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GRAPH PRODUCTS AND THE ABSENCE OF PROPERTY (AR)

Part of: Semigroups Selfadjoint operator algebras Special aspects of infinite or finite groups

Published online by Cambridge University Press:  16 October 2017

NICOLAI STAMMEIER
NICOLAI STAMMEIER*
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway email [email protected]
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Abstract

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We discuss the internal structure of graph products of right LCM semigroups and prove that there is an abundance of examples without property (AR). Thereby we provide the first examples of right LCM semigroups lacking this seemingly common feature. The results are particularly sharp for right-angled Artin monoids.

Keywords

right LCM monoidsright-angled Artin monoidssemigroup C∗ -algebrasKMS-statesgeneralised scales

MSC classification

Primary: 20M10: General structure theory
Secondary: 20F36: Braid groups; Artin groups 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 2 , April 2018 , pp. 274 - 288
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Afsar, Z., Brownlowe, N., Larsen, N. S. and Nicolai, S., ‘Equilibrium states on right LCM semigroup $C^{\ast }$ -algebras’, Int. Math. Res. Not. IMRN, to appear, Preprint, 2016, arXiv:1611.01052.Google Scholar
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(1) (2012), 35–62; doi:10.1017/S0143385710000830.Google Scholar
Brownlowe, N., Larsen, N. S. and Stammeier, N., ‘On C ∗ -algebras associated to right LCM semigroups’, Trans. Amer. Math. Soc. 369(1) (2017), 31–68; doi:10.1090/tran/6638.Google Scholar
Brownlowe, N., Larsen, N. S. and Stammeier, N., ‘ $C^{\ast }$ -algebras of algebraic dynamical systems and right LCM semigroups’, Indiana Univ. Math. J., to appear, available as IUMJ/Preprints/7527.Google Scholar
Brownlowe, N., Ramagge, J., Robertson, D. and Whittaker, M. F., ‘Zappa–Szép products of semigroups and their C ∗ -algebras’, J. Funct. Anal. 266(6) (2014), 3937–3967; doi:10.1016/j.jfa.2013.12.025.CrossRefGoogle Scholar
Brownlowe, N. and Stammeier, N., ‘The boundary quotient for algebraic dynamical systems’, J. Math. Anal. Appl. 438(2) (2016), 772–789; doi:10.1016/j.jmaa.2016.02.015.CrossRefGoogle Scholar
Charney, R., ‘An introduction to right-angled Artin groups’, Geom. Dedicata 125 (2007), 141–158; doi:10.1007/s10711-007-9148-6.CrossRefGoogle Scholar
Clark, L. O., an Huef, A. and Raeburn, I., ‘Phase transitions on the Toeplitz algebras of Baumslag–Solitar semigroups’, Indiana Univ. Math. J. 65(6) (2016), 2137–2173; doi:10.1512/iumj.2016.65.5934.Google Scholar
Crisp, J. and Laca, M., ‘On the Toeplitz algebras of right-angled and finite-type Artin groups’, J. Aust. Math. Soc. 72(2) (2002), 223–245; doi:10.1017/S1446788700003876.CrossRefGoogle Scholar
Crisp, J. and Laca, M., ‘Boundary quotients and ideals of Toeplitz C ∗ -algebras of Artin groups’, J. Funct. Anal. 242(1) (2007), 127–156; doi:10.1016/j.jfa.2006.08.001.CrossRefGoogle Scholar
Diekert, V., Combinatorics on Traces, Lecture Notes in Computer Science, 454 (Springer, Berlin, 1990), with a foreword by Wilfried Brauer.Google Scholar
Eilers, S., Li, X. and Ruiz, E., ‘The isomorphism problem for semigroup C ∗ -algebras of right-angled Artin monoids’, Doc. Math. 21 (2016), 309–343; electronic version.Google Scholar
Exel, R. and Pardo, E., ‘The tight groupoid of an inverse semigroup’, Semigroup Forum 92(1) (2016), 274–303; doi:10.1007/s00233-015-9758-5.Google Scholar
Exel, R. and Pardo, E., ‘Self-similar graphs, a unified treatment of Katsura and Nekrashevych C ∗ -algebras’, Adv. Math. 306(1) (2017), 1046–1129; doi:10.1016/j.aim.2016.10.030.Google Scholar
Fountain, J. and Kambites, M., ‘Graph products of right cancellative monoids’, J. Aust. Math. Soc. 87(2) (2009), 227–252; doi:10.1017/S144678870900010X.Google Scholar
Green, E. R., Graph Products of Groups (University of Leeds, 1990).Google Scholar
Laca, M. and Raeburn, I., ‘Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers’, Adv. Math. 225(2) (2010), 643–688; doi:10.1016/j.aim.2010.03.007.Google Scholar
Laca, M., Raeburn, I. and Ramagge, J., ‘Phase transition on Exel crossed products associated to dilation matrices’, J. Funct. Anal. 261(12) (2011), 3633–3664; doi:10.1016/j.jfa.2011.08.015.CrossRefGoogle Scholar
Laca, M., Raeburn, I., Ramagge, J. and Whittaker, M. F., ‘Equilibrium states on the Cuntz–Pimsner algebras of self-similar actions’, J. Funct. Anal. 266(11) (2014), 6619–6661; doi:10.1016/j.jfa.2014.03.003.CrossRefGoogle Scholar
Li, X., ‘Semigroup C ∗ -algebras and amenability of semigroups’, J. Funct. Anal. 262(10) (2012), 4302–4340; doi:10.1016/j.jfa.2012.02.020.Google Scholar
Li, X., ‘Nuclearity of semigroup C ∗ -algebras and the connection to amenability’, Adv. Math. 244 (2013), 626–662; doi:10.1016/j.aim.2013.05.016.Google Scholar
Spielberg, J., ‘ C ∗ -algebras for categories of paths associated to the Baumslag–Solitar groups’, J. Lond. Math. Soc. (2) 86(3) (2012), 728–754; doi:10.1112/jlms/jds025.Google Scholar
Stammeier, N., ‘A boundary quotient diagram for right LCM semigroups’, Semigroup Forum (2017), 1–16; doi:10.1007/s00233-017-9850-0.Google Scholar
Starling, C., ‘Boundary quotients of C ∗ -algebras of right LCM semigroups’, J. Funct. Anal. 268(11) (2015), 3326–3356; doi:10.1016/j.jfa.2015.01.001.Google Scholar
Steinberg, B., ‘Simplicity, primitivity and semiprimitivity of étale groupoid algebras with applications to inverse semigroup algebras’, J. Pure Appl. Algebra 220(3) (2016), 1035–1054; doi:10.1016/j.jpaa.2015.08.006.Google Scholar
Veloso da Costa, A., ‘Graph products of monoids’, Semigroup Forum 63(2) (2001), 247–277; doi:10.1007/s002330010075.Google Scholar