice; von Neumann regular ring; locally matricial

Domaines: Mathématiques générales [math.GM]

Friedrich Wehrung. A $K_0$-avoiding dimension group with an order-unit of index two. Journal of Algebra, 2006, 301 (2), pp.728--747. ⟨10.1016/j.jalgebra.2005.06.003⟩. ⟨hal-00004942v2⟩