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Reconstructing directed graphs from generalized gauge actions on their Toeplitz algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  13 June 2019

Nathan Brownlowe ,
Marcelo Laca ,
Dave Robertson  and
Aidan Sims
Nathan Brownlowe
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW2006, Australia (nathan.brownlowe@sydney.edu.au)
Marcelo Laca
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BCV8W 3P4, Canada (laca@math.uvic.ca)
Dave Robertson
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan2308, Australia (dave84robertson@gmail.com)
Aidan Sims
Affiliation:
School of Mathematics and Applied Statistics,University of Wollongong, NSW2522, Australia (asims@uow.edu.au)
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Abstract

We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalized gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.

Keywords

Graph C*-algebraToeplitz algebraKMS state

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2632 - 2641
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Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Carlsen, T. M. and Larsen, N. S.. Partial actions and KMS states on relative graph C*-algebras. J. Funct. Anal. 271 (2016), 2090–2132.CrossRefGoogle Scholar
2Eilers, S. and Tomforde, M.. On the classification of nonsimple graph C*-algebras. Math. Ann. 346 (2010), 393–418.Google Scholar
3Eilers, S., Ruiz, E. and Sørensen, A. P. W.. Amplified graph C*-algebras. Münster J. Math. 5 (2012), 121–150.Google Scholar
4Eilers, S., Restorff, G. and Ruiz, E.. Classification of graph C*-algebras with no more than four primitive ideals, Springer Proc. Math. Stat.,vol. 58, Operator algebra and dynamics,pp. 89–129 (Heidelberg: Springer, 2013).Google Scholar
5Eilers, S., Restorff, G., Ruiz, E. and Sørensen, A.. The complete classification of unital graph C*-algebras: Geometric and strong, preprint (2016), (arXiv: 1611.07120 [math.OA]).Google Scholar
6Eilers, S., Restorff, G., Ruiz, E. and Sørensen, A. P. W.. Invariance of the Cuntz splice. Math. Ann. 369 (2017), 1061–1080.Google Scholar
7Eilers, S., Restorff, G., Ruiz, E. and Sørensen, A. P. W.. Geometric classification of graph C*-algebras over finite graphs. Canad. J. Math. 70 (2018), 294–353.CrossRefGoogle Scholar
8Fowler, N. J. and Raeburn, I.. The Toeplitz algebra of a Hilbert bimodule. Indiana Univ. Math. J. 48 (1999), 155–181.CrossRefGoogle Scholar
9an Huef, A., Laca, M., Raeburn, I. and Sims, A.. KMS states on the C*-algebras of finite graphs. J. Math. Anal. Appl. 405 (2013), 388–399.CrossRefGoogle Scholar
10an Huef, A., Laca, M., Raeburn, I. and Sims, A.. KMS states on the C*-algebras of reducible graphs. Ergodic Theory Dynam. Systems 35 (2015), 2535–2558.CrossRefGoogle Scholar
11Kajiwara, T. and Watatani, Y.. KMS states on finite-graph C*-algebras. Kyushu J. Math. 67 (2013), 83–104.CrossRefGoogle Scholar
12Katsoulis, E. and Kribs, D. W.. Isomorphisms of algebras associated with directed graphs. Math. Ann. 330 (2004), 709–728.CrossRefGoogle Scholar
13Laca, M. and Neshveyev, S.. KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211 (2004), 457–482.Google Scholar
14Pimsner, M. V.. A class of C*-algebras generalizing both Cuntz–Krieger algebras and crossed products by Z. Fields Inst. Commun.,vol. 12, Free probability theory (Waterloo, ON, 1995),pp. 189–212 (Providence, RI:, Amer. Math. Soc., 1997).Google Scholar
15Raeburn, I.. Graph algebras, Published for the Conference Board of the Mathematical Sciences, Washington, DC, vi+113, (2005).CrossRefGoogle Scholar
16Solel, B.. You can see the arrows in a quiver operator algebra. J. Aust. Math. Soc. 77 (2004), 111–122.CrossRefGoogle Scholar
17Sørensen, A. P. W.. Geometric classification of simple graph algebras. Ergodic Theory Dynam. Systems 33 (2013), 1199–1220.CrossRefGoogle Scholar
18Thomsen, K.. KMS weights on graph C*-algebras. Adv. Math. 309 (2017), 334–391.CrossRefGoogle Scholar