Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-06-05T23:38:44.781Z Has data issue: false hasContentIssue false
  • English
  • Français

Groupoid Fell bundles for product systems over quasi-lattice ordered groups

Published online by Cambridge University Press:  16 March 2017

ADAM RENNIE ,
DAVID ROBERTSON  and
AIDAN SIMS
ADAM RENNIE
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia. e-mail: renniea@uow.edu.au
DAVID ROBERTSON
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, 2308, Australia. e-mail: dave84robertson@gmail.com
AIDAN SIMS
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2522, Australia. e-mail: asims@uow.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a product system over the positive cone of a quasi-lattice ordered group. We construct a Fell bundle over an associated groupoid so that the cross-sectional algebra of the bundle is isomorphic to the Nica–Toeplitz algebra of the product system. Under the additional hypothesis that the left actions in the product system are implemented by injective homomorphisms, we show that the cross-sectional algebra of the restriction of the bundle to a natural boundary subgroupoid coincides with the Cuntz–Nica–Pimsner algebra of the product system. We apply these results to improve on existing sufficient conditions for nuclearity of the Nica–Toeplitz algebra and the Cuntz–Nica–Pimsner algebra, and for the Cuntz–Nica–Pimsner algebra to coincide with its co-universal quotient.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 561 - 580
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Anantharaman-Delaroche, C. and Renault, J. Amenable groupoids. l'Ensignement Mathématique 36 (2000).Google Scholar
[2] Blackadar, B. Operator algebras: Theory of C*-algebras and von Neumann algebras. Encyclopaedia of Mathematical Sciences 122 (2006).CrossRefGoogle Scholar
[3] Brown, N.P. and Ozawa, N. C*-algebras and finite-dimensional approximations. Amer. Math. Soc. (Providence, RI, 2008), xvi+509.Google Scholar
[4] Brownlowe, N., an Huef, A., Laca, M. and Raeburn, I. Boundary quotients of the Toeplitz algebra of the affine semigroup over the natural numbers. Ergodic Theory Dynam. Systems 32 (2012), 35–62.CrossRefGoogle Scholar
[5] Carlsen, T., Larsen, N.S., Sims, A. and Vittadello, S. Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems. Proc. Lond. Math. Soc. 103 (2011), 563–600.CrossRefGoogle Scholar
[6] Crisp, J. and Laca, M. Boundary quotients and ideals of Toeplitz C*-algebras of Artin groups. J. Funct. Anal. 242 (2007), 127–156.CrossRefGoogle Scholar
[7] Davidson, K.R., Fuller, A.H. and Kakariadis, E.T.A. Semicrossed products of operator algebras by semigroups. Mem. Amer. Math. Soc., to appear.Google Scholar
[8] Dixmier, J. C*-algebras, Translated from the French by Francis Jellett. North-Holland Mathematical Library, vol. 15 (North-Holland Publishing Co., Amsterdam, 1977), xiii+492.Google Scholar
[9] 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
[10] Exel, R. Inverse semigroups and combinatorial C*-algebras. Bull. Braz. Math. Soc. (N.S.) 39 (2008), 191–313.CrossRefGoogle Scholar
[11] Exel, R., Laca, M. and Quigg, J. Partial dynamical systems and C*-algebras generated by partial isometries. J. Operator Theory 47 (2002), 169–186.Google Scholar
[12] Fell, J.M.G. and Doran, R.S. Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles. Pure Appl. Math. 178 (1988).Google Scholar
[13] Fowler, N.J. Discrete product systems of Hilbert bimodules. Pacific J. Math. 204 (2002), 335–375.CrossRefGoogle Scholar
[14] an Huef, A., Raeburn, I. and Williams, D.P. Properties preserved under Morita equivalence of C*-algebras. Proc. Amer. Math. Soc. 135 (2007), 1495–1503.CrossRefGoogle Scholar
[15] Kaliszewski, S. and Quigg, J. Erratum to “Full and reduced C*-coactions”. Math. Proc. Camb. Phil. Soc. 116 (1994), 435–450, Math. Proc. Camb. Phil. Soc. 161 (2016), 379–380.Google Scholar
[16] Kwasniewski, B. and Szymański, W. Topological aperiodicity for product systems over semi-groups of Ore type. J. Funct. Anal. 270 (2016), 3453–3504.CrossRefGoogle Scholar
[17] Laca, M. and Raeburn, I. Semigroup crossed products and the Toeplitz algebras of nonabelian groups. J. Funct. Anal. 139 (1996), 415–440.CrossRefGoogle Scholar
[18] Muhly, P.S. and Renault, J. C*-algebras of multivariable Wiener–Hopf operators. Trans. Amer. Math. Soc. 274 (1982), 1–44.Google Scholar
[19] Muhly, P.S. and Williams, D.P. Equivalence and disintegration theorems for Fell bundles and their C*-algebras. Dissertationes Mathematicae, Warszawa (2008).CrossRefGoogle Scholar
[20] Nica, A. C*-algebras generated by isometries and Weiner–Hopf operators. J. Operator Theory 27 (1992), 17–52.Google Scholar
[21] Pimsner, M.V. A class of C*-algebras generalizing both Cuntz-Krieger algebras and crossed products by Z . Fields Inst. Commun. 12, Free probability theory (Waterloo, ON, 1995) (Amer. Math. Soc., Providence, RI, 1997), 189–212.Google Scholar
[22] Quigg, J. Full and reduced C*-coactions. Math. Proc. Camb. Phil. Soc. 116 (1994), 435–450.CrossRefGoogle Scholar
[23] Quigg, J. Discrete C*-coactions and C*-algebraic bundles. J. Austral. Math. Soc. Ser. A 60 (1996), 204–221.CrossRefGoogle Scholar
[24] Raeburn, I. and Sims, A. Product systems of graphs and the Toeplitz algebras of higher-rank graphs. J. Operator Theory 53 (2005), no. 2, 399–429.Google Scholar
[25] Rørdam, M. Classification of nuclear, simple C*-algebras. Classification of nuclear C*-algebras. Entropy in operator algebras. Encyclopaedia Math. Sci. 126, (Springer, Berlin, 2002), 1–145.CrossRefGoogle Scholar
[26] Sims, A. and Williams, D.P. Amenability for Fell bundles over groupoids. Illinois J. Math. 57 (2013), 429–444.CrossRefGoogle Scholar
[27] Sims, A. and Yeend, T. C*-algebras associated to product systems of Hilbert bimodules. J. Operator Theory 64 (2010), 349–376.Google Scholar
[28] Takeishi, T. On nuclearity of C*-algebras of Fell bundles over étale groupoids. Publ. Res. Inst. Math. Sci. 50 (2014), 251–268.CrossRefGoogle Scholar