Weak $C^*$-simplicity without the weak unique trace property, just-infinite-type structure theory, and an ASH model for $C^*(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2)$ (Questions 4.1-4.3 of Bruce-Scarparo)

Bruce and Scarparo proved that, for multiplicatively independent integers $p,q\ge 2$, Furstenberg's $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on $C^*(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2)$, and they computed the primitive ideal space and K-theory of this algebra. In Section 4 of that paper they posed three open questions, considered here as one problem: (1) does there exist an icc group with only countably many subgroups that is weakly $C^*$-simple but does not have the weak unique trace property (their ad hoc weak analogues of $C^*$-simplicity and of the unique trace property)? (2) can (some of) the theory of just-infinite $C^*$-algebras be extended to the class of $C^*$-algebras for which any irreducible representation is either faithful or has finite-dimensional image, a class containing crossed products of faithful almost minimal actions of countable abelian groups? (3) is there a unital approximately subhomogeneous (ASH) $C^*$-algebra in this class with the same primitive ideal space and K-theory as $C^*(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2)$, and if so, what is its trace simplex?

Background

Furstenberg's $\times p,\times q$ conjecture (1967) predicts that for multiplicatively independent integers $p,q\ge 2$ (no $r,s>0$ with $p^{r}=q^{s}$), the only ergodic probability measure on the circle $\mathbb{T}$ that is invariant under both $z\mapsto z^{p}$ and $z\mapsto z^{q}$ and has infinite (equivalently full) support is Lebesgue measure; Furstenberg himself proved the topological analogue that the only infinite closed $\times p,\times q$-invariant subset of $\mathbb{T}$ is $\mathbb{T}$ itself. A natural operator-algebraic home for the conjecture is the group $\Gamma_{p,q}=\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2$, in which the two generators of $\mathbb{Z}^2$ act on $\mathbb{Z}[1/pq]$ by multiplication by $p$ and by $q$; it has the presentation $\langle s,t,z\mid st=ts,\ szs^{-1}=z^{p},\ tzt^{-1}=z^{q}\rangle$. Huang and Wu connected ergodic invariant measures for the $\times p,\times q$ action with irreducible representations of the crossed product, and Bruce and Scarparo proved the tracial characterization: the canonical trace is the only faithful extreme tracial state on $C^*(\Gamma_{p,q})$ if and only if Furstenberg's conjecture holds. Here a tracial state is a state $\tau$ with $\tau(ab)=\tau(ba)$; 'extreme' means extreme in the compact convex set of tracial states; 'faithful' means $\tau(a^{*}a)=0$ forces $a=0$. Bruce and Scarparo also computed $\operatorname{Prim}(C^*(\Gamma_{p,q}))$, the set of kernels of irreducible representations equipped with the hull-kernel topology, and its K-theory (the standard homotopy-invariant functor from $C^*$-algebras to abelian groups): $K_{i}(C^*(\Gamma_{p,q}))\cong\mathbb{Z}^{2}\oplus\mathbb{Z}/\gcd(p-1,q-1)\mathbb{Z}$ for $i=0,1$. Their framework is the theory of almost minimal actions: an action $\Gamma\curvearrowright X$ of a discrete group on a compact space is almost minimal if every proper closed invariant subset is finite. For faithful, almost minimal actions of countable abelian groups they showed that $\operatorname{Prim}(C(X)\rtimes\Gamma)$ is a countable, explicitly described space and that every irreducible representation of $C(X)\rtimes\Gamma$ is either faithful (injective) or has finite-dimensional image.

This irreducible-representation dichotomy is exactly the one satisfied by just-infinite $C^*$-algebras, i.e. unital $C^*$-algebras every nonzero ideal of which has finite codimension (equivalently, every proper quotient is finite-dimensional): the kernel of an irreducible representation of a just-infinite algebra is either zero or finite-codimensional. Grigorchuk, Musat and Rordam developed the structure theory of just-infinite $C^*$-algebras, and Rordam computed their invariants, showing for instance that every infinite-dimensional metrizable Choquet simplex arises as the trace simplex of a residually finite-dimensional just-infinite AF-algebra (a Choquet simplex is a compact convex set all of whose finite-dimensional faces are simplices; the trace simplex $T(A)$ of a unital $C^*$-algebra $A$ is the Choquet simplex of its tracial states, metrizable whenever $A$ is separable). The three questions considered here concern the class of $C^*$-algebras for which any irreducible representation is either faithful or has finite-dimensional image, a class containing both the just-infinite algebras and the crossed products of faithful almost minimal actions of countable abelian groups.

For the first question one needs the following standard background from the theory of $C^*$-simplicity. A group is icc if every nontrivial conjugacy class is infinite. A discrete group $\Gamma$ is $C^*$-simple if its reduced group $C^*$-algebra $C^{*}_{r}(\Gamma)$ (the norm-closure of the image of the left regular representation) is simple, and has the unique trace property if the canonical trace is the only tracial state on $C^{*}_{r}(\Gamma)$. Breuillard, Kalantar, Kennedy and Ozawa proved that $C^*$-simplicity implies the unique trace property, that the unique trace property is equivalent to triviality of the amenable radical (the largest amenable normal subgroup), and that for groups with only countably many amenable subgroups, in particular groups with only countably many subgroups, $C^*$-simplicity and the unique trace property are equivalent. Bruce and Scarparo then introduce, explicitly for the purpose of these questions, weak analogues: an icc group $\Gamma$ is weakly $C^*$-simple if every nonzero ideal of $C^{*}_{r}(\Gamma)$ intersects the group algebra $\mathbb{C}\Gamma$ nontrivially, and has the weak unique trace property if the canonical trace is the only faithful extreme tracial state on $C^{*}_{r}(\Gamma)$. Scarparo had shown that $\Gamma_{p,q}$ is weakly $C^*$-simple; Becker, Lubotzky and Thom showed that $\Gamma_{p,q}$ has only countably many subgroups; and by the tracial characterization, $\Gamma_{p,q}$ has the weak unique trace property exactly if Furstenberg's conjecture is true. The lamplighter group $\mathbb{Z}_{2}\wr\mathbb{Z}$ is weakly $C^*$-simple but fails the weak unique trace property (Bernoulli product measures on $\{0,1\}^{\mathbb{Z}}$ give faithful extreme tracial states distinct from the canonical one, an observation recorded by Bruce-Scarparo and attributed there to Ozawa and independently to Vaes), yet it has uncountably many subgroups; so the countable-subgroup regime, where the non-weak properties collapse to equivalence by the result of Breuillard-Kalantar-Kennedy-Ozawa, is genuinely open territory for the weak ones.

For the third question: a $C^*$-algebra is subhomogeneous if it is isomorphic to a sub-$C^*$-algebra of some matrix algebra $M_{n}(C_{0}(X))$ over a locally compact Hausdorff space $X$, and approximately subhomogeneous (ASH) if it is an inductive limit of a sequence of subhomogeneous $C^*$-algebras; ASH algebras are among the building blocks of the Elliott classification program. Question 4.3 asks whether the invariant pair (primitive ideal space, K-theory) of $C^*(\Gamma_{p,q})$ can be realized by a unital ASH algebra inside the faithful-or-finite-dimensional class, so that the trace simplex, the object governing the Furstenberg tracial characterization, could be computed with ASH techniques as in Rordam's just-infinite realization theorem.

Problem Statement

The problem is the three-part agenda posed as Questions 4.1-4.3 of Bruce-Scarparo, which in the source arise naturally in light of their results: the weak analogues of $C^*$-simplicity and of the unique trace property for icc groups with only countably many subgroups, the extension of just-infinite structure theory to the class of C*-algebras for which any irreducible representation is either faithful or has finite-dimensional image (a class containing the crossed products of faithful almost minimal actions of countable abelian groups), and an approximately subhomogeneous model for the test case $C^*(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2)$:

(1) Does there exist an icc group with only countably many subgroups that is weakly $C^*$-simple (every nonzero ideal of $C^{*}_{r}(\Gamma)$ intersects $\mathbb{C}\Gamma$ nontrivially) but does not have the weak unique trace property (the canonical trace is not the only faithful extreme tracial state on $C^{*}_{r}(\Gamma)$)? In other words, can the lamplighter-type failure of the implication 'weakly $C^*$-simple implies weak unique trace' persist in the countable-subgroup regime, the regime in which $C^*$-simplicity and the unique trace property are equivalent?

(2) Can (some of) the theory of just-infinite C*-algebras, namely the structure theory of Grigorchuk-Musat-Rordam, be extended to the class of C*-algebras for which any irreducible representation is either faithful or has finite-dimensional image?

(3) Let $p,q\ge 2$ be multiplicatively independent integers. Is there a unital approximately subhomogeneous (ASH) C*-algebra $A$ with the property that any of its irreducible representations are either faithful or finite-dimensional, and such that $A$ has the same K-theory and the same primitive ideal space as $C^*(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2)$? If yes, can one compute the trace simplex of $A$?

The three questions are linked as one program: (1) probes at the group level whether the dichotomy underlying the tracial characterization of Furstenberg's conjecture is robust in the countable-subgroup regime (a negative answer would imply Furstenberg's $\times p,\times q$ conjecture); (2) is the structural program for the class containing both just-infinite algebras and these crossed products; and (3) is its concrete realization problem for the Furstenberg algebra, whose trace-simplex component is the object directly governing the tracial characterization.

References

1. Chris Bruce and Eduardo Scarparo, A tracial characterization of Furstenberg's $\times p,\times q$ conjecture, Canadian Mathematical Bulletin 67 (2024), no. 1, 244-256. DOI: 10.4153/S0008439523000693, https://doi.org/10.4153/S0008439523000693 (arXiv:2304.04456)
2. Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Mathematical Systems Theory 1 (1967), 1-49. DOI: 10.1007/BF01692494, https://doi.org/10.1007/BF01692494
3. Emmanuel Breuillard, Mehrdad Kalantar, Matthew Kennedy and Narutaka Ozawa, C*-simplicity and the unique trace property for discrete groups, Publications Mathématiques de l'IHÉS 126 (2017), 35-71. DOI: 10.1007/s10240-017-0091-2, https://doi.org/10.1007/s10240-017-0091-2 (arXiv:1410.2518)
4. Eduardo Scarparo, A torsion-free algebraically C*-unique group, Rocky Mountain Journal of Mathematics 50 (2020), no. 5, 1813-1815. DOI: 10.1216/rmj.2020.50.1813, https://doi.org/10.1216/rmj.2020.50.1813
5. Oren Becker, Alexander Lubotzky and Andreas Thom, Stability and invariant random subgroups, Duke Mathematical Journal 168 (2019), no. 12, 2207-2234. DOI: 10.1215/00127094-2019-0024, https://doi.org/10.1215/00127094-2019-0024
6. Rostislav Grigorchuk, Magdalena Musat and Mikael Rørdam, Just-infinite C*-algebras, Commentarii Mathematici Helvetici 93 (2018), no. 1, 157-201. DOI: 10.4171/CMH/432, https://doi.org/10.4171/CMH/432
7. Mikael Rørdam, Just-infinite C*-algebras and their invariants, International Mathematics Research Notices 2019 (2019), no. 12, 3621-3645. DOI: 10.1093/imrn/rnx227, https://doi.org/10.1093/imrn/rnx227
8. Huichi Huang and Jianchao Wu, Ergodic invariant states and irreducible representations of crossed product C*-algebras, Journal of Operator Theory 78 (2017), no. 1, 159-172. DOI: 10.7900/jot.2016jun11.2141, https://doi.org/10.7900/jot.2016jun11.2141 (arXiv:1505.06633)
9. Robert Valente and Dilian Yang, Semigroups of self-similar actions and higher rank Baumslag-Solitar semigroups, arXiv:2405.07062 (2024), to appear in Proceedings of the Royal Society of Edinburgh Section A. https://arxiv.org/abs/2405.07062
10. Peter Burton and Jane Panangaden, Formulations of Furstenberg's $\times 2\ \times 3$ conjecture in complex analysis and operator algebras, Annals of Functional Analysis 17 (2026), no. 3, Paper No. 40. DOI: 10.1007/s43034-026-00512-1, https://doi.org/10.1007/s43034-026-00512-1 (arXiv:2410.22701)
11. Kevin Brix, Chris Bruce, Kang Li and Eduardo Scarparo, Maximal ideals of reduced group C*-algebras and Thompson's groups, Transactions of the American Mathematical Society 379 (2026), no. 6, 4307-4321. DOI: 10.1090/tran/9627, https://doi.org/10.1090/tran/9627 (arXiv:2403.13645)
12. Alistair Miller and Eduardo Scarparo, Invariant measures and traces on groupoid C*-algebras, arXiv:2603.04020 (2026). https://arxiv.org/abs/2603.04020