A dichotomy for groupoid $\text{C}^{\ast }$ -algebras

@article{Rainone2017ADF,
  title={A dichotomy for groupoid \$\text\{C\}^\{\ast \}\$ -algebras},
  author={Timothy Rainone and Aidan Sims},
  journal={Ergodic Theory and Dynamical Systems},
  year={2017},
  volume={40},
  pages={521 - 563}
}
  • T. RainoneA. Sims
  • Published 14 July 2017
  • Mathematics
  • Ergodic Theory and Dynamical Systems
We study the finite versus infinite nature of C $^{\ast }$ -algebras arising from étale groupoids. For an ample groupoid $G$ , we relate infiniteness of the reduced C $^{\ast }$ -algebra $\text{C}_{r}^{\ast }(G)$ to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid $S(G)$ which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid characterizes the finite/infinite nature… 

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References

SHOWING 1-10 OF 61 REFERENCES

Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$ -algebras

In this paper, we study the ideal structure of reduced $C^{\ast }$ -algebras $C_{r}^{\ast }(G)$ associated to étale groupoids $G$ . In particular, we characterize when there is a one-to-one

Purely infinite $C^{\ast }$-algebras associated to étale groupoids

Let $G$ be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if all the non-zero positive elements of

Crossed products of nuclear C*-algebras by free groups and their traces

We study the matricial field (MF) property for certain reduced crossed product C*-algebras and their traces. Using classification techniques and induced K-theoretic dynamics, we show that reduced

Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras

We present a classification theorem for a class of unital simple separable amenable ${\cal Z}$-stable $C^*$-algebras by the Elliott invariant. This class of simple $C^*$-algebras exhausts all

On the classification of simple amenable C*-algebras with finite decomposition rank

Let $A$ be a unital simple separable C*-algebra satisfying the UCT. Assume that $\mathrm{dr}(A)<+\infty$, $A$ is Jiang-Su stable, and $\mathrm{K}_0(A)\otimes \mathbb{Q}\cong \mathbb{Q}$. Then $A$ is

Nonstable K-theory for Graph Algebras

We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property and

A classification theorem for nuclear purely infinite simple $C^*$-algebras

Starting from Kirchberg's theorems announced at the operator algebra conference in Gen eve in 1994, namely O2 A = O2 for separable unital nuclear simple A and O1 A = A for separable unital nuclear

Unbounded quasitraces, stable finiteness and pure infiniteness

We prove that if A is a \sigma-unital exact C*-algebra of real rank zero, then every state on K_0(A) is induced by a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure

Purely infinite C*-algebras associated to etale groupoids

Let G be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that C∗ r (G) is purely infinite simple if and only if all the nonzero positive elements of C0(G ) are

Classification of Nuclear, Simple C*-algebras

The possibility that nuclear (or amenable) C*-algebras should be classified up to isomorphism by their K-theory and related invariants was raised in an article by Elliott [48] (written in 1989) in
...