Pure group C*-algebras – Hannes Thiel

Question: Is $C^*_r(G)$ pure if (and only if) $G$ is non-amenable?

Here, $C^*_r(G)$ denotes the reduced group C*-algebra of a discrete group $G$ . A C*-algebra $A$ is said to be \emph{pure} if it is Jiang-Su stable at the level of Cuntz semigroups, in the sense that

$\operatorname{Cu}(A) \cong \operatorname{Cu}(A)\otimes \operatorname{Cu}(\mathcal{Z}).$

Equivalently, the Cuntz semigroup $\operatorname{Cu}(A)$

is almost unperforated and almost divisible (see Theorem 7.3.11 in [1]). This notion was introduced by Winter in his seminal work on regularity properties of simple, nuclear C-algebras [2]. More recently, it was shown in [3,4] that pure C*-algebras form a robust class: they are closed under ideals, quotients, extensions, and inductive limits.

Jiang-Su stability implies purity, but reduced group C*-algebras are rarely Jiang-Su stable. Another important source of examples comes from Robert’s notion of \emph{self-less} C*-algebras [5], which are also pure.

If $G$ is amenable, then $C^*_r(G)$ admits a quotient isomorphic to $\mathbb{C}$ , arising from the trivial representation. This provides an obstruction to purity. In particular, if $C^*_r(G)$ is pure, then $G$ must be non-amenable.

There is some evidence for the converse: A recent breakthrough by Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell showed that $C^*_r(G)$ is self-less (and hence pure) whenever $G$ is an acylindrically hyperbolic group with trivial finite radical and rapid decay. The assumption of rapid decay was later removed by Ozawa [7], and the expectation is that all C*-simple groups (that is, groups for which $C^*_r(G)$ is simple) have self-less and thus pure reduced group C*-algebra.

Things are more complicated for non-amenable groups for which $C^*_r(G)$ is not simple. A first step was obtained by extending the results of [6,7] to the twisted case, which implies that $C^*_r(G)$ is pure for every acylindrically hyperbolic group [8.9]. One might also expect that a minimal tensor product $A \otimes B$ is pure whenever $A$ or $B$ is pure (see this problem), which would handle groups like $\mathbb{F}_2 \times \mathbb{Z}$ , or more generally $G \times H$ for $G$ an acylindrically hyperbolic group and $H$ any other group. Update (February 2026): Seth and Vilalta [10] showed that $A \otimes B$ is pure whenever $A$ is simple and pure and $B$ is an ASH algebra. This handles in particular $\mathbb{F}_m \times \mathbb{Z}^n$ with $m \geq 2$ , and more generally groups of the form $G \times H$ with $G$ an acylindrically hyperbolic group (so that $C^*_r(G)$ is a finite direct sum of simple, pure C*-algebras) and $H$ virtually abelian (so that $C^*_r(H)$ is subhomogeneous).

A necessary condition for pureness is that the C*-algebra is nowhere scattered, meaning that none of its ideal-quotients are elementary (that is, isomorphic to the compact operators on some Hilbert spaces). It is known that $C^*_r(G)$ has no finite-dimensional irreducible representations (and hence no elementary quotients) whenever $G$ is non-amenable. However, it is unknown if $C^*_r(G)$ is nowhere scattered whenever $G$ is non-amenable.

[1] R. Antoine, F. Perera, H. Thiel. Tensor products and regularity properties of Cuntz semigroups. Mem. Amer. Math. Soc. 251 (2018).

[2] W. Winter. Nuclear dimension and Z-stability of pure C*-algebras. Invent. Math. 187 (2012), 259-342.

[3] R. Antoine, F. Perera, H. Thiel, E. Vilalta. Pure C*-algebras. preprint arXiv:2406.11052 (2024).

[4] F. Perera, H. Thiel, E. Vilalta. Extensions of pure C*-algebras. preprint arXiv:2506.10529 (2025).

[5] L. Robert. Selfless C*-algebras. Adv. Math. 478 (2025), Article ID 110409, 28 p.

[6] T. Amrutam, D. Gao, S. Kunnawalkam Elayavalli, G. Patchell. Strict comparison in reduced group C*-algebras. Invent. Math. 242 (2025), 639-657.

[7] N. Ozawa. Proximality and selflessness for group C*-algebras. preprint arXiv:2508.07938 (2025).

[8] S. Raum, H. Thiel, E. Vilalta. Strict comparison for twisted group C*-algebras. arXiv:2505.18569 (2025).

[9] F. Flores, M. Klisse, M. Cobhthaigh, M. Pagliero. Pureness and stable rank one for reduced twisted group C*-algebras of certain group extensions. preprint arXiv:2601.19758 (2026).

[10] A. Seth, E. Vilalta. Continuous functions over a pure C*-algebra. preprint arXiv:2602.14809 (2026).

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