Abstract
We prove that the full C ∗-algebra of a second-countable, Hausdorff, étale, amenable groupoid is simple if and only if the groupoid is both topologically principal and minimal. We also show that if G has totally disconnected unit space, then the complex ∗-algebra of its inverse semigroup of compact open bisections, as introduced by Steinberg, is simple if and only if G is both effective and minimal.
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Notes
See Remark 2.3 regarding terminology.
We prefer the notation A(G) because Steinberg’s notation \(\mathbb {C}G\) suggests the free \(\mathbb {C}\)-module with basis G, which is substantially larger. To avoid clashing with Steinberg’s notation, we use \(\mathbb {F}(W)\) for the free complex module with basis W.
Exel uses the term essentially principal for what we call effective (see [10, p. 897])
G U embeds properly into G since G acts properly on itself.
Our convention for G(X,T) is slightly different than in [11] for compatibility with Example 7.3(2).
References
Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J. Algebra 293, 319–334 (2005)
Anantharaman-Delaroche, C.: Purely infinite C ∗-algebras arising from dynamical systems. Bull. Soc. Math. Fr. 125, 199–225 (1997)
Anantharaman-Delaroche, C., Renault, J.: Amenable Groupoids. Monographies de L’Enseignement Mathématique, vol. 36. L’Enseignement Mathématique, Geneva (2000)
Aranda Pino, G., Clark, J., an Huef, A., Raeburn, I.: Kumjian-Pask algebras of higher-rank graphs. Trans. Am. Math. Soc. 365, 3613–3641 (2013)
Archbold, R.J., Spielberg, J.S.: Topologically free actions and ideals in discrete C ∗-dynamical systems. Proc. Edinb. Math. Soc. (2) 37, 119–124 (1994)
Carlsen, T., Silvestrov, S.: On the Exel crossed product of topological covering maps. Acta Appl. Math. 108, 573–583 (2009)
Clark, L.O., Farthing, C., Sims, A., Tomforde, M.: A groupoid generalization of Leavitt path algebras. Preprint (2011). arXiv:1110.6198v2 [math.RA]
Deaconu, V.: Groupoids associated with endomorphisms. Trans. Am. Math. Soc. 347, 1779–1786 (1995)
Exel, R.: A new look at the crossed product of a C ∗-algebra by a semigroup of endomorphisms. Ergod. Theory Dyn. Syst. 28, 749–789 (2008)
Exel, R.: Non-Hausdorff étale groupoids. Proc. Am. Math. Soc. 139, 897–907 (2011)
Exel, R., Renault, J.: Semigroups of local homeomorphisms and interaction groups. Ergod. Theory Dyn. Syst. 27, 1737–1771 (2007)
Exel, R., Vershik, A.: C ∗-algebras of irreversible dynamical systems. Can. J. Math. 58, 39–63 (2006)
Farthing, C., Muhly, P.S., Yeend, T.: Higher-rank graph C ∗-algebras: an inverse semigroup and groupoid approach. Semigroup Forum 71, 159–187 (2005)
Giordano, T., Putnam, I.F., Skau, C.F.: Topological orbit equivalence and C ∗-crossed products. J. Reine Angew. Math. 469, 51–111 (1995)
Ionescu, M., Muhly, P.S.: Groupoid methods in wavelet analysis. In: Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey. Contemp. Math., vol. 449, pp. 193–208. Amer. Math. Soc., Providence (2008)
Kumjian, A., Pask, D.: Higher rank graph C ∗-algebras. N.Y. J. Math. 6, 1–20 (2000)
Kumjian, A., Pask, D., Raeburn, I.: Cuntz-Krieger algebras of directed graphs. Pac. J. Math. 184, 161–174 (1998)
Laca, M.: From endomorphisms to automorphisms and back: dilations and full corners. J. Lond. Math. Soc. (2) 61, 893–904 (2000)
Muhly, P.S., Renault, J., Williams, D.P.: Continuous trace groupoid C ∗-algebras, III. Trans. Am. Math. Soc. 348, 3621–3641 (1996)
Paterson, A.L.T.: Groupoids, Inverse Semigroups, and Their Operator Algebras p. xvi+274. Birkhäuser, Boston (1999)
Powers, R.T.: Simplicity of the C ∗-algebra associated with the free group on two generators. Duke Math. J. 42, 151–156 (1975)
Quigg, J., Sieben, N.: C ∗-actions of r-discrete groupoids and inverse semigroups. J. Aust. Math. Soc. A 66, 143–167 (1999)
Renault, J.: A Groupoid Approach to C ∗-Algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)
Renault, J.: The ideal structure of groupoid crossed product C ∗-algebras. J. Oper. Theory 25, 3–36 (1991)
Renault, J.: Cartan subalgebras in C ∗-algebras. Ir. Math. Soc. Bull. 61, 29–63 (2008)
Renault, J.: Examples of masas in C ∗-algebras. In: Operator Structures and Dynamical Systems. Contemp. Math., vol. 503, pp. 259–265. Amer. Math. Soc., Providence (2009)
Robertson, D.I., Sims, A.: Simplicity of C ∗-algebras associated to higher-rank graphs. Bull. Lond. Math. Soc. 39, 337–344 (2007)
Steinberg, B.: A groupoid approach to inverse semigroup algebras. Adv. Math. 223, 689–727 (2010)
Thomsen, K.: Semi-étale groupoids and applications. Ann. Inst. Fourier (Grenoble) 60, 759–800 (2010)
Yeend, T.: Groupoid models for the C*-algebras of topological higher-rank graphs. J. Oper. Theory 57, 95–120 (2007)
Acknowledgements
Thanks to Iain Raeburn, Astrid an Huef, and Dana Williams for a number of helpful conversations. Further thanks to Dana for his helpful and constructive comments on a preprint of the paper. Thanks also to Alex Kumjian and Paul Muhly for very helpful email correspondence.
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Communicated by Mark V. Lawson.
This research has been supported by the Australian Research Council.
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Brown, J., Clark, L.O., Farthing, C. et al. Simplicity of algebras associated to étale groupoids. Semigroup Forum 88, 433–452 (2014). https://doi.org/10.1007/s00233-013-9546-z
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DOI: https://doi.org/10.1007/s00233-013-9546-z