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The Brauer semigroup of a groupoid and a symmetric imprimitivity theorem


Authors: Jonathan Henry Brown and Geoff Goehle
Journal: Trans. Amer. Math. Soc. 366 (2014), 1943-1972
MSC (2010): Primary 46L55, 22A22
DOI: https://doi.org/10.1090/S0002-9947-2013-05953-3
Published electronically: October 31, 2013
MathSciNet review: 3152718
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Abstract: In this paper we define a monoid called the Brauer semigroup for a locally compact Hausdorff groupoid $ E$ whose elements consist of Morita equivalence classes of $ E$-dynamical systems. This construction generalizes both the equivariant Brauer semigroup for transformation groups and the Brauer group for a groupoid. We show that groupoid equivalence induces an isomorphism of Brauer semigroups and that this isomorphism preserves the Morita equivalence classes of the respective crossed products, thus generalizing Raeburn's symmetric imprimitivity theorem.


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  • [1] Claire Anantharaman-Delaroche and Jean Renault, Amenable groupoids, Monographies de L'Enseignement Mathématique [Monographs of L'Enseignement Mathématique], vol. 36, L'Enseignement Mathématique, Geneva, 2000, With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683 (2001m:22005)
  • [2] Jonathan H. Brown, Proper actions of groupoids on $ {C}^*$-algebras, J. Operator Theory 67 (2012), no. 2, 437-467.
  • [3] Jonathan H. Brown, Geoff Goehle, and Dana P. Williams, Groupoid equivalence and the associated iterated crossed product, to appear in the Houst. J. Math, arXiv:1206.2066v1 [math.OA].
  • [4] Étienne Blanchard, Déformations de $ C^*$-algèbres de Hopf, Bull. Soc. Math. France 124 (1996), no. 1, 141-215 (French, with English and French summaries). MR 1395009 (97f:46092)
  • [5] David Crocker, Alexander Kumjian, Iain Raeburn, and Dana P. Williams, An equivariant Brauer group and actions of groups on $ C^*$-algebras, J. Funct. Anal. 146 (1997), no. 1, 151-184. MR 1446378 (98j:46076), https://doi.org/10.1006/jfan.1996.3010
  • [6] Siegfried Echterhoff and Dana P. Williams, Crossed products by $ C_0(X)$-actions, J. Funct. Anal. 158 (1998), no. 1, 113-151. MR 1641562 (2001b:46105), https://doi.org/10.1006/jfan.1998.3295
  • [7] James M. G. Fell and Robert S. Doran, Representations of $ \sp *$-algebras, locally compact groups, and Banach $ \sp *$-algebraic bundles. Vol. 1, Pure and Applied Mathematics, vol. 125, Academic Press Inc., Boston, MA, 1988, Basic representation theory of groups and algebras. MR 936628 (90c:46001)
  • [8] Gerald B. Folland, Fourier analysis and its applications, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. MR 1145236 (93f:42001)
  • [9] Geoff Goehle, The Mackey machine for crossed products by regular groupoids. I, Houston J. Math. 36 (2010), no. 2, 567-590. MR 2661261 (2011i:47111)
  • [10] Geoff Goehle, Locally unitary groupoid crossed products, Indiana Univ. Math. J. 60 (2011), no. 2, 411-441. MR 2963781, https://doi.org/10.1512/iumj.2011.60.4143
  • [11] Geoff Goehle, The Mackey machine for crossed products by regular groupoids. II, Rocky Mountain J. of Math., to appear, arXiv:0908.1434v2 [math.OA].
  • [12] Astrid an Huef, Iain Raeburn, and Dana P. Williams, An equivariant Brauer semigroup and the symmetric imprimitivity theorem, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4759-4787. MR 1709774 (2001b:46107), https://doi.org/10.1090/S0002-9947-00-02618-0
  • [13] Alexander Kumjian, Paul S. Muhly, Jean N. Renault, and Dana P. Williams, The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998), no. 5, 901-954. MR 1646047 (2000b:46122)
  • [14] Alexander Kumjian, Iain Raeburn, and Dana P. Williams, The equivariant Brauer groups of commuting free and proper actions are isomorphic, Proc. Amer. Math. Soc. 124 (1996), no. 3, 809-817. MR 1301034 (96f:46107), https://doi.org/10.1090/S0002-9939-96-03146-2
  • [15] Paul S. Muhly, Jean N. Renault, and Dana P. Williams, Equivalence and isomorphism for groupoid $ C^\ast $-algebras, J. Operator Theory 17 (1987), no. 1, 3-22. MR 873460 (88h:46123)
  • [16] Paul S. Muhly and Dana P. Williams, Groupoid cohomology and the Dixmier-Douady class, Proc. London Math. Soc. (3) 71 (1995), no. 1, 109-134. MR 1327935 (97d:46082), https://doi.org/10.1112/plms/s3-71.1.109
  • [17] Paul S. Muhly and Dana P. Williams, Equivalence and disintegration theorems for Fell bundles and their $ C^*$-algebras, Dissertationes Math. (Rozprawy Mat.) 456 (2008), 1-57. MR 2446021 (2010b:46146), https://doi.org/10.4064/dm456-0-1
  • [18] Paul S. Muhly and Dana P. Williams, Renault's equivalence theorem for groupoid crossed products, New York Journal of Mathematics Monographs 3 (2008), 1-87.
  • [19] Iain Raeburn, Induced $ C^*$-algebras and a symmetric imprimitivity theorem, Math. Ann. 280 (1988), no. 3, 369-387. MR 936317 (90k:46144), https://doi.org/10.1007/BF01456331
  • [20] Iain Raeburn and Dana P. Williams, Pull-backs of $ C^\ast $-algebras and crossed products by certain diagonal actions, Trans. Amer. Math. Soc. 287 (1985), no. 2, 755-777. MR 768739 (86m:46054), https://doi.org/10.2307/1999675
  • [21] 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 (2000c:46108)
  • [22] Marc A. Rieffel, Applications of strong Morita equivalence to transformation group $ C^{\ast } $-algebras, Operator algebras and applications, Part I (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 299-310. MR 679709 (84k:46046)
  • [23] Marc A. Rieffel, Proper actions of groups on $ C^*$-algebras, Mappings of operator algebras (Philadelphia, PA, 1988) Progr. Math., vol. 84, Birkhäuser Boston, Boston, MA, 1990, pp. 141-182. MR 1103376 (92i:46079)
  • [24] Marc A. Rieffel, Integrable and proper actions on $ C^*$-algebras, and square-integrable representations of groups, Expo. Math. 22 (2004), no. 1, 1-53. MR 2166968 (2006g:46108), https://doi.org/10.1016/S0723-0869(04)80002-1
  • [25] Anthony K. Seda, On the continuity of Haar measure on topological groupoids, Proc. Amer. Math. Soc. 96 (1986), no. 1, 115-120, DOI 10.2307/2045664. MR 813822 (87m:46146)
  • [26] André Weil, Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211 (French). MR 0165033 (29 #2324)
  • [27] Dana P. Williams, Crossed products of $ C{^\ast }$-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954 (2007m:46003)

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Additional Information

Jonathan Henry Brown
Affiliation: Department of Mathematics, 138 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506-2602
Email: brownjh@math.kansas.edu

Geoff Goehle
Affiliation: Mathematics and Computer Science Department, Stillwell 426, Western Carolina University, Cullowhee, North Carolina 28723
Email: grgoehle@email.wcu.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05953-3
Keywords: Groupoids, crossed products, equivalence theorem, symmetric imprimitivity theorem
Received by editor(s): June 12, 2012
Published electronically: October 31, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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