## Amenability and functoriality of right-LCM semigroup C*-algebras

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- by Marcelo Laca and Boyu Li PDF
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## Abstract:

We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups having no nontrivial units is inherited by the factors.## References

- Egbert Brieskorn and Kyoji Saito,
*Artin-Gruppen und Coxeter-Gruppen*, Invent. Math.**17**(1972), 245–271 (German). MR**323910**, DOI 10.1007/BF01406235 - 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 - C. Coleman, R. Corran, J. Crisp, D. Easdown, R. Howlett, D. Jackson, and A. Ram,
*Artin groups and Coxeter groups*, A translation, with notes, of the paper Artin-Gruppen und Coxeter-Gruppen, by E. Brieskorn and K. Saito, University of Sydney, 1996. - L. A. Coburn,
*The $C^{\ast }$-algebra generated by an isometry*, Bull. Amer. Math. Soc.**73**(1967), 722–726. MR**213906**, DOI 10.1090/S0002-9904-1967-11845-7 - John Crisp,
*Injective maps between Artin groups*, Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin, 1999, pp. 119–137. MR**1714842** - John Crisp and Marcelo Laca,
*On the Toeplitz algebras of right-angled and finite-type Artin groups*, J. Aust. Math. Soc.**72**(2002), no. 2, 223–245. MR**1887134**, DOI 10.1017/S1446788700003876 - 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,
*Simple $C^*$-algebras generated by isometries*, Comm. Math. Phys.**57**(1977), no. 2, 173–185. MR**467330** - Joachim Cuntz,
*$K$-theory for certain $C^{\ast }$-algebras*, Ann. of Math. (2)**113**(1981), no. 1, 181–197. MR**604046**, DOI 10.2307/1971137 - 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** - Mahlon M. Day,
*Amenable semigroups*, Illinois J. Math.**1**(1957), 509–544. MR**92128** - R. G. Douglas,
*On the $C^{\ast }$-algebra of a one-parameter semigroup of isometries*, Acta Math.**128**(1972), no. 3-4, 143–151. MR**394296**, DOI 10.1007/BF02392163 - 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** - Ruy Exel,
*Amenability for Fell bundles*, J. Reine Angew. Math.**492**(1997), 41–73. MR**1488064**, DOI 10.1515/crll.1997.492.41 - John Fountain and Mark Kambites,
*Graph products of right cancellative monoids*, J. Aust. Math. Soc.**87**(2009), no. 2, 227–252. MR**2551120**, DOI 10.1017/S144678870900010X - L. C. Grove and C. T. Benson,
*Finite reflection groups*, 2nd ed., Graduate Texts in Mathematics, vol. 99, Springer-Verlag, New York, 1985. MR**777684**, DOI 10.1007/978-1-4757-1869-0 - Astrid an Huef, Iain Raeburn, and Ilija Tolich,
*HNN extensions of quasi-lattice ordered groups and their operator algebras*, Doc. Math.**23**(2018), 327–351. MR**3846056** - A. an Huef, B. Nucinkis, C. F. Sehnem, and D. Yang,
*Nuclearity of semigroup C*-algebras*, (preprint) arXiv:1910.04898 (2019). - Nikolay A. Ivanov,
*The $K$-theory of Toeplitz $C^*$-algebras of right-angled Artin groups*, Trans. Amer. Math. Soc.**362**(2010), no. 11, 6003–6027. MR**2661506**, DOI 10.1090/S0002-9947-2010-05162-1 - 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 - 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 - X. Li, T. Omland, and J. Spielberg,
*C*-algebras of right LCM one-relator monoids and Artin-Tits groups of finite type*, (preprint) arXiv:1807.08288 (2018). - S. Neshveyev and N. Stammeier. The groupoid approach to equilibrium states on right LCM semigroup C*-algebras, (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 - Luis Paris,
*Artin monoids inject in their groups*, Comment. Math. Helv.**77**(2002), no. 3, 609–637. MR**1933791**, DOI 10.1007/s00014-002-8353-z - Gelu Popescu,
*Isometric dilations for infinite sequences of noncommuting operators*, Trans. Amer. Math. Soc.**316**(1989), no. 2, 523–536. MR**972704**, DOI 10.1090/S0002-9947-1989-0972704-3 - 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

## Additional Information

**Marcelo Laca**- Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 1700 STN CSC, Victoria, B.C., Canada V8W 2Y2
- MR Author ID: 335785
- Email: laca@uvic.ca
**Boyu Li**- Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 1700 STN CSC, Victoria, B.C., Canada V8W 2Y2
- MR Author ID: 1169918
- ORCID: 0000-0003-2484-1851
- Email: boyuli@uvic.ca
- Received by editor(s): January 13, 2020
- Received by editor(s) in revised form: March 30, 2020, and April 13, 2020
- Published electronically: September 4, 2020
- Additional Notes: The second author was supported by a fellowship from the Pacific Institute for the Mathematical Sciences.
- Communicated by: Adrian Ioana
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5209-5224 - MSC (2010): Primary 46L05; Secondary 20F36, 47D03
- DOI: https://doi.org/10.1090/proc/15139
- MathSciNet review: 4163833