Semi-étale groupoids and applications

Thomsen, Klaus

Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 759-800.

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.

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.

[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., Volume 122 (1994) no. 4, pp. 1133-1141 | DOI | MR | Zbl

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

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

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

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

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

[7] Carlsen, Toke Meier Cuntz-Pimsner $C^{*}$-algebras associated with subshifts, Internat. J. Math., Volume 19 (2008) no. 1, pp. 47-70 | DOI | MR | Zbl

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

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

[10] Cuntz, Joachim; Krieger, Wolfgang A class of $C^{*}$-algebras and topological Markov chains, Invent. Math., Volume 56 (1980) no. 3, pp. 251-268 | DOI | MR | Zbl

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

[12] Deaconu, Valentin; Shultz, Fred $C^{*}$-algebras associated with interval maps, Trans. Amer. Math. Soc., Volume 359 (2007) no. 4, p. 1889-1924 (electronic) | DOI | MR | Zbl

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

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

[15] Exel, R.; Vershik, A. $C^{*}$-algebras of irreversible dynamical systems, Canad. J. Math., Volume 58 (2006) no. 1, pp. 39-63 | DOI | MR | Zbl

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

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

[18] Matsumoto, Kengo On $C^{*}$-algebras associated with subshifts, Internat. J. Math., Volume 8 (1997) no. 3, pp. 357-374 | DOI | MR | Zbl

[19] Matsumoto, Kengo $K$-theory for $C^{*}$-algebras associated with subshifts, Math. Scand., Volume 82 (1998) no. 2, pp. 237-255 | MR | Zbl

[20] Matsumoto, Kengo Relations among generators of $C^{*}$-algebras associated with subshifts, Internat. J. Math., Volume 10 (1999) no. 3, pp. 385-405 | DOI | MR | Zbl

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

[22] Matsumoto, Kengo Stabilized $C^{*}$-algebras constructed from symbolic dynamical systems, Ergodic Theory Dynam. Systems, Volume 20 (2000) no. 3, pp. 821-841 | DOI | MR | Zbl

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

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

[25] Raeburn, Iain; Williams, Dana P. Morita equivalence and continuous-trace $C^{*}$-algebras, Mathematical Surveys and Monographs, 60, American Mathematical Society, Providence, RI, 1998 | MR | Zbl

[26] Renault, Jean A groupoid approach to $C^{*}$-algebras, Lecture Notes in Mathematics, 793, Springer, Berlin, 1980 | MR | Zbl

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

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

[29] Stacey, P. J. Crossed products of $C^{*}$-algebras by $*$-endomorphisms, J. Austral. Math. Soc. Ser. A, Volume 54 (1993) no. 2, pp. 204-212 | DOI | MR | Zbl