Semi-étale groupoids and applications
Annales de l'Institut Fourier, Volume 60 (2010) no. 3, p. 759-800

We associate a C * -algebra to a locally compact Hausdorff groupoid with the property that the range map is locally injective. The construction generalizes J. Renault’s reduced groupoid C * -algebra of an étale groupoid and has the advantage that it works for the groupoid arising from a locally injective dynamical system by the method introduced in increasing generality by Renault, Deaconu and Anantharaman-Delaroche. We study the C * -algebras of such groupoids and give necessary and sufficient conditions for simplicity, and show that many of them contain a Cartan subalgebra as defined by Renault. In particular, this holds when the dynamical system is a shift space, in which case the C * -algebra coincides with the one introduced by Matsumoto and Carlsen.

Nous associons une C * -algèbre à un groupoïde de Hausdorff localement compact avec la propriété pour l’application but d’être localement injective. La construction généralise celle de la C * -algèbre réduite de J. Renault pour un groupoïde étale. Elle possède l’avantage de fonctionner pour le groupoïde résultant d’un système dynamique localement injectif par la méthode introduite par Renault, Deaconu et Anantharaman-Delaroche, en augmentant la généralité. Nous étudions les C * -algèbres de tels groupoïdes et nous donnons des conditions nécessaires et suffisantes pour la simplicité. Nous montrons qu’un grand nombre d’entre elles contiennent une sous-algèbre de Cartan au sens de Renault. En particulier, cela est valable pour un système dynamique symbolique, auquel cas la C * -algèbre coïncide avec celle introduite par Matsumoto et Carlsen.

DOI : https://doi.org/10.5802/aif.2539
Classification:  37B05,  46L55
Keywords: Groupoids, C * -algebras, dynamical systems
@article{AIF_2010__60_3_759_0,
     author = {Thomsen, Klaus},
     title = {Semi-\'etale groupoids and applications},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {3},
     year = {2010},
     pages = {759-800},
     doi = {10.5802/aif.2539},
     mrnumber = {2680816},
     zbl = {1209.46042},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_3_759_0}
}
Thomsen, Klaus. Semi-étale groupoids and applications. Annales de l'Institut Fourier, Volume 60 (2010) no. 3, pp. 759-800. doi : 10.5802/aif.2539. http://www.numdam.org/item/AIF_2010__60_3_759_0/

[1] Adji, Sriwulan; Laca, Marcelo; Nilsen, May; Raeburn, Iain Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups, Proc. Amer. Math. Soc., Tome 122 (1994) no. 4, pp. 1133-1141 | Article | MR 1215024 | Zbl 0818.46071

[2] Anantharaman-Delaroche, Claire Purely infinite C * -algebras arising from dynamical systems, Bull. Soc. Math. France, Tome 125 (1997) no. 2, pp. 199-225 | Numdam | MR 1478030 | Zbl 0896.46044

[3] Blackadar, Bruce K-theory for operator algebras, Springer-Verlag, New York, Mathematical Sciences Research Institute Publications, Tome 5 (1986) | MR 859867 | Zbl 0597.46072

[4] Boyd, Sarah; Keswani, Navin; Raeburn, Iain Faithful representations of crossed products by endomorphisms, Proc. Amer. Math. Soc., Tome 118 (1993) no. 2, pp. 427-436 | Article | MR 1126190 | Zbl 0785.46051

[5] Brown, Lawrence G. Stable isomorphism of hereditary subalgebras of C * -algebras, Pacific J. Math., Tome 71 (1977) no. 2, pp. 335-348 | MR 454645 | Zbl 0362.46042

[6] Brown, Lawrence G.; Green, Philip; Rieffel, Marc A. Stable isomorphism and strong Morita equivalence of C * -algebras, Pacific J. Math., Tome 71 (1977) no. 2, pp. 349-363 | MR 463928 | Zbl 0362.46043

[7] Carlsen, Toke Meier Cuntz-Pimsner C * -algebras associated with subshifts, Internat. J. Math., Tome 19 (2008) no. 1, pp. 47-70 | Article | MR 2380472 | Zbl 1153.46038

[8] Carlsen, Toke Meier; Matsumoto, Kengo Some remarks on the C * -algebras associated with subshifts, Math. Scand., Tome 95 (2004) no. 1, pp. 145-160 | MR 2091486 | Zbl 1067.46045

[9] Carlsen, Toke Meier; Silvestrov, Sergei C * -crossed products and shift spaces, Expo. Math., Tome 25 (2007) no. 4, pp. 275-307 | MR 2360917 | Zbl 1154.46039

[10] Cuntz, Joachim; Krieger, Wolfgang A class of C * -algebras and topological Markov chains, Invent. Math., Tome 56 (1980) no. 3, pp. 251-268 | Article | MR 561974 | Zbl 0434.46045

[11] Deaconu, Valentin Groupoids associated with endomorphisms, Trans. Amer. Math. Soc., Tome 347 (1995) no. 5, pp. 1779-1786 | Article | MR 1233967 | Zbl 0826.46058

[12] Deaconu, Valentin; Shultz, Fred C * -algebras associated with interval maps, Trans. Amer. Math. Soc., Tome 359 (2007) no. 4, p. 1889-1924 (electronic) | Article | MR 2272154 | Zbl 1117.46049

[13] Elliott, George A.; Gong, Guihua; Li, Liangqing On the classification of simple inductive limit C * -algebras. II. The isomorphism theorem, Invent. Math., Tome 168 (2007) no. 2, pp. 249-320 | Article | MR 2289866 | Zbl 1129.46051

[14] Exel, R.; Renault, J. Semigroups of local homeomorphisms and interaction groups, Ergodic Theory Dynam. Systems, Tome 27 (2007) no. 6, pp. 1737-1771 | Article | MR 2371594 | Zbl 1133.46040

[15] Exel, R.; Vershik, A. C * -algebras of irreversible dynamical systems, Canad. J. Math., Tome 58 (2006) no. 1, pp. 39-63 | Article | MR 2195591 | Zbl 1104.46037

[16] Granas, Andrzej; Dugundji, James Fixed point theory, Springer-Verlag, New York, Springer Monographs in Mathematics (2003) | MR 1987179 | Zbl 1025.47002

[17] Lind, Douglas; Marcus, Brian An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge (1995) | MR 1369092 | Zbl 1106.37301

[18] Matsumoto, Kengo On C * -algebras associated with subshifts, Internat. J. Math., Tome 8 (1997) no. 3, pp. 357-374 | Article | MR 1454478 | Zbl 0885.46048

[19] Matsumoto, Kengo K-theory for C * -algebras associated with subshifts, Math. Scand., Tome 82 (1998) no. 2, pp. 237-255 | MR 1646513 | Zbl 0921.46078

[20] Matsumoto, Kengo Relations among generators of C * -algebras associated with subshifts, Internat. J. Math., Tome 10 (1999) no. 3, pp. 385-405 | Article | MR 1688137 | Zbl 1039.46505

[21] Matsumoto, Kengo On automorphisms of C * -algebras associated with subshifts, J. Operator Theory, Tome 44 (2000) no. 1, pp. 91-112 | MR 1774695 | Zbl 0994.46021

[22] Matsumoto, Kengo Stabilized C * -algebras constructed from symbolic dynamical systems, Ergodic Theory Dynam. Systems, Tome 20 (2000) no. 3, pp. 821-841 | Article | MR 1764930 | Zbl 0964.46035

[23] Paschke, William L. The crossed product of a C * -algebra by an endomorphism, Proc. Amer. Math. Soc., Tome 80 (1980) no. 1, pp. 113-118 | MR 574518 | Zbl 0435.46045

[24] Paterson, Alan L. T. Groupoids, inverse semigroups, and their operator algebras, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 170 (1999) | MR 1724106 | Zbl 0913.22001

[25] Raeburn, Iain; Williams, Dana P. Morita equivalence and continuous-trace C * -algebras, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 60 (1998) | MR 1634408 | Zbl 0922.46050

[26] Renault, Jean A groupoid approach to C * -algebras, Springer, Berlin, Lecture Notes in Mathematics, Tome 793 (1980) | MR 584266 | Zbl 0433.46049

[27] Renault, Jean Cuntz-like algebras, Operator theoretical methods (Timişoara, 1998), Theta Found., Bucharest (2000), pp. 371-386 | MR 1770333 | Zbl 1032.46535

[28] Renault, Jean Cartan subalgebras in C * -algebras, Irish Math. Soc. Bull. (2008) no. 61, pp. 29-63 | MR 2460017 | Zbl 1175.46050

[29] Stacey, P. J. Crossed products of C * -algebras by *-endomorphisms, J. Austral. Math. Soc. Ser. A, Tome 54 (1993) no. 2, pp. 204-212 | Article | MR 1200792 | Zbl 0785.46053