A torsion-free algebraically \(C^*\)-unique group. (English) Zbl 1454.22003

A discrete group \(G\) is called algebraically \(C^*\)-unique if the complex group ring \(\mathbb C[G]\) admits a unique \(C^*\)-norm. The family of algebraically \(C^*\)-unique groups is evidently contained in the class of all amenable discrete groups, but the converse is shown not to be true, see V. Alekseev and D. Kyed [Pac. J. Math. 298, No. 2, 257–266 (2019; Zbl 1452.16028)]. It is also proved by R. Grigorchuk et al. [Comment. Math. Helv. 93, No. 1, 157–201 (2018; Zbl 1396.46044)] that every locally finite group is algebraically \(C^*\)-unique. The converse of this fact dose not hold as well. Indeed the group \((\oplus_{\mathbb Z}\mathbb Z_2) \rtimes\mathbb Z\), which is given by Ozawa, is a \(C^*\)-unique group which is not locally finite.
In the paper under review, the author proves that for two multiplicatively independent integers \(p\) and \(q\), the torsion-free group \(\mathbb Z[1/pq]\rtimes \mathbb Z^2 \) is algebraically \(C^*\)-unique. Providing this example, he answers a question asked by Vadim Alekseev.


22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
Full Text: DOI arXiv Euclid


[1] V. Alekseev, “(Non)-uniqueness of \(\text{C}^\ast \)-norms on group rings of amenable groups”, pp. 2292-2293 in \(\text{C}^\ast \)-algebras, Oberwolfach Report 37/2019 16, European Mathematical Society Publishing House, 2019.
[2] V. Alekseev and D. Kyed, “Uniqueness questions for \(C^*\)-norms on group rings”, Pacific J. Math. 298:2 (2019), 257-266. Mathematical Reviews (MathSciNet): MR3936017
Zentralblatt MATH: 07050829
Digital Object Identifier: doi:10.2140/pjm.2019.298.257
· Zbl 1452.16028
[3] M. D. Boshernitzan, “Elementary proof of Furstenberg”s Diophantine result”, Proc. Amer. Math. Soc. 122:1 (1994), 67-70. Mathematical Reviews (MathSciNet): MR1195714
Zentralblatt MATH: 0815.11036
· Zbl 0815.11036
[4] M. Caspers and A. Skalski, “On \(\rm C^*\)-completions of discrete quantum group rings”, Bull. Lond. Math. Soc. 51:4 (2019), 691-704. Mathematical Reviews (MathSciNet): MR3990385
Zentralblatt MATH: 07118831
Digital Object Identifier: doi:10.1112/blms.12267
· Zbl 1447.46052
[5] C. Eckhardt, “A note on strongly quasidiagonal groups”, J. Operator Theory 73:2 (2015), 417-424. Mathematical Reviews (MathSciNet): MR3346129
Zentralblatt MATH: 1389.46045
Digital Object Identifier: doi:10.7900/jot.2014jan22.2034
· Zbl 1389.46045
[6] R. Exel, Partial dynamical systems, Fell bundles and applications, Mathematical Surveys and Monographs 224, American Mathematical Society, Providence, RI, 2017. Mathematical Reviews (MathSciNet): MR3699795
Zentralblatt MATH: 1405.46003
· Zbl 1405.46003
[7] H. Furstenberg, “Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation”, Math. Systems Theory 1 (1967), 1-49. Mathematical Reviews (MathSciNet): MR213508
Zentralblatt MATH: 0146.28502
Digital Object Identifier: doi:10.1007/BF01692494
· Zbl 0146.28502
[8] R. Grigorchuk, M. Musat, and M. Rørdam, “Just-infinite \(C^*\)-algebras”, Comment. Math. Helv. 93:1 (2018), 157-201. Mathematical Reviews (MathSciNet): MR3777128
Zentralblatt MATH: 1396.46044
Digital Object Identifier: doi:10.4171/CMH/432
· Zbl 1396.46044
[9] H. · Zbl 1399.37003
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