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Amenable actions of inverse semigroups


We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup ${\mathcal{S}}$ , the action of ${\mathcal{S}}$ on its spectrum is amenable if and only if every action of ${\mathcal{S}}$ is amenable.

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[AD02] Anantharaman-Delaroche, C.. Amenability and exactness for dynamical systems and their C*-algebras. Trans. Amer. Math. Soc. 354 (2002), 41534178.
[ADR00] Anantharaman-Delaroche, C. and Renault, J.. Amenable Groupoids (Monographie de l’Enseignement Mathématique, 36) . l’Enseignement Mathématique, Geneva, 2000.
[DP85] Duncan, J. and Paterson, A.. C*-algebras of inverse semigroups. Proc. Edinb. Math. Soc. 28 (1985), 4158.
[Exe08] Exel, R.. Inverse semigroups and combinatorial C*-algebras. Bull. Braz. Math. Soc. (N.S.) 39(2) (2008), 191313.
[Mil10] Milan, D.. C*-algebras of inverse semigroups: amenability and weak containment. J. Operator Theory 63(2) (2010), 317332.
[Pat99] Paterson, A.. Groupoids, Inverse Semigroups, and Their Operator Algebras. Birkhäuser, Boston, MA, 1999.
[Ren80] Renault, J.. A Groupoid Approach to C*-algebras (Lecture Notes in Mathematics, 793) . Springer, Berlin, 1980.
[Wil15] Willett, R.. A non-amenable groupoid whose maximal and reduced C*-algebras are the same. Preprint, 2015, arXiv:1504.05615 [math.OA].
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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