Kwaśniewski, Bartosz Kosma; Meyer, Ralf

Essential Crossed Products for Inverse Semigroup Actions: Simplicity and Pure Infiniteness

Doc. Math. 26, 271-335 (2021)
DOI: 10.25537/dm.2021v26.271-335
Communicated by Wilhelm Winter

Summary

We study simplicity and pure infiniteness criteria for \(\mathrm{C}^*\)-algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of \textit{R. Exel} and \textit{D. R. Pitts} [``Characterizing groupoid \(\mathrm{C}^*\)-algebras of non-Hausdorff étale groupoids'', Preprint (2019); \url{arXiv: 1901.09683}], we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced \(\mathrm{C}^*\)-algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic \(\mathrm{C}^*\)-inclusions equipped with supportive generalised expectations. We thoroughly discuss the relationship between aperiodicity, detection of ideals, purely outer inverse semigroup actions, and non-triviality conditions for dual groupoids.

Mathematics Subject Classification

46L55, 46L05, 20M18, 22A22

Keywords/Phrases

Fell bundles, inverse semigroups, étale groupoids, simplicity, pure infiniteness

References

  • 1. Abadie, Fernando; Abadie, Beatriz, Ideals in cross sectional \(C^*\)-algebras of Fell bundles, Rocky Mountain J. Math., 47, 2, 351-381 (2017); DOI 10.1216/RMJ-2017-47-2-351; zbl 1373.46042; MR3635363; arxiv 1503.07094.
  • 2. Anantharaman-Delaroche, Claire, Purely infinite \(C^*\)-algebras arising from dynamical systems, Bull. Soc. Math. France, 125, 2, 199-225 (1997); DOI 10.24033/bsmf.2304; zbl 0896.46044; MR1478030; http://www.numdam.org/item?id={BSMF_1997__125_2_199_0}.
  • 3. Mathieu, Martin; Ara, Pere, Local multipliers of \(C^*\)-algebras, Springer Monographs in Mathematics (2003), Springer-Verlag London, Ltd., London; DOI 10.1007/978-1-4471-0045-4; zbl 1015.46001; MR1940428.
  • 4. Spielberg, John S.; Archbold, Robert J., Topologically free actions and ideals in discrete \(C^*\)-dynamical systems, Proc. Edinburgh Math. Soc. (2), 37, 1, 119-124 (1994); DOI 10.1017/S0013091500018733; zbl 0799.46076; MR1258035.
  • 5. Sims, Aidan; Farthing, Cynthia; Clark, Lisa Orloff; Brown, Jonathan Henry, Simplicity of algebras associated to étale groupoids, Semigroup Forum, 88, 2, 433-452 (2014); DOI 10.1007/s00233-013-9546-z; zbl 1304.46046; MR3189105; arxiv 1204.3127.
  • 6. Ozawa, Narutaka; Brown, Nathanial P., \(C^*\)-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88 (2008), Providence, RI: Amer. Math. Soc; zbl 1160.46001; MR2391387.
  • 7. Vittadello, Sean T.; Raeburn, Iain; Brownlowe, Nathan, Exel's crossed product for non-unital \(C^*\)-algebras, Math. Proc. Cambridge Philos. Soc., 149, 3, 423-444 (2010); DOI 10.1017/S030500411000037X; zbl 1210.46049; MR2726727; arxiv 0908.2671.
  • 8. Exel, Ruy; Buss, Alcides, Fell bundles over inverse semigroups and twisted étale groupoids, J. Operator Theory, 67, 1, 153-205 (2012); zbl 1249.46053; MR2881538; arxiv 0903.3388.
  • 9. Exel, Ruy; Buss, Alcides, Inverse semigroup expansions and their actions on \(C^*\)-algebras, Illinois J. Math., 56, 4, 1185-1212 (2012); zbl 1298.46053; MR3231479; arxiv 1112.0771.
  • 10. Meyer, Ralf; Exel, Ruy; Buss, Alcides, Reduced \(C^*\)-algebras of Fell bundles over inverse semigroups, Israel J. Math., 220, 1, 225-274 (2017); DOI 10.1007/s11856-017-1516-9; zbl 1377.46037; MR3666825; arxiv 1512.05570.
  • 11. Meyer, Ralf; Buss, Alcides, Inverse semigroup actions on groupoids, Rocky Mountain J. Math., 47, 1, 53-159 (2017); DOI 10.1216/RMJ-2017-47-1-53; zbl 1404.46058; MR3619758; arxiv 1410.2051.
  • 12. Choi, Man Duen, A Schwarz inequality for positive linear maps on \(C^*\)-algebras, Illinois J. Math., 18, 565-574 (1974); DOI 10.1215/ijm/1256051007; zbl 0293.46043; MR0355615.
  • 13. Starling, Charles; Sims, Aidan; Pardo, Enrique; Exel, Ruy; Orloff Clark, Lisa, Simplicity of algebras associated to non-Hausdorff groupoids, Trans. Amer. Math. Soc., 372, 5, 3669-3712 (2019); DOI 10.1090/tran/7840; zbl 07089873; MR3988622; arxiv 1806.04362.
  • 14. Cuntz, Joachim, Dimension functions on simple \(C^*\)-algebras, Math. Ann., 233, 2, 145-153 (1978); DOI 10.1007/BF01421922; zbl 0354.46043; MR0467332.
  • 15. Ramazan, Birant; Kumjian, Alex; Deaconu, Valentin, Fell bundles associated to groupoid morphisms, Math. Scand., 102, 2, 305-319 (2008); DOI 10.7146/math.scand.a-15064; zbl 1173.46034; MR2437192; arxiv math/0612746.
  • 16. Dixmier, Jacques, Sur certains espaces consideres par M. H. Stone, Summa Bras. Math., 2, 151-181 (1951); DOI 10.24033/bsmf.1545; zbl 0045.38002; MR0121674.
  • 17. Dixmier, Jacques, \(C^*\)-algebras, North-Holland Mathematical Library, 15 (1977), Amsterdam: North-Holland Publishing Co; zbl 0372.46058; MR0458185.
  • 18. Exel, Ruy, Amenability for Fell bundles, J. Reine Angew. Math., 492, 41-73 (1997); DOI 10.1515/crll.1997.492.41; zbl 0881.46046; MR1488064; arxiv funct-an/9604009.
  • 19. Exel, Ruy, Inverse semigroups and combinatorial \(C^*\)-algebras, Bull. Braz. Math. Soc. (N.S.), 39, 2, 191-313 (2008); DOI 10.1007/s00574-008-0080-7; zbl 1173.46035; MR2419901; arxiv math/0703182.
  • 20. Exel, Ruy, Noncommutative Cartan subalgebras of \(C^*\)-algebras, New York J. Math., 17, 331-382 (2011); zbl 1228.46061; MR2811068; arxiv 0806.4143.
  • 21. Exel, Ruy, Non-Hausdorff étale groupoids, Proc. Amer. Math. Soc., 139, 3, 897-907 (2011); DOI 10.1090/S0002-9939-2010-10477-X; zbl 1213.46064; MR2745642.
  • 22. Exel, Ruy, Partial dynamical systems, Fell bundles and applications, Mathematical Surveys and Monographs, 224 (2017), Providence, RI: Amer. Math. Soc; DOI 10.1090/surv/224; zbl 1405.46003; MR3699795; arxiv 1511.04565.
  • 23. Pardo, Enrique; Exel, Ruy, The tight groupoid of an inverse semigroup, Semigroup Forum, 92, 1, 274-303 (2016); DOI 10.1007/s00233-015-9758-5; zbl 1353.20040; MR3448414; arxiv 1408.5278.
  • 24. Pardo, Enrique; Exel, Ruy, Self-similar graphs, a unified treatment of Katsura and Nekrashevych \(C^*\)-algebras, Adv. Math., 306, 1046-1129 (2017); DOI 10.1016/j.aim.2016.10.030; zbl 1390.46050; MR3581326; arxiv 1409.1107.
  • 25. Pitts, David R.; Exel, Ruy, Characterizing groupoid \(C^*\)-algebras of non-Hausdorff étale groupoids, eprint; arxiv 1901.09683.
  • 26. Frank, Michael, Injective envelopes and local multiplier algebras of \(C^*\)-algebras, Int. Math. J., 1, 6, 611-620 (2002); zbl 1221.46057; MR1860642; arxiv math/9910109.
  • 27. Sierakowski, Adam; Giordano, Thierry, Purely infinite partial crossed products, J. Funct. Anal., 266, 9, 5733-5764 (2014); DOI 10.1016/j.jfa.2014.02.025; zbl 1305.46058; MR3182957; arxiv 1303.4483.
  • 28. Gonshor, Harry, Injective hulls of \(C^*\) algebras. II, Proc. Amer. Math. Soc., 24, 486-491 (1970); DOI 10.2307/2037393; zbl 0188.44801; MR0287318.
  • 29. Hamana, Masamichi, Injective envelopes of \(C^*\)-algebras, J. Math. Soc. Japan, 31, 1, 181-197 (1979); DOI 10.2969/jmsj/03110181; zbl 0395.46042; MR519044.
  • 30. Robertson, Guyan; Jolissaint, Paul, Simple purely infinite \(C^*\)-algebras and \(n\)-filling actions, J. Funct. Anal., 175, 1, 197-213 (2000); DOI 10.1006/jfan.2000.3608; zbl 0993.46033; MR1774856; arxiv math/0004052.
  • 31. Tomiyama, Jun; Kawamura, Shinzō, Properties of topological dynamical systems and corresponding \(C^*\)-algebras, Tokyo J. Math., 13, 2, 251-257 (1990); DOI 10.3836/tjm/1270132260; zbl 0724.54037; MR1088230.
  • 32. Schafhauser, Christopher; Kennedy, Matthew, Noncommutative boundaries and the ideal structure of reduced crossed products, Duke Math. J., 168, 17, 3215-3260 (2019); DOI 10.1215/00127094-2019-0032; zbl 07154925; MR4030364.
  • 33. Skandalis, Georges; Khoshkam, Mahmood, Regular representation of groupoid \(C^*\)-algebras and applications to inverse semigroups, J. Reine Angew. Math., 546, 47-72 (2002); DOI 10.1515/crll.2002.045; zbl 1029.46082; MR1900993.
  • 34. Skandalis, Georges; Khoshkam, Mahmood, Crossed products of \(C^*\)-algebras by groupoids and inverse semigroups, J. Operator Theory, 51, 2, 255-279 (2004); zbl 1061.46047; MR2074181.
  • 35. Rørdam, Mikael; Kirchberg, Eberhard, Non-simple purely infinite \(C^*\)-algebras, Amer. J. Math., 122, 3, 637-666 (2000); DOI 10.1353/ajm.2000.0021; zbl 0968.46042; MR1759891.
  • 36. Rørdam, Mikael; Kirchberg, Eberhard, Infinite non-simple \(C^*\)-algebras: absorbing the Cuntz algebra \(\mathcal O_\infty \), Adv. Math., 167, 2, 195-264 (2002); DOI 10.1006/aima.2001.2041; zbl 1030.46075; MR1906257.
  • 37. Sierakowski, Adam; Kirchberg, Eberhard, Strong pure infiniteness of crossed products, Ergodic Theory Dynam. Systems, 38, 1, 220-243 (2018); DOI 10.1017/etds.2016.25; zbl 1030.46075; MR3742544; arxiv 1312.5195.
  • 38. Kishimoto, Akitaka, Outer automorphisms and reduced crossed products of simple \(C^*\)-algebras, Comm. Math. Phys., 81, 3, 429-435 (1981); DOI 10.1007/BF01209077; zbl 0467.46050; MR634163.
  • 39. Kumjian, Alexander, On \(C^*\)-diagonals, Canad. J. Math., 38, 4, 969-1008 (1986); DOI 10.4153/CJM-1986-048-0; zbl 0627.46071; MR854149.
  • 40. Kwaśniewski, Bartosz Kosma, Topological freeness for Hilbert bimodules, Israel J. Math., 199, 2, 641-650 (2014); DOI 10.1007/s11856-013-0057-0; zbl 1304.46053; MR3219552; arxiv 1212.0361.
  • 41. Kwaśniewski, Bartosz Kosma, Crossed products by endomorphisms of \(C_0(X)\)-algebras, J. Funct. Anal., 270, 6, 2268-2335 (2016); DOI 10.1016/j.jfa.2016.01.015; zbl 1353.46052; MR3460241; arxiv 1412.8240.
  • 42. Kwaśniewski, Bartosz Kosma, Exel's crossed product and crossed products by completely positive maps, Houston J. Math., 43, 2, 509-567 (2017); zbl 1395.46038; MR3690127; arxiv 1404.4929.
  • 43. Meyer, Ralf; Kwaśniewski, Bartosz Kosma, Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles, Studia Math., 241, 3, 257-303 (2018); DOI 10.4064/sm8762-5-2017; zbl 1398.46057; MR3756105; arxiv 1611.06954.
  • 44. Meyer, Ralf; Kwaśniewski, Bartosz Kosma, Stone duality and quasi-orbit spaces for generalised \(C^*\)-inclusions, Proc. Lond. Math. Soc. (3), 121, 4, 788-827 (2020); DOI 10.1112/plms.12332; zbl 07242516; MR4105787; arxiv 1804.09387.
  • 45. Meyer, Ralf; Kwaśniewski, Bartosz Kosma, Aperiodicity, the almost extension property and uniqueness of pseudo-expectations (2020); arxiv 2007.05409.
  • 46. Szymański, Wojciech; Kwaśniewski, Bartosz Kosma, Pure infiniteness and ideal structure of \(C^*\)-algebras associated to Fell bundles, J. Math. Anal. Appl., 445, 1, 898-943 (2017); DOI 10.1016/j.jmaa.2013.10.078; zbl 1444.46046; MR3543802; arxiv 1505.05202.
  • 47. Spielberg, Jack; Laca, Marcelo, Purely infinite \(C^*\)-algebras from boundary actions of discrete groups, J. Reine Angew. Math., 480, 125-139 (1996); DOI 10.1515/crll.1996.480.125; zbl 0863.46044; MR1420560.
  • 48. Nilsen, May, \(C^*\)-bundles and \(C_0(X)\)-algebras, Indiana Univ. Math. J., 45, 2, 463-477 (1996); DOI 10.1512/iumj.1996.45.1086; zbl 0872.46029; MR1414338.
  • 49. Pedersen, Gert K.; Olesen, Dorte, Applications of the Connes spectrum to \(C^*\)-dynamical systems. II, J. Funct. Anal., 36, 1, 18-32 (1980); DOI 10.1016/0022-1236(80)90104-4; zbl 0422.46054; MR515224.
  • 50. Phillips, N. Christopher; Pasnicu, Cornel, Crossed products by spectrally free actions, J. Funct. Anal., 269, 4, 915-967 (2015); DOI 10.1016/j.jfa.2015.04.020; zbl 1334.46046; MR3352760; arxiv 1308.4921.
  • 51. Paterson, Alan L. T., Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, 170 (1999), Boston, MA: Birkhäuser Boston Inc; DOI 10.1007/978-1-4612-1774-9; zbl 0913.22001; MR1724106.
  • 52. Pitts, David R., Structure for regular inclusions. I, J. Operator Theory, 78, 2, 357-416 (2017); DOI 10.7900/jot.2016sep15.2128; zbl 1424.46078; MR3725511; arxiv 1202.6413.
  • 53. Zarikian, Vrej; Pitts, David R., Unique pseudo-expectations for \(C^*\)-inclusions, Illinois J. Math., 59, 2, 449-483 (2015); zbl 1351.46056; MR3499520; arxiv 1508.05048.
  • 54. Sieben, Nándor; Quigg, John, \(C^*\)-actions of \(r\)-discrete groupoids and inverse semigroups, J. Austral. Math. Soc. Ser. A, 66, 2, 143-167 (1999); DOI 10.1017/S1446788700039288; zbl 0992.46051; MR1671944; arxiv math/9801002.
  • 55. Renault, Jean, The ideal structure of groupoid crossed product \(C^*\)-algebras, J. Operator Theory, 25, 1, 3-36 (1991); zbl 0786.46050; MR1191252.
  • 56. Renault, Jean, Cartan subalgebras in \(C^*\)-algebras, Irish Math. Soc. Bull., 61, 29-63 (2008); zbl 1175.46050; MR2460017; arxiv 0803.2284.
  • 57. Sierakowski, Adam; Rørdam, Mikael, Purely infinite \(C^*\)-algebras arising from crossed products, Ergodic Theory Dynam. Systems, 32, 1, 273-293 (2012); DOI 10.1017/S0143385710000829; zbl 1252.46061; MR2873171; arxiv 1006.1304.
  • 58. Sierakowski, Adam, The ideal structure of reduced crossed products, Münster J. Math., 3, 237-261 (2010); zbl 1378.46050; MR2775364; arxiv 0804.3772.
  • 59. Szakács, Nóra; Steinberg, Benjamin, Simplicity of inverse semigroup and étale groupoid algebras, Adv. Math., 380, article ID 107611, 56 p. pp. (2021); DOI 10.1016/j.aim.2021.107611; zbl 07309960; MR4205706; arxiv 2006.13787.
  • 60. Tomiyama, Jun, On the projection of norm one in \(W^*\)-algebras, Proc. Japan Acad., 33, 608-612 (1957); DOI 10.3792/pja/1195524885; zbl 0081.11201; MR96140.
  • 61. Zarikian, Vrej, Unique expectations for discrete crossed products, Ann. Funct. Anal., 10, 1, 60-71 (2019); DOI 10.1215/20088752-2018-0008; zbl 07045485; MR3899956; arxiv 1707.09339.

Affiliation

Kwaśniewski, Bartosz Kosma
Faculty of Mathematics, University of Białystok ul. K. Ciołkowskiego 1M, 15-245 Białystok, Poland
Meyer, Ralf
Mathematisches Institut, Universität Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany

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