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

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

2005-11-10

arxiv.org/abs/math/0511272v2

K-Theory 37 (2006) 1--23

Mathematics

Operator Algebras

K-Theory, to appear

Scientific paper

10.1007/s10977-006-0005-4

We give an elementary characterization of those (abelian) semigroups $M$ that are direct limits of countable sequences of finite direct products of monoids of the form $C\cup\{0\}$ for monogenic groups $C$. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of subgroups of $M$, and it makes it possible to express $M$ as a certain submonoid of a direct product $S\times G$, where $S$ 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 products of full matrix algebras over either Cuntz algebras $O\_n$, where $2\leq n<\infty$, or corners of $O\_{\infty}$ by projections, thus extending to the case including $O\_{\infty}$ earlier work by the authors together with K.R. Goodearl.

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