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A dichotomy for groupoid C*-algebras

Journal Article


Abstract


  • We study the finite versus infinite nature of C -algebras arising from étale groupoids. For an ample groupoid, we relate infiniteness of the reduced C -algebra to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid 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 -algebra of in the sense that if is ample, minimal, topologically principal, and is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph -algebras as well.

UOW Authors


  •   Rainone, Timothy (external author)
  •   Sims, Aidan

Publication Date


  • 2018

Citation


  • Rainone, T. & Sims, A. (2018). A dichotomy for groupoid C*-algebras. Ergodic Theory and Dynamical Systems, Online First 1-43.

Scopus Eid


  • 2-s2.0-85052709705

Number Of Pages


  • 42

Start Page


  • 1

End Page


  • 43

Volume


  • Online First

Place Of Publication


  • United Kingdom

Abstract


  • We study the finite versus infinite nature of C -algebras arising from étale groupoids. For an ample groupoid, we relate infiniteness of the reduced C -algebra to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid 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 -algebra of in the sense that if is ample, minimal, topologically principal, and is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph -algebras as well.

UOW Authors


  •   Rainone, Timothy (external author)
  •   Sims, Aidan

Publication Date


  • 2018

Citation


  • Rainone, T. & Sims, A. (2018). A dichotomy for groupoid C*-algebras. Ergodic Theory and Dynamical Systems, Online First 1-43.

Scopus Eid


  • 2-s2.0-85052709705

Number Of Pages


  • 42

Start Page


  • 1

End Page


  • 43

Volume


  • Online First

Place Of Publication


  • United Kingdom