Article contents
A dichotomy for groupoid
$\text{C}^{\ast }$-algebras
Published online by Cambridge University Press: 13 August 2018
Abstract
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 of the reduced groupoid C
$^{\ast }$-algebra of
$G$ in the sense that if
$G$ is ample, minimal, topologically principal, and
$S(G)$ is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for
$\text{C}_{r}^{\ast }(G)$. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph
$\text{C}^{\ast }$-algebras as well.
MSC classification
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- Original Article
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- © Cambridge University Press, 2018
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