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Decomposition of Thompson group representations arising from Cuntz algebras

  • Nuno Carneiro and Paulo R. Pinto ORCID logo EMAIL logo
Published/Copyright: November 6, 2024
Journal of Group Theory
From the journal Journal of Group Theory Volume 28 Issue 3

Abstract

For every integer n ≥ 2 , we consider a family { π w } w ∈ { 0 , 1 , … , n − 1 } N of irreducible representations of the Cuntz algebra O n . All these representations (except one) are shown to be equivalent to those arising from the orbits of the interval map dynamical system ( I , f ) with f ⁢ ( x ) = n ⁢ x ⁢ ( mod ⁢ 1 ) . We consider the embeddings V = V 2 ⊂ V n ⊂ O n of the Thompson group 𝑉 in the Higman–Thompson group V n obtained by Birget and then from V n into O n which was obtained independently by Birget and Nekrashevych. The restriction of π w to V n is still irreducible; however, the restriction of π w to 𝑉 is no longer irreducible, and we obtain the corresponding irreducible decomposition in terms of quasi-regular representations.

Funding source: Fundação Calouste Gulbenkian

Award Identifier / Grant number: Novos Talentos/New Talents

Funding source: Fundação para a Ciência e a Tecnologia

Award Identifier / Grant number: UIDB/04459/2020

Award Identifier / Grant number: UIDP/04459/2020

Funding statement: The first author was partially supported by Fundação Calouste Gulbenkian Foundation under the Novos Talentos/New Talents project. The second author was partially supported by FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.

Acknowledgements

We thank the referee for their careful reading of the manuscript and constructive suggestions.

  1. Communicated by: Adrian Ioana

References

[1] V. Aiello, R. Conti and V. F. R. Jones, The Homflypt polynomial and the oriented Thompson group, Quantum Topol. 9 (2018), no. 3, 461–472. 10.4171/qt/112Search in Google Scholar

[2] F. Araújo and P. R. Pinto, Representations of Higman–Thompson groups from Cuntz algebras, J. Math. Anal. Appl. 509 (2022), no. 2, Article ID 125970. 10.1016/j.jmaa.2021.125970Search in Google Scholar

[3] M. Barata and P. R. Pinto, Representations of Thompson groups from Cuntz algebras, J. Math. Anal. Appl. 478 (2019), no. 1, 212–228. 10.1016/j.jmaa.2019.05.028Search in Google Scholar

[4] J.-C. Birget, The groups of Richard Thompson and complexity, Internat. J. Algebra Comput. 14 (2004), 569–626. 10.1142/S0218196704001980Search in Google Scholar

[5] J.-C. Birget, New embeddings between the Higman–Thompson groups, Comm. Algebra 48 (2020), no. 8, 3429–3438. 10.1080/00927872.2020.1739290Search in Google Scholar

[6] N. M. Bon, Rigidity properties of full groups of pseudogroups over the Cantor set, preprint (2018), https://arxiv.org/abs/1801.10133. Search in Google Scholar

[7] O. Bratteli and P. E. T. Jorgensen, Iterated function systems and permutation representations of the Cuntz algebra, Mem. Amer. Math. Soc. 663 (1999), 1–89. 10.1090/memo/0663Search in Google Scholar

[8] A. Brothier, On Jones’ connections between subfactors, conformal field theory, Thompson’s groups and knots, preprint (2019), https://arxiv.org/abs/1912.07140. Search in Google Scholar

[9] A. Brothier and V. F. R. Jones, Pythagorean representations of Thompson’s groups, J. Funct. Anal. 277 (2019), no. 7, 2442–2469. 10.1016/j.jfa.2019.02.009Search in Google Scholar

[10] A. Brothier and D. Wijesena, Decomposition of Pythagorean representations of R. Thompson’s groups, preprint (2023), https://arxiv.org/abs/2302.04458v2. Search in Google Scholar

[11] A. Brothier and D. Wijesena, Irreducible Pythagorean representations of R. Thompson’s groups and of the Cuntz algebra, Adv. Math. 454 (2024), Article ID 109871. 10.1016/j.aim.2024.109871Search in Google Scholar

[12] K. S. Brown, Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987), no. 1–3, 45–75. 10.1016/0022-4049(87)90015-6Search in Google Scholar

[13] J. Burillo, S. Cleary and M. I. Stein, Metrics and embeddings of generalizations of Thompson’s group 𝐹, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1677–1689. 10.1090/S0002-9947-00-02650-7Search in Google Scholar

[14] J. W. Cannon, W. J. Floyd and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3–4, 215–256. Search in Google Scholar

[15] C. Correia Ramos, N. Martins and P. R. Pinto, On graph algebras from interval maps, Ann. Funct. Anal. 10 (2019), no. 2, 203–217. 10.1215/20088752-2018-0019Search in Google Scholar

[16] C. Correia Ramos, N. Martins, P. R. Pinto and J. Sousa Ramos, Cuntz–Krieger algebras representations from orbits of interval maps, J. Math. Anal. Appl. 341 (2008), no. 2, 825–833. 10.1016/j.jmaa.2007.10.059Search in Google Scholar

[17] J. Cuntz, Simple C ∗ -algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. 10.1007/BF01625776Search in Google Scholar

[18] J. Dixmier, Traces sur les C ∗ -algèbres. II, Bull. Sci. Math. (2) 88 (1964), 39–57. Search in Google Scholar

[19] A. Guimarães and P. R. Pinto, On a family of representations of the Higman–Thompson groups, J. Group Theory 25 (2022), 997–1015. 10.1515/jgth-2021-0190Search in Google Scholar

[20] U. Haagerup and K. K. Olesen, Non-inner amenability of the Thompson groups 𝑇 and 𝑉, J. Funct. Anal. 272 (2017), no. 11, 4838–4852. 10.1016/j.jfa.2017.02.003Search in Google Scholar

[21] G. Higman, Finitely Presented Infinite Simple Groups, Notes Pure Math. 8, Australian National University, Canberra, 1974. Search in Google Scholar

[22] K. Kawamura, Universal algebra of sectors, Internat. J. Algebra Comput. 19 (2009), no. 3, 347–371. 10.1142/S0218196709005172Search in Google Scholar

[23] V. V. Nekrashevych, Cuntz–Pimsner algebras of group actions, J. Operator Theory 52 (2004), no. 2, 223–249. Search in Google Scholar

[24] N. Obata, Some remarks on induced representations of infinite discrete groups, Math. Ann. 284 (1989), no. 1, 91–102. 10.1007/BF01443507Search in Google Scholar

[25] G. K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, 2nd ed., London Math. Soc. Monogr. 14, Academic Press, London, 1979. Search in Google Scholar

[26] X. Sheng, Quasi-isometric embeddings from generalised Thompson’s groups to Thompson’s group 𝑇, Tokyo J. Math. 45 (2022), no. 2, 451–465. 10.3836/tjm/1502179371Search in Google Scholar

[27] M. Stein, Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc. 332 (1992), no. 2, 477–514. 10.1090/S0002-9947-1992-1094555-4Search in Google Scholar
Received: 2023-11-13
Revised: 2024-10-02
Published Online: 2024-11-06
Published in Print: 2025-05-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Rigid stabilizers and local prosolubility for boundary-transitive actions on tree
  3. Twisted conjugacy and separability
  4. Mixed identities, hereditarily separated actions and oscillation
  5. Decomposition of Thompson group representations arising from Cuntz algebras
  6. Liftable automorphisms of right-angled Artin groups
  7. Uncountable groups in which permutability is a transitive relation
  8. On semiabelian groups
  9. Critical classes of power graphs and reconstruction of directed power graphs
  10. Weights for 𝜋-partial characters of 𝜋-separable groups
https://doi.org/10.1515/jgth-2023-0245

Articles in the same Issue

  1. Frontmatter
  2. Rigid stabilizers and local prosolubility for boundary-transitive actions on tree
  3. Twisted conjugacy and separability
  4. Mixed identities, hereditarily separated actions and oscillation
  5. Decomposition of Thompson group representations arising from Cuntz algebras
  6. Liftable automorphisms of right-angled Artin groups
  7. Uncountable groups in which permutability is a transitive relation
  8. On semiabelian groups
  9. Critical classes of power graphs and reconstruction of directed power graphs
  10. Weights for 𝜋-partial characters of 𝜋-separable groups
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