The structure of the Brauer group and crossed products of -linear group actions on

Authors:
Siegfried Echterhoff and Ryszard Nest

Journal:
Trans. Amer. Math. Soc. **353** (2001), 3685-3712

MSC (1991):
Primary 46L55; Secondary 22D25

Published electronically:
May 4, 2001

MathSciNet review:
1837255

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

For a second countable locally compact group and a second countable locally compact space let denote the equivariant Brauer group (for the trivial -space ) consisting of all Morita equivalence classes of spectrum fixing actions of on continuous-trace -algebras with spectrum . Extending recent results of several authors, we give a complete description of in terms of group cohomology of and Cech cohomology of . Moreover, if has a splitting group in the sense of Calvin Moore, we give a complete description of the -bundle structure of the crossed product in terms of the topological data associated to the given action and the bundle structure of the group -algebra of .

**[B]**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****[C]**F. Combes,*Crossed products and Morita equivalence*, Proc. London Math. Soc. (3)**49**(1984), no. 2, 289–306. MR**748991**, 10.1112/plms/s3-49.2.289**[CKRW]**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**, 10.1006/jfan.1996.3010**[D]**Jacques Dixmier,*𝐶*-algebras*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. MR**0458185****[E]**Siegfried Echterhoff,*Crossed products with continuous trace*, Mem. Amer. Math. Soc.**123**(1996), no. 586, viii+134. MR**1371090**, 10.1090/memo/0586**[ER]**Siegfried Echterhoff and Jonathan Rosenberg,*Fine structure of the Mackey machine for actions of abelian groups with constant Mackey obstruction*, Pacific J. Math.**170**(1995), no. 1, 17–52. MR**1359971****[EW1]**Siegfried Echterhoff and Dana P. Williams,*Locally inner actions on**-algebras*, preprint, June 1997; to appear in J. Operator Theory.**[EW2]**Siegfried Echterhoff and Dana P. Williams,*Crossed products by 𝐶₀(𝑋)-actions*, J. Funct. Anal.**158**(1998), no. 1, 113–151. MR**1641562**, 10.1006/jfan.1998.3295**[G]**Phil Green,*The local structure of twisted covariance algebras*, Acta. Math.**140**(1978), 191-250. 58:12376**[HORR]**Steven Hurder, Dorte Olesen, Iain Raeburn, and Jonathan Rosenberg,*The Connes spectrum for actions of abelian groups on continuous-trace algebras*, Ergodic Theory Dynam. Systems**6**(1986), no. 4, 541–560. MR**873431**, 10.1017/S0143385700003680**[M1]**Calvin C. Moore,*Extensions and low dimensional cohomology theory of locally compact groups. I, II*, Trans. Amer. Math. Soc.**113**(1964), 40–63. MR**0171880**, 10.1090/S0002-9947-1964-0171880-5**[M2]**Calvin C. Moore,*Extensions and low dimensional cohomology theory of locally compact groups. I, II*, Trans. Amer. Math. Soc.**113**(1964), 40–63. MR**0171880**, 10.1090/S0002-9947-1964-0171880-5**[M3]**Calvin C. Moore,*Group extensions and cohomology for locally compact groups. III*, Trans. Amer. Math. Soc.**221**(1976), no. 1, 1–33. MR**0414775**, 10.1090/S0002-9947-1976-0414775-X**[M4]**Calvin C. Moore,*Group extensions and cohomology for locally compact groups. IV*, Trans. Amer. Math. Soc.**221**(1976), no. 1, 35–58. MR**0414776**, 10.1090/S0002-9947-1976-0414776-1**[N]**May Nielsen,*-bundles and**-algebras*, Indiana Univ. Math. J.**45**(1996), 463-477.**[OR]**Dorte Olesen and Iain Raeburn,*Pointwise unitary automorphism groups*, J. Funct. Anal.**93**(1990), no. 2, 278–309. MR**1073288**, 10.1016/0022-1236(90)90130-D**[P]**Judith A. Packer,*Moore cohomology and central twisted crossed product 𝐶*-algebras*, Canad. J. Math.**48**(1996), no. 1, 159–174. MR**1382480**, 10.4153/CJM-1996-007-6**[PR1]**Judith A. Packer and Iain Raeburn,*Twisted crossed products of 𝐶*-algebras*, Math. Proc. Cambridge Philos. Soc.**106**(1989), no. 2, 293–311. MR**1002543**, 10.1017/S0305004100078129**[PR2]**Judith A. Packer and Iain Raeburn,*On the structure of twisted group 𝐶*-algebras*, Trans. Amer. Math. Soc.**334**(1992), no. 2, 685–718. MR**1078249**, 10.1090/S0002-9947-1992-1078249-7**[PRW]**Judith A. Packer, Iain Raeburn, and Dana P. Williams,*The equivariant Brauer group of principal bundles*, J. Operator Theory**36**(1996), no. 1, 73–105. MR**1417188****[Pal]**Richard S. Palais,*On the existence of slices for actions of non-compact Lie groups*, Ann. of Math. (2)**73**(1961), 295–323. MR**0126506****[Pe]**Gert K. Pedersen,*𝐶*-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006****[PhR]**-,*Crossed products by locally unitary automorphism groups and principal bundles*, J. Operator Theory**11**(1984), 215-241.**[RR]**Iain Raeburn and Jonathan Rosenberg,*Crossed products of continuous-trace 𝐶*-algebras by smooth actions*, Trans. Amer. Math. Soc.**305**(1988), no. 1, 1–45. MR**920145**, 10.1090/S0002-9947-1988-0920145-6**[RW]**Iain Raeburn and Dana P. Williams,*Pull-backs of 𝐶*-algebras and crossed products by certain diagonal actions*, Trans. Amer. Math. Soc.**287**(1985), no. 2, 755–777. MR**768739**, 10.1090/S0002-9947-1985-0768739-2**[RW1]**Iain Raeburn and Dana P. Williams,*Moore cohomology, principal bundles, and actions of groups on 𝐶*-algebras*, Indiana Univ. Math. J.**40**(1991), no. 2, 707–740. MR**1119194**, 10.1512/iumj.1991.40.40032**[RW2]**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****[R]**Jonathan Rosenberg,*Some results on cohomology with Borel cochains, with applications to group actions on operator algebras*, Advances in invariant subspaces and other results of operator theory (Timişoara and Herculane, 1984) Oper. Theory Adv. Appl., vol. 17, Birkhäuser, Basel, 1986, pp. 301–330. MR**901077****[R1]**Jonathan Rosenberg,*𝐶*-algebras and Mackey’s theory of group representations*, 𝐶*-algebras: 1943–1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 150–181. MR**1292014**, 10.1090/conm/167/1292014

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
46L55,
22D25

Retrieve articles in all journals with MSC (1991): 46L55, 22D25

Additional Information

**Siegfried Echterhoff**

Affiliation:
Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany

Email:
echters@math.uni-muenster.de

**Ryszard Nest**

Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Email:
rnest@math.ku.dk

DOI:
https://doi.org/10.1090/S0002-9947-01-02794-5

Received by editor(s):
March 9, 1999

Received by editor(s) in revised form:
March 3, 2000

Published electronically:
May 4, 2001

Article copyright:
© Copyright 2001
American Mathematical Society