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15.7 (2002)

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(Well-known problem). Is it true that the reduced $C^*$-algebra of a countable group without amenable normal subgroups distinct from $\{1\}$ is always a simple $C^*$-algebra with unique trace? See (M. B. Bekka, P. de la Harpe, Expos. Math., 18, no. 3 (2000), 215–230).

Contributor: P. de la Harpe

Progress

Comment of 2009: This was proved for linear groups (T. Poznansky, https://arxiv.org/pdf/0812.2486.pdf).

Yes, it is true (E. Breuillard, M. Kalantar, M. Kennedy, N. Ozawa, Publ. Math. IHÉS, 126, no. 1 (2017), 35–71). If $\Gamma$ is assumed to be linear, then $C^*_{\text{red}}(\Gamma)$ is simple if and only if it has a unique trace (Theorem 1.6 ibid., and also Poznansky's 2009 preprint). However, there are infinite countable groups $\Gamma$ having the following two properties: the only amenable normal subgroup is $\{1\}$, and $C^*_{\text{red}}(\Gamma)$ is not simple, as shown in (A. Le Boudec, Invent. Math., 209, no. 1 (2017), 159–174).

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