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Group actions on topological graphs

Published online by Cambridge University Press:  16 September 2011

VALENTIN DEACONU ,
ALEX KUMJIAN  and
JOHN QUIGG
VALENTIN DEACONU
Affiliation:
Department of Mathematics, University of Nevada, Reno, NV 89557-0084, USA (email: vdeaconu@unr.edu, alex@unr.edu)
ALEX KUMJIAN
Affiliation:
Department of Mathematics, University of Nevada, Reno, NV 89557-0084, USA (email: vdeaconu@unr.edu, alex@unr.edu)
JOHN QUIGG
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA (email: quigg@asu.edu)
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Abstract

We define the action of a locally compact group G on a topological graph E. This action induces a natural action of G on the C*-correspondence ℋ(E) and on the graph C*-algebra C*(E). If the action is free and proper, we prove that C*(E)⋊rG is strongly Morita equivalent to C*(E/G) . We define the skew product of a locally compact group G by a topological graph E via a cocycle c:E1 →G. The group acts freely and properly on this new topological graph E×cG. If G is abelian, there is a dual action on C* (E)such that . We also define the fundamental group and the universal covering of a topological graph.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5: Daniel J. Rudolph – in Memoriam , October 2012 , pp. 1527 - 1566
Copyright
Copyright © Cambridge University Press 2011

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References

[ADR00]Anantharaman-Delaroche, C. and Renault, J.. Amenable Groupoids (Monographs of L’Enseignement Mathématique, 36). L’Enseignement Mathématique, Geneva, 2000, with a foreword by Georges Skandalis and Appendix B by E. Germain.Google Scholar
[Bau93]Baumslag, G.. Topics in Combinatorial Group Theory (Lectures in Mathematics ETH Zürich). Birkhäuser, Basel, 1993.CrossRefGoogle Scholar
[BH99]Bridson, M. R. and Haefliger, A.. Metric spaces of non-positive curvature (Grundlehren der Mathematischen Wissenschaften, 319). Springer, Berlin, 1999.CrossRefGoogle Scholar
[BO08]Brown, N. P. and Ozawa, N.. C *-algebras and finite-dimensional approximations (Graduate Studies in Mathematics, 88). American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
[EKQR06]Echterhoff, S., Kaliszewski, S., Quigg, J. and Raeburn, I.. A Categorical Approach to Imprimitivity Theorems for C *-Dynamical Systems, Vol. 180 (Memoires of the American Mathematical Society, 850). American Mathematical Society, Providence, RI, 2006.Google Scholar
[FLR00]Fowler, N. J., Laca, M. and Raeburn, I.. The C *-algebras of infinite graphs. Proc. Amer. Math. Soc. 128 (2000), 2319–2327.CrossRefGoogle Scholar
[GT01]Gross, J. L. and Tucker, T. W.. Topological Graph Theory. Dover Publications Inc., Mineola, NY, 2001, reprint of the 1987 original [Wiley, New York] with a new preface and supplementary bibliography.Google Scholar
[HN08]Hao, G. and Ng, C.-K.. Crossed products of C *-correspondences by amenable group actions. J. Math. Anal. Appl. 345(2) (2008), 702–707.CrossRefGoogle Scholar
[HRW05]an Huef, A., Raeburn, I. and Williams, D. P.. A symmetric imprimitivity theorem for commuting proper actions. Canad. J. Math. 57 (2005), 983–1011.CrossRefGoogle Scholar
[Hus94]Husemoller, D.. Fibre Bundles, 3rd edn(Graduate Texts in Mathematics, 20). Springer, New York, 1994.CrossRefGoogle Scholar
[KQR97]Kaliszewski, S., Quigg, J. and Raeburn, I.. Duality of restriction and induction for C *-coactions. Trans. Amer. Math. Soc. 349 (1997), 2085–2113.CrossRefGoogle Scholar
[KQR08]Kaliszewski, S., Quigg, J. and Raeburn, I.. Proper actions, fixed-point algebras and naturality in nonabelian duality. J. Funct. Anal. 254 (2008), 2949–2968.CrossRefGoogle Scholar
[KQR11]Kaliszewski, S., Quigg, J. and Raeburn, I.. Skew products and coactions for topological graphs. Preprint, 2011.Google Scholar
[Kas88]Kasparov, G. G.. Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91(1) (1988), 147–201.CrossRefGoogle Scholar
[Kat02]Katsura, T.. Continuous graphs and crossed products of Cuntz algebras. Sūrikaisekikenkyūsho Kōkyūroku No. 1291 (2002), 73–83, recent aspects of C *-algebras (in Japanese) (Kyoto, 2002).Google Scholar
[Kat04]Katsura, T.. A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras. I. Fundamental results. Trans. Amer. Math. Soc. 356(11) (2004), 4287–4322.CrossRefGoogle Scholar
[Kat06]Katsura, T.. A class of C *-algebras generalizing both graph algebras and homeomorphism C *-algebras. III. Ideal structures. Ergod. Th. & Dynam. Syst. 26(6) (2006), 1805–1854.CrossRefGoogle Scholar
[Kat08]Katsura, T.. A class of C *-algebras generalizing both graph algebras and homeomorphism C *-algebras. IV. Pure infiniteness. J. Funct. Anal. 254(5) (2008), 1161–1187.CrossRefGoogle Scholar
[KK97]Kishimoto, A. and Kumjian, A.. Crossed products of Cuntz algebras by quasi-free automorphisms. Operator Algebras and their Applications (Waterloo, ON, 1994/1995) (Fields Institute Communications, 13). American Mathematical Society, Providence, RI, 1997, pp. 173–192.Google Scholar
[KP99]Kumjian, A. and Pask, D.. C *-algebras of directed graphs and group actions. Ergod. Th. & Dynam. Sys. 19 (1999), 1503–1519.CrossRefGoogle Scholar
[LS77]Lyndon, R. C. and Schupp, P. E.. Combinatorial Group Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete, 89). Springer, Berlin-New York, 1977.Google Scholar
[Mas91]Massey, W. S.. A Basic Course in Algebraic Topology (Graduate Texts in Mathematics, 127). Springer, New York, 1991.CrossRefGoogle Scholar
[OP78]Olesen, D. and Pedersen, G. K.. Applications of the Connes spectrum to C *-dynamical systems. J. Funct. Anal. 30(2) (1978), 179–197.CrossRefGoogle Scholar
[OP80]Olesen, D. and Pedersen, G. K.. Applications of the Connes spectrum to C *-dynamical systems. II. J. Funct. Anal. 36(1) (1980), 18–32.CrossRefGoogle Scholar
[Pal61]Palais, R. S.. On the existence of slices for actions of non-compact Lie groups. Ann. of Math. (2) 73 (1961), 295–323.CrossRefGoogle Scholar
[QR95]Quigg, J. C. and Raeburn, I.. Induced C *-algebras and Landstad duality for twisted coactions. Trans. Amer. Math. Soc. 347 (1995), 2885–2915.Google Scholar
[Qui92]Quigg, J. C.. Landstad duality for C *-coactions. Math. Scand. 71 (1992), 277–294.CrossRefGoogle Scholar
[Rae05]Raeburn, I.. Graph Algebras (CBMS Regional Conference Series in Mathematics, 103). Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2005.CrossRefGoogle Scholar
[Rie90]Rieffel, M. A.. Proper actions of groups on C*-algebras. Mappings of Operator Algebras (Philadelphia, PA, 1988). Birkhäuser Boston, Boston, MA, 1990.Google Scholar
[Rie04]Rieffel, M. A.. Integrable and proper actions on C*-algebras, and square-integrable representations of groups. Expo. Math. 22 (2004), 1–53.CrossRefGoogle Scholar
[RW98]Raeburn, I. and Williams, D. P.. Morita equivalence and continuous-trace C *-algebras (Mathematical Surveys and Monographs, 60). American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar