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Scarparo, Eduardo

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

Rocky Mt. J. Math. 50, No. 5, 1813-1815 (2020).
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.
Reviewer: Akram Yousofzadeh (Isfahan)

MSC:

22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations

Keywords:

\(C^*\)-norm; complex group ring

Citations:

Zbl 1396.46044; Zbl 1452.16028
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\textit{E. Scarparo}, Rocky Mt. J. Math. 50, No. 5, 1813--1815 (2020; Zbl 1454.22003)
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References:

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