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Article Dans Une Revue Journal für die reine und angewandte Mathematik Année : 2005

Semilattices of groups and inductive limits of Cuntz algebras

Ken Goodearl
  • Fonction : Auteur
  • PersonId : 829318
Enrique Pardo
  • Fonction : Auteur
  • PersonId : 829318

Résumé

We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form $(Z/nZ)\cup\{0\}$ (where $0$ is a new zero element), for positive integers $n$. The key properties are the Riesz refinement property and the requirement that each element $x$ has finite order, that is, $(n+1)x=x$ for some positive integer $n$. Such monoids are necessarily semilattices of abelian groups, and part of our approach yields a characterization of the Riesz refinement property among semilattices of abelian groups. Further, we describe the monoids in question as certain submonoids of direct products $\Lambda\times G$ for semilattices $\Lambda$ and torsion abelian groups $G$. When applied to the monoids $V(A)$ appearing in the non-stable K-theory of C*-algebras, our results yield characterizations of the monoids $V(A)$ for C* inductive limits $A$ of sequences of finite direct products of matrix algebras over Cuntz algebras $O_n$. In particular, this completely solves the problem of determining the range of the invariant in the unital case of R\o rdam's classification of inductive limits of the above type.
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Dates et versions

hal-00002856 , version 1 (16-09-2004)

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Friedrich Wehrung, Ken Goodearl, Enrique Pardo. Semilattices of groups and inductive limits of Cuntz algebras. Journal für die reine und angewandte Mathematik, 2005, 588, pp.1--25. ⟨10.1515/crll.2005.2005.588.1⟩. ⟨hal-00002856⟩
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