NUCLEAR DIMENSION OF SUBHOMOGENEOUS TWISTED GROUPOID C*-ALGEBRAS AND DYNAMIC ASYMPTOTIC DIMENSION

. In this paper, we characterise subhomogeneity for twisted ´etale groupoid C ∗ -algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted ´etale groupoid C ∗ - algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the inﬁnite dihedral group D ∞ on an inﬁnite compact Hausdorﬀ space X is always one. So if we further assume that X is second-countable and has ﬁnite covering dimension, then C ( X ) ⋊ r D ∞ has ﬁnite nuclear dimension and is classiﬁable by its Elliott invariant.


Introduction
The nuclear dimension for C * -algebras is a non-commutative version of Lebesgue covering dimension introduced by Winter and Zacharias [42].It played a crucial role in the Elliott classification programme for simple nuclear C * -algebras.More precisely, all non-elementary separable simple C * -algebras with finite nuclear dimension satisfying the Universal Coefficient Theorem are classifiable by their Elliott invariant (see [37,15,16,13]).In the sequel, we refer to this class as "classifiable C * -algebras".
In this article, we focus on the nuclear dimension of C * -algebras arising from twisted étale groupoids.This class of C * -algebras is on the one hand large enough to cover many examples of interest.In fact, all classifiable C * -algebras admit a twisted étale groupoid model [26].More generally, every C * -algebra admitting a Cartan subalgebra has a twisted étale groupoid model [33,31].On the other hand, the nuclear dimension of twisted étale groupoid C * -algebras is naturally related to the dimensions of the underlying groupoids.More specifically, Guentner, Willett, and Yu introduced in [18] the concept of the dynamic asymptotic dimension of an étale groupoid G (denoted dad(G)), and proved that the nuclear dimension of the reduced groupoid C * -algebra C * r (G) is bounded above by a number depending on the dynamic asymptotic dimension of G and the covering dimension of its unit space G 0 provided that G is principal (i.e., when the isotropy groups G x x = {g ∈ G | s(g) = x = r(g)} are trivial for all x ∈ G 0 ).Recently, this result has been generalized to twisted étale groupoids [10] still under the principality assumption.
The strategy of the proofs for the main results in [18,10] is as follows: when G has finite dynamic asymptotic dimension d and Σ is any twist over G, then C * r (G; Σ) can be locally approximated by d + 1 sub-C * -algebras associated with certain open precompact subgroupoids H i of G, 0 ≤ i ≤ d.This reduces the problem of bounding the nuclear dimension of C * r (G; Σ) to bounding the nuclear dimension of each of these d + 1 sub-C * -algebras associated with the groupoids H i .As each H i is an open precompact subgroupoid of G, it turns out that these sub-C * -algebras are all subhomogeneous 1 .Luckily, the nuclear dimension of subhomogeneous C * -algebras has already been analysed and computed by Winter in [41].
In our main result, we characterise subhomogeneity for twisted étale groupoid C * -algebras and then use Winter's result in [41] to give an estimate on their nuclear dimension (see Proposition 2.5, Theorem 2.12 and Proposition 2.14): Theorem A. Let G be a locally compact, second-countable, Hausdorff étale groupoid and let Σ be a twist over G.For each x ∈ G 0 let σ x : G x x × G x x → T denote the 2-cocycle associated with the restriction of the twist Σ to x.Then C * (G; Σ) is subhomogeneous if and only if In this case, the nuclear dimension of C * (G; Σ) can be bounded above as follows: where asdim(G x x ) denotes Gromov's asymptotic dimension of the countable group G x x .
Replacing [10,Proposition 4.3] about the nuclear dimension of subhomogenous principal twisted étale groupoid C * -algebras in the argument outlined above with our Theorem A, allows us to drop the principality assumption from the main results of [18,10] (see Theorem 3.2): Theorem B. Let G be a second-countable, locally compact, Hausdorff étale groupoid and let Σ be a twist over G. Then [18,Theorem 4.7] with Theorem B, we obtain the following corollary: Corollary C. Let Γ be a finitely generated virtually nilpotent group acting on a compact metric space of finite covering dimension.If all stabilizer subgroups are finite, then both the dynamic asymptotic dimension and the nuclear dimension are finite 2 .
Note that having finite dynamic asymptotic dimension forces all the isotropy groups G x x to be locally finite.Nevertheless, there are many interesting examples with finite isotropy groups such as minimal actions of the infinite dihedral group (see Theorem 3.3 and Example 3.4): 1 Recall that a C * -algebra A is subhomogeneous if there is a finite upper bound on the dimension of the irreducible representations of A.
2 If we remove the condition "all stabilizer subgroups are finite", then the dynamic asymptotic dimension is often infinite but the nuclear dimension is always finite by [20,Corollary 10.6].
Theorem D. The dynamic asymptotic dimension of any minimal action D ∞ X of the infinite dihedral group D ∞ on an infinite compact Hausdorff space X is one.
If we further assume that X is second-countable and has finite covering dimension, then C(X) ⋊ D ∞ is classifiable by its Elliott invariant and has nuclear dimension at most one.
Finally, as an application we obtain the following corollary (see Corollary 3.5): Corollary E. Let X be an infinite compact Hausdorff space.If Γ is a virtually cyclic group acting minimally on X, then the dynamic asymptotic dimension of Γ X is one and The article is organised as follows: In section 2 we present our characterisation of subhomogeneity for twisted étale groupoid C * -algebras and prove Theorem A. In Section 3 we discuss several applications, including the proofs of Theorem B, Theorem D and Corollary E.

Subhomogeneous twisted groupoid C*-algebras
Throughout the article G will denote an étale, locally compact Hausdorff groupoid.We will denote its space of units by G 0 and the source and range maps by s, r : G → G 0 , respectively.We will use the standard notation x for the range fibre, the source fibre, and the isotropy group at x ∈ G 0 , respectively.Recall, that a twist over G is a central groupoid extension j is a continuous and open surjection, and (3) the extension is central in the sense that i(r(σ), z)σ = σi(s(σ), z).
To construct the twisted groupoid C * -algebra one forms the associated line bundle L Σ = C × Σ/∼ → G and considers the set Γ c (G, L Σ ) of continuous, compactly supported sections of this bundle.Equipped with a suitably defined involution and convolution it becomes a * -algebra that can be completed with respect to suitable norms to form the full and reduced twisted groupoid C * -algebras C * (G; Σ) and C * r (G; Σ), respectively (see for example [33] for the details).
For every x ∈ G 0 the twist Σ over G restricts to a twist T → Σ x x → G x x over the discrete group G x x .Such a twist gives rise to a 2-cocycle σ x on G x x .In fact a bit more is true, so let us review the construction.
For each x ∈ G 0 let θ x : G x → Σ x be a section of the restriction of j to Σ x such that θ x (x) = x.Then j(θ s(h) (gh) −1 θ s(g) (g)θ s(h) (h)) = (gh) −1 gh = s(h) for all (g, h) ∈ G (2) .It follows that θ s(h) (gh) −1 θ s(g) (g)θ s(h) (h) actually lives in the image of T ∼ = {s(h)} × T inside Σ s(h) .So identifying T with its image in Σ x allows us to define a maps One can easily check that σ x satisfies the cocycle identity x it is also routine to verify that σ x is normalised in the sense that σ x (g, x) = σ x (x, h) = 1 for all g ∈ G x and h ∈ G x .
2.1.A characterisation of subhomogeneity.Recall, that a σ x -representation of G x x is a map π : G x x → B(H π ) such that σ x (g, h)π(gh) = π(g)π(h).The set of (equivalence classes of) irreducible σ x -representations will be denoted by (G x  x , σ x ) Central to our analysis is the procedure of inducing σ x -representations of an isotropy group G x x to a representation of C * (G; Σ).There are two approaches to this which we shall now describe.
The first of these is more concrete and inspired by Mackey's classical theory of induced representations for groups (compare [7,Chapter 8], or [9, Section 3] for a related construction in the groupoid case).Since none of the references describe exactly what we need, we spell out the constructions but omit the proofs as they are all classical.Consider the complex vector space of all functions ξ : G x → H π such that ξ(gh) = σ x (g, h)π * h ξ(g) for all g ∈ G x and h ∈ G x x and such that There is an inner product on this vector space given by ξ, η := Note that this is well-defined since . The completion with respect to this inner product is a Hilbert space and will be denoted Ind H π in the sequel.On this Hilbert space one can define the induced representation of π One of the advantages of this concrete approach is that it allows for a simple proof of the following result: x → B(H π ) be a σ x -representation.Then the Hilbert space Ind H π for the induced representation is finite-dimensional if and only if H π is finite dimensional and G x /G x x is finite.In this case we have Let (ξ i ) i∈I be an orthonormal basis for H π and let (g j ) j∈J be a complete system of representatives of the orbit space G x /G x x .For each i ∈ I and j ∈ J define a function ϕ i,j : G x → H π by setting Then there exists a unique j ∈ J such that g = g j h ′ , so we get gh = g j h ′ h.Hence we can use the cocycle identity to compute g) It follows, that ϕ i,j ∈ Ind H π , and in fact, (ϕ i,j ) i∈I,j∈J is easily seen to be an orthonormal family in Ind H π .
Conversely, if ϕ : G x → H π is any function as in the definition of Ind H π , write ϕ(g j ) = j∈J λ i,j ξ i .Then ϕ = i∈I j∈J λ i,j ϕ i,j .It follows that (ϕ i,j ) i,j is an orthonormal basis, and the claims in the statement follow.
An alternative route to obtain the induced representation is via Hilbert modules.To describe it, recall the definition of the maximal twisted group C * -algebra C * (G x x ; σ x ): consider the complex vector space ℓ 1 (G x x ; σ x ) of all summable complex functions on G x x equipped with the twisted convolution and involution given by .
x .The following description of induction via Hilbert modules is taken from [36, Section 4.1] applied to the special case of line bundles and the subgroupoid G x x .Consider the complex vector space C c (G x ) and define a C * (G x x ; σ x )-valued inner product for ξ, η ∈ C c (G x ) by ξ, η := ξ * * σx η, where the latter denotes the twisted convolution.Completing C c (G x ) with respect to this inner product one obtains a Hilbert C * (G x x ; σ x )-module X on which C * (G; Σ) acts from the left by adjointable operators.To induce a σ x -representation π of G x x one first considers the integrated form π : C * (G x x ; σ x ) → B(H π ) of π and then forms the balanced tensor product X ⊗ π H π .The latter is then a Hilbert space on which one can define the induced representation The following result is well-known in the untwisted setting and it is routine to extend it to the twisted one.It shows that the two approaches to induction explained above yield the same result up to unitary equivalence: Then there exists a unitary operator U : Induction via the Hilbert module X is the preferred method to prove the following result, which is (a special case of) the main result of [23,Theorem 6.3] (we only use the case of line bundles).
Theorem 2.3.Let x ∈ G 0 and π ∈ (G x x ; σ x ) be an irreducible representation.Then Ind X π is an irreducible representation of C * (G; Σ).Thus, twisted irreducible representations of the isotropy groups give rise to irreducible representations of C * (G; Σ).Before we turn to our characterisation of subhomogeneity, we prove some sufficient conditions for a twisted groupoid C * -algebra to be GCR or CCR. 3 In the untwisted setting Clark gave a nice characterisation for C * (G) to be GCR (resp.CCR) in [8, Theorems 1.3 and 1.4] and this was later improved by van Wyk in [39,Theorem 4.3] and [40,Theorem 3.6].A full analogue of this result is not available in the literature for the twisted case.There are partial results in [9, Proposition 3.3] that deal with the case of twists over a principal groupoid, but our main interest is really beyond the principal case.However, for twists over étale groupoids we can exploit an algebraic characterisation of GCR/CCR twisted group C * -algebras to at least obtain sufficient conditions for C * (G; Σ) to be GCR or CCR as follows: Proposition 2.4.Let G be a second-countable, locally compact, Hausdorff étale groupoid and Σ be a twist over G. Then the following hold: ( Proof.We only prove the first item as the second one can be proven in exactly the same way by replacing GCR by CCR and T 0 by T 1 throughout the proof.By [21, Theorem 1.1] the twisted group (G x x , σ x ) is GCR, or equivalently, type I if and only if there exists an abelian subgroup Λ x G x x of finite index on which σ x is symmetric.It is then clear that the 2-cocycle σ n x given by σ n x (g, h) := σ x (g, h) n is also symmetric on Λ x for all n ∈ Z and hence (G x x , σ n x ) is type I for all n ∈ Z by invoking [21, Theorem 1.1] again.It follows that C ) and since quotients of GCR-algebras are GCR, we get that C * (G; Σ) is GCR.
The converse of this result is not clear to us as there is no longer an obvious continuous inclusion G 0 /G → C * (G; Σ) that allows to pull back topological information from the spectrum to the orbit space.
We are now ready for the main result of this section: Proposition 2.5.Let G be a second-countable, locally compact, Hausdorff étale groupoid and Σ be a twist over G. Then C * (G; Σ) is subhomogeneous if and only if the following hold: (1) Then there exists a uniform upper bound N ∈ N for the dimensions of irreducible representations of C * (G; Σ).Let x ∈ G 0 and π be a twisted irreducible representation of G x x .Then Ind G G x x π is an irreducible representation of C * (G; Σ) and Lemma 2.1 implies x π) ≤ N. Since x and π were arbitrary, this implies conditions (1) and (2).
Conversely, we know that for every x ∈ G 0 there is a bijection between the orbit Gx and the quotient space G x /G x x .Using this, condition (1) implies that every orbit is a finite subset of G 0 .Since G 0 is Hausdorff this implies that every orbit is closed.It follows in particular that G 0 /G is T 1 .Moreover, the second condition implies that C * (G x x ; σ x ) is subhomogeneous and so in particular CCR for every x ∈ G 0 .Consequently, C * (G; Σ) is CCR by Proposition 2.4.We need to find an upper bound on the dimensions of irreducible representations of C * (G; Σ).Since the orbit space of G is T 1 we know that every irreducible representation of C * (G; Σ) is is induced from an isotropy group, i.e. it is of the form Ind G G x x π for some irreducible σ xrepresentation π of G x x .Again, it can be interpreted as an irreducible representation on the Hilbert space Ind H π .By (2) we have dim(H π ) < ∞ and hence another application of Lemma 2.1 implies that Ind H π is finite dimensional if and only if the quotient space G x /G x x is finite and in that case dim(Ind . Thus, the result follows. 2.2.Nuclear dimension of subhomogeneous twisted groupoid C * -algebras.Now that we have a satisfactory description of subhomogeneous groupoid C *algebras we want to estimate their nuclear dimension.The main tool to achieve this is the following result of Winter.To use Winter's result effectively, we first need a better understanding of the topology of Prim(C * (G; Σ)).Recall, that the quasi-orbit space Q(G) of G is the quotient of G 0 by the equivalence relation that identifies two points x, y ∈ G 0 if their orbit closures agree, i.e.Gx = Gy.
As a first step we will see that the map p : Prim(C * (G; Σ)) → Q(G) that associates to every kernel of an irreducible representation the closure of the orbit it lives on is continuous.To do this we need to make our colloquial description of p a bit more precise.The key ingredient is the following Lemma: Lemma 2.7.Let G be an étale groupoid and Σ a twist over G. Then there exists a canonical non-degenerate homomorphism Then M ϕ acts as a double centralizer on C c (G; Σ) and hence extends to an element in M(C * (G; Σ)).It is easy to see that this is a * -homomorphism.It is nondegenerate since C 0 (G 0 ) contains an approximate unit for C * (G; Σ).
Proposition 2.9.The restriction of the map Res M to the set of primitive ideals of C * (G; Σ) gives rise to a continuous map onto the quasi-orbit space of G.
Proof.If I is a primitive ideal, then it is the kernel of some irreducible representation ρ of C * (G; Σ).We know that every such representation is induced from a stabiliser, i.e. there exists an x ∈ G 0 and an irreducible σ But the subspace of I(C 0 (G 0 )) of ideals of the form C 0 (G 0 \ Gx) is homeomorphic to the quasi-orbit space.So the composition of Res M (restricted and corestricted to a map Prim(C * (G; Σ)) → {I ∈ I(C 0 (G 0 )) | I = I Gx }) with this homeomorphism is the desired continuous map p.

Now let X
x | = k}, and X k its image in G 0 /G.Lemma 2.10.Let G be an étale groupoid.Then X k and X k are locally compact Hausdorff spaces in the respective relative topology.

Proof. Let us first show that X
x | ≤ k} is closed.To see this we show that G 0 \ X ≤k is open.Let x ∈ G 0 be a point whose orbit has at least k + 1 distinct elements.Then there exist g 1 , . . ., g k ∈ G such that x, g 1 x, . . ., g k x are pairwise distinct.We also let g 0 := x to have a coherent notation.For each 0 ≤ i ≤ k choose an open bisection U i around g i .Using that G 0 is Hausdorff, we can shrink the U i if necessary to assume without loss of generality that the ranges r(U i ) are pairwise disjoint.But then V := k i=0 s(U i ) is an open neighbourhood of x in G 0 .By construction, the orbit of every element in y ∈ V has at least k +1 elements, namely the images of y under the partial homeomorphisms V ⊆ s(U i ) → r(U i ) given by the bisections We have shown that X k is locally closed in G 0 and hence locally compact.
Since X ≤k and X k are both G-invariant subspaces of G 0 the same conclusions easily follow for X≤k and Xk (note that G 0 /G is locally compact since the quotient map is open).
Finally, we show that Xk is Hausdorff in the relative topology.So suppose Gx = Gy.Then in fact all the points in Gx are distinct from all the points in Gy.Let us write Gx = {x 1 , . . ., x k } and Gy = {y 1 , . . ., y k }.Since all these points are distinct we can find open neighbourhoods U i of x i and V i of y i such that the U i and V j are all pairwise disjoint.Let U := k i=1 GU i and V := k i=1 GV i .Then U and V are open, G-invariant, and we have Gx ⊆ U and Gy ⊆ V .Moreover, Next, we restrict our attention to those primitive ideals that arise as kernels of irreducible representations of dimension k ∈ N.
Proposition 2.11.The canonical map sending a representation to the orbit on which it lives is continuous.Moreover, if x ∈ X n for n ≤ k such that n divides k, then we obtain a homeomorphism Proof.Let x ∈ G 0 such that |Gx| = n.Then Gx is a closed G-invariant subset of G 0 and hence we have a canonical quotient map q : C * (G; Σ) → C * (G| Gx ; Σ| Gx ).Let I = ker q.Then a classical result of Kaplansky (see [11,Proposition 3.2.1])provides a homeomorphism where Prim I (C * (G; Σ)) is the set of two-sided primitive ideals in C * (G; Σ) containing I. Let further Z be the Hilbert module implementing the Morita equivalence between C * (G| Gx ; Σ| Gx ) and C * (G x x ; σ x ).The Rieffel correspondence tells us that Z induces a homeomoprhism between the corresponding primitive ideal spaces [32,Corollary 3.33] that we will denote by Ind Z .It follows that the composition ) is a homeomorphism.The inverse is given by q * • Ind Z = Ind X .We note that x ; σ x )) which finishes our proof.We can now collect our findings from this section to prove: Theorem 2.12.Let G be a second-countable étale groupoid and Σ be a twist over G such that (1) sup Then the nuclear dimension of C * (G; Σ) can be estimated as follows: Proof.We may assume that sup x dim nuc (C * (G x x ; σ x )) < ∞ and dim(G 0 ) < ∞ as otherwise there is nothing to show.The assumption in line (1) implies that C * (G; Σ) is subhomogeneous by Proposition 2.5.Hence we are in a position to apply Winter's theorem 2.6 and the remaining task is to estimate the dimensions of the spaces Prim k (C * (G; Σ)) for all k ∈ N. Since every k-dimensional irreducible representation of C * (G; Σ) is supported on some finite orbit, we have a decomposition Prim Since the inverse image of every point under the quotient map X n → X n is finite, and using the definition of these spaces, we have dim( Moreover, the algebra C * (G x x ; σ x ) is separable and subhomogeneous for all x ∈ G 0 .Consequently, another application of Winter's theorem 2.6 implies which concludes the proof of the first inequality.
For the proof of the second inequality, recall that the general assumption that orbits are finite implies that the orbit space G 0 /G is T 1 .Hence C * (G x x ; σ x ) is stably isomorphic to a quotient of C * (G; Σ).The claimed inequality then follows from the known permanence properties of dim nuc in [42].
Remark 2.13.Suppose we are in the situation of Theorem 2.12.If the twist Σ is trivial, or G is principal, we also have the estimate Indeed, in either case inducing the trivial representation of G x x gives rise to a continuous map l : G 0 → Prim(C * (G; Σ)) (see [8,9]).The preimage of the representation Ind G G x x 1 is precisely the orbit Gx, which is finite, and hence zero-dimensional.It follows that for each k ∈ N we have ).As we have seen in the proof of Lemma 2.10, the space X ≤k is a finite union of the (relatively) open sets X n , n ≤ k and hence dim(X ≤k ) ≤ dim nuc (C * (G; Σ)) by the sum theorem for open sets [30, 3.5.10].Each of the sets X ≤k in turn is closed in G 0 and G 0 = k∈N X k , and hence dim(G 0 ) ≤ dim nuc (C * (G; Σ)) by the countable sum theorem [30, 2.2.5].
Observe, that under the assumptions of Theorem 2.12, the twisted C * -algebras of the isotropy groups C * (G x x ; σ x ) are all subhomogeneous.In particular, G x x is a countable, discrete, virtually abelian group and σ x is type I by [25,Theorem 2].As the next proposition demonstrates, we can use this to replace dim nuc (C * (G x x ; σ x )) by the asymptotic dimension of G x x , which completes the proof of Theorem A.
Proposition 2.14.Let Γ be a countable, discrete, virtually abelian group with a normalized 2-cocycle σ with values in T. If σ is type I, then we have Proof.As σ is type I, there is an abelian subgroup Λ ⊆ Γ of finite index such that σ is trivial on Λ (see [25,Theorem 2]).Since the normalizer subgroup N Γ (Λ) contains Λ, there are only finite number of conjugates which is a subgroup of Λ.It is not hard to see that N is an abelian, normal, finite-index subgroup of Γ such that σ is trivial on N .

Applications to nuclear dimension of non-principal twisted groupoid C * -algebras
In this section we apply our results from Section 2 to show that the principality assumptions in [18,Theorem 8.6] and its recent generalisation in the twisted case (see [10,Theorem 4.1]) are redundant.Let us first recall the definition of dynamic asymptotic dimension.The reader can find further information on this notion and many examples in [18,10,3].We will go straight to our main application: Theorem 3.2.Let G be a second-countable, locally compact, Hausdorff étale groupoid and let j : Σ → G be a twist over G. Then Proof.The main idea is to follow the the proof of [10, Theorem 4.1] and replace the invocation of [10,Proposition 4.3] therein by our Theorem 2.12.We may suppose without loss of generality that d := dad(G) < ∞ and N := dim(G 0 ) < ∞ as otherwise there is nothing to show.
It is furthermore sufficient to consider the case where G 0 is compact.Indeed, if G 0 is not compact, we consider the Alexandrov groupoid G of G and the Alexandrov twist Σ over G.It follows from [10, Proposition 3.13, Lemma 2.6 and Lemma 3.8] that dad(G) = dad( G), dim( G0 ) = dim(G 0 ) and C * r ( G; Σ) is the minimal unitization of C * r (G; Σ).Therefore, C * r ( G; Σ) and C * r (G; Σ) have the same nuclear dimension (see [42,Remark 2.11]).
Let us now assume that the unit space G 0 is actually compact.Let F be a finite subset of C c (G; Σ)\{0} and let ǫ > 0. There exists a compact subset K of Σ such that f ∈ F implies suppf ⊆ K. Since both K −1 and j −1 (j(K)) are compact sets (see [10,Lemma 2.2]), we may assume that K = K −1 and that K = j −1 (j(K)).Since G 0 = Σ 0 is compact and open in Σ, we may also assume that G 0 ⊆ K and hence that 1 ∈ F .
Let V ⊆ G be an open and precompact neighborhood of j(K), and let δ be as in [10,   x have at most M elements for all x ∈ H 0 i .By the twisted version of the Peter-Weyl Theorem (see for example [7,Corollary 7.15]), it follows that each H i satisfies the assumptions of Theorem 2.12 and hence that C * (H i ; j −1 (H i )) has nuclear dimension at most the covering dimension of H 0 i , which is bounded by N .As each H i is an amenable groupoid by [17,Lemma A.10], its full and reduced twisted groupoid C * -algebras coincide.We can thus follow the argument in [ The following result provides us with a class of not necessarily free actions to which one can apply Theorem 3.2.
Theorem 3.3.The dynamic asymptotic dimension of any minimal action D ∞ X of the infinite dihedral group D ∞ on an infinite compact Hausdorff space X is one.
If we further assume that X is second-countable and has finite covering dimension, then C(X) ⋊ D ∞ is classifiable by its Elliott invariant and has nuclear dimension at most one.
Proof.By [1, Theorem 2.2 and Definition 3.1], it suffices to show that any minimal action D ∞ X has the marker property.More precisely, we aim to show that for any finite subset F ⊆ D ∞ there exists a non-empty open subset U ⊆ X such that As D ∞ U is a non-empty open invariant subset of X and the action is minimal, (2) holds automatically.To verify (1), we write As every non-trivial subgroup of Z has the form nZ, we obtain Hence, it follows easily that either Stab(x) = {e, s n t} for some n ∈ Z or Stab(x) = {e}.Case I: If Stab(x) = {e}, then gx = g ′ x for all g = g ′ in F .As X is Hausdorff, we may find open sets U g ⊆ X for each g ∈ F such that gx ∈ U g and U g ∩ U g ′ = ∅ for any two distinct g, g ′ ∈ F .Then the non-empty open subset U := ∩ g∈F g −1 U g does the job.
Case II: If Stab(x) = {e, s n t} for some n ∈ Z, there exists k ∈ Z such that Stab(s k x) ∩ (F −1 F \{e}) = ∅.To see this, we write F = {s i , s j t | i ∈ I, j ∈ J} for some finite subsets I, J ⊆ Z.Using the relations t 2 = e and tst −1 = s −1 , we see that In particular g(s k x) = g ′ (s k x) for all g = g ′ ∈ F .Applying the same argument as in Case I to s k x instead of x, we obtain the desired non-empty open subset U .Hence, the action has the marker property.
If we also assume that X is second-countable and has finite covering dimension, then C(X) ⋊ D ∞ has finite nuclear dimension by Theorem 3.2.By [24, Proposition 2.6] and [38], C(X)⋊D ∞ is a simple C * -algebra in the UCT class.Therefore, it is classifiable.By [6, Theorem A and Theorem B] and [37, Theorem A], C(X)⋊D ∞ has decomposition rank (hence also nuclear dimension) at most one.
Example 3.4.We refer the reader to [35] for certain non-free D ∞ -odometers, which were shown to be counterexamples to the HK conjecture.Since they are all Cantor minimal D ∞ -systems, it follows from Theorem 3.3 that their dynamic asymptotic dimension is one and the nuclear dimension of the associated crossed product is bounded by one.It follows in particular, that these crossed products are classifiable.
The following corollary complements [1, Corollary 2.5]: Corollary 3.5.Let X be an infinite compact Hausdorff space.If Γ is a virtually cyclic group acting minimally (not necessarily topologically free) on X, then the dynamic asymptotic dimension of Γ X is one and Proof.Since X is infinite and the action is minimal, the group Γ must be infinite.As Γ is an infinite virtually cyclic group, it has a finite normal subgroup N ⊆ Γ such that Γ/N is either Z or D ∞ .It is easy to deduce that the dynamic asymptotic dimension of Γ X is bounded by the dynamic asymptotic dimension of the minimal action Γ/N X/N , which is equal to one by [18, Theorem 3.1] for Γ/N = Z and Theorem 3.3 for Γ/N = D ∞ .The dynamic asymptotic dimension equals 0 only for actions of locally finite groups, and Γ contains an infinite-order element.Thus, we have completed the proof of the first statement.
If X is second-countable, it follows directly from Theorem 3.2 that the inequality (2) holds.If X is not second-countable, the inequality (2) follows from the secondcountable case of X via a direct limit argument (see [19,Lemma 1.3 ] and [42, Proposition 2.3 (iii)]).Remark 3.6.In Corollary 3.5, the minimal action may not be topologically free and the C * -algebra may not be simple.Indeed, if we consider any minimal action of Z on X and any non-trivial finite group F .Then Z × F acts minimally on X when F acts trivially on X.However, this minimal action of Z × F is not topologically free because X g = X for all non-trivial g ∈ F .
We end the paper by providing some further positive evidence towards the following open question: Question 3.7.Does every separable nuclear Z-stable C * -algebra A have finite nuclear dimension?Proposition 3.8.Let G be a second-countable, locally compact, Hausdorff and étale groupoid and let Σ be a twist over G. Suppose that G has dynamic asymptotic dimension d.Then the nuclear dimension of C * r (G; Σ) ⊗ Z is at most 3d + 2, where Z is the Jiang-Su algebra.
Proof.The proof is a slight variant of the one for Theorem 3.2.Indeed, let {H i } 0≤i≤d be the second-countable étale open precompact subgroupoids of G as constructed in the proof of Theorem 3.2.By Proposition 2.5, the twisted C *algebras C * r (H i ; j −1 (H i )) are all subhomogenuous.As Z is locally subhomogeneous, it follows from [14, Theorem A] that C * r (π −1 (H i ); H i ) ⊗ Z has nuclear dimension at most 2. Therefore, dim nuc (C * r (G; Σ) ⊗ Z) ≤ (2 + 1)(d + 1) − 1 = 3d + 2 by following the argument in the proof of Theorem 3.2.Remark 3.9.So far we have an affirmative answer to Question 3.7 in the following three cases: • if A is simple (see [6,5]); • if A is traceless (see [4,34]); • if A is a twisted étale groupoid C * -algebra with finite dynamic asymptotic dimension (see Proposition 3.8).

Definition 3 . 1 .
[18, Definition 5.1]  Let G be a locally compact, Hausdorff and étale groupoid and d ∈ N 0 .Then G has dynamic asymptotic dimension at most d, if for every open and precompact subset K ⊆ G, there exists a cover ofs(K) ∪ r(K) by d + 1 open subsets U 0 , . . ., U d of G 0 such that for each 0 ≤ i ≤ d, the open subgroupoid K ∩ G| Ui is precompact in G.We write dad(G) for the minimal d ∈ N 0 satisfying the above and call it the dynamic asymptotic dimension of G.
continuous map between second countable, locally compact Hausdorff spaces (see Lemma 2.10).Via this map, we can view C 0 (p −1 k ( X n )) as a C 0 ( X n )-algebra and hence we can apply [19, Lemma 3.3] and Proposition 2.11 to get dim +1 a twist over H i by [10, Lemma 2.5].Since each H i is a second-countable étale open subgroupoid which is precompact in G, we can cover H i by finitely many open bisections U 1 , . . ., U M .It follows that (H i ) x and (H i ) x