Semilattices of groups and nonstable K-theory of extended Cuntz limits

Abstract

We give an elementary characterization of those abelian monoidsM that are direct limits of countable sequences of finite direct sums of monoids of the form either (Z/nZ) ⊔ {0} or Z ⊔ {0}. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of all subgroups of M (viewed as a semigroup), and it makes it pos- sible to express M as a certain submonoid of a direct product ×G, where is a distributive semilattice with zero and G is an abelian group. When applied to the monoids V (A) appearing in the nonstable K-theory of C*-algebras, our results yield a full description of V (A) for C*-inductive limits A of finite sums of full matrix algebras over either Cuntz algebras On, where 2 ≤ n < ∞, or corners of O1 by projections, thus extending to the case including O1 earlier work by the authors together with K.R. Goodearl.