𝐶*-algebras of right LCM monoids and their equilibrium states

Brownlowe, Nathan; Larsen, Nadia; Ramagge, Jacqui; Stammeier, Nicolai

doi:10.1090/tran/8097

$C^*$-algebras of right LCM monoids and their equilibrium states
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by Nathan Brownlowe, Nadia S. Larsen, Jacqui Ramagge and Nicolai Stammeier PDF

Trans. Amer. Math. Soc. 373 (2020), 5235-5273 Request permission

Abstract:

We study the internal structure of $C^*$-algebras of right LCM monoids by means of isolating the core semigroup $C^*$-algebra as the coefficient algebra of a Fock-type module on which the full semigroup $C^*$-algebra admits a left action. If the semigroup has a generalised scale, we classify the KMS-states for the associated time evolution on the semigroup $C^*$-algebra and provide sufficient conditions for uniqueness of the KMS$_\beta$-state at inverse temperature $\beta$ in a critical interval.

References

S. I. Adjan, Defining relations and algorithmic problems for groups and semigroups, Proceedings of the Steklov Institute of Mathematics, No. 85 (1966), American Mathematical Society, Providence, R.I., 1966. Translated from the Russian by M. Greendlinger. MR 0218434
Zahra Afsar, Nathan Brownlowe, Nadia S. Larsen, and Nicolai Stammeier, Equilibrium states on right LCM semigroup $C^*$-algebras, Int. Math. Res. Not. IMRN 6 (2019), 1642–1698. MR 3932591, DOI 10.1093/imrn/rnx162
Valeriano Aiello, Roberto Conti, Stefano Rossi, and Nicolai Stammeier, The inner structure of boundary quotients of right LCM semigroups, Indiana Univ. Math. J. (to appear), arXiv:1709.08839, 2017.
Zahra Afsar, Nadia S. Larsen, and Sergey Neshveyev, KMS states on Nica-Toeplitz C*-algebras, Comm. Math. Phys. (2020)., DOI 10.1007/s00220-020-03711-6
Selçuk Barlak, Tron Omland, and Nicolai Stammeier, On the $K$-theory of $C^{\ast }$-algebras arising from integral dynamics, Ergodic Theory Dynam. Systems 38 (2018), no. 3, 832–862. MR 3784245, DOI 10.1017/etds.2016.63
Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199–201. MR 142635, DOI 10.1090/S0002-9904-1962-10745-9
Ievgen V. Bondarenko and Rostyslav V. Kravchenko, Finite-state self-similar actions of nilpotent groups, Geom. Dedicata 163 (2013), 339–348. MR 3032698, DOI 10.1007/s10711-012-9752-y
Ola Bratteli and Derek W. Robinson, Operator algebras and quantum statistical mechanics. 2, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. Equilibrium states. Models in quantum statistical mechanics. MR 1441540, DOI 10.1007/978-3-662-03444-6
Nathan Brownlowe, Astrid An Huef, Marcelo Laca, and Iain Raeburn, Boundary quotients of the Toeplitz algebra of the affine semigroup over the natural numbers, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 35–62. MR 2873157, DOI 10.1017/S0143385710000830
Nathan Brownlowe, Nadia S. Larsen, and Nicolai Stammeier, On $C^*$-algebras associated to right LCM semigroups, Trans. Amer. Math. Soc. 369 (2017), no. 1, 31–68. MR 3557767, DOI 10.1090/tran/6638
Nathan Brownlowe, Nadia S. Larsen, and Nicolai Stammeier, $C^*$-algebras of algebraic dynamical systems and right LCM semigroups, Indiana Univ. Math. J. 67 (2018), no. 6, 2453–2486. MR 3900375, DOI 10.1512/iumj.2018.67.7527
Nathan Brownlowe, Jacqui Ramagge, David Robertson, and Michael F. Whittaker, Zappa-Szép products of semigroups and their $C^\ast$-algebras, J. Funct. Anal. 266 (2014), no. 6, 3937–3967. MR 3165249, DOI 10.1016/j.jfa.2013.12.025
Nathan Brownlowe and Nicolai Stammeier, The boundary quotient for algebraic dynamical systems, J. Math. Anal. Appl. 438 (2016), no. 2, 772–789. MR 3466063, DOI 10.1016/j.jmaa.2016.02.015
Chris Bruce, Marcelo Laca, Jacqui Ramagge, and Aidan Sims, Equilibrium states and growth of quasi-lattice ordered monoids, Proc. Amer. Math. Soc. 147 (2019), no. 6, 2389–2404. MR 3951419, DOI 10.1090/proc/14108
Joan Claramunt and Aidan Sims, Preferred traces on $C^\ast$-algebras of self-similar groupoids arising as fixed points, J. Math. Anal. Appl. 466 (2018), no. 1, 806–818. MR 3818146, DOI 10.1016/j.jmaa.2018.06.019
Lisa Orloff Clark, Astrid an Huef, and Iain Raeburn, Phase transitions on the Toeplitz algebras of Baumslag-Solitar semigroups, Indiana Univ. Math. J. 65 (2016), no. 6, 2137–2173. MR 3595491, DOI 10.1512/iumj.2016.65.5934
Alain Connes and Matilde Marcolli, $\Bbb Q$-lattices: quantum statistical mechanics and Galois theory, J. Geom. Phys. 56 (2006), no. 1, 2–23. MR 2170598, DOI 10.1016/j.geomphys.2005.04.010
John Crisp and Marcelo Laca, Boundary quotients and ideals of Toeplitz $C^*$-algebras of Artin groups, J. Funct. Anal. 242 (2007), no. 1, 127–156. MR 2274018, DOI 10.1016/j.jfa.2006.08.001
Joachim Cuntz, $C^*$-algebras associated with the $ax+b$-semigroup over $\Bbb N$, $K$-theory and noncommutative geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 201–215. MR 2513338, DOI 10.4171/060-1/8
Joachim Cuntz, Christopher Deninger, and Marcelo Laca, $C^*$-algebras of Toeplitz type associated with algebraic number fields, Math. Ann. 355 (2013), no. 4, 1383–1423. MR 3037019, DOI 10.1007/s00208-012-0826-9
Joachim Cuntz, Siegfried Echterhoff, Xin Li, and Guoliang Yu, $K$-theory for group $C^*$-algebras and semigroup $C^*$-algebras, Oberwolfach Seminars, vol. 47, Birkhäuser/Springer, Cham, 2017. MR 3618901, DOI 10.1007/978-3-319-59915-1
Patrick Dehornoy, François Digne, Eddy Godelle, Daan Krammer, and Jean Michel, Foundations of Garside theory, EMS Tracts in Mathematics, vol. 22, European Mathematical Society (EMS), Zürich, 2015. Author name on title page: Daan Kramer. MR 3362691, DOI 10.4171/139
Søren Eilers, Xin Li, and Efren Ruiz, The isomorphism problem for semigroup $C^*$-algebras of right-angled Artin monoids, Doc. Math. 21 (2016), 309–343. MR 3505132
Astrid an Huef, Marcelo Laca, Iain Raeburn, and Aidan Sims, KMS states on the $C^*$-algebra of a higher-rank graph and periodicity in the path space, J. Funct. Anal. 268 (2015), no. 7, 1840–1875. MR 3315580, DOI 10.1016/j.jfa.2014.12.006
Yosuke Kubota and Takuya Takeishi, Reconstructing the Bost-Connes semigroup actions from K-theory, Adv. Math. 366 (2020), 107070, 33. MR 4070304, DOI 10.1016/j.aim.2020.107070
Marcelo Laca and Sergey Neshveyev, KMS states of quasi-free dynamics on Pimsner algebras, J. Funct. Anal. 211 (2004), no. 2, 457–482. MR 2056837, DOI 10.1016/j.jfa.2003.08.008
Marcelo Laca and Iain Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Funct. Anal. 139 (1996), no. 2, 415–440. MR 1402771, DOI 10.1006/jfan.1996.0091
Marcelo Laca and Iain Raeburn, Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers, Adv. Math. 225 (2010), no. 2, 643–688. MR 2671177, DOI 10.1016/j.aim.2010.03.007
Marcelo Laca, Iain Raeburn, and Jacqui Ramagge, Phase transition on Exel crossed products associated to dilation matrices, J. Funct. Anal. 261 (2011), no. 12, 3633–3664. MR 2838037, DOI 10.1016/j.jfa.2011.08.015
Marcelo Laca, Iain Raeburn, Jacqui Ramagge, and Michael F. Whittaker, Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions, J. Funct. Anal. 266 (2014), no. 11, 6619–6661. MR 3192463, DOI 10.1016/j.jfa.2014.03.003
Boyu Li, Regular dilation and Nica-covariant representation on right LCM semigroups, Integral Equations Operator Theory 91 (2019), no. 4, Paper No. 36, 35. MR 3988115, DOI 10.1007/s00020-019-2534-2
Xin Li, Semigroup $\textrm {C}^*$-algebras and amenability of semigroups, J. Funct. Anal. 262 (2012), no. 10, 4302–4340. MR 2900468, DOI 10.1016/j.jfa.2012.02.020
Xin Li, Nuclearity of semigroup $C^*$-algebras and the connection to amenability, Adv. Math. 244 (2013), 626–662. MR 3077884, DOI 10.1016/j.aim.2013.05.016
Xin Li, On K-theoretic invariants of semigroup $\textrm {C}^*$-algebras attached to number fields, Adv. Math. 264 (2014), 371–395. MR 3250289, DOI 10.1016/j.aim.2014.07.014
Xin Li, Tron Omland, and Jack Spielberg, C*-algebra of right LCM one-relator monoids and Artin-Tits monoids of finite type, preprint, arXiv:1807.08288v1, 2018.
Volodymyr Nekrashevych, Virtual endomorphisms of groups, Algebra Discrete Math. 1 (2002), 88–128. MR 2048651
Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, American Mathematical Society, Providence, RI, 2005. MR 2162164, DOI 10.1090/surv/117
Volodymyr Nekrashevych, $C^*$-algebras and self-similar groups, J. Reine Angew. Math. 630 (2009), 59–123. MR 2526786, DOI 10.1515/CRELLE.2009.035
Volodymyr Nekrashevych, Palindromic subshifts and simple periodic groups of intermediate growth, Ann. of Math. (2) 187 (2018), no. 3, 667–719. MR 3779956, DOI 10.4007/annals.2018.187.3.2
Sergey Neshveyev, KMS states on the $C^\ast$-algebras of non-principal groupoids, J. Operator Theory 70 (2013), no. 2, 513–530. MR 3138368, DOI 10.7900/jot.2011sep20.1915
Sergey Neshveyev and Nicolai Stammeier, The groupoid approach to equilibrium states on right LCM semigroup $C^*$-algebras,\nopunct. preprint, arXiv:1912.03141, 2019.
A. Nica, $C^*$-algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory 27 (1992), no. 1, 17–52. MR 1241114
Magnus Dahler Norling, Inverse semigroup $C^*$-algebras associated with left cancellative semigroups, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 2, 533–564. MR 3200323, DOI 10.1017/S0013091513000540
Jack Spielberg, $C^\ast$-algebras for categories of paths associated to the Baumslag-Solitar groups, J. Lond. Math. Soc. (2) 86 (2012), no. 3, 728–754. MR 3000828, DOI 10.1112/jlms/jds025
Nicolai Stammeier, The nature of generalized scales, Internat. J. Algebra Comput. 29 (2019), no. 6, 1035–1062. MR 3996984, DOI 10.1142/S0218196719500401
Charles Starling, Boundary quotients of $\rm C^*$-algebras of right LCM semigroups, J. Funct. Anal. 268 (2015), no. 11, 3326–3356. MR 3336727, DOI 10.1016/j.jfa.2015.01.001