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The representation theory of C*-algebras associated to groupoids
Published online by Cambridge University Press: 27 February 2012
Abstract
Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of such that G: = E/
is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E; G, λ) is a quotient of the C*-algebra of E obtained by completing the space of
-equivariant functions on E. We show that C*(E; G, λ) is postliminal if and only if the orbit space of G is T0 and that C*(E; G, λ) is liminal if and only if the orbit space is T1. We also show that C*(E; G, λ) has bounded trace if and only if G is integrable and that C*(E; G, λ) is a Fell algebra if and only if G is Cartan.
Let be a second-countable, locally compact, Hausdorff groupoid with Haar system λ and continuously varying, abelian isotropy groups. Let
be the isotropy groupoid and
: =
/
. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(
, λ) has bounded trace if and only if
is integrable and that C*(
, λ) is a Fell algebra if and only if
is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 1 , July 2012 , pp. 167 - 191
- Copyright
- Copyright © Cambridge Philosophical Society 2012
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