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KMS states on the C*-algebras of Fell bundles over groupoids

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  19 November 2019

ZAHRA AFSAR  and
AIDAN SIMS
ZAHRA AFSAR
Affiliation:
Quadrangle, Camperdown Campus, University of Sydney, A14, L4.45, City Road, Sydney, NSW 2006, Australia e-mail: [email protected]
AIDAN SIMS
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, 39C. 195, Northfields Avenue, Wollongong, NSW 2522, Australia. e-mail: [email protected]
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Abstract

We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo–Martin–Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of states on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a description of the trace space of the cross-sectional algebra. We apply our result to generalise Neshveyev’s main theorem to twisted groupoid C*-algebras, and then apply this to twisted C*-algebras of strongly connected finite k-graphs.

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L55: Noncommutative dynamical systems
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 2 , March 2021 , pp. 221 - 246
Copyright
© Cambridge Philosophical Society 2019

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References

Bost, J.-B. and Connes, A.. Hecke algebras type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Math. (New Series) 1 (1995), 411–457.CrossRefGoogle Scholar
Bratteli, O. and Robinson., D.W. Operator algebras and quantum statistical mechanics II (2nd edition). (Springer-Verlag, Berlin, 1997.)CrossRefGoogle Scholar
Carlsen, 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
Enomoto, M., Fujii, M. and Watatani., Y. KMS states for gauge action on O A. Math. Japon. 29 (1984), 607–619.Google Scholar
Fell, J. M. G. and Doran, R. S.. Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles, Basic representation theory of groups and algebras, Pure and Applied Mathematics 1 (Academic Press Inc., Boston, MA, 1988).Google Scholar
An Huef, A., Laca, M., Raeburn, I. and Sims, A.. KMS states on the C*-algebra of a higher-rank graph and periodicity in the path space. J. Funct. Anal. 268 (2015), 1840–1875.CrossRefGoogle Scholar
Kakariadis, E. T. A.. KMS states on Pimsner algebras associated with C*-dynamical systems. J. Funct. Anal. 269 (2015), 325–354.CrossRefGoogle Scholar
Kumjian, A.. On C*-diagonals. Canad. J. Math. 38 (1986), 969–1008.CrossRefGoogle Scholar
Kumjian, A. and Pask, D.. Higher-rank graph C*-algebras. New York J. Math. 6 (2000), 1–20.Google Scholar
Kumjian, A. and Renault, J.. KMS states on C*-algebras associated to expansive maps. Proc. Amer. Math. Soc. 134 (2006), 2067–2078.CrossRefGoogle Scholar
Kumjian, A., Pask, D. and Sims, A.. On twisted higher-rank graph C*-algebras. Trans. Amer. Math. Soc. 367 (2015), 5177–5216.CrossRefGoogle Scholar
Kumjian, A., Pask, D. and Sims, A.. Simplicity of twisted C*-algebras of higher-rank graphs and crossed products by quasifree actions. J. Noncommut. Geom. 10 (2016), 515–549.CrossRefGoogle Scholar
Laca, M., Larsen, N. S., Neshveyev, S., Sims, A. and Webster, S. B. G.. Von Neumann algebras of strongly connected higher-rank graphs. Math. Ann. 363 (2015), 657–678.CrossRefGoogle Scholar
Muhly, P. S. and Williams, D. P.. Equivalence and disintegration theorems for Fell bundles and their C*-algebras. Dissertationes Math. 456 (2008), 1–57.CrossRefGoogle Scholar
Neshveyev, S.. KMS states on the C*-algebras of non-principal groupoids. J. Operator Theory 70 (2013), 513–530.CrossRefGoogle Scholar
Olesen, D., Pedersen, G. K. and Takesaki, M.. Ergodic actions of compact abelian groups. J. Operator Theory 3 (1980), 237–269.Google Scholar
Renault, J.. A groupoid approach to C*-algebras. Lecture Notes in Mathematics 793 (Springer-Verlag, New York, 1980).CrossRefGoogle Scholar
Raeburn, I. and Williams, D. P.. Morita equivalence and continuous-trace C*-algebras. Math. Surv. Monog. 60 (Amer. Math. Soc., Providence, 1998).CrossRefGoogle Scholar
Sims, A., Whitehead, B. and Whittaker, M. F.. Twisted C*-algebras associated to finitely aligned higher-rank graphs. Documenta Math. 19 (2014), 831–866.Google Scholar
Thomsen, K.. KMS weights on graph C*-algebras. Adv. Math. 309 (2017), 334–391.CrossRefGoogle Scholar
Williams, D. P.. Crossed products of C*-algebras. Math. Surv. Monog. 134 (Amer. Math. Soc., Providence, 2007).Google Scholar
Yang, D.. Type III von Neumann algebras associated with 2-graphs. Bull. Lond. Math. Soc. 44 (2012), 675–686.CrossRefGoogle Scholar