Almost elementary groupoid models for $C^{*}$-algebras

Recently, spurred by the rapid progress in the theory of classification and structure of $C^{*}$-algebras, there has been a growing recognition of the deep connections between the study of $C^{*}$-algebras and that of topological groupoids. A large part of these recent developments build on top of the fulfillment of the main goal in the Elliott classification program, that is, a classification theorem which states that there is a class of so-called classifiable $C^{*}$-algebras wherein any two members are $*$-isomorphic if and only if they have the same Elliott invariant (see, e.g., [EGLN15], [GLN15], [TWW17], [CET⁺21] and [CGS⁺23]). Here the Elliott invariant comprises of the (ordered) K-theory of the $C^{*}$-algebras as well as the tracial information associated with them.

Since the discovery of non-classifiable simple separable nuclear $C^{*}$-algebras , a significant aspect of the Elliott classification program is concerned with characterizing the class of classifiable $C^{*}$-algebras. These efforts result in a rich theory of regularity properties for simple separable nuclear $C^{*}$-algebras, with the centerpiece being the Toms-Winter conjecture. Using the successful verification of the first half of this conjecture (between finite nuclear dimension and $\mathcal{Z}$-stability) [CET⁺21], one can characterize a classifiable $C^{*}$-algebra $A$ as:

• a simple separable nuclear $C^{*}$-algebra that is

  1. (1)

     $\mathcal{Z}$-stable, i.e., $A$ tensorially absorbs the Jiang-Su algebra $\mathcal{Z}$, and

  2. (2)

     in the UCT class, i.e., $A$ satisfies the hypothesis of the universal coefficient theorem for $KK$-theory.

While it has long been known that the condition of $\mathcal{Z}$-stability is necessary since taking tensor products with $\mathcal{Z}$ does not change the Elliott invariant [JS99], the necessity of the UCT condition remains largely a mystery and is known as (or more precisely, equivalent to) the UCT problem (see, e.g., [BBWW22]).

However, combining some recent results regarding Cartan subalgebras, one can circumvent the UCT problem and equivalently characterize a classifiable $C^{*}$-algebra $A$ as:

• a twisted groupoid $C^{*}$-algebra of an amenable minimal effective second countable locally compact Hausdorff étale groupoid so that $A$ is $\mathcal{Z}$-stable.

Indeed, on the one hand, Li [Li20] showed that every classifiable $C^{*}$-algebra has a Cartan subalgebra, which, by the work of Renault [Ren80], means that it is a twisted groupoid $C^{*}$-algebra of an effective locally compact Hausdorff étale groupoid (see also [CFaH20], [DPS18], [DPS24], and [Spi07] for various other constructions of groupoid models); on the other hand, by extending the method of Tu [Tu99], Barlak and Li [BL17] showed that twisted groupoid $C^{*}$-algebras of amenable locally compact Hausdorff étale groupoids are in the UCT class. Finally, by a number of more classical results linking properties of C*-algebras and those of groupoids [ADR00, BCFS14], we know that a twisted groupoid $C^{*}$-algebra is nuclear, separable, and simple if and only if the groupoid is amenable, second countable, minimal, and effective.

Once we take this groupoid perspective towards classifiability, a natural question arises:

Research in this direction, particularly in the special case of group actions on compact metrizable spaces, has been a staple topic in the Elliott classification program long before the aforementioned recent progress. While earlier positive results often place strong assumptions on the acting group, the underlying space, and the action itself, recent results have reached a much greater generality. Inspired by the construction of non-classifiable crossed product $C^{*}$-algebras and results on a conjecture of Toms relating regularity of crossed product $C^{*}$-algebras and mean dimension of dynamical systems, Kerr [Ker20] adapted Matui’s notion [Mat12] of almost finiteness for ample groupoids to the setting of actions of amenable groups on compact metric spaces and showed that almost finite free minimal actions give rise to $\mathcal{Z}$-stable (simple, separable, nuclear, and stably finite) crossed products. Almost finiteness was later related to the notion of small boundary property in topological dynamics and applied to produce many positive results on the classifiability of crossed product $C^{*}$-algebras. In the setting of infinite $C^{*}$-algebras, the notion of pure infiniteness for ample groupoids, first introduced by Matui in [Mat15], was extended to general étale groupoids by the first author in [Ma22], where it was demonstrated that a purely infinite groupoid (see Definition 4.1 below) gives rise to a groupoid $C^{*}$-algebra that is (strongly) purely infinite and thus $\mathcal{Z}$-stable under some additional natural assumptions.

More recently, generalizing the stably finite setting above from group actions to groupoids and unifying it with the infinite setting, the authors introduced in [MW20] a new approximation property called almost elementariness for étale groupoids with compact unit spaces (see 2.4 below), and showed it guarantees (tracial) $\mathcal{Z}$-stability of groupoid $C^{*}$-algebras. This generalizes the results above. In our framework, the divide between finite and infinite $C^{*}$-algebras is governed by what we call fiberwise amenability for étale groupoids, which we introduced along with a coarse geometric framework on the groupoids. In the special case of a group action, fiberwise amenability corresponds to the amenability of the acting group.

Hence, combining the results in [MW20] and the aforementioned classical facts, we know that if $\mathcal{G}$ is an amenable minimal effective second countable locally compact Hausdorff étale groupoid that has a compact unit space and is also almost elementary, then its groupoid $C^{*}$-algebra is classifiable. Based on the known results and examples so far, it is tempting to ask the following partial converse to this result:

We first remark that one cannot expect all groupoid models of a classifiable $C^{*}$-algebra to be almost elementary. Indeed, Joseph constructed in [Jos24] minimal topologically free non-almost elementary dynamical systems (to be more precise, it was shown in [Jos24] that those dynamical systems are not almost finite; on the other hand, it was proved in [MW20] that the almost finiteness is equivalent to the almost elementariness for actions by amenable groups). It was then proved in [HW23] and [GGG⁺24] that some of the non-almost elementary dynamical systems constructed this way still yield classifiable crossed product $C^{*}$-algebras.

In this paper, we provide a positive answer to Question II for a large class of Elliott invariants. Denote by $\mathcal{C}$ the class of $C^{*}$-algebras $A$ whose Elliott invariant $\operatorname{Ell}(A)$ coincides with that of an $C^{*}$-algebra in either of the following classes.

1. (1)

   unital AF-algebras;

2. (2)

   unital $C^{*}$-algebras with $\operatorname{Ell}(A)\cong((G_{0},1_{G_{0}}),G_{1}),$ where $G_{0},G_{1}$ are countable abelian groups.

3. (3)

   unital $C^{*}$-algebras $A$ whose Elliott’s invariant

   $$\operatorname{Ell}(A)\cong(({\mathbb{Z}}\oplus G_{0},{\mathbb{Z}}_{>0}\oplus G_{0},[1]=(k,0)),G_{1},\Delta,r),$$

   in which $G_{0},G_{1}$ are countable abelian groups, $k\in{\mathbb{Z}}_{>0}$, $\Delta$ is a finite-dimensional Choquet simplex and $r:\Delta\to S({\mathbb{Z}}\oplus G_{0})$ is defined by $\tau\mapsto((n,g)\mapsto n/k)$.

Now, we state our main theorem as a combination of 3.15 and 4.12.

Since the above class covers all the strongly self-absorbing $C^{*}$-algebras in the UCT class, namely $\mathcal{Z}$, $\mathcal{O}_{2}$, $\mathcal{O}_{\infty}$, $\operatorname{UHF}$ and $\operatorname{UHF}\otimes\mathcal{O}_{\infty}$, where $\operatorname{UHF}$ denotes an arbitrary UHF-algebra of infinite type, we obtain the following corollary.

These results suggests that almost elementariness may, to a certain extent, be considered as a groupoid analog of $\mathcal{Z}$-stability. The method employed to establish A involves investigating groupoids in [DPS18], [DPS24], [Spi07], and [CFaH20], and proving that these groupoids satisfy the property of being almost elementary as defined in Definition 2.4. Notably, the groupoids in [DPS18] and [DPS24] can be expressed as certain ${\mathbb{Z}}$-partial dynamical systems. By utilizing the concept of almost finiteness for global actions, as discussed in [Ker20], the almost elementariness of the partial transformation groupoids associated with these partial actions is established.

In the case of purely infinite $C^{*}$-algebras, even stronger conclusions can be drawn, extending beyond the unital cases. Specifically, it is demonstrated that all groupoids constructed from a combination of $k$-graphs in [Spi07] and [CFaH20] are purely infinite in Definition 4.1. Furthermore, the equivalence between pure infiniteness and almost elementariness is established in cases where the unit space is compact and there are no invariant measures, as shown in Proposition 4.2. It follows that all Kirchberg algebras satisfying the UCT have a purely infinite groupoid model, as stated in 4.12.

We conclude the introduction with an exploratory discussion of the following related question:

It is indeed known that UHF algebras, or more generally, AF algebras, can be represented as crossed products of minimal free actions of locally finite groups on the Cantor set, as mentioned in [GKPT18, Example 8.1.24]. Moreover, in [Ma22, Proposition 3.24], it was shown that $\mathcal{O}_{2}$ can be expressed as a minimal purely infinite dynamical system of $\mathbb{Z}_{2}*\mathbb{Z}_{3}$. But in general, this question is still widely open.

A prominent unresolved case regarding this question is the Jiang-Su algebra $\mathcal{Z}$. It is well-known that $\mathcal{Z}$ cannot be realized as a $\mathbb{Z}$-crossed product using Pimsner-Voiculescu exact sequences. In this paper, using the bijectivity of the Baum-Connes assembly for amenable groups and $KK$-theory with real coefficients, we demonstrate that there is an obstruction on the acting group for $\mathcal{Z}$ to be written in the form of a crossed product. See Theorem 3.16 below.

Since these conditions on $\Gamma$ rules out almost all naturally occurring examples, this result supports the view that in the study of classifiable $C^{*}$-algebras, it is necessary to work with groupoids rather than just group actions.

In this section, we recall some basic backgrounds on locally compact Hausdorff étale groupoids and their $C^{*}$-algebras. We refer to [Ren80] and [Sim17] as standard references for these topics. Using the terminology in [Sim17], we denote by $\mathcal{G}^{(0)}$ the unit space of $\mathcal{G}$. We write source and range maps $s,r:\mathcal{G}\rightarrow\mathcal{G}^{(0)}$, respectively and they are defined by $s(\gamma)=\gamma^{-1}\gamma$ and $r(\gamma)=\gamma\gamma^{-1}$ for $\gamma\in\mathcal{G}$. When a groupoid $\mathcal{G}$ is endowed with a locally compact Hausdorff topology under which the product and inverse maps are continuous, the groupoid $\mathcal{G}$ is called a locally compact Hausdorff groupoid. A locally compact Hausdorff groupoid $\mathcal{G}$ is called étale if the range map $r$ is a local homeomorphism from $\mathcal{G}$ to itself, which means for any $\gamma\in\mathcal{G}$ there is an open neighborhood $U$ of $\gamma$ such that $r(U)$ is open and $r|_{U}$ is a homeomorphism. A set $B$ is called an $s$-section (respectively, $r$-section) if there is an open set $U$ in $\mathcal{G}$ such that $B\subset U$ and the restriction of the source map $s|_{U}:U\rightarrow s(U)$ (respectively, the range map $r|_{U}:U\rightarrow r(U)$) on $U$ is a homeomorphism onto an open subset of $\mathcal{G}^{(0)}$. The set $B$ is called a bisection if it is both an $s$-section and an $r$-section at the same time. It is not hard to see a locally compact Hausdorff groupoid is étale if and only if its topology has a basis consisting of open bisections. We say a locally compact Hausdorff étale groupoid $\mathcal{G}$ is ample if its topology has a basis consisting of compact open bisections.

For any set $D\subset\mathcal{G}^{(0)}$, Denote by

Note that $\mathcal{G}|_{D}\coloneqq\mathcal{G}_{D}^{D}$ is a subgroupoid of $\mathcal{G}$ with the unit space $D$. For the singleton case $D=\{u\}$, we write $\mathcal{G}_{u}$, $\mathcal{G}^{u}$ and $\mathcal{G}_{u}^{u}$ instead for simplicity. In this situation, we call $\mathcal{G}_{u}$ a source fiber and $\mathcal{G}^{u}$ a range fiber. In addition, each $\mathcal{G}_{u}^{u}$ is a group, which is called the isotropy at $u$. We also denote by

the isotropy of the groupoid $\mathcal{G}$. We say a groupoid $\mathcal{G}$ is principal if $\operatorname{Iso}(\mathcal{G})=\mathcal{G}^{(0)}$. A groupoid $\mathcal{G}$ is called topologically principal if the set $\{u\in\mathcal{G}^{(0)}:\mathcal{G}_{u}^{u}=\{u\}\}$ is dense in $\mathcal{G}^{(0)}$. The groupoid $\mathcal{G}$ is also said to be effective if $\operatorname{Iso}(\mathcal{G})^{o}=\mathcal{G}^{(0)}$. Recall that effectiveness is equivalent to topological principalness if $\mathcal{G}$ is second countable (see [Sim17, Lemma 4.2.3]). A subset $D$ in $\mathcal{G}^{(0)}$ is called $\mathcal{G}$-invariant if $r(\mathcal{G}D)=D$, which is equivalent to the condition $\mathcal{G}^{D}=\mathcal{G}_{D}$. A groupoid $\mathcal{G}$ is called minimal if there are no proper non-trivial closed $\mathcal{G}$-invariant subsets in $\mathcal{G}^{(0)}$.

The following definition of multisections was introduced by Nekrashevych in [Nek19, Definition 3.1], which serves as the concrete interpretation of towers introduced in [MW20, Section 6] for groupoids.

We may also explicitly write $\mathcal{C}=\{C_{i,j}^{l}:i,j\in F_{l},l\in I\}$, which satisfies the following

1. (i)

   $\{C_{i,j}^{l}:i,j\in F_{l}\}$ is a multisection;

2. (ii)

   $C_{i,j}^{l}C_{i^{\prime},j^{\prime}}^{l^{\prime}}=\emptyset$ if $l\neq l^{\prime}$.

Let $\mathcal{C}=\{C_{i,j}^{l}:i,j\in F_{l},l\in I\}$ be a castle. Any certain level in a multisection in $\mathcal{C}$ is usually referred to as a $\mathcal{C}$-level. Analogously, any ladder in a multisection in $\mathcal{C}$ is usually referred as a $\mathcal{C}$-ladder. Let $\mathcal{C}$ and $\mathcal{D}$ be two castles in a groupoid $\mathcal{G}$ described above and $K$ a set in $\mathcal{G}$. We say $\mathcal{C}$ is $K$-extendable to $\mathcal{D}$ if $K\cdot\bigcup\mathcal{C}\subset\bigcup\mathcal{D}$.

We then recall the groupoid subequivalence relation.

We then recall the definition of almost elementariness of groupoids with compact unit spaces.

Finally, throughout the paper, we write $B\sqcup C$ to indicate that the union of sets $B$ and $C$ is a disjoint union. In addition, we denote by $\bigsqcup_{i\in I}B_{i}$ for the disjoint union of the family $\{B_{i}:i\in I\}$. From now on, we only consider locally compact, Hausdorff, $\sigma$-compact, étale, topological groupoids. The word “groupoid” below is reserved for this kind of groupoids. We remark that we do NOT always assume the unit space $\mathcal{G}^{(0)}$ is compact.

In this section, we study transformation groupoids of partial dynamical systems, which form groupoid models for stably finite $C^{*}$-algebras. We first recall the definition of partial dynamical systems and refer to [Exe17] as a standard reference for this topic.

In the topological dynamical setting throughout this section, we additionally assume the following throughout the section.

1. (i)

   The acting group $\Gamma$ is countable and discrete.

2. (ii)

   The underlying space $X$ is a compact metrizable space.

3. (iii)

   Each $D_{g}$ is an open set in $X$.

4. (iv)

   Each $\beta(g):D_{g^{-1}}\to D_{g}$ is a homeomorphism.

One can define groupoids from partial dynamical systems. Similar to the global action case, we write $gx$ for $\beta(g)(x)$ for simplicity if the context is clear.

The following lemma is straightforward but useful.

We then describe towers and castles in the setting of partial dynamical systems.

In the following sense, the towers and castles for a partial dynamical system coincide with the same notions in the transformation groupoid (Definition 2.1).

A natural way to define a partial dynamical system is to restrict a (global) group action on a certain family of open sets (see, e.g., [Exe17, section 3]). Let $\alpha:\Gamma\curvearrowright X$ be a global action of a countable amenable discrete group $\Gamma$ on a compact Hausdorff space $X$. We now focus on minimal partial dynamical systems $\beta:\Gamma\curvearrowright X$ obtained by restricting a global minimal action $\alpha$ to certain open sets $D_{g^{-1}}=X\setminus\bigcup_{j\in I_{g}}Y_{j,g}$, where $J_{g}$ is a finite index set (with $J_{e}=\emptyset$) and each $Y_{j,g}$ is compact set in $X$ intersecting every $\alpha$-orbit at most once. We first record some basic properties of such sets.

We also need the following lemma.

Using the almost finiteness introduced in [Ker20, Definition 8.2] in the setting of global dynamical systems, we have the following.

The following is a consequence of the proposition by using [MW20, Theorem 15.10].

Now, we turn to the case $\alpha:{\mathbb{Z}}\curvearrowright X$ be a global minimal (free) action induced by a homeomorphism $\varphi:X\to X$. For each integer $n$, write

A simple but useful observation for any integers and $n,m\in{\mathbb{Z}}$ are following.

1. (1)

   $I_{-n}+n=I_{n}$ and

2. (2)

   $I_{n+m}-n\subset I_{-n}\cup I_{m}$.

Let $Y$ be a closed set in $X$ meeting each $\alpha$-orbit at most once. For any $g\in{\mathbb{Z}}$, define $D_{g}=X\setminus\bigsqcup_{n\in I_{g}}\varphi^{n}(Y)$. Then define partial action $\beta$ by restricting $\alpha$ on all of these $D_{g^{-1}}$. One may verify directly that the partial system $\beta$ is well-defined. Since $X$ is compact, the one-sided half $\alpha$-orbits $\{\varphi^{n}(x):n\geq 1\}$ and $\{\varphi^{n}(x):n\leq 0\}$ are both dense in $X$ for any $x\in X$, from which one concludes that the partial action $\beta$ is minimal as well. In addition, $\beta$ is free because $\alpha$ is free. Then the transformation groupoid $\mathcal{G}=X\rtimes_{\beta}{\mathbb{Z}}$ is minimal and principal. Actually, the transformation groupoid $\mathcal{G}=X\rtimes_{\beta}{\mathbb{Z}}$ for $\beta$ is nothing but the equivalence relation

which is exactly the so-called orbit-breaking equivalence relation considered in [DPS24]. For this kind of groupoids, we have the following result as a corollary of Proposition 3.10 and Theorem 3.11.

It was proved in [DPS18] and [DPS24] that the orbit-breaking equivalence relation $R_{Y}$ defined above provides groupoid model for a large class of classifiable $C^{*}$-algebras including Jiang-Su algebra $\mathcal{Z}$.

Therefore, we have arrived at the main result in this section by using Corollary 3.12, Theorem 3.13 and Remark 3.14.