Morita equivalence of partial group actions and globalization

Abadie, F.; Dokuchaev, M.; Exel, R.; Simón, J.

doi:10.1090/tran/6525

Morita equivalence of partial group actions and globalization
HTML articles powered by AMS MathViewer

by F. Abadie, M. Dokuchaev, R. Exel and J. J. Simón PDF

Trans. Amer. Math. Soc. 368 (2016), 4957-4992 Request permission

Abstract:

We consider a large class of partial actions of groups on rings, called regular, which contains all $s$-unital partial actions as well as all partial actions on $C^{\ast }$-algebras. For them the notion of Morita equivalence is introduced, and it is shown that any regular partial action is Morita equivalent to a globalizable one and that the globalization is essentially unique. It is also proved that Morita equivalent $s$-unital partial actions on rings with orthogonal local units are stably isomorphic. In addition, we show that Morita equivalent $s$-unital partial actions on commutative rings must be isomorphic, and an analogous result for $C^{\ast }$-algebras is also established.

References

F. Abadie, Sobre ações parciais, fibrados de Fell e grupóides, PhD Thesis, São Paulo, 1999.
Fernando Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (2003), no. 1, 14–67. MR 1957674, DOI 10.1016/S0022-1236(02)00032-0
Fernando Abadie, On partial actions and groupoids, Proc. Amer. Math. Soc. 132 (2004), no. 4, 1037–1047. MR 2045419, DOI 10.1090/S0002-9939-03-07300-3
Marcelo Muniz S. Alves and Eliezer Batista, Enveloping actions for partial Hopf actions, Comm. Algebra 38 (2010), no. 8, 2872–2902. MR 2730285, DOI 10.1080/00927870903095582
Dirceu Bagio, Wagner Cortes, Miguel Ferrero, and Antonio Paques, Actions of inverse semigroups on algebras, Comm. Algebra 35 (2007), no. 12, 3865–3874. MR 2371262, DOI 10.1080/00927870701511905
Dirceu Bagio and Antonio Paques, Partial groupoid actions: globalization, Morita theory, and Galois theory, Comm. Algebra 40 (2012), no. 10, 3658–3678. MR 2982887, DOI 10.1080/00927872.2011.592889
L. Bemm, Ações parciais de grupos sobre anéis semiprimos, PhD Thesis, Porto Alegre, 2011.
Laerte Bemm, Wagner Cortes, Saradia Della Flora, and Miguel Ferrero, Partial crossed products and Goldie rings, Comm. Algebra 43 (2015), no. 9, 3705–3724. MR 3360844, DOI 10.1080/00927872.2014.923895
Laerte Bemm and Miguel Ferrero, Globalization of partial actions on semiprime rings, J. Algebra Appl. 12 (2013), no. 4, 1250202, 9. MR 3037276, DOI 10.1142/S0219498812502027
Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928
S. Caenepeel and E. De Groot, Galois corings applied to partial Galois theory, Proceedings of the International Conference on Mathematics and its Applications (ICMA 2004), Kuwait Univ. Dep. Math. Comput. Sci., Kuwait, 2005, pp. 117–134. MR 2143295
S. Caenepeel and K. Janssen, Partial (co)actions of Hopf algebras and partial Hopf-Galois theory, Comm. Algebra 36 (2008), no. 8, 2923–2946. MR 2440292, DOI 10.1080/00927870802110334
Paula A. A. B. Carvalho, Wagner Cortes, and Miguel Ferrero, Partial skew group rings over polycyclic by finite groups, Algebr. Represent. Theory 14 (2011), no. 3, 449–462. MR 2785917, DOI 10.1007/s10468-009-9197-7
Wagner Cortes, Partial skew polynomial rings and Jacobson rings, Comm. Algebra 38 (2010), no. 4, 1526–1548. MR 2656590, DOI 10.1080/00927870902973904
Wagner Cortes and Miguel Ferrero, Partial skew polynomial rings: prime and maximal ideals, Comm. Algebra 35 (2007), no. 4, 1183–1199. MR 2313659, DOI 10.1080/00927870601142017
Wagner Cortes and Miguel Ferrero, Globalization of partial actions on semiprime rings, Groups, rings and group rings, Contemp. Math., vol. 499, Amer. Math. Soc., Providence, RI, 2009, pp. 27–35. MR 2581923, DOI 10.1090/conm/499/09788
W. Cortes, M. Ferrero, and E. Marcos, Partial Actions on Categories, Preprint.
Wagner Cortes, Miguel Ferrero, and Hidetoshi Marubayashi, Partial skew polynomial rings and Goldie rings, Comm. Algebra 36 (2008), no. 11, 4284–4295. MR 2460417, DOI 10.1080/00927870802107645
M. Dokuchaev, Partial actions: a survey, Groups, algebras and applications, Contemp. Math., vol. 537, Amer. Math. Soc., Providence, RI, 2011, pp. 173–184. MR 2799098, DOI 10.1090/conm/537/10573
M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. Amer. Math. Soc. 357 (2005), no. 5, 1931–1952. MR 2115083, DOI 10.1090/S0002-9947-04-03519-6
M. Dokuchaev, R. Exel, and J. J. Simón, Crossed products by twisted partial actions and graded algebras, J. Algebra 320 (2008), no. 8, 3278–3310. MR 2450727, DOI 10.1016/j.jalgebra.2008.06.023
M. Dokuchaev, R. Exel, and J. J. Simón, Globalization of twisted partial actions, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4137–4160. MR 2608399, DOI 10.1090/S0002-9947-10-04957-3
Michael Dokuchaev, Miguel Ferrero, and Antonio Paques, Partial actions and Galois theory, J. Pure Appl. Algebra 208 (2007), no. 1, 77–87. MR 2269829, DOI 10.1016/j.jpaa.2005.11.009
Michael Dokuchaev, Ángel Del Río, and Juan Jacobo Simón, Globalizations of partial actions on nonunital rings, Proc. Amer. Math. Soc. 135 (2007), no. 2, 343–352. MR 2255280, DOI 10.1090/S0002-9939-06-08503-0
Ruy Exel, Circle actions on $C^*$-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence, J. Funct. Anal. 122 (1994), no. 2, 361–401. MR 1276163, DOI 10.1006/jfan.1994.1073
Ruy Exel, The Bunce-Deddens algebras as crossed products by partial automorphisms, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 2, 173–179. MR 1306559, DOI 10.1007/BF01321306
Ruy Exel, Approximately finite $C^\ast$-algebras and partial automorphisms, Math. Scand. 77 (1995), no. 2, 281–288. MR 1379271, DOI 10.7146/math.scand.a-12566
Ruy Exel, Twisted partial actions: a classification of regular $C^*$-algebraic bundles, Proc. London Math. Soc. (3) 74 (1997), no. 2, 417–443. MR 1425329, DOI 10.1112/S0024611597000154
Ruy Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3481–3494. MR 1469405, DOI 10.1090/S0002-9939-98-04575-4
Ruy Exel, Hecke algebras for protonormal subgroups, J. Algebra 320 (2008), no. 5, 1771–1813. MR 2437631, DOI 10.1016/j.jalgebra.2008.05.001
Ruy Exel, Marcelo Laca, and John Quigg, Partial dynamical systems and $C^*$-algebras generated by partial isometries, J. Operator Theory 47 (2002), no. 1, 169–186. MR 1905819
R. Exel, T. Giordano, and D. Gonçalves, Enveloping algebras of partial actions as groupoid $C^*$-algebras, J. Operator Theory 65 (2011), no. 1, 197–210. MR 2765764
Miguel Ferrero, Partial actions of groups on semiprime rings, Groups, rings and group rings, Lect. Notes Pure Appl. Math., vol. 248, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 155–162. MR 2226190, DOI 10.1201/9781420010961.ch14
Jesús Ávila, Miguel Ferrero, and João Lazzarin, Partial actions and partial fixed rings, Comm. Algebra 38 (2010), no. 6, 2079–2091. MR 2675523, DOI 10.1080/00927870903023139
Miguel Ferrero and João Lazzarin, Partial actions and partial skew group rings, J. Algebra 319 (2008), no. 12, 5247–5264. MR 2423825, DOI 10.1016/j.jalgebra.2007.12.009
Peter A. Fillmore, A user’s guide to operator algebras, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1996. A Wiley-Interscience Publication. MR 1385461
Daiane Freitas and Antonio Paques, On partial Galois Azumaya extensions, Algebra Discrete Math. 11 (2011), no. 2, 64–77. MR 2858129
Kent R. Fuller, On rings whose left modules are direct sums of finitely generated modules, Proc. Amer. Math. Soc. 54 (1976), 39–44. MR 393133, DOI 10.1090/S0002-9939-1976-0393133-6
José Luis García and Juan Jacobo Simón, Morita equivalence for idempotent rings, J. Pure Appl. Algebra 76 (1991), no. 1, 39–56. MR 1140639, DOI 10.1016/0022-4049(91)90096-K
N. D. Gilbert, Actions and expansions of ordered groupoids, J. Pure Appl. Algebra 198 (2005), no. 1-3, 175–195. MR 2132881, DOI 10.1016/j.jpaa.2004.11.006
Daniel Gonçalves and Danilo Royer, Leavitt path algebras as partial skew group rings, Comm. Algebra 42 (2014), no. 8, 3578–3592. MR 3196063, DOI 10.1080/00927872.2013.790038
Jesús Ávila Guzmán and João Lazzarin, A Morita context related to finite groups acting partially on a ring, Algebra Discrete Math. 3 (2009), 49–61. MR 2640387
J. Kellendonk and Mark V. Lawson, Partial actions of groups, Internat. J. Algebra Comput. 14 (2004), no. 1, 87–114. MR 2041539, DOI 10.1142/S0218196704001657
Mark V. Lawson, Inverse semigroups, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. The theory of partial symmetries. MR 1694900, DOI 10.1142/9789812816689
Kevin McClanahan, $K$-theory for partial crossed products by discrete groups, J. Funct. Anal. 130 (1995), no. 1, 77–117. MR 1331978, DOI 10.1006/jfan.1995.1064
Michael Megrelishvili and Lutz Schröder, Globalization of confluent partial actions on topological and metric spaces, Topology Appl. 145 (2004), no. 1-3, 119–145. MR 2100868, DOI 10.1016/j.topol.2004.06.006
Antonio Paques, Virgínia Rodrigues, and Alveri Sant’ana, Galois correspondences for partial Galois Azumaya extensions, J. Algebra Appl. 10 (2011), no. 5, 835–847. MR 2847502, DOI 10.1142/S0219498811004999
Antonio Paques and Alveri Sant’Ana, When is a crossed product by a twisted partial action Azumaya?, Comm. Algebra 38 (2010), no. 3, 1093–1103. MR 2650393, DOI 10.1080/00927870902888250
John Quigg and Iain Raeburn, Characterisations of crossed products by partial actions, J. Operator Theory 37 (1997), no. 2, 311–340. MR 1452280
Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace $C^*$-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. MR 1634408, DOI 10.1090/surv/060
Nándor Sieben, Morita equivalence of $C^\ast$-crossed products by inverse semigroup actions and partial actions, Rocky Mountain J. Math. 31 (2001), no. 2, 661–686. MR 1840961, DOI 10.1216/rmjm/1020171582
Benjamin Steinberg, Partial actions of groups on cell complexes, Monatsh. Math. 138 (2003), no. 2, 159–170. MR 1964463, DOI 10.1007/s00605-002-0521-0