Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T12:09:45.882Z Has data issue: false hasContentIssue false
  • English
  • Français

COVERINGS OF SKEW-PRODUCTS AND CROSSED PRODUCTS BY COACTIONS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  01 June 2009

DAVID PASK ,
JOHN QUIGG  and
AIDAN SIMS
DAVID PASK
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW, 2522, Australia (email: dpask@uow.edu.au)
JOHN QUIGG
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona, 85287, USA (email: quigg@asu.edu)
AIDAN SIMS*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW, 2522, Australia (email: asims@uow.edu.au)
*
✉For correspondence; e-mail: asims@uow.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider a projective limit G of finite groups Gn. Fix a compatible family δn of coactions of the Gn on a C*-algebra A. From this data we obtain a coaction δ of G on A. We show that the coaction crossed product of A by δ is isomorphic to a direct limit of the coaction crossed products of A by the δn. If A=C*(Λ) for some k-graph Λ, and if the coactions δn correspond to skew-products of Λ, then we can say more. We prove that the coaction crossed product of C*(Λ) by δ may be realized as a full corner of the C*-algebra of a (k+1)-graph. We then explore connections with Yeend’s topological higher-rank graphs and their C*-algebras.

Keywords

C*-algebracoactioncoveringcrossed-productgraph algebrak-graph

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 3 , June 2009 , pp. 379 - 398
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This research was supported by the ARC.

References

[1]Allen, S., ‘A gauge invariant uniqueness theorem for corners of higher rank graph algebras’, Rocky Mountain J. Math. 38 (2008), 1887–1907.CrossRefGoogle Scholar
[2]Bates, T., Hong, J., Raeburn, I. and Szymański, W., ‘The ideal structure of the C *-algebras of infinite graphs’, Illinois J. Math. 46 (2002), 1159–1176.Google Scholar
[3]Drinen, D. and Tomforde, M., ‘The C *-algebras of arbitrary graphs’, Rocky Mountain J. Math. 35 (2005), 105–135.Google Scholar
[4]Echterhoff, S., Kaliszewski, S., Quigg, J. and Raeburn, I., ‘A categorical approach to imprimitivity theorems for C*-dynamical systems’, Mem. Amer. Math. Soc. 180 (2006), viii+169 .Google Scholar
[5]Enomoto, M. and Watatani, Y., ‘A graph theory for C *-algebras’, Math. Japon. 25 (1980), 435–442.Google Scholar
[6]Fillmore, P. A., A User’s Guide to Operator Algebras, Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley & Sons, New York, 1996), pp. xiv+223.Google Scholar
[7]Kaliszewski, S. and Quigg, J., ‘Mansfield’s imprimitivity theorem for full crossed products’, Trans. Amer. Math. Soc. 357 (2005), 2021–2042.Google Scholar
[8]Kaliszewski, S. and Quigg, J., ‘Landstad’s characterisation for full crossed-products’, New York J. Math. 13 (2007), 1–10.Google Scholar
[9]Kumjian, A. and Pask, D., ‘Higher rank graph C *-algebras’, New York J. Math. 6 (2000), 1–20.Google Scholar
[10]Kumjian, A., Pask, D. and Raeburn, I., ‘Cuntz–Krieger algebras of directed graphs’, Pacific J. Math. 184 (1998), 161–174.Google Scholar
[11]Kumjian, A., Pask, D., Raeburn, I. and Renault, J., ‘Graphs, groupoids and Cuntz–Krieger algebras’, J. Funct. Anal. 144 (1997), 505–541.CrossRefGoogle Scholar
[12]Kumjian, A., Pask, D. and Sims, A., ‘C *-algebras associated to coverings of k-graphs’, Documenta Math. 13 (2008), 161–205.Google Scholar
[13]Landstad, M. B., ‘Duality for dual C *-covariance algebras over compact groups’, Preprint, 1978.Google Scholar
[14]Landstad, M., ‘Duality theory for covariant systems’’, Trans. Amer. Math. Soc. 248 (1979), 223–267.Google Scholar
[15]Pask, D., Quigg, J. and Raeburn, I., ‘Coverings of k-graphs’, J. Algebra 289 (2005), 161–191.CrossRefGoogle Scholar
[16]Pask, D., Raeburn, I., Rørdam, M. and Sims, A., ‘Rank-2 graphs whose C *-algebras are direct limits of circle algebras’, J. Funct. Anal. 239 (2006), 137–178.Google Scholar
[17]Quigg, J., ‘Landstad duality for C *-coactions’, Math. Scand. 71 (1992), 277–294.Google Scholar
[18]Robertson, D. I. and Sims, A., ‘Simplicity of C *-algebras associated to higher-rank graphs’, Bull. London Math. Soc. 39 (2007), 337–344.Google Scholar
[19]Robertson, G. and Steger, T., ‘Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras’, J. Reine Angew. Math. 513 (1999), 115–144.CrossRefGoogle Scholar
[20]Yeend, T., ‘Topological higher-rank graphs and the C*-algebras of topological 1-graphs’, Contemp. Math. 414 (2006), 231–244.Google Scholar
[21]Yeend, T., ‘Groupoid models for the C *-algebras of topological higher-rank graphs’, J. Operator Theory 57 (2007), 95–120.Google Scholar