## Finite decomposition rank for virtually nilpotent groups

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- by Caleb Eckhardt, Elizabeth Gillaspy and Paul McKenney PDF
- Trans. Amer. Math. Soc.
**371**(2019), 3971-3994 Request permission

## Abstract:

We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the group C*-algebra of a finitely generated virtually nilpotent group $G$ is bounded by $2\cdot h(G)!-1$, where $h(G)$ is the Hirsch length of $G.$ This extends and sharpens results of the first and third authors on finitely generated nilpotent groups. It then follows that if a C*-algebra generated by an irreducible representation of a virtually nilpotent group satisfies the universal coefficient theorem, it is classified by its Elliott invariant.## References

- Selçuk Barlak and Gábor Szabó,
*Rokhlin actions of finite groups on UHF-absorbing $\textrm {C}^*$-algebras*, Trans. Amer. Math. Soc.**369**(2017), no. 2, 833–859. MR**3572256**, DOI 10.1090/tran6697 - B. Blackadar,
*Operator algebras*, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of $C^*$-algebras and von Neumann algebras; Operator Algebras and Non-commutative Geometry, III. MR**2188261**, DOI 10.1007/3-540-28517-2 - Joan Bosa, Nathanial Brown, Yasuhiko Sato, Aaron Tikuisis, Stuart White, and Wilhelm Winter,
*Covering dimension of C*-algebras and 2-coloured classification*, arXiv:1506.03974 [math.OA], 2016. - Nathanial P. Brown and Narutaka Ozawa,
*$C^*$-algebras and finite-dimensional approximations*, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR**2391387**, DOI 10.1090/gsm/088 - José R. Carrión,
*Classification of a class of crossed product $C^\ast$-algebras associated with residually finite groups*, J. Funct. Anal.**260**(2011), no. 9, 2815–2825. MR**2772352**, DOI 10.1016/j.jfa.2011.02.002 - Jacques Dixmier,
*$C^*$-algebras*, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR**0458185** - Caleb Eckhardt,
*Quasidiagonal representations of nilpotent groups*, Adv. Math.**254**(2014), 15–32. MR**3161088**, DOI 10.1016/j.aim.2013.12.011 - Caleb Eckhardt,
*A note on strongly quasidiagonal groups*, J. Operator Theory**73**(2015), no. 2, 417–424. MR**3346129**, DOI 10.7900/jot.2014jan22.2034 - Caleb Eckhardt and Elizabeth Gillaspy,
*Irreducible representations of nilpotent groups generate classifiable $C^*$-algebras*, Münster J. Math.**9**(2016), no. 1, 253–261. MR**3549551**, DOI 10.17879/35209675696 - Caleb Eckhardt, Craig Kleski, and Paul McKenney,
*Classification of $\textrm {C}^*$-algebras generated by representations of the unitriangular group $UT(4,\Bbb {Z})$*, J. Funct. Anal.**271**(2016), no. 4, 1022–1042. MR**3507997**, DOI 10.1016/j.jfa.2016.05.003 - Caleb Eckhardt and Paul McKenney,
*Finitely generated nilpotent group $\rm C^*$-algebras have finite nuclear dimension*, J. Reine Angew. Math.**738**(2018), 281–298. MR**3794894**, DOI 10.1515/crelle-2015-0049 - G. A. Elliott,
*On the $K$-theory of the $C^{\ast }$-algebra generated by a projective representation of a torsion-free discrete abelian group*, Operator algebras and group representations, Vol. I (Neptun, 1980) Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp. 157–184. MR**731772** - George A. Elliott, Guihua Gong, Huaxin Lin, and Zhuang Niu,
*On the classification of simple amenable C*-algebras with finite decomposition rank*, II, arXiv:1507.03437v3, 2016. - George A. Elliott and Andrew S. Toms,
*Regularity properties in the classification program for separable amenable $C^*$-algebras*, Bull. Amer. Math. Soc. (N.S.)**45**(2008), no. 2, 229–245. MR**2383304**, DOI 10.1090/S0273-0979-08-01199-3 - Julien Giol and David Kerr,
*Subshifts and perforation*, J. Reine Angew. Math.**639**(2010), 107–119. MR**2608192**, DOI 10.1515/CRELLE.2010.012 - Guihua Gong, Huaxin Lin, and Zhuang Niu,
*Classification of finite simple amenable $\mathcal {Z}$-stable C*-algebras*, arXiv:1501.00135v6, 2016. - Philip Green,
*The local structure of twisted covariance algebras*, Acta Math.**140**(1978), no. 3-4, 191–250. MR**493349**, DOI 10.1007/BF02392308 - Don Hadwin,
*Strongly quasidiagonal $C^*$-algebras*, J. Operator Theory**18**(1987), no. 1, 3–18. With an appendix by Jonathan Rosenberg. MR**912809** - Ilan Hirshberg and Jianchao Wu,
*The nuclear dimension of $C^*$-algebras associated to homeomorphisms*, Adv. Math.**304**(2017), 56–89. MR**3558205**, DOI 10.1016/j.aim.2016.08.022 - Roger E. Howe,
*On representations of discrete, finitely generated, torsion-free, nilpotent groups*, Pacific J. Math.**73**(1977), no. 2, 281–305. MR**499004** - Xinhui Jiang and Hongbing Su,
*On a simple unital projectionless $C^*$-algebra*, Amer. J. Math.**121**(1999), no. 2, 359–413. MR**1680321** - Eberhard Kaniuth,
*Ideals in group algebras of finitely generated FC-nilpotent discrete groups*, Math. Ann.**248**(1980), no. 2, 97–108. MR**573341**, DOI 10.1007/BF01421950 - Eberhard Kirchberg and Wilhelm Winter,
*Covering dimension and quasidiagonality*, Internat. J. Math.**15**(2004), no. 1, 63–85. MR**2039212**, DOI 10.1142/S0129167X04002119 - Christopher Lance,
*On nuclear $C^{\ast }$-algebras*, J. Functional Analysis**12**(1973), 157–176. MR**0344901**, DOI 10.1016/0022-1236(73)90021-9 - Hiroki Matui and Yasuhiko Sato,
*Decomposition rank of UHF-absorbing $\mathrm {C}^*$-algebras*, Duke Math. J.**163**(2014), no. 14, 2687–2708. MR**3273581**, DOI 10.1215/00127094-2826908 - Hiroki Matui and Yasuhiko Sato,
*$\scr Z$-stability of crossed products by strongly outer actions II*, Amer. J. Math.**136**(2014), no. 6, 1441–1496. MR**3282978**, DOI 10.1353/ajm.2014.0043 - Calvin C. Moore and Jonathan Rosenberg,
*Groups with $T_{1}$ primitive ideal spaces*, J. Functional Analysis**22**(1976), no. 3, 204–224. MR**0419675**, DOI 10.1016/0022-1236(76)90009-4 - May Nilsen,
*$C^*$-bundles and $C_0(X)$-algebras*, Indiana Univ. Math. J.**45**(1996), no. 2, 463–477. MR**1414338**, DOI 10.1512/iumj.1996.45.1086 - Judith A. Packer,
*$C^*$-algebras generated by projective representations of the discrete Heisenberg group*, J. Operator Theory**18**(1987), no. 1, 41–66. MR**912812** - Judith A. Packer and Iain Raeburn,
*Twisted crossed products of $C^*$-algebras*, Math. Proc. Cambridge Philos. Soc.**106**(1989), no. 2, 293–311. MR**1002543**, DOI 10.1017/S0305004100078129 - Judith A. Packer and Iain Raeburn,
*Twisted crossed products of $C^*$-algebras. II*, Math. Ann.**287**(1990), no. 4, 595–612. MR**1066817**, DOI 10.1007/BF01446916 - Judith A. Packer and Iain Raeburn,
*On the structure of twisted group $C^*$-algebras*, Trans. Amer. Math. Soc.**334**(1992), no. 2, 685–718. MR**1078249**, DOI 10.1090/S0002-9947-1992-1078249-7 - Gert K. Pedersen,
*$C^{\ast }$-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006** - Detlev Poguntke,
*Discrete nilpotent groups have a $T_{1}$ primitive ideal space*, Studia Math.**71**(1981/82), no. 3, 271–275. MR**667315**, DOI 10.4064/sm-71-3-271-275 - Marc A. Rieffel,
*$C^{\ast }$-algebras associated with irrational rotations*, Pacific J. Math.**93**(1981), no. 2, 415–429. MR**623572** - Marc A. Rieffel,
*Projective modules over higher-dimensional noncommutative tori*, Canad. J. Math.**40**(1988), no. 2, 257–338. MR**941652**, DOI 10.4153/CJM-1988-012-9 - M. Rørdam,
*Classification of nuclear, simple $C^*$-algebras*, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. MR**1878882**, DOI 10.1007/978-3-662-04825-2_{1} - Jonathan Rosenberg and Claude Schochet,
*The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized $K$-functor*, Duke Math. J.**55**(1987), no. 2, 431–474. MR**894590**, DOI 10.1215/S0012-7094-87-05524-4 - Daniel Segal,
*Polycyclic groups*, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR**713786**, DOI 10.1017/CBO9780511565953 - M. Takesaki,
*Theory of operator algebras. I*, Encyclopaedia of Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition; Operator Algebras and Non-commutative Geometry, 5. MR**1873025** - Aaron Tikuisis, Stuart White, and Wilhelm Winter,
*Quasidiagonality of nuclear $C^\ast$-algebras*, Ann. of Math. (2)**185**(2017), no. 1, 229–284. MR**3583354**, DOI 10.4007/annals.2017.185.1.4 - Dana P. Williams,
*Crossed products of $C{^\ast }$-algebras*, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR**2288954**, DOI 10.1090/surv/134 - Wilhelm Winter,
*Nuclear dimension and $\scr {Z}$-stability of pure $\rm C^*$-algebras*, Invent. Math.**187**(2012), no. 2, 259–342. MR**2885621**, DOI 10.1007/s00222-011-0334-7 - Wilhelm Winter and Joachim Zacharias,
*The nuclear dimension of $C^\ast$-algebras*, Adv. Math.**224**(2010), no. 2, 461–498. MR**2609012**, DOI 10.1016/j.aim.2009.12.005

## Additional Information

**Caleb Eckhardt**- Affiliation: Department of Mathematics, Miami University, Oxford, Ohio, 45056
- MR Author ID: 885145
- Email: eckharc@miamioh.edu
**Elizabeth Gillaspy**- Affiliation: Department of Mathematical Sciences, University of Montana, 32 Campus Drive # 0864, Missoula, Montana, 59812-0864
- MR Author ID: 1107754
- Email: elizabeth.gillaspy@mso.umt.edu
**Paul McKenney**- Affiliation: Department of Mathematics, Miami University, Oxford, Ohio, 45056
- MR Author ID: 1024792
- Email: mckennp2@miamioh.edu
- Received by editor(s): July 21, 2017
- Received by editor(s) in revised form: October 30, 2017
- Published electronically: December 3, 2018
- Additional Notes: The first author was partially supported by a grant from the Simons Foundation.

The second author was primarily supported by the Deutsches Forschungsgemeinschaft via SFB 878 (awarded to the Universität Münster, Germany). - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 3971-3994 - MSC (2010): Primary 46L05; Secondary 20F19, 46L35, 46L55, 46L80
- DOI: https://doi.org/10.1090/tran/7453
- MathSciNet review: 3917214