Every classifiable simple C*-algebra has a Cartan subalgebra

Classification of C*-algebras has seen tremendous advances recently. In the unital case, the classification of unital separable simple nuclear \({\mathcal {Z}}\)-stable C*-algebras satisfying the UCT is by now complete. This is the culmination of work by many mathematicians. The reader may consult [12, 20, 24, 34, 44] and the references therein. In the stably projectionless case, classification results are being developed (see [13,14,15, 18, 19]). It is expected that—once the stably projectionless case is settled—the final result will classify all separable simple nuclear \({\mathcal {Z}}\)-stable C*-algebras satisfying the UCT by their Elliott invariants. This class of C*-algebras is what we refer to as “classifiable C*-algebras”.

To complete these classification results, it is important to construct concrete models realizing all possible Elliott invariants by classifiable C*-algebras. Such models have been constructed—in the greatest possible generality—in [11] (see also [43] which covers special cases). In the stably finite unital case, the reader may also find such range results in [20], where the construction follows the ideas in [11] (with slight modifications, so that the models belong to the special class considered in [20]). In the stably projectionless case, models have been constructed in a slightly different way in [19] (again to belong to the special class of algebras considered) under the additional assumption of a trivial pairing between K-theory and traces.

Recently, the notion of Cartan subalgebras in C*-algebras [25, 36] has attracted attention, due to connections to topological dynamics [26,27,28] and the UCT question [3, 4]. In particular the reformulation of the UCT question in [3, 4] raises the following natural question (see [29, Question 5.9], [42, Question 16] and [5, Problems 1 and 2]):

Question 1.1

Which classifiable C*-algebras have Cartan subalgebras?

By [25, 36], we can equally well ask for groupoid models for classifiable C*-algebras. In the purely infinite case, groupoid models and hence Cartan subalgebras have been constructed in [41] (see also [29, § 5]). For special classes of stably finite unital C*-algebras, groupoid models have been constructed in [8, 35] using topological dynamical systems. Using a new approach, the goal of this paper is to answer Question 1.1 by constructing Cartan subalgebras in all the C*-algebra models constructed in [11, 19, 20], covering all classifiable stably finite C*-algebras, in particular in all classifiable unital C*-algebras. Generally speaking, Cartan subalgebras allow us to introduce ideas from geometry and dynamical systems to the study of C*-algebras. More concretely, in view of [3, 4], we expect that our answer to Question 1.1 will lead to progress on the UCT question.

The following two theorems are the main results of this paper. The reader may consult [25, 36] for the definition of twisted groupoids and their relation to Cartan subalgebras, and [38, § 2.2], [32, § 8.4], [18,19,20] for the precise definition of the Elliott invariant.

Theorem 1.2

(unital case) Given

• a weakly unperforated, simple scaled ordered countable abelian group \((G_0,G_0^+,u)\),

• a non-empty metrizable Choquet simplex T,

• a surjective continuous affine map \(r: \, T \rightarrow S(G_0)\),

• a countable abelian group \(G_1\),

there exists a twisted groupoid \((G,\Sigma )\) such that

• G is a principal étale second countable locally compact Hausdorff groupoid,

• \(C^*_r(G,\Sigma )\) is a simple unital C*-algebra which can be described as the inductive limit of subhomogeneous C*-algebras whose spectra have dimension at most 3,

• the Elliott invariant of \(C^*_r(G,\Sigma )\) is given by

  $$\begin{aligned}&(K_0(C^*_r(G,\Sigma )), K_0(C^*_r(G,\Sigma ))^+, [1_{C^*_r(G,\Sigma )}], T(C^*_r(G,\Sigma )), r_{C^*_r(G,\Sigma )},\\&K_1(C^*_r(G,\Sigma ))) \cong (G_0,G_0^+,u,T,r,G_1). \end{aligned}$$

Theorem 1.3

(stably projectionless case) Given

• countable abelian groups \(G_0\) and \(G_1\),

• a non-empty metrizable Choquet simplex T,

• a homomorphism \({\rho }: \, G_0 \rightarrow \mathrm{Aff}(T)\) which is weakly unperforated in the sense that for all \(g \in G_0\), there is \(\tau \in T\) with \(\rho (g)(\tau ) = 0\)

there exists a twisted groupoid \((G,\Sigma )\) such that

• G is a principal étale second countable locally compact Hausdorff groupoid,

• \(C^*_r(G,\Sigma )\) is a simple stably projectionless C*-algebra with continuous scale in the sense of [18, 19, 30, 31] which can be described as the inductive limit of subhomogeneous C*-algebras whose spectra have dimension at most 3,

• the Elliott invariant of \(C^*_r(G,\Sigma )\) is given by

  $$\begin{aligned}&(K_0(C^*_r(G,\Sigma )), K_0(C^*_r(G,\Sigma ))^+, T(C^*_r(G,\Sigma )), \rho _{C^*_r(G,\Sigma )},\\&K_1(C^*_r(G,\Sigma ))) \cong (G_0,\left\{ 0 \right\} ,T,\rho ,G_1). \end{aligned}$$

The condition on \(\rho \) means that the pairing between K-theory and traces is weakly unperforated, in the sense of [11]. It has been shown in [14, § A.1] that this condition of weak unperforation is necessary in the classifiable setting (i.e., it follows from finite nuclear dimension, or \({\mathcal {Z}}\)-stability).

It is worth pointing out that in the main theorems, the twisted groupoids are constructed explicitly in such a way that the inductive limit structure with subhomogeneous building blocks will become visible at the groupoid level.

Remark 1.4

The original building blocks in [11] have spectra with dimension at most two. The reason three-dimensional spectra are needed in this paper is because it is not clear how to realize all possible connecting maps at the level of \(K_1\) by Cartan-preserving homomomorphisms using the building blocks in [11]. Therefore, the building blocks have to be modified (see Sect. 3). Roughly speaking, the idea is to realize all possible connecting maps in \(K_1\) at the level of topological spaces. This however requires three-dimensional spectra because “nice” topological spaces (say CW-complexes) of dimension two or lower have torsion-free \(K^1\) (because cohomology is torsion-free in all odd degrees for these spaces). The dimension can be reduced to two if \(K_1\) is torsion-free (see Corollary 1.8 and Remark 3.9).

In particular, together with the classification results in [12, 20, 24, 34, 44], the groupoid models in [41], and [3, Theorem 3.1], we obtain the following

Corollary 1.5

A unital separable simple C*-algebra with finite nuclear dimension has a Cartan subalgebra if and only if it satisfies the UCT.

The only reason we restrict to the unital case here is that classification in the stably projectionless case has not been completed yet.

The constructions of the twisted groupoids in Theorems 1.2 and 1.3 yield the following direct consequences:

Corollary 1.6

In the situation of Theorem 1.2, suppose that in addition to \((G_0, G_0^+, u)\), T, r and \(G_1\), we are given a topological cone \(\tilde{T}\) with base T and a lower semicontinuous affine map \(\tilde{\gamma }: \, \tilde{T} \rightarrow [0,\infty ]\). Then there exists a twisted groupoid \((\tilde{G},\tilde{\Sigma })\) such that

• \(\tilde{G}\) is a principal étale second countable locally compact Hausdorff groupoid,

• \(C^*_r(\tilde{G},\tilde{\Sigma })\) is a non-unital hereditary sub-C*-algebra of \(C^*_r(G,\Sigma ) \otimes {\mathcal {K}}\),

• the Elliott invariant of \(C^*_r(\tilde{G},\tilde{\Sigma })\) is given by

  $$\begin{aligned}&(K_0(C^*_r(\tilde{G},\tilde{\Sigma })), K_0(C^*_r(\tilde{G},\tilde{\Sigma }))^+, \tilde{T}(C^*_r(G,\Sigma )), \Sigma _{C^*_r(\tilde{G},\tilde{\Sigma })}, r_{C^*_r(\tilde{G},\tilde{\Sigma })},\\&K_1(C^*_r(\tilde{G},\tilde{\Sigma }))) \cong (G_0,G_0^+,\tilde{T},\tilde{\gamma },r,G_1). \end{aligned}$$

Corollary 1.7

In the situation of Theorem 1.3, suppose that in addition to \(G_0\), \(G_1\), T and \(\rho \), we are given a topological cone \(\tilde{T}\) with base T and a lower semicontinuous affine map \(\tilde{\gamma }: \, \tilde{T} \rightarrow [0,\infty ]\). Then there exists a twisted groupoid \((\tilde{G},\tilde{\Sigma })\) such that

• \(\tilde{G}\) is a principal étale second countable locally compact Hausdorff groupoid,

• \(C^*_r(\tilde{G},\tilde{\Sigma })\) is a hereditary sub-C*-algebra of \(C^*_r(G,\Sigma ) \otimes {\mathcal {K}}\),

• the Elliott invariant of \(C^*_r(\tilde{G},\tilde{\Sigma })\) is given by

  $$\begin{aligned}&(K_0(C^*_r(\tilde{G},\tilde{\Sigma })), K_0(C^*_r(\tilde{G},\tilde{\Sigma }))^+, \tilde{T}(C^*_r(\tilde{G},\tilde{\Sigma })), \Sigma _{C^*_r(\tilde{G},\tilde{\Sigma })}, \rho _{C^*_r(\tilde{G},\tilde{\Sigma })},\\&K_1(C^*_r(\tilde{G},\tilde{\Sigma }))\cong (G_0,\left\{ 0 \right\} ,\tilde{T},\tilde{\gamma },\rho ,G_1). \end{aligned}$$

Note that all the groupoids in Theorems 1.2, 1.3 and Corollaries 1.6, 1.7 are necessarily minimal and amenable. Theorem 1.2 and Corollary 1.6, together with [41], imply that every classifiable C*-algebra which is not stably projectionless has a Cartan subalgebra. Once the classification of stably projectionless C*-algebras is completed, Theorem 1.3 and Corollary 1.7 will imply that every classifiable stably projectionless C*-algebra has a Cartan subalgebra. Actually, using \({\mathcal {Z}}\)-stability, we see that all of the above-mentioned classifiable C*-algebras have infinitely many non-isomorphic Cartan subalgebras (compare [29, Proposition 5.1]). Moreover, the constructions in this paper show that in every classifiable stably finite C*-algebra, we can even find C*-diagonals (and even infinitely many non-isomorphic ones).

Moreover, more can be said about the twist, and also about the dimension of the spectra of our Cartan subalgebras.

Corollary 1.8

The twisted groupoids \((G,\Sigma )\) constructed in the proofs of Theorems 1.2 and 1.3 have the following additional properties:

1. (i)

   If \(G_0\) is torsion-free, then the twist \(\Sigma \) is trivial, i.e., \(\Sigma = {\mathbb {T}}\times G\).

2. (ii)

   If \(G_1\) has torsion, then \(C^*_r(G,\Sigma )\) is an inductive limit of subhomogeneous C*-algebras whose spectra are three-dimensional, and \(\mathrm{dim}\,(G^{(0)}) = 3\).

3. (iii)

   If \(G_1\) is torsion-free and \(G_0\) has torsion, \(C^*_r(G,\Sigma )\) is an inductive limit of subhomogeneous C*-algebras whose spectra are two-dimensional, and \(\mathrm{dim}\,(G^{(0)}) = 2\).

4. (iv)

   If both \(G_0\) and \(G_1\) are torsion-free with \(G_1 \ncong \left\{ 0 \right\} \), then \(C^*_r(G,\Sigma )\) is an inductive limit of subhomogeneous C*-algebras whose spectra are one-dimensional, and \(\mathrm{dim}\,(G^{(0)}) = 1\).

5. (v)

   If \(G_0\) is torsion-free and \(G_1 \cong \left\{ 0 \right\} \), then \(C^*_r(G,\Sigma )\) is an inductive limit of one-dimensional non-commutative finite CW-complexes, with \(\mathrm{dim}\,(G^{(0)}) \le 1\) in Theorem 1.2 and \(\mathrm{dim}\,(G^{(0)}) = 1\) in Theorem 1.3.

In particular, Corollary 1.8 implies the following:

Corollary 1.9

The Jiang–Su algebra \({\mathcal {Z}}\), the Razak–Jacelon algebra \({\mathcal {W}}\) and the stably projectionless version \({\mathcal {Z}}_0\) of the Jiang–Su algebra of [19, Definition 7.1] have C*-diagonals with one-dimensional spectra. The corresponding twisted groupoids \((G,\Sigma )\) can be chosen so that \(\Sigma \) is trivial, i.e., \(\Sigma = {\mathbb {T}}\times G\).

Concrete groupoid models for \({\mathcal {Z}}\), \({\mathcal {W}}\) and \({\mathcal {Z}}_0\) are described in Sect. 8. It is worth pointing out that a groupoid model has been constructed for \({\mathcal {Z}}\) in [8] using a different construction (but the precise dimension of the unit space has not been determined in [8]). Moreover, G. Szabó and S. Vaes independently found groupoid models for \({\mathcal {W}}\), again using constructions different from ours. Furthermore, independently from [4] and the present paper, similar tools to the ones in [4, § 3] were developed in [2], which give rise to groupoid models for \({\mathcal {Z}}\) and \({\mathcal {W}}\) as well as other examples.

The key tool for all the results in this paper is an improved version of [4, Theorem 3.6], which allows us to construct Cartan subalgebras in inductive limit C*-algebras. The C*-algebraic formulation reads as follows.

Theorem 1.10

Let \((A_n,B_n)\) be Cartan pairs with normalizers \(N_n \mathrel {:=}N_{A_n}(B_n)\) and faithful conditional expectations \(P_n: \, A_n \twoheadrightarrow B_n\). Let \(\varphi _n: \, A_n \rightarrow A_{n+1}\) be injective *-homomorphisms with \(\varphi _n(B_n) \subseteq B_{n+1}\), \(\varphi _n(N_n) \subseteq N_{n+1}\) and \(P_{n+1} \circ \varphi _n = \varphi _n \circ P_n\) for all n. Then \(\varinjlim \left\{ B_n;\varphi _n \right\} \) is a Cartan subalgebra of \(\varinjlim \left\{ A_n;\varphi _n \right\} \).

If all \(B_n\) are C*-diagonals, then \(\varinjlim \left\{ B_n;\varphi _n \right\} \) is a C*-diagonal of \(\varinjlim \left\{ A_n;\varphi _n \right\} \).

A special case of this theorem is proved in [6].

Actually, in addition to Theorem 1.10, much more is accomplished: Groupoid models are developed for *-homomorphisms such as \(\varphi _n\), and the twisted groupoid corresponding to \(\left( \varinjlim \left\{ A_n;\varphi _n \right\} , \varinjlim \left\{ B_n;\varphi _n \right\} \right) \) as in Theorem 1.10 is described explicitly. These results (in Sect. 5) might be of independent interest.

Applications of these explicit descriptions of groupoid models (for homomorphisms and Cartan pairs) and Theorem 1.10 include a unified approach to Theorems 1.2, 1.3, and explicit constructions of the desired twisted groupoids. The strategy is as follows: C*-algebras with prescribed Elliott invariant have been constructed in [11] (see also [20, § 13] for the unital case). These C*-algebras have all the desired properties as in Theorems 1.2 and 1.3 and are constructed as inductive limits of subhomogeneous C*-algebras. However, the connecting maps in [11] and [20, § 13] do not preserve the canonical Cartan subalgebras in these building blocks in general. Therefore, a careful choice or modification of the building blocks and connecting maps in the constructions in [11, 20] is necessary in order to allow for an application of Theorem 1.10. The modification explained in Remark 4.1 is particularly important. Actually, a more general result is established in Sect. 4.2, where a class of inductive limits of subhomogeneous C*-algebras is identified, which encompasses all the C*-algebras in Theorems 1.2, 1.3 and Corollaries 1.6, 1.7, where we can apply Theorem 1.10.

I am grateful to the organizers Selçuk Barlak, Wojciech Szymański and Wilhelm Winter of the Oberwolfach Mini-Workshop “MASAs and Automorphisms of C*-Algebras” for inviting me, and for the discussions in Oberwolfach with Selçuk Barlak which eventually led to this paper. I also thank Selçuk Barlak and Gábor Szabó for helpful comments on earlier drafts. Moreover, I would like to thank the referee for very helpful comments which led to an improved version of Theorem 1.3 (previous versions of this theorem only covered classifiable stably projectionless C*-algebras with trivial pairing between K-theory and traces).

Let us briefly recall the constructions in [11] (see also [16] for simplifications and further explanations) and [20, § 13].

2.1 The unital case

Let us describe the construction in [20, § 13], which is based on [11] (with slight modifications). Given \((G_0,G_0^+,u,T,r,G_1)\) as in Theorem 1.2, write \(G = G_0\), \(K = G_1\), and let \(\rho : \, G \rightarrow \mathrm{Aff}(T)\) be the dual map of r. Choose a dense subgroup \(G' \subseteq \mathrm{Aff}(T)\). Set \(H \mathrel {:=}G \oplus G'\),

and view u in G as an element of H. Then \((H,H^+,u)\) becomes a simple ordered group, inducing the structure of a dimension group on \(H / \mathrm{Tor}(H)\). Now construct a commutative diagram

where:

• \(H_n\) is a finitely generated abelian group with \(H_n = \bigoplus _i H_n^i\), where for one distinguished index \(\varvec{i}\), \(H_n^{\varvec{i}} = {\mathbb {Z}}\oplus \mathrm{Tor}(H_n)\), and for all other indices, \(H_n^i = {\mathbb {Z}}\);

• with \((H_n^{\varvec{i}})^+ \mathrel {:=}\left\{ (0,0) \right\} \cup ({\mathbb {Z}}_{>0} \oplus \mathrm{Tor}(H_n))\), \((H_n^i)^+ \mathrel {:=}{\mathbb {Z}}_{\ge 0}\) for all \(i \ne \varvec{i}\), \(H_n^+ \mathrel {:=}\bigoplus _i (H_n^i)^+ \subseteq H_n^{\varvec{i}} \oplus \bigoplus _{i \ne \varvec{i}} H_n^i = H_n\) and \(u_n = (([n,\varvec{i}],\tau _n),([n,i])_{i \ne \varvec{i}}) \in H_n^+\), we have

  $$\begin{aligned} \varinjlim \left\{ (H_n,H_n^+,u_n); \gamma _n \right\} \cong (H,H^+,u); \end{aligned}$$

  (1)

• with \(G_n \mathrel {:=}(\gamma _n^{\infty })^{-1}(G)\), where \(\gamma _n^{\infty }: \, H_n \rightarrow H\) is the map provided by (1), and \(G_n^+ \mathrel {:=}G_n \cap H_n^+\), we have \(u_n \in G_n \subseteq H_n\), and (1) induces \(\varinjlim \left\{ (G_n,G_n^+,u_n); \gamma _n \right\} \cong (G,G^+,u)\);

• the vertical maps are the canonical ones.

Let \({\hat{\gamma }}_n: \, H_n / \mathrm{Tor}(H_n) \mathrel {=:}{\hat{H}}_n = \bigoplus _i {\hat{H}}_n^i \rightarrow \bigoplus _j {\hat{H}}_{n+1}^j = {\hat{H}}_{n+1} \mathrel {:=}H_{n+1} / \mathrm{Tor}(H_{n+1})\) be the homomorphism induced by \(\gamma _n\), where \({\hat{H}}_n^i = {\mathbb {Z}}= {\hat{H}}_{n+1}^j\) for all i and j. For fixed n, the map \({\hat{\gamma }} = {\hat{\gamma }}_n\) is given by a matrix \(({\hat{\gamma }}_{ji})\), where we can always assume that \({\hat{\gamma }}_{ji} \in {\mathbb {Z}}_{>0}\) (considered as a map \({\hat{H}}_n^i = {\mathbb {Z}}\rightarrow {\mathbb {Z}}= {\hat{H}}_{n+1}^j\)). Then \(\gamma _n = {\hat{\gamma }} + \tau + t\) for homomorphisms \(\tau : \, \mathrm{Tor}(H_n) \rightarrow \mathrm{Tor}(H_{n+1})\) and \(t: \, {\hat{H}}_n \rightarrow \mathrm{Tor}(H_{n+1})\). Here we think of \({\hat{H}}_n\) as a subgroup (actually a direct summand) of \(H_n\). As explained in [20, § 6], given a positive constant \(\Gamma _n\) depending on n, we can always arrange that

Also, let \(K_n\) be finitely generated abelian groups and \(\chi _n: \, K_n \rightarrow K_{n+1}\) homomorphisms such that \(K \cong \varinjlim \left\{ K_n; \chi _n \right\} \).

Let \(F_n = \bigoplus _i F_n^i\) be C*-algebras, where \(F_n^{\varvec{i}}\) is a homogeneous C*-algebra of the form \(F_n^{\varvec{i}} = P_n^{\varvec{i}} M_{\infty }(C(Z_n^{\varvec{i}})) P_n^{\varvec{i}}\) for a connected compact space \(Z_n^{\varvec{i}}\) with base point \(\theta _n^{\varvec{i}}\) and a projection \(P_n^{\varvec{i}} \in M_{\infty }(C(Z_n^{\varvec{i}}))\), while for all other indices \(i \ne \varvec{i}\), \(F_n^i\) is a matrix algebra, \(F_n^i = M_{[n,i]}\). We require that \((K_0(F_n^{\varvec{i}}), K_0(F_n^{\varvec{i}})^+,[1_{F_n^{\varvec{i}}}]) \cong (H_n^{\varvec{i}},(H_n^{{\mathbf {i}}})^+,([n,\varvec{i}],\tau _n))\) and \(K_1(F_n^{\varvec{i}}) \cong K_n\), so that \((K_0(F_n), K_0(F_n)^+,[1_{F_n}], K_1(F_n)) \cong (H_n,H_n^+,u_n,K_n)\).

Let \(\psi _n\) be a unital homomorphism \(F_n \rightarrow F_{n+1}\) which induces \(\gamma _n\) in \(K_0\) and \(\chi _n\) in \(K_1\). We write \(F_n = P_n C(Z_n) P_n\) where \(Z_n = Z_n^{\varvec{i}} \amalg \coprod _{i \ne \varvec{i}} \{ \theta _n^i \}\), and \(P_n = (P_n^{\varvec{i}},(1_{[n,i]})_{i \ne \varvec{i}}) \in M_{\infty }(C(Z_n^{\varvec{i}})) \oplus \bigoplus _{i \ne \varvec{i}} M_{[n,i]}(C(\{ \theta _n^i \}))\). Thus evaluation in \(\theta _n^i\) induces a quotient map \(\pi _n: \, F_n \rightarrow {\hat{F}}_n \mathrel {:=}\bigoplus _i {\hat{F}}_n^i\), where \({\hat{F}}_n^i = M_{[n,i]}\). We require that \(\psi _n\) induce homomorphisms \({\hat{\psi }}_n: \, {\hat{F}}_n \rightarrow {\hat{F}}_{n+1}\) so that we obtain a commutative diagram

which induces in \(K_0\)

where the vertical arrows are the canonical projections. As \(\mathrm{Tor}(H_n) \subseteq G_n\), \(H_n / G_n\) is torsion-free, and there is a canonical projection \(H_n / \mathrm{Tor}(H_n) \rightarrow H_n / G_n\). Now let \(E_n \mathrel {:=}\bigoplus _p E_n^p\), \(E_n^p = M_{\left\{ n,p \right\} }\), so that \(K_0(E_n) \cong H_n / G_n\), and for fixed n, let \(\beta _0, \, \beta _1: \, {\hat{F}}_n \rightarrow E_n\) be unital homomorphisms which yield the commutative diagram

We can assume \(\beta _0 \oplus \beta _1: \, {\hat{F}}_n \rightarrow E_n \oplus E_n\) to be injective, because only the difference \((\beta _0)_* - (\beta _1)_*\) matters.

Define

As \(\beta _0 \oplus \beta _1\) is injective, we view \({\hat{A}}_n\) as a subalgebra of \(C([0,1],E_n)\) via \((f,{\hat{a}}) \mapsto f\).

Choose for each n a unital homomorphism \({\hat{\varphi }}_n: \, {\hat{A}}_n \rightarrow {\hat{A}}_{n+1}\) such that the composition with the map \(C([0,1],E_{n+1}) \twoheadrightarrow C([0,1],E_{n+1}^q)\) induced by the canonical projection \(E_{n+1} \twoheadrightarrow E_{n+1}^q\),

is of the form

where u is a continuous path of unitaries \([0,1] \rightarrow U(E_{n+1}^q)\),

for some \(\pi _{\bullet }\) of the form \(\pi _{\bullet }: \, {\hat{A}}_n \rightarrow {\hat{F}}_n \twoheadrightarrow {\hat{F}}_n^i\), where the first map is given by \((f,{\hat{a}}) \mapsto {\hat{a}}\) and the second map is the canonical projection, and

for some continuous maps \(\lambda _{\bullet }: \, [0,1] \rightarrow [0,1]\) with \(\lambda _{\bullet }^{-1}(\left\{ 0,1 \right\} ) \subseteq \left\{ 0,1 \right\} \). We require that the diagram

commute, where the vertical maps are given by \((f,{\hat{a}}) \mapsto {\hat{a}}\).

Then there exists a unique homomorphism \({\varphi _n}: \, A_n \rightarrow A_{n+1}\) which fits into the commutative diagram

where all the unlabelled arrows are given by the canonical maps.

By construction, \(\varinjlim \left\{ A_n;\varphi _n \right\} \) has the desired Elliott invariant (in particular, the canonical map \(\varinjlim \left\{ A_n;\varphi _n \right\} \rightarrow {\hat{F}} \mathrel {:=}\varinjlim \left\{ {\hat{F}}_n; {\hat{\psi }}_n \right\} \) induces \(T(\varinjlim \left\{ A_n;\varphi _n \right\} ) \cong T({\hat{F}})\)). However, this is not a simple C*-algebra. Thus a further modification is needed to enforce simplicity. To this end, choose \(\varvec{I}_n \subseteq (0,1)\) and \(\varvec{Z}_n^{\varvec{i}} \subseteq Z_n^{\varvec{i}}\)\(\frac{1}{n}\)-dense and replace \(\varphi _n: \, A_n \rightarrow A_{n+1}\) by the unital homomorphism \(\xi _n: \, A_n \rightarrow A_{n+1}\) such that:

• the compositions

  $$\begin{aligned}&A_n \overset{\xi _n}{\longrightarrow } A_{n+1} \rightarrow F_{n+1} \twoheadrightarrow F_{n+1}^j\ \ \ \mathrm{and} \\&A_n \overset{\varphi _n}{\longrightarrow } A_{n+1} \rightarrow F_{n+1} \twoheadrightarrow F_{n+1}^j \ \text {coincide except for one index} \ j_{\xi } \ne \varvec{j}; \end{aligned}$$

• the composition

  $$\begin{aligned} A_n \overset{\xi _n}{\longrightarrow } A_{n+1} \rightarrow F_{n+1} \twoheadrightarrow F_{n+1}^{j_{\xi }} \end{aligned}$$

  is of the form

  $$\begin{aligned} A_n \ni (f,a) \mapsto u^* \begin{pmatrix} I(f) &{} &{} \\ &{} Z(a) &{} \\ &{} &{} P(a) \end{pmatrix} u, \end{aligned}$$

  where u is a permutation matrix in \(M_{[n+1,j_{\xi }]}\),

  $$\begin{aligned} I(f) = \begin{pmatrix} f^{p_1}(t_1) &{} &{} \\ &{} f^{p_2}(t_2) &{} \\ &{} &{} \ddots \end{pmatrix} \end{aligned}$$

  for indices \(p_{\bullet }\) and \(t_{\bullet } \in \varvec{I}_n\) such that all possible pairs \(p_{\bullet }, t_{\bullet }\) appear (\(f^p\) is the component of f in \(C([0,1],E_n^p)\)),

  $$\begin{aligned} Z(a) = \begin{pmatrix} \tau _1(a(z_1)) &{} &{} \\ &{} \tau _2(a(z_2)) &{} \\ &{} &{} \ddots \end{pmatrix} \end{aligned}$$

  (4)

  for \(z_{\bullet } \in \varvec{Z}_n\) and isomorphisms \(\tau _{\bullet }: \, P_n^{\varvec{i}}(z_{\bullet })M_{\infty }P_n^{\varvec{i}}(z_{\bullet }) \cong {\hat{F}}_n^{\varvec{i}} = M_{[n,\varvec{i}]}\), and

  $$\begin{aligned} P(a) = \begin{pmatrix} \pi _n^{i_1}(a) &{} &{} \\ &{} \pi _n^{i_2}(a) &{} \\ &{} &{} \ddots \end{pmatrix}, \end{aligned}$$

  where \(\pi _n^i\) is the canonical projection \(F_n \twoheadrightarrow {\hat{F}}_n \twoheadrightarrow {\hat{F}}_n^i\);

• for every q, the composition

  $$\begin{aligned} A_n \overset{\xi _n}{\longrightarrow } A_{n+1} \rightarrow C([0,1],E_{n+1}) \twoheadrightarrow C([0,1],E_{n+1}^q) \end{aligned}$$

  is of the form

  $$\begin{aligned} A_n \ni (f,a) \mapsto u^* \begin{pmatrix} \Phi (f) &{} \\ &{} \Xi (a) \end{pmatrix} u, \end{aligned}$$

  where u is a continuous path of unitaries \([0,1] \rightarrow U(E_{n+1}^q)\), \(\Phi (f)\) is of the same form

  $$\begin{aligned} \begin{pmatrix} V(f) &{} \\ &{} D(f) \end{pmatrix} \end{aligned}$$

  as in (3),

  $$\begin{aligned} \Xi (a)(t) = \begin{pmatrix} \tau _1(t)(a(z_1(t))) &{} &{} \\ &{} \tau _2(t)(a(z_2(t))) &{} \\ &{} &{} \ddots \end{pmatrix} \end{aligned}$$

  for continuous maps \(z_{\bullet }: \, [0,1] \rightarrow Z_n^{\varvec{i}}\), each of which is either a constant map with value in \(\varvec{Z}_n\) or connects \(\theta _n^{\varvec{i}}\) with \(z_{\bullet } \in \varvec{Z}_n\), and isomorphisms \(\tau _{\bullet }(t): \, P_n^{\varvec{i}}(z_{\bullet }(t)) M_{\infty }P_n^{\varvec{i}}(z_{\bullet }(t)) \cong {\hat{F}}_n^{\varvec{i}}\) depending continuously on \(t \in [0,1]\) such that for \(t \in \left\{ 0,1 \right\} \), \(\tau _{\bullet }(t) = \mathrm{id}\) if \(z_{\bullet }(t) = \theta _n^{\varvec{i}}\) and \(\tau _{\bullet }(t) = \tau _{\bullet }\) if \(z_{\bullet }(t) = z_{\bullet }\), where \(\tau _{\bullet }\) is as in (4).

Then \(\varinjlim \left\{ A_n; \xi _n \right\} \) is a simple unital C*-algebra with prescribed Elliott invariant.

2.2 The stably projectionless case

We follow [11] (see also [16]), with slight modifications as in the unital case. Let \((G_0,T,\rho ,G_1)\) be as in Theorem 1.3.

Write \(G = G_0\) and \(K = G_1\). Choose a dense subgroup \(G' \subseteq \mathrm{Aff}(T)\). Set \(H \mathrel {:=}G \oplus G'\),

Then \((H,H^+)\) becomes a simple ordered group, inducing the structure of a dimension group on \(H / \mathrm{Tor}(H)\). Now construct a commutative diagram

where

• \(H_n\) is a finitely generated abelian group with \(H_n = \bigoplus _i H_n^i\), where for one distinguished index \(\varvec{i}\), \(H_n^{\varvec{i}} = {\mathbb {Z}}\oplus \mathrm{Tor}(H_n)\), and for all other indices, \(H_n^i = {\mathbb {Z}}\);

• with \((H_n^{\varvec{i}})^+ \mathrel {:=}\left\{ (0,0) \right\} \cup ({\mathbb {Z}}_{>0} \oplus \mathrm{Tor}(H_n))\), \((H_n^i)^+ \mathrel {:=}{\mathbb {Z}}_{\ge 0}\) for all \(i \ne \varvec{i}\), and \(H_n^+ \mathrel {:=}\bigoplus _i (H_n^i)^+ \subseteq H_n^{\varvec{i}} \oplus \bigoplus _{i \ne \varvec{i}} H_n^i = H_n\) we have

  $$\begin{aligned} \varinjlim \left\{ (H_n,H_n^+); \gamma _n \right\} \cong (H,H^+); \end{aligned}$$

  (5)

• with \(G_n \mathrel {:=}(\gamma _n^{\infty })^{-1}(G)\), where \(\gamma _n^{\infty }: \, H_n \rightarrow H\) is the map provided by (5), we have \(G_n \cap H_n^+ = \left\{ 0 \right\} \), and (1) induces \(\varinjlim \left\{ G_n; \gamma _n \right\} \cong G\);

• the vertical maps are the canonical ones.

Let \({\hat{\gamma }}_n: \, H_n / \mathrm{Tor}(H_n) \mathrel {=:}{\hat{H}}_n = \bigoplus _i {\hat{H}}_n^i \rightarrow \bigoplus _j {\hat{H}}_{n+1}^j = {\hat{H}}_{n+1} \mathrel {:=}H_{n+1} / \mathrm{Tor}(H_{n+1})\) be the homomorphism induced by \(\gamma _n\), where \({\hat{H}}_n^i = {\mathbb {Z}}= {\hat{H}}_{n+1}^j\) for all i and j. For fixed n, the map \({\hat{\gamma }} = {\hat{\gamma }}_n\) is given by a matrix \(({\hat{\gamma }}_{ji})\), where we can always assume that \({\hat{\gamma }}_{ji} \in {\mathbb {Z}}_{>0}\) (considered as a map \({\hat{H}}_n^i = {\mathbb {Z}}\rightarrow {\mathbb {Z}}= {\hat{H}}_{n+1}^j\)). Then \(\gamma _n = {\hat{\gamma }} + \tau + t\) for homomorphisms \(\tau : \, \mathrm{Tor}(H_n) \rightarrow \mathrm{Tor}(H_{n+1})\) and \(t: \, {\hat{H}}_n \rightarrow \mathrm{Tor}(H_{n+1})\). Here we think of \({\hat{H}}_n\) as a subgroup of \(H_n\). As in the unital case (see [20, § 6]), given a positive constant \(\Gamma _n\) depending on n, we can always arrange that

Also, let \(K_n\) be finitely generated abelian groups and \(\chi _n: \, K_n \rightarrow K_{n+1}\) homomorphisms such that \(K \cong \varinjlim \left\{ K_n; \chi _n \right\} \).

Let \(F_n = \bigoplus _i F_n^i\) be C*-algebras, where \(F_n^{\varvec{i}}\) is a homogeneous C*-algebra of the form \(F_n^{\varvec{i}} = P_n^{\varvec{i}} M_{\infty }(C(Z_n^{\varvec{i}})) P_n^{\varvec{i}}\) for a connected compact space \(Z_n^{\varvec{i}}\) with base point \(\theta _n^{\varvec{i}}\) and a projection \(P_n^{\varvec{i}} \in M_{\infty }(C(Z_n^{\varvec{i}}))\), while for all other indices \(i \ne \varvec{i}\), \(F_n^i\) is a matrix algebra, \(F_n^i = M_{[n,i]}\). We require that \((K_0(F_n^{\varvec{i}}), K_0(F_n^{\varvec{i}})^+) \cong (H_n^{\varvec{i}},(H_n^{{\mathbf {i}}})^+)\) and \(K_1(F_n^{\varvec{i}}) \cong K_n\), so that \((K_0(F_n), K_0(F_n)^+, K_1(F_n)) \cong (H_n,H_n^+,K_n)\).

Let \(\psi _n\) be a unital homomorphism \(F_n \rightarrow F_{n+1}\) which induces \(\gamma _n\) in \(K_0\) and \(\chi _n\) in \(K_1\). We write \(F_n = P_n C(Z_n) P_n\) where \(Z_n = Z_n^{\varvec{i}} \amalg \coprod _{i \ne \varvec{i}} \{ \theta _n^i \}\), and \(P_n = (P_n^{\varvec{i}},(1_{[n,i]})_{i \ne \varvec{i}}) \in M_{\infty }(C(Z_n^{\varvec{i}})) \oplus \bigoplus _{i \ne \varvec{i}} M_{[n,i]}(C(\{ \theta _n^i \}))\). Thus, evaluation in \(\theta _n^i\) induces a quotient map \(\pi _n: \, F_n \rightarrow {\hat{F}}_n \mathrel {:=}\bigoplus _i {\hat{F}}_n^i\), where \({\hat{F}}_n^i = M_{[n,i]}\). We require that \(\psi _n\) induce homomorphisms \({\hat{\psi }}_n: \, {\hat{F}}_n \rightarrow {\hat{F}}_{n+1}\) so that we obtain a commutative diagram

which induces in \(K_0\)

where the vertical arrows are the canonical projections. As \(\mathrm{Tor}(H_n) \subseteq G_n\), \(H_n / G_n\) is torsion-free, and there is a canonical projection \(H_n / \mathrm{Tor}(H_n) \rightarrow H_n / G_n\). Now let \(E_n \mathrel {:=}\bigoplus _p E_n^p\), \(E_n^p = M_{\left\{ n,p \right\} }\), such that \(K_0(E_n) \cong H_n / G_n\), and for fixed n, let \(\beta _0, \, \beta _1: \, {\hat{F}}_n \rightarrow E_n\) be (necessarily non-unital) homomorphisms which yield the commutative diagram

As in the unital case, we may assume \(\beta _0 \oplus \beta _1: \, {\hat{F}}_n \rightarrow E_n \oplus E_n\) to be injective.

Define

As \(\beta _0 \oplus \beta _1\) is injective, we view \({\hat{A}}_n\) as a subalgebra of \(C([0,1],E_n)\) via \((f,{\hat{a}}) \mapsto f\).

Choose for each n a homomorphism \({\hat{\varphi }}_n: \, {\hat{A}}_n \rightarrow {\hat{A}}_{n+1}\) such that the composition with the map \(C([0,1],E_{n+1}) \twoheadrightarrow C([0,1],E_{n+1}^q)\) induced by the canonical projection \(E_{n+1} \twoheadrightarrow E_{n+1}^q\),

is of the form

where u is a continuous path of unitaries \([0,1] \rightarrow U(E_{n+1}^q)\),

for some \(\pi _{\bullet }\) of the form \(\pi _{\bullet }: \, {\hat{A}}_n \rightarrow {\hat{F}}_n \twoheadrightarrow {\hat{F}}_n^i\), where the first map is given by \((f,{\hat{a}}) \mapsto {\hat{a}}\) and the second map is the canonical projection, and

for some continuous maps \(\lambda _{\bullet }: \, [0,1] \rightarrow [0,1]\) with \(\lambda _{\bullet }^{-1}(\left\{ 0,1 \right\} ) \subseteq \left\{ 0,1 \right\} \). We require that

commutes, where the vertical maps are given by \((f,{\hat{a}}) \mapsto {\hat{a}}\).

Then there exists a unique homomorphism \(\varphi _n: \, A_n \rightarrow A_{n+1}\) which fits into the commutative diagram

where all the unlabelled arrows are given by the canonical maps.

By construction, \(\varinjlim \left\{ A_n;\varphi _n \right\} \) has the desired Elliott invariant (the details are as in the unital case, see [20, § 13]). The same modification as in the unital case produces new connecting maps \(\xi _n: \, A_n \rightarrow A_{n+1}\) such that \(\varinjlim \left\{ A_n;\xi _n \right\} \) is a simple (stably projectionless) C*-algebra with prescribed Elliott invariant. Moreover, choosing \(\xi _n\) with the property that strictly positive elements are sent to strictly positive elements, \(\varinjlim \left\{ A_n;\xi _n \right\} \) will have continuous scale by [18, Theorem 9.3] (compare also [19, § 6]). In addition, we choose \(\xi _n\) such that full elements are sent to full elements.

Remark 2.1

In an earlier version of this paper, we modified the construction in [19, § 6] instead, which covers all Elliott invariants for stably projectionless C*-algebras with trivial pairing between K-theory and traces (\(\rho = 0\)). I would like to thank the referee for pointing out that [11] (see also [16]) describes a general construction exhausting all possible Elliott invariants with weakly unperforated pairing between K-theory and traces (in the stably projectionless case, this is precisely the condition that \(\rho \) is weakly unperforated as in Theorem 1.3).

We start with the following standard fact.

Lemma 3.1

Given an integer \(N > 1\), let \(\mu _N: \, S^1 \rightarrow S^1, \, z \mapsto z^N\), and set \(X_N \mathrel {:=}D^2 \cup _{\mu _N} S^1\), where we identify \(z \in S^1 = \partial D^2\) with \(\mu _N(z) \in S^1\). Then

Moreover, \((K_0(C(X_N)),K_0(C(X_N))^+,[1_{C(X_N)}],K_1(C(X_N))) \cong ({\mathbb {Z}}\oplus {\mathbb {Z}}/ N, \left\{ (0,0) \right\} \cup ({\mathbb {Z}}_{>0} \oplus {\mathbb {Z}}/ N), (1,0), \left\{ 0 \right\} )\).

In the following, we view \(S^2\) as the one point compactification of \(\mathring{D}^2\), \(S^2 = \mathring{D}^2 \cup \left\{ \infty \right\} \).

Lemma 3.2

Let \(X_N \twoheadrightarrow S^2\) be the continuous map sending \(\mathring{D}^2 \subseteq D^2\) identically to \(\mathring{D}^2 \subseteq S^2\), \(\partial D^2\) to \(\infty \) and \(S^1\) to \(\infty \). Let \(p_{X_N}\) be the pullback of the Bott line bundle on \(S^2\) (see for instance [39, § 6.2]) to \(X_N\) via this map. We view \(p_{X_N}\) as a projection in \(M_2(C(X_N))\). Then there is an isomorphism \(K_0(C(X_N)) \cong {\mathbb {Z}}\oplus {\mathbb {Z}}/ N\) identifying the class of \(1_{C(X_N)}\) with the generator of \({\mathbb {Z}}\) and the class of \(p_{X_N}\) with (1, 1).

Proof

Just analyse the K-theory exact sequence attached to \(0 \rightarrow C_0(\mathring{D}^2) \rightarrow C(X_N) \rightarrow C(S^1) \rightarrow 0\). \(\square \)

We recall another standard fact.

Lemma 3.3

Given an integer \(N > 1\), let \(Y_N \mathrel {:=}\Sigma X_N \cong D^3 \cup _{\Sigma \mu _N} S^2\), where we identify \(z \in S^2 = \partial D^3 \cong \Sigma S^1\) with \((\Sigma \mu _N)(z) \in \Sigma S^1 \cong S^2\). (Here \(\Sigma \) stands for suspension.) Then

Moreover, \(K_0(C(Y_N)) = {\mathbb {Z}}[1_{C(Y_N)}]\) and \(K_1(C(Y_N)) \cong {\mathbb {Z}}/ N\).

In the following, we view \(S^3\) as the one point compactification of \(\mathring{D}^3\), \(S^3 = \mathring{D}^3 \cup \left\{ \infty \right\} \).

Lemma 3.4

Let \(Y_N \twoheadrightarrow S^3\) be the continuous map sending \(\mathring{D}^3 \subseteq D^3\) identically to \(\mathring{D}^3 \subseteq S^3\), \(\partial D^3\) to \(\infty \) and \(S^2\) to \(\infty \). Then the dual map \(C(S^3) \rightarrow C(Y_N)\) induces in \(K_1\) a surjection \(K_1(C(S^3)) \cong {\mathbb {Z}}\twoheadrightarrow {\mathbb {Z}}/ N \cong K_1(C(Y_N))\).

Proof

Just analyse the K-theory exact sequence attached to \(0 \rightarrow C_0(\mathring{D}^3) \rightarrow C(Y_N) \rightarrow C(S^2) \rightarrow 0\).

Analysing K-theory exact sequences, the following is a straightforward observation.

Lemma 3.5

Let \(N, N' \in {\mathbb {Z}}_{>1}\) and \(m \in {\mathbb {Z}}_{>0}\) with \(N' \mid m \cdot N\), say \(m \cdot N = m' \cdot N'\). Define a continuous map

by sending \(x \in D^2\) to \(x^m \in D^2\) and \(z \in S^1\) to \(z^{m'} \in S^1\). Then the dual map \(\Psi _m: \, C(X_N) \rightarrow C(X_{N'})\) induces in \(K_0\) the homomorphism

Naturality of suspension yields

Lemma 3.6

Let \(N, N' \in {\mathbb {Z}}_{>1}\) and \(m \in {\mathbb {Z}}_{>0}\) with \(N' \mid m \cdot N\), say \(m \cdot N = m' \cdot N'\). Let \(\Sigma \Psi _m: \, C(Y_N) \rightarrow C(Y_{N'})\) be the map dual to \(\Sigma \Psi _m^*: \, Y_{N'} \cong \Sigma X_{N'} \rightarrow \Sigma X_N \cong Y_N\). Then \(\Sigma \Psi _m\) induces in \(K_1\) the homomorphism

In the following, we view \(X_N\) and \(Y_N\) as pointed spaces, with base point \(1 = (1,0) \in S^1 = \partial D^2 \subseteq D^2\) in \(X_N\) and base point \((1,0,0) \in S^2 = \partial D^3 \subseteq D^3\) in \(Y_N\). Note that \(\Psi _m\) and \(\Sigma \Psi _m\) preserve base points. Moreover, if \(\theta \) denotes the base point of \(X_N\), then the projection \(p_{X_N}\) in Lemma 3.2 satisfies

Now let \(H_n = H_n^{\varvec{i}} \oplus \bigoplus _{i \ne \varvec{i}} H_n^i\), \(H_{n+1} = H_{n+1}^{\varvec{j}} \oplus \bigoplus _{j \ne \varvec{j}} H_{n+1}^j\) be abelian groups with \(H_n^{\varvec{i}} = {\mathbb {Z}}\oplus T_n\), \(H_{n+1}^{\varvec{j}} = {\mathbb {Z}}\oplus T_{n+1}\) for finitely generated torsion groups \(T_n\), \(T_{n+1}\), and \(H_n^i = {\mathbb {Z}}\), \(H_{n+1}^j = {\mathbb {Z}}\) for all \(i \ne \varvec{i}\), \(j \ne \varvec{j}\). Let \((H_n^{\varvec{i}})^+ \mathrel {:=}\left\{ (0,0) \right\} \cup ({\mathbb {Z}}_{>0} \oplus T_n)\), \((H_n^i)^+ \mathrel {:=}{\mathbb {Z}}_{\ge 0}\) for all \(i \ne \varvec{i}\), \(H_n^+ \mathrel {:=}\bigoplus _i (H_n^i)^+ \subseteq H_n^{\varvec{i}} \oplus \bigoplus _{i \ne \varvec{i}} H_n^i = H_n\) and \(u_n = (([n,\varvec{i}],\tau _n),([n,i])_{i \ne \varvec{i}}) \in H_n^+\). Similarly, define \((H_{n+1}^j)^+\), \(H_{n+1}^+ \mathrel {:=}\bigoplus _i (H_{n+1}^j)^+\), and let \(u_{n+1} = (([n+1,\varvec{j}],\tau _{n+1}),([n+1,j])_{j \ne \varvec{j}}) \in H_{n+1}^+\). Let \(T_n = \bigoplus _k T_n^k\), where \(T_n^k = {\mathbb {Z}}/ N_n^k\), and \(T_{n+1} = \bigoplus _k T_{n+1}^l\), where \(T_{n+1}^l = {\mathbb {Z}}/ N_{n+1}^l\). Let \(\gamma _n: \, H_n \rightarrow H_{n+1}\) be a homomorphism with \(\gamma _n(u_n) = u_{n+1}\). (In the stably projectionless case, these order units are not part of the given data, but we can always choose such order units.) Let us fix n, and suppose that \(\gamma = \gamma _n\) induces a homomorphism \({\hat{\gamma }}: \, H_n / \mathrm{Tor}(H_n) = {\hat{H}}_n = \bigoplus _i {\hat{H}}_n^i \rightarrow \bigoplus _j {\hat{H}}_{n+1}^j = {\hat{H}}_{n+1} = H_{n+1} / \mathrm{Tor}(H_{n+1})\), where \({\hat{H}}_n^i = {\mathbb {Z}}= {\hat{H}}_{n+1}^j\) for all i and j. Viewing \({\hat{H}}_n\) as a subgroup (actually a direct summand) of \(H_n\), we obtain that \(\gamma _n = {\hat{\gamma }} + \tau + t\) for homomorphisms \(\tau : \, \mathrm{Tor}(H_n) \rightarrow \mathrm{Tor}(H_{n+1})\) and \(t: \, {\hat{H}}_n \rightarrow \mathrm{Tor}(H_{n+1})\). \({\hat{\gamma }}\) is given by an integer matrix \(({\hat{\gamma }}_{ji})\). Similarly, \(\tau \) is given by an integer matrix \((\tau _{lk})\), where we view \(\tau _{lk}\) as a homomorphism \(T_n^k \rightarrow T_{n+1}^l\). Also, t is given by an integer matrix \((t_{li})\), where we view \(t_{li}\) as a homomorphism \(H_n^i \rightarrow T_{n+1}^l\). Clearly, we can always arrange \(\tau _{lk}, t_{li} > 0\) for all l, k, i, and because of (2) and (6), we can also arrange

Here \(\#_0(k)\) is the number of direct summands in \(T_n\) (i.e., the number of indices k).

We have the following direct consequence of Lemma 3.1.

Lemma 3.7

Let \(X_n^{\varvec{i}} \mathrel {:=}\bigvee _k X_{N_n^k}\), where we take the wedge sum with respect to the base points of the individual \(X_{N_n^k}\). Denote the base point of \(X_n^{\varvec{i}}\) by \(\theta _n^{\varvec{i}}\). Set \(X_n \mathrel {:=}X_n^{\varvec{i}} \amalg \coprod _{i \ne \varvec{i}} \{ \theta _n^i \}\). Then

Define \(X_{n+1}\) in an analogous way, i.e., \(X_{n+1}^{\varvec{j}} \mathrel {:=}\bigvee _l X_{N_{n+1}^l}\), and \(X_{n+1} \mathrel {:=}X_{n+1}^{\varvec{j}} \amalg \coprod _{j \ne \varvec{j}} \{ \theta _{n+1}^j \}\). Now, for fixed n, our goal is to construct a homomorphism \(\psi \) realizing the homomorphism \(\gamma \) in \(K_0\).

The map \(\bigvee _l \Psi _{\tau _{lk}}^*: \, \bigvee _l X_{N_{n+1}^l} \rightarrow X_{N_n^k}\) induces the dual homomorphism \(\psi _{\tau }^k: \, C(X_{N_n^k}) \rightarrow C(X_{n+1}^{\varvec{j}})\). Here \(\Psi _{\tau _{lk}}\) are the maps from Lemma 3.5. The direct sum \(\bigoplus _k \psi _{\tau }^k: \, \bigoplus _k C(X_{N_n^k}) \rightarrow M_{\#_0(k)}(C(X_{n+1}^{\varvec{j}}))\) restricts to a homomorphism \(\psi _{\tau }: \, C(X_n^{\varvec{i}}) = C(\bigvee _k X_{N_n^k}) \rightarrow M_{\#_0(k)}(C(X_{n+1}^{\varvec{j}}))\).

Let \(p^{(\varvec{i})} \in M_2(C(X_{n+1}^{\varvec{j}})) = M_2(C(\bigvee _l X_{N_{n+1}^l}))\) be given by \(p^{(\varvec{i})} \vert _{C(X_{N_{n+1}^l})} = M_2(\Psi _{t_{l \varvec{i}}})(p_{X_{N_{n+1}^l}})\). Define \(\psi _t\) as the composite

Moreover, define \(\psi _{\varvec{j} \varvec{i}}: \, C(X_n^{\varvec{i}}) \rightarrow M_{{\hat{\gamma }}_{\varvec{j} \varvec{i}} + 1}(C(X_{n+1}^{\varvec{j}}))\) by setting

where we put \({\hat{\gamma }}_{\varvec{j} \varvec{i}} - \#_0(k) - 1\) copies of \(f(\theta _n^{\varvec{i}})\) on the diagonal.

For \(i \ne \varvec{i}\), let \(p^{(i)} \in M_2(C(X_{n+1}^{\varvec{j}})) = M_2(C(\bigvee _l X_{N_{n+1}^l}))\) be given by \(p^{(i)} \vert _{C(X_{N_{n+1}^l})} = M_2(\Psi _{t_{l i}})(p_{X_{N_{n+1}^l}})\). Define

where we put \({\hat{\gamma }}_{\varvec{j} i} - 1\) copies of 1 on the diagonal.

For \(j \ne \varvec{j}\), define

where we put \({\hat{\gamma }}_{j \varvec{i}}\) copies of \(f(\theta _n^{\varvec{i}})\) on the diagonal.

For \(i \ne \varvec{i}\) and \(j \ne \varvec{j}\), define

where we put \({\hat{\gamma }}_{ji}\) copies of 1 on the diagonal.

To unify notation, let us set \(X_n^i = \{ \theta _n^i \}\), \(X_{n+1}^j = \{ \theta _{n+1}^j \}\).

For \(u_n = (([n,\varvec{i}],\tau _n),([n,i])_{i \ne \varvec{i}}) \in H_n^+\), let \(s(n,\varvec{i})\) be a positive integer and \(P_n^{\varvec{i}} \in M_{s(n,\varvec{i})}(C(X_n^{\varvec{i}}))\) a projection such that:

• \(P_n^{\varvec{i}}\) is a sum of line bundles;

• \([P_n^{\varvec{i}}]\) corresponds to \(([n,\varvec{i}],\tau _n)\) under the identification in Lemma 3.7;

• \(P_n^{\varvec{i}}(\theta _n^{\varvec{i}}) = 1_{[n,\varvec{i}]}\) is of the form

  $$\begin{aligned} u^* \begin{pmatrix} 1 &{} &{} &{} &{} &{} \\ &{} \ddots &{} &{} &{} &{} \\ &{} &{} 1 &{} &{} &{} \\ &{} &{} &{} 0 &{} &{} \\ &{} &{} &{} &{} \ddots &{} \\ &{} &{} &{} &{} &{} 0 \end{pmatrix} u, \ \ \ \mathrm{where} \ u \ \text {is a permutation matrix}. \end{aligned}$$

\(P_n^{\varvec{i}}\) exists because of Lemma 3.2. Moreover, we can extend \(P_n^{\varvec{i}}\) by \(1_{[n,i]}\) to a projection in \(\bigoplus _i M_{s(n,i)}(C(X_n^i))\) such that \([P_n]\) corresponds to \(u_n\) under the isomorphism in Lemma 3.7. Here \(s(n,i) = [n,i]\) whenever \(i \ne \varvec{i}\). Then

sends \(P_n\) to \(P_{n+1}\), where \(P_{n+1}\) is of the same form as \(P_n\), with \([P_{n+1}]\) corresponding to \(u_{n+1}\) under the isomorphism from Lemma 3.7. Hence the map in (9) restricts to a unital homomorphism

which in \(K_0\) induces \(\gamma \) by Lemma 3.5.

Now we turn to \(K_1\). Assume \(K_n = \bigoplus _i K_n^i\) is an abelian group, where for a distinguished index \(\varvec{i}\), \(K_n^{\varvec{i}} = T_n\) is a finitely generated torsion group \(T_n = \bigoplus _k T_n^k\), \(T_n^k = {\mathbb {Z}}/ N_n^k\), and \(K_n^i = {\mathbb {Z}}\) for all \(i \ne \varvec{i}\). Similarly, let \(K_{n+1} = \bigoplus _j K_{n+1}^j\) be an abelian group, where for a distinguished index \(\varvec{j}\), \(K_{n+1}^{\varvec{j}} = T_{n+1}\) is a finitely generated torsion group \(T_{n+1} = \bigoplus _l T_{n+1}^l\), \(T_{n+1}^l = {\mathbb {Z}}/ N_{n+1}^l\), and \(K_{n+1}^j = {\mathbb {Z}}\) for all \(j \ne \varvec{j}\). For fixed n, let \(\chi : \, K_n \rightarrow K_{n+1}\) be a homomorphism which is a sum \(\chi = {\hat{\chi }} + \tau + t\), where \({\hat{\chi }}: \, \bigoplus _{i \ne \varvec{i}} K_n^i \rightarrow \bigoplus _{j \ne \varvec{j}} K_{n+1}^j\) is given by an integer matrix \(({\hat{\chi }}_{ji})\) (viewing \({\hat{\chi }}_{ji}\) as a homomorphism \(K_n^i \rightarrow K_{n+1}^j\)), \(\tau : \, T_n \rightarrow T_{n+1}\) is given by an integer matrix \((\tau _{lk})\) (viewing \(\tau _{lk}\) as a homomorphism \(T_n^k \rightarrow T_{n+1}^l\)), and \(t: \, \bigoplus _{i \ne \varvec{i}} K_n^i \rightarrow T_{n+1}\) is given by an integer matrix \((t_{li})\) (viewing \(t_{li}\) as a homomorphism \(K_n^i \rightarrow T_{n+1}^l\)). We can always arrange that all the entries of these matrices are positive integers.

The following is a direct consequence of Lemma 3.3.

Lemma 3.8

Let \(Y_n^{\varvec{i}} = \bigvee _k Y_{N_n^k}\) and \(Y_n = Y_n^{\varvec{i}} \vee \bigvee _{i \ne \varvec{i}} S^3\). Then \(K_0(C(Y_n)) \cong {\mathbb {Z}}\) and \(K_1(C(Y_n)) \cong K_n\).

We view \(Y_n\) as a pointed space, and let \(\theta _n\) be the base point of \(Y_n\). Now let \(\psi _{\tau }^k: \, C(Y_{N_n^k}) \rightarrow C(\bigvee _l Y_{N_{n+1}^l}) = C(Y_{n+1}^{\varvec{j}})\) be the dual homomorphism of the map \(\bigvee _l \Sigma (\Psi _{\tau _{lk}}^*): \, Y_{n+1}^{\varvec{j}} = \bigvee _l Y_{N_{n+1}^l} \rightarrow Y_{N_n^k}\). Here \(\Sigma (\Psi _{\tau _{lk}}^*)\) are the maps from Lemma 3.6. The direct sum \(\bigoplus _k \psi _{\tau }^k: \, \bigoplus _k C(Y_{N_n^k}) \rightarrow M_{\#_1(k)}(C(Y_{n+1}^{\varvec{j}}))\) restricts to a homomorphism \(\psi _{\varvec{i}}: \, C(Y_n^{\varvec{i}}) = C(\bigvee _k Y_{N_n^k}) \rightarrow M_{\#_1(k)}(C(Y_{n+1}^{\varvec{j}})) \hookrightarrow M_{\#_1(k)}(C(Y_{n+1}))\).

For \(i \ne \varvec{i}\), define \(\psi _{\varvec{j}i}: \, C(Y_n^i) = C(S^3) \rightarrow C(Y_{n+1}^{\varvec{j}})\) as the dual map of the composite

where \(\Omega ^*_l\) is the map \(Y_{N_{n+1}^l} \rightarrow S^3\) constructed in Lemma 3.4.

For \(i \ne \varvec{i}\) and \(j \ne \varvec{j}\), define \(\psi _{ji}: \, C(Y_n^i) = C(S^3) \rightarrow C(S^3) = C(Y_{n+1}^j)\) as the dual map of \(\Sigma \Sigma \mu _{{\hat{\chi }}_{ji}}: \, S^3 \cong \Sigma \Sigma S^1 \rightarrow \Sigma \Sigma S^1 \cong S^3\), where \(\mu _{{\hat{\chi }}_{ji}}\) is the map from Lemma 3.1.

For every \(i \ne \varvec{i}\), we thus obtain the direct sum \(\bigoplus _j \psi _{ji}: \, C(Y_n^i) \rightarrow \bigoplus _j C(Y_{n+1}^j)\) with image in \(C(Y_{n+1}) = C(\bigvee _j Y_{n+1}^j) \subseteq \bigoplus _j C(Y_{n+1}^j)\). Hence we obtain a homomorphism \(\psi _i: \, C(Y_n^i) \rightarrow C(Y_{n+1})\).

Now let \(\#_1(i)\) be the number of summands of \(K_n\). Then let \(\psi : \, C(Y_n) \rightarrow M_{\#_1(k) + \#_1(i) - 1}(C(Y_{n+1}))\) be the restriction of \(\bigoplus _i \psi _i\) to \(C(Y_n) = C(\bigvee _i Y_n^i) \subseteq \bigoplus _i C(Y_n^i)\). By construction, and using Lemmas 3.4 and 3.6 , \(\psi \) induces \(\chi \) in \(K_1\).

We now combine our two constructions. Define \(Z_n = X_n \vee Y_n\), where we identify the base point \(\theta _n^{\varvec{i}} \in X_n^{\varvec{i}} \subseteq X_n\) with \(\theta _n \in Y_n\). We extend \(P_n\) from \(X_n\) constantly to \(Y_n\) (with value \(P_n(\theta _n^{\varvec{i}})\)). Note that \(\mathrm{rk}\,(P_{n+1}(\theta _{n+1}^{\varvec{j}})) = {\hat{\gamma }}_{\varvec{j} \varvec{i}} \cdot \mathrm{rk}\,(P_n (\theta _n^{\varvec{i}}))\). Because of (2) and (6), we can arrange \({\hat{\gamma }}_{\varvec{j} \varvec{i}} \ge \#_1(k) + \#_1(i) - 1\). By adding \({\text {ev}}_{\theta _n}\) on the diagonal if necessary, we can modify \(\psi \) to a homomorphism \(\psi : \, C(Y_n) \rightarrow M_{{\hat{\gamma }}_{\varvec{j} \varvec{i}}}(C(Y_{n+1}))\) which induces \(\gamma \) in \(K_1\). We can thus think of \(M_{[n,\varvec{i}]}(\psi )\) as a unital homomorphism \(P_n(\theta _n^{\varvec{i}}) M_{s(n,\varvec{i})}(C(Y_n)) P_n(\theta _n^{\varvec{i}}) \rightarrow P_{n+1}(\theta _{n+1}^{\varvec{j}}) M_{s(n+1,\varvec{j})}(C(Y_{n+1})) P_{n+1}(\theta _{n+1}^{\varvec{j}})\), i.e., as a unital homomorphism \(P_n M_{s(n,\varvec{i})}(C(Y_n)) P_n \rightarrow P_{n+1} M_{s(n+1,\varvec{j})}(C(Y_{n+1})) P_{n+1}\). In combination with the homomorphism (10), we obtain a unital homomorphism

which induces \(\gamma \) in \(K_0\), sending \(u_n\) to \(u_{n+1}\), and \(\chi \) in \(K_1\).

Evaluation at \(\theta _n^{\varvec{i}} = \theta _n\) and \(\theta _n^i\) (for \(i \ne \varvec{i}\)) induces a quotient homomorphism which fits into a commutative diagram

(11)

which induces in \(K_0\)

Remark 3.9

If all \(K_n\) are torsion-free, then we can replace \(S^3\) by \(S^1\) in our construction of \(Y_n\).

4.1 The general construction with concrete models

Applying our construction in Sect. 3, we obtain concrete models for \(F_n\), \({\hat{F}}_n\), \(\gamma _n\) and \({\hat{\gamma }}_n\) which we now plug into the general construction in Sects. 2.1 and  2.2. Note that it is crucial that we work with these concrete models from Sect. 3. The reason is that only for these models can we provide groupoid descriptions of the C*-algebras and their homomorphisms which arise in the general construction (see Sect. 6).

Note that with these concrete models, the composition

where the first and third maps are the canonical ones, is of the form

for a permutation matrix u.

Apart from inserting these concrete models, we keep the same construction as in Sects. 2.1 and 2.2.

4.2 Summary of the construction

In both the unital and stably projectionless cases, the C*-algebra with the prescribed Elliott invariant which we constructed is an inductive limit whose building blocks are of the form

where:

• \(E_n\) is finite dimensional;

• \(F_n\) is homogeneous of the form \(P_n M_{\infty }(Z_n) P_n\), where \(P_n\) is a sum of line bundles, and there are points \(\theta _n^i \in Z_n\), one for each connected component, and all connected components just consist of \(\theta _n^i\) with the only possible exception being the component of a distinguished point \(\theta _n^{\varvec{i}}\);

• both \(\beta _0\) and \(\beta _1\) are compositions of the form \(F_n \rightarrow \bigoplus _i M_{[n,i]} \rightarrow E_n\), where the first homomorphism is given by evaluation in \(\theta _n^i \in Z_n\) and the second homomorphism is determined by the composites \(M_{[n,i]} \hookrightarrow \bigoplus _i M_{[n,i]} \rightarrow E_n \twoheadrightarrow E_n^p\) (where \(E_n^p\) is a matrix block of \(E_n\)), which are of the form

  $$\begin{aligned} x \mapsto v^* \begin{pmatrix} x &{} &{} &{} &{} &{} \\ &{} \ddots &{} &{} &{} &{} \\ &{} &{} x &{} &{} &{} \\ &{} &{} &{} 0 &{} &{} \\ &{} &{} &{} &{} \ddots &{} \\ &{} &{} &{} &{} &{} 0 \end{pmatrix} v \end{aligned}$$

  for a permutation matrix v.

The connecting maps \(\varphi _n\) of our inductive limit can be described as two parts:

Both parts send \((f,a) \in A_n\) to an element which is in diagonal form up to permutation, i.e.,

where for the entries on the diagonal, there are the following possibilities:

• a map of the form

  $$\begin{aligned}&[0,1] \ni t \mapsto f^p(\lambda (t)), \ \mathrm{for} \ \mathrm{a} \ \mathrm{continuous} \ \mathrm{map} \ \lambda : \, [0,1] \rightarrow [0,1] \nonumber \\&\mathrm{with} \ \lambda ^{-1}(\left\{ 0,1 \right\} ) \subseteq \left\{ 0,1 \right\} , \end{aligned}$$

  (16)

  where \(f^p\) is the image of f under the canonical projection \(C([0,1],E_n) \twoheadrightarrow C([0,1],E_n^p)\);

• a map of the form

  $$\begin{aligned}{}[0,1] \ni t \mapsto \tau (t) a(x(t)), \end{aligned}$$

  (17)

  where \(x: \, [0,1] \rightarrow Z_n\) is continuous and \(\tau (t): \, P_n(x(t)) M_{\infty } P_n(x(t)) \cong P_n(\theta _n^i) M_{\infty }P_n(\theta _n^i)\) is an isomorphism depending continuously on t, with \(\theta _n^i\) in the same connected component as x(t), and \(\tau (t) = \mathrm{id}\) if \(x(t) = \theta _n^i\);

• an element of \(P_{n+1} M_{\infty }(C(Z_{n+1})) P_{n+1}\) with support in an isolated point \(\theta _{n+1}^j\), which is of the form

  $$\begin{aligned} f^p(\varvec{t}), \ \mathrm{for} \ \mathrm{some} \ \varvec{t} \in (0,1), \end{aligned}$$

  (18)

  where \(f^p\) is the image of f under the canonical projection \(C([0,1],E_n) \twoheadrightarrow C([0,1],E_n^p)\);

• an element of \(P_{n+1} M_{\infty }(C(Z_{n+1})) P_{n+1}\) with support in an isolated point \(\theta _{n+1}^j\), which is of the form

  $$\begin{aligned}&\tau (a(x)) \ \ \ \text {for some} \ x \in Z_n \ \mathrm{with} \ x \notin \{ \theta _n^i \}_i \ \nonumber \\&\text {and an isomorphism} \ \tau : \, P_n(x) M_{\infty } P_n(x) \cong P_n(\theta _n^i) M_{\infty } P_n(\theta _n^i),\qquad \end{aligned}$$

  (19)

  where \(\theta _n^i\) is in the same connected component as x;

• an element of \(P_{n+1} M_{\infty }(C(Z_{n+1})) P_{n+1}\) with support in an isolated point \(\theta _{n+1}^j\), which is of the form

  $$\begin{aligned} a(\theta _n^i), \ \mathrm{where} \ \theta _n^i \ \text {is an isolated point in} \ Z_n; \end{aligned}$$

  (20)

• an element of \(P_{n+1} M_{\infty }(C(Z_{n+1})) P_{n+1}\), which is of the form

  $$\begin{aligned} (a_{ij} \cdot q)_{ij}, \ \mathrm{where} \ q \ \text {is a line bundle over} \ Z_{n+1}, \ \mathrm{and} \ (a_{ij}) = a(\theta _n^i);\qquad \end{aligned}$$

  (21)

• an element of \(P_{n+1} M_{\infty }(C(Z_{n+1})) P_{n+1}\) of the form

  $$\begin{aligned} a \circ \lambda , \end{aligned}$$

  (22)

  where \(\lambda : \, Z_{n+1} \rightarrow Z_n\) is a continuous map whose image is only contained in one wedge summand of \(Z_n\) (see our constructions in Sect. 3).

Note that in (17) and (19), we identify \(P_n(\theta _n^{\varvec{i}}) M_{\infty } P_n(\theta _n^{\varvec{i}})\) with \(M_{[n,\varvec{i}]}\) via a fixed isomorphism.

Let \(P^{\varvec{a}} \in M(A_{n+1})\) be projections, with \(\sum _{\varvec{a}} P^{\varvec{a}} = 1\), giving rise to the diagonal form in (15), and let \(\varphi ^{\varvec{a}}\) be the homomorphism \(A_n \rightarrow P_{\varvec{a}} A_{n+1} P_{\varvec{a}}, \, x \mapsto P^{\varvec{a}} u \varphi (a) u^* P^{\varvec{a}}\). Since each of the \(P^{\varvec{a}}\) either lies in \(C([0,1],E_{n+1}^q)\) or \(F_{n+1}\), we have \(\mathrm{im\,}(\varphi ^{\varvec{a}}) \subseteq P^{\varvec{a}} C([0,1],E_{n+1}^q) P^{\varvec{a}}\) or \(\mathrm{im\,}(\varphi ^{\varvec{a}}) \subseteq P^{\varvec{a}} F_{n+1} P^{\varvec{a}}\). Then both maps in (13), (14) are of the form \(u^* (\bigoplus _{\varvec{a}} \varphi ^{\varvec{a}}) u\). The unitary u is a permutation matrix for the map in (13) and is a unitary in \(C([0,1],E_{n+1})\) such that u(0) and u(1) are permutation matrices for the map in (14).

Remark 4.1

Let us write \(C_n \mathrel {:=}C([0,1],E_n)\) and \(u_{n+1} \in C_{n+1}\) for the unitary for the map in (14). The only reason we need \(u_{n+1}\) is to ensure that we send (f, a) to an element satisfying the right boundary conditions at \(t=0\) and \(t=1\). For this, only the values \(u_{n+1,t} \mathrel {:=}u_{n+1}(t)\) at \(t \in \left\{ 0,1 \right\} \) matter. Therefore, by an iterative process, we can change \(\beta _t\) in order to arrange \(u_{n+1}=1\) for the map in (14): First of all, it is easy to see that \(\varphi _n\) extends uniquely to a homomorphism \(\Phi _n: \, C_n \oplus F_n \rightarrow C_{n+1} \oplus F_{n+1}\). Let us write \(\Phi _n^C\) and \(\Phi _n^F\) for the composites

As \(\varphi _n\) sends strictly positive elements to strictly positive elements, \(\Phi _n\) is unital. Now, for all n, let \(\Lambda _n(t) \subseteq [0,1]\) be a finite set such that for all \((f_n,a_n) \in A_n\) with \(\varphi (f_n,a_n) = (f_{n+1},a_{n+1}) \in A_{n+1}\), \(f_n \vert _{\Lambda _n(t)} \equiv 0\) implies \(f_{n+1}(t) = 0\). In other words, \(\Lambda _n(t)\) are the evaluation points for \(f_{n+1}(t)\). Similarly, let \(T_n \subseteq (0,1)\) be such that for all \((f_n,a_n) \in A_n\) with \(\varphi (f_n,a_n) = (f_{n+1},a_{n+1}) \in A_{n+1}\), \(f_n \vert _{T_n} \equiv 0\) and \(a_n = 0\) imply \(a_{n+1} = 0\). Now we choose inductively on n unitaries \(v_n \in U(C_n)\) and \(u_{n+1} \in U(C_{n+1})\) such that, for all n, \(v_n(s) = 1\) for all \(s \in (\Lambda _n(0) \cup \Lambda _n(1) \cup T_n) \setminus \left\{ 0,1 \right\} \), \(u_{n+1}(t) = u_{n+1,t}\) for \(t \in \left\{ 0,1 \right\} \), and \(v_{n+1} = \Phi _n^C(v_n,1) u_{n+1}^*\): Simply start with \(v_1 \mathrel {:=}1\), and if \(v_n\) and \(u_n\) have been chosen, choose \(u_{n+1} \in U(C_{n+1})\) such that \(u_{n+1}(t) = u_{n+1,t}\) for all \(t \in \left\{ 0,1 \right\} \) and \(u_{n+1}(s) = \Phi _n^C(v_n,1)(s)\) for all \(s \in (\Lambda _n(0) \cup \Lambda _n(1) \cup T_n) \setminus \left\{ 0,1 \right\} \), and set \(v_{n+1} \mathrel {:=}\Phi _n^C(v_n,1) u_{n+1}^*\). If we now take this \(u_{n+1}\) for the map in (14) giving rise to \(\varphi _n\) and \(\Phi _n\), then we obtain a commutative diagram

which restricts to

where the unitary \({\bar{u}}_{n+1}\) for the map in (14) for \({\bar{\varphi }}_n\) is now trivial, \({\bar{u}}_{n+1} = 1\), and \({\bar{A}}_n\) is of the same form (12) as \(A_n\), with \({\bar{\beta }}_t = v_n(t)^* \beta _t v_n(t)\) of the same form as \(\beta _t\) for \(t = 0, 1\) (the point being that \(v_n(t)\) is a permutation matrix). Obviously, we have \(\varinjlim \left\{ {\bar{A}}_n;{\bar{\varphi }}_n \right\} \cong \varinjlim \left\{ A_n;\varphi _n \right\} \).

Remark 4.2

Note that the construction described in Sect. 4.2 also encompasses (a slight modification of) the C*-algebra construction in [19, § 6]. (In particular, one obtains model algebras of rational generalized tracial rank one, in the sense of [19].)

In this section, we improve the main result in [4, § 3] and give a C*-algebraic interpretation. Let us first recall [4, Theorem 3.6]. We use the same notations and definitions as in [4, 36]. We start with the following

Remark 5.1

We can drop the assumptions of second countability for groupoids and separability for C*-algebras in [36] if we replace “topologically principal” by “effective” throughout. In other words, given a twisted étale effective groupoid \((G,\Sigma )\), i.e., a twisted étale groupoid \((G,\Sigma )\) where G is effective (not necessarily second countable), \((C^*_r(G,\Sigma ),C_0(G^{(0)}))\) is a Cartan pair; and conversely, given a Cartan pair (A, B) (where A is not necessarily separable), the Weyl twist \((G(A,B),\Sigma (A,B))\) from [36] is a twisted étale effective groupoid. These constructions are inverse to each other, i.e., there are canonical isomorphisms \((G,\Sigma ) \cong (G(C^*_r(G,\Sigma ),C_0(G^{(0)})),\Sigma (C^*_r(G,\Sigma ),C_0(G^{(0)})))\) (provided by [36, 4.13, 4.15, 4.16]) and \((A,B) \cong (C^*_r(G(A,B),\Sigma (A,B)),C_0(G(A,B)^{(0)}))\) (provided by [36, 5.3, 5.8, 5.9]). Similarly, everything in [4, § 3] works without the assumption of second countability. In particular, [4, Theorem 3.6] holds for general twisted étale groupoids if we replace “topologically principal” by “effective”. This is why in this section, we formulate everything for twisted étale effective groupoids and general Cartan pairs. In our applications later on, however, we will only consider second countable groupoids and separable C*-algebras.

Now suppose that \((A_n,B_n)\) are Cartan pairs, let \((G_n,\Sigma _n)\) be their Weyl twists, and set \(X_n \mathrel {:=}G_n^{(0)}\). Let \(\varphi _n: \, A_n \rightarrow A_{n+1}\) be injective *-homomorphisms. Assume that there are twisted groupoids \((H_n,T_n)\), with \(Y_n \mathrel {:=}H_n^{(0)}\), together with twisted groupoid homomorphisms \((i_n,\imath _n): \, (H_n,T_n) \rightarrow (G_{n+1},T_{n+1})\) and \(({\dot{p}}_n,p_n): \, (H_n,T_n) \rightarrow (G_n,T_n)\) such that \(i_n: \, H_n \rightarrow G_{n+1}\) is an embedding with open image, and \({\dot{p}}_n: \, H_n \rightarrow G_n\) is surjective, proper, and fibrewise bijective (i.e., for every \(y \in Y_n\), \({\dot{p}}_n \vert _{(H_n)_y}\) is a bijection onto \((G_n)_{{\dot{p}}_n(y)}\)). Suppose that \(\varphi _n = (\imath _n)_* \circ p_n^*\) for all n. Further assume that condition (LT) is satisfied, i.e., for every continuous section \(\rho : \, U \rightarrow \rho (U)\) for the canonical projection \(\Sigma _n \twoheadrightarrow G_n\), where \(U \subseteq G_n\) is open, there is a continuous section \(\tilde{\rho }: \, {\dot{p}}_n^{-1}(U) \rightarrow \tilde{\rho }({\dot{p}}_n^{-1}(U))\) for the canonical projection \(T_n \twoheadrightarrow H_n\) such that \(\tilde{\rho }({\dot{p}}_n^{-1}(U)) \subseteq {\dot{p}}_n^{-1}(\rho (U))\) and \(p_n \circ \tilde{\rho } = \rho \circ {\dot{p}}_n\). Also assume that condition (E) is satisfied, i.e., for every continuous section \(t: \, U \rightarrow t(U)\) for the source map of \(G_n\), where \(U \subseteq X_n\) and \(t(U) \subseteq G_n\) are open, there is a continuous section \(\tilde{t}: \, {\dot{p}}_n^{-1}(U) \rightarrow \tilde{t}({\dot{p}}_n^{-1}(U))\) for the source map of \(H_n\) such that \(\tilde{t}({\dot{p}}_n^{-1}(U)) \subseteq {\dot{p}}_n^{-1}(t(U))\) and \({\dot{p}}_n \circ \tilde{t} = t \circ {\dot{p}}_n\).

In this situation, define

Then [4, Theorem 3.6] tells us that

• (a) \(({\bar{G}}_n,{\bar{\Sigma }}_n)\) are twisted groupoids, and \((i_n,\imath _n)\) induce twisted groupoid homomorphisms \(({\bar{i}}_n,{\bar{\imath }}_n): \, ({\bar{G}}_n,{\bar{\Sigma }}_n) \rightarrow ({\bar{G}}_{n+1},{\bar{\Sigma }}_{n+1})\) such that \({\bar{i}}_n\) is an embedding with open image for all n, and

  $$\begin{aligned} {\bar{\Sigma }} \mathrel {:=}\varinjlim \left\{ {\bar{\Sigma }}_n; {\bar{\imath }}_n \right\} \ \mathrm{and} \ {\bar{G}} \mathrel {:=}\varinjlim \left\{ {\bar{G}}_n; {\bar{i}}_n \right\} \end{aligned}$$

  (24)

  defines a twisted étale groupoid \(({\bar{G}},{\bar{\Sigma }})\),

• (b) & (c) \((\varinjlim \left\{ A_n;\varphi _n \right\} ,\varinjlim \left\{ B_n;\varphi _n \right\} )\) is a Cartan pair whose Weyl twist is given by \(({\bar{G}},{\bar{\Sigma }})\).

Remark 5.2

It is clear that the proof of [4, Theorem 3.6] shows that if all \(B_n\) are C*-diagonals, i.e., all \(G_n\) are principal, then \({\bar{G}}\) is principal, i.e., \(\varinjlim \left\{ B_n;\varphi _n \right\} \) is a C*-diagonal.

It turns out that conditions (LT) and (E) are redundant.

Lemma 5.3

In the situation above, conditions (LT) and (E) are automatically satisfied.

Proof

To prove condition (LT), let \(\rho : \, U \rightarrow \rho (U)\) be a continuous section for the canonical projection \(\pi _n: \, \Sigma _n \twoheadrightarrow G_n\), where \(U \subseteq G_n\) is open. Let \(\pi _{n+1}: \, \Sigma _{n+1} \twoheadrightarrow G_{n+1}\) be the canonical projection. Then \(\pi _{n+1} \vert _{p_n^{-1}(\rho (U))}: \, p_n^{-1}(\rho (U)) \rightarrow {\dot{p}}_n^{-1}(U)\) is bijective. Indeed, given \(\tau _1, \tau _2 \in p_n^{-1}(\rho (U))\) with \(\pi _{n+1}(\tau _1) = \pi _{n+1}(\tau _2) \mathrel {=:}\eta \in H_n\), we must have \(\tau _2 = z \cdot \tau _1\) for some \(z \in {\mathbb {T}}\). Also, \(\pi _n(p_n(\tau _1)) = {\dot{p}}_n(\eta ) = \pi _n(p_n(\tau _1))\). As \(\pi _n \vert _{\rho (U)}: \, \rho (U) \rightarrow U\) is bijective (with inverse \(\rho \)), we deduce \(p_n(\tau _1) = p_n(\tau _2)\). Hence \(p_n(\tau _1) = p_n(\tau _2) = z \cdot p_n(\tau _1)\), which implies \(z = 1\), i.e., \(\tau _2 = \tau _1\). This proves injectivity, and surjectivity is easy to see. As \(\pi _{n+1}\) is open, \(\tilde{\rho } \mathrel {:=}(\pi _{n+1} \vert _{p_n^{-1}(\rho (U))})^{-1}: \, {\dot{p}}_n^{-1}(U) \rightarrow p_n^{-1}(\rho (U))\) is the continuous section we are looking for.

To verify (E), let \(t: \, U \rightarrow t(U)\) be a continuous section for the source map \(s_n\) of \(G_n\), where \(U \subseteq X_n\) and \(t(U) \subseteq G_n\) are open. Let \(s_{n+1}\) be the source map of \(H_n\). Then \(s_{n+1} \vert _{{\dot{p}}_n^{-1}(t(U))}: \, {\dot{p}}_n^{-1}(t(U)) \rightarrow {\dot{p}}_n^{-1}(U)\) is bijective. Indeed, given \(\eta _1, \eta _2 \in {\dot{p}}_n^{-1}(t(U))\) with \(s_{n+1}(\eta _1) = s_{n+1}(\eta _2) \mathrel {=:}y \in Y_n\), we must have \(s_n({\dot{p}}_n(\eta _1)) = {\dot{p}}_n(y) = s_n({\dot{p}}_n(\eta _2))\). As \(s_n \vert _{t(U)}: \, t(U) \rightarrow U\) is bijective (with inverse t), we deduce \({\dot{p}}_n(\eta _1) = {\dot{p}}_n(\eta _2)\). Since \({\dot{p}}_n\) is fibrewise bijective, this implies \(\eta _1 = \eta _2\). This proves injectivity, and surjectivity is easy to see. As \({\dot{p}}_n^{-1}(t(U))\) is open and \(s_{n+1}\) is open, \(\tilde{t} \mathrel {:=}(s_{n+1} \vert _{{\dot{p}}_n^{-1}(t(U))})^{-1}: \, {\dot{p}}_n^{-1}(U) \rightarrow {\dot{p}}_n^{-1}(t(U))\) is the continuous section we are looking for.

Let us now determine which *-homomorphisms are of the form \(\imath _* \circ p^*\). Let (A, B) and \(({\hat{A}},{\hat{B}})\) be Cartan pairs with normalizers \(N \mathrel {:=}N_A(B)\), \({\hat{N}} \mathrel {:=}N_{{\hat{A}}}({\hat{B}})\) and faithful conditional expectations \(P: \, A \twoheadrightarrow B\), \({\hat{P}}: \, {\hat{A}} \twoheadrightarrow {\hat{B}}\). Let \((G,\Sigma )\) and \(({\hat{G}},{\hat{\Sigma }})\) be the Weyl twists of (A, B) and \(({\hat{A}},{\hat{B}})\). Suppose that \(\varphi : \, A \rightarrow {\hat{A}}\) is an injective *-homomorphism.

Proposition 5.4

The following are equivalent:

1. (i)

   \(\varphi (B) \subseteq {\hat{B}}\), \(\varphi (N) \subseteq {\hat{N}}\), \({\hat{P}} \circ \varphi = \varphi \circ P\);

2. (ii)

   There exists a twisted étale effective groupoid (H, T) and twisted groupoid homomorphisms \((i,\imath ): \, (H,T) \rightarrow ({\hat{G}},{\hat{\Sigma }})\), \(({\dot{p}},p): \, (H,T) \rightarrow (G,\Sigma )\), where i is an embedding with open image and \({\dot{p}}\) is surjective, proper and fibrewise bijective, such that \(\varphi = \imath _* \circ p^*\).

Proof

(ii) \(\Rightarrow \) (i): It is easy to see that \((\imath _* \circ p^*)(B) \subseteq {\hat{B}}\). Given an open bisection S of G, \({\dot{p}}^{-1}(S)\) is an open bisection of H, and then \(i({\dot{p}}^{-1}(S))\) is an open bisection of \({\hat{G}}\). Therefore, \((\imath _* \circ p^*)(N) \subseteq {\hat{N}}\). Finally, we have \({\hat{P}} \circ (\imath _* \circ p^*) = (\imath _* \circ p^*) \circ P\) because \({\dot{p}}^{-1}(G^{(0)}) = H^{(0)}\).

(i) \(\Rightarrow \) (ii): Let \(\breve{B}\) be the ideal of \({\hat{B}}\) generated by \(\varphi (B)\), and \(\breve{A} \mathrel {:=}C^*(\varphi (A),\breve{B})\). Then \((\breve{A},\breve{B})\) is a Cartan pair: It is clear that \(\breve{B}\) contains an approximate unit for \(\breve{A}\). To see that \(\breve{B}\) is maximal abelian, take \(a \in \breve{A} \cap (\breve{B})'\). Let \(b \in {\hat{B}}\), and take an approximate unit \((h_{\lambda }) \subseteq \breve{B}\) for \(\breve{A}\). Then \(ba = \lim _{\lambda } b h_{\lambda } a = \lim _{\lambda } a b h_{\lambda } = \lim _{\lambda } a h_{\lambda } b = ab\). Hence \(a \in \breve{A} \cap ({\hat{B}})' = \breve{A} \cap {\hat{B}} = \breve{B}\) (the last equality holds because \(\breve{B}\) contains an approximate unit for \(\breve{A}\), and \(\breve{B} \cdot {\hat{B}} \subseteq \breve{B}\)). This shows \(\breve{A} \cap (\breve{B})' = \breve{B}\). Moreover, we have \(\varphi (N) \subseteq \breve{N} \mathrel {:=}N_{\breve{A}}(\breve{B})\): Let \(n \in \varphi (N)\), \(b \in \breve{B}\), and \((h_{\lambda }) \subseteq B\) be an approximate unit for A. Then \(nbn^* \in {\hat{B}}\) as \(n \in \varphi (N) \subseteq {\hat{N}}\), and thus \(nbn^* = \lim _{\lambda } \varphi (h_{\lambda }) n b n^* \subseteq \overline{\varphi (B) \cdot {\hat{B}}} \subseteq \breve{B}\). Finally, it is clear that \(\breve{P} \mathrel {:=}{\hat{P}} \vert _{\breve{A}}\) is a faithful conditional expectation onto \(\breve{B}\).

Let (H, T) be the Weyl twist attached to \((\breve{A},\breve{B})\), and write \(X \mathrel {:=}G^{(0)}\), \(Y \mathrel {:=}H^{(0)}\) and \({\hat{X}} \mathrel {:=}{\hat{G}}^{(0)}\). It is easy to see that \(\breve{N} \subseteq {\hat{N}}\). Hence we may define maps

Clearly, i and \(\imath \) are continuous groupoid homomorphisms. i is injective since \([x,\alpha _n,y] = [x',\alpha _{n'},y']\) in \({\hat{G}}\) implies \(x = x'\), \(y = y'\) and \(\alpha _n = \alpha _{n'}\) on a neighbourhood \(U \subseteq {\hat{X}}\) of y, so that \(\alpha _n = \alpha _{n'}\) on \(U \cap Y\), which is a neighbourhood of y in Y, and hence \([x,\alpha _n,y] = [x',\alpha _{n'},y']\) in H. The image of i is given by \(\bigcup _{n \in \breve{N}} \left\{ [\alpha _n(y),\alpha _n,y] \text {: }y \in \mathrm{dom\,}(n) \right\} \) which is clearly open in \({\hat{G}}\). Finally, it is easy to see that we have a commutative diagram

where the upper horizontal map is given by inclusion, and the vertical isomorphisms are as in [36, Definition 5.4].

We now proceed to construct \(({\dot{p}},p)\). Since \(A = C^*(N)\) and \(\varphi (N) \subseteq \breve{N}\), it is easy to see that \(\breve{A} = {\overline{\mathrm{span}}}(\varphi (N) \cdot \breve{B})\). It follows that for every \(\breve{n} \in \breve{N}\) and \(y \in \mathrm{dom\,}(\breve{n})\), there is \(n \in \varphi (N)\) such that \(y \in \mathrm{dom\,}(n)\) and \([x,\breve{n},y] = [x,n,y]\) in T. Indeed, for \(a \in \mathrm{span}(\varphi (N) \cdot \breve{B})\) it is clear that \(a \equiv 0\) on \(T \setminus \big ( \bigcup _{n \in \varphi (N)} \left\{ [\alpha _n(y),n,y] \text {: }y \in \mathrm{dom\,}(n) \right\} \big )\). As the latter set is closed in T, we must have \(a \equiv 0\) on \(T \setminus \big ( \bigcup _{n \in \varphi (N)} \left\{ [\alpha _n(y),n,y] \text {: }y \in \mathrm{dom\,}(n) \right\} \big )\) for all \(a \in \breve{A}\). Hence \(T = \bigcup _{n \in \varphi (N)} \left\{ [\alpha _n(y),n,y] \text {: }y \in \mathrm{dom\,}(n) \right\} \). This observation allows us to define the maps

where \(\varphi ^*: \, Y \rightarrow X\) is the map dual to \(B \rightarrow \breve{B}, \, b \mapsto \varphi (b)\) determined by \(\varphi (b) = b \circ \varphi ^*\) for all \(b \in B\). Note that \(\varphi ^*\) exists since \(\varphi (B)\) is full in \(\breve{B}\). p is well-defined because \([x,\varphi (m),y] = [x,\varphi (n),y]\) implies \({\hat{P}}(\varphi (n)^*\varphi (m))(y) > 0\), so that \(P(n^*m)(\varphi ^*(y)) = \varphi (P(n^*m))(y) = {\hat{P}}(\varphi (n)^* \varphi (m))(y) > 0\), which in turn yields \([\varphi ^*(x), \alpha _m, \varphi ^*(y)] = [\varphi ^*(x), \alpha _n, \varphi ^*(y)]\). Similarly, \({\dot{p}}\) is well-defined. Clearly, \(({\dot{p}},p)\) is a twisted groupoid homomorphism. As \(\varphi \) is injective, \(\varphi ^*\) is surjective, so that \({\dot{p}}\) is surjective.

To see that \({\dot{p}}\) is proper, let \(K \subseteq G\) be compact. Given \(n \in N\), write \(U(n) \mathrel {:=}\left\{ [\alpha _n(y),\alpha _n,y] \text {: }y \in \mathrm{dom\,}(n) \right\} \) and \(K(n) \mathrel {:=}(s \vert _{U(n)})^{-1}(s(K))\). As K is compact, there exists a finite set \(\left\{ n_i \right\} \subseteq N\) such that \(K \subseteq \bigcup _i U(n_i)\), so that \(K = \bigcup _i U(n_i) \cap K \subseteq \bigcup _i K(n_i)\). Now given \(m \in N\), \({\dot{p}}([x,\alpha _{\varphi (m)},y]) \in K(n)\) implies \(\varphi ^*(y) \in s(K)\), i.e., \(y \in (\varphi ^*)^{-1}(s(K))\), \({\dot{p}}([x,\alpha _{\varphi (m)},y]) = [\varphi ^*(x), \alpha _m, \varphi ^*(y)] = [\varphi ^*(x), \alpha _n, \varphi ^*(y)]\), so that \(P(n^*m)(\varphi ^*(y)) \ne 0\), which yields \({\hat{P}}(\varphi (n)^* \varphi (m))(y) = \varphi (P(n^* m))(y) \ne 0\), thus \([x,\alpha _{\varphi (m)},y] = [x,\alpha _{\varphi (n)},y]\). Hence \({\dot{p}}^{-1}(K(n)) \subseteq \left\{ [\alpha _{\varphi (n)}(y), \varphi (n), y] \text {: }y \in (\varphi ^*)^{-1}(s(K)) \right\} = (s \vert _{U(\varphi (n))})^{-1}((\varphi ^*)^{-1}(s(K)) \mathrel {=:}\breve{K}(n)\). As \(\varphi ^*\) is proper, \(\breve{K}(n)\) is compact for all \(n \in N\). Hence \({\dot{p}}^{-1}(K) \subseteq \bigcup _i {\dot{p}}^{-1}(K(n_i)) \subseteq \bigcup _i \breve{K}(n_i)\) is a closed subset of a compact set, thus compact itself.

Moreover, given \(y \in Y\), \({\dot{p}}([w,\alpha _{\varphi (m)},y]) = {\dot{p}}([x,\alpha _{\varphi (n)},y])\) implies \([\varphi ^*(w), \alpha _m, \varphi ^*(y)] = [\varphi ^*(x), \alpha _n, \varphi ^*(y)]\), so that \({\hat{P}}(\varphi (n)^* \varphi (m))(y) = P(n^* m)(\varphi ^*(y)) \ne 0\), so that \([w,\alpha _{\varphi (m)},y] = [x,\alpha _{\varphi (n)},y]\). This shows injectivity of \({\dot{p}} \vert _{H_y}\), and it is clear that \({\dot{p}}(H_y) = G_{{\dot{p}}(y)}\). Thus \({\dot{p}}\) is fibrewise bijective.

Finally, it is easy to see that we have a commutative diagram

where the vertical isomorphisms are as in [36, Definition 5.4].

Remark 5.5

In Proposition 5.4, \(\varphi \) sends full elements to full elements if and only if we have \(i(H^{(0)}) = {\hat{G}}^{(0)}\).

Theorem 1.10 now follows from [4, Theorem 3.6], Lemma 5.3, Proposition 5.4 and Remark 5.2.

Remark 5.6

The Weyl twist of \((\varinjlim \left\{ A_n; \varphi _n \right\} , \varinjlim \left\{ B_n; \varphi _n \right\} )\) in the situation of Theorem 1.10 is given by \(({\bar{G}},{\bar{\Sigma }})\) as given in (23) and (24).

If, in Theorem 1.10, all \(\varphi _n\) send full elements to full elements, then \(G_{n,m+1}^{(0)} = H_{n+m}^{(0)} = G_{n+m+i}^{(0)}\) (where we identify \(H_{n+m}^{(0)}\) with \(i_{n+m}(H_{n+m}^{(0)})\)), so that \({\bar{G}}_n^{(0)} = \varprojlim _m \{ G_{n,m}^{(0)};{\dot{p}}_{n+m} \} \cong \varprojlim \{ G_l^{(0)}; {\dot{p}}_l \}\) for all n, and thus \({\bar{i}}_n({\bar{G}}_n^{(0)}) = {\bar{G}}_{n+1}^{(0)}\) for all n, which implies \({\bar{G}}^{(0)} \cong \varprojlim \{ G_n^{(0)}; {\dot{p}}_n \}\).

If all \(B_n\) in Theorem 1.10 are C*-diagonals, i.e., all \(G_n\) are principal, then \({\bar{G}}\) is principal.

6.1 The building blocks

We first present groupoid models for the building blocks that give rise to our AH-algebras (see Sect. 3). Let Z be a second countable compact Hausdorff space and let \(p_i \in M_{\infty }(C(Z))\) be a finite collection of line bundles over Z. Let \(P = \sum _i{}^{\oplus } \ p_i \in M_{\infty }(C(Z))\). The following is easy to check:

Lemma 6.1

\(\bigoplus _i p_i M_{\infty }(C(Z)) p_i\) is a Cartan subalgebra of \(P M_{\infty }(C(Z)) P\).

Thus, by [36, Theorem 5.9], there exists a twisted groupoid \(({\dot{{\mathcal {F}}}},{\mathcal {F}})\) (the Weyl twist) such that

Let us now describe \(({\dot{{\mathcal {F}}}},{\mathcal {F}})\) explicitly. Let R be the full equivalence relation on the finite set \(\left\{ p_i \right\} \) (just a set with the same number of elements as the number of line bundles). Let \({\dot{{\mathcal {F}}}} = Z \times R\), which is a groupoid in the canonical way. For every \(p_i\), let \(T_i\) be a circle bundle over Z such that \(p_i = {\mathbb {C}}\times _{{\mathbb {T}}} T_i\). We form the circle bundles \(T_j \cdot T_i^*\), which are given as follows: For each index i, let \(\left\{ V_{i,a} \right\} _a\) be an open cover of Z, and let \(v_{i,a}\) be a trivialization of \(T_i \vert _{V_{i,a}}\). We view \(v_{i,a}\) as a continuous map \(v_{i,a}: \, V_{i,a} \rightarrow M_{\infty }\) with values in partial isometries such that \(v_{i,a}(z)\) has source projection \(e_{11}\) and range projection \(p_i(z)\), so that \(v_{i,a}(z) = p_i(z) v_{i,a}(z) e_{11}\). Here \(e_{11}\) is the rank one projection in \(M_{\infty }\) which has zero entry everywhere except in the upper left (1, 1)-entry, where the value is 1. Then

where we define \((z,x) \sim (z',x')\) if \(x = x'\), and if \(x \in V_{j,c} \cap V_{i,a}\), \(x' \in V_{j,d} \cap V_{i,b}\), then \(z' = v_{i,b} v_{j,d}^* v_{j,c} v_{i,a}^* z\).

We set

Note that \(T_i \cdot T_i^*\) is just the trivial circle bundle \({\mathbb {T}}\times Z\). We define a multiplication on \({\mathcal {F}}\): For ([z, x], (j, i)) and \(([z',x'],(j',i'))\) in \({\mathcal {F}}\), we can only multiply these elements if \(x = x'\) and \(i = j'\). In that case, write \(h \mathrel {:=}i'\) and assume that \(x \in V_{j,c} \cap V_{i,b}\) and \(x' = x \in V_{i,b} \cap V_{h,a}\). Then we define the product as

Moreover, \({\mathcal {F}}\) becomes a twist of \({\dot{{\mathcal {F}}}}\) via the map

It is now straightforward to check (compare [36]) that the twisted groupoid \(({\dot{{\mathcal {F}}}},{\mathcal {F}})\) is precisely the Weyl twist of \((P M_{\infty }(C(Z)) P, \, \bigoplus _i p_i M_{\infty }(C(Z)) p_i)\). More precisely, we have the following

Lemma 6.2

We have a C(Z)-linear isomorphism \(C^*_r({\dot{{\mathcal {F}}}},{\mathcal {F}}) \cong P M_{\infty }(C(Z)) P\) sending \(\tilde{f} \in C_c({\dot{{\mathcal {F}}}},{\mathcal {F}})\) with \(\mathrm{supp}(\tilde{f}) \subseteq \left( V_{j,c} \cap V_{i,a} \right) \times \left\{ (j,i) \right\} \subseteq {\dot{{\mathcal {F}}}}\) to \(f v_{j,c} v_{i,a}^*\), where \(f \in C(Z)\) is determined by \(\tilde{f}(([z,x],(j,i)) = {\bar{z}} f(x)\). Moreover, this C(Z)-linear isomorphism identifies \(C({\dot{{\mathcal {F}}}}^{(0)})\) with \(\bigoplus _i p_i M_{\infty }(C(Z)) p_i\).

Let us now fix n, and apply the result above to the homogeneous C*-algebra \(F \mathrel {:=}F_n\) from Sect. 4.2 to obtain a twisted groupoid \(({\dot{{\mathcal {F}}}},{\mathcal {F}})\) such that \(C^*_r({\dot{{\mathcal {F}}}},{\mathcal {F}}) \cong F\). More precisely, we apply our construction above to the summand of F corresponding to the component of \(\theta _n^{\varvec{i}}\). Note that in the construction above, all our line bundles satisfy

because of (7). For the other summands, it is easy to construct a groupoid model, as these are just matrix algebras, so that we can just take the full equivalence relation on finite sets.

Now our goal is to present a groupoid model for the building block \(A \mathrel {:=}A_n\) in Sect. 4.2. Let \({\mathcal {R}}\) be an equivalence relation (on a finite set) such that \(C^*({\mathcal {R}}) \cong E \mathrel {:=}E_n\). Write \({\mathcal {R}}= \coprod _p {\mathcal {R}}^p\) for subgroupoids \({\mathcal {R}}^p\) such that the isomorphism \(C^*({\mathcal {R}}) \cong E\) restricts to isomorphisms \(C^*({\mathcal {R}}^p) \cong E^p \mathrel {:=}E_n^p\). Set \({\dot{{\mathcal {C}}}} \mathrel {:=}[0,1] \times {\mathcal {R}}\). Then \(C^*_r({\dot{{\mathcal {C}}}})\) is canonically isomorphic to \(C \mathrel {:=}C([0,1],E)\). Consider the trivial twist \({\mathcal {C}}\mathrel {:=}{\mathbb {T}}\times {\dot{{\mathcal {C}}}}\) of \({\dot{{\mathcal {C}}}}\). Clearly, we have \(C^*_r({\dot{{\mathcal {C}}}} \amalg {\dot{{\mathcal {F}}}}, {\mathcal {C}}\amalg {\mathcal {F}}) \cong C \oplus F\).

For \(t=0,1\) and \(\beta _t\) as in Sect. 4.2, write

as the composition

where each of the components \(F \rightarrow M_{n_l} \otimes {\mathbb {C}}^{I_l^p}\) of the first map is given by

with \(\# I_l^p\) copies of \(a(\theta ^l)\) on the diagonal, and the components \(M_{n_l} \otimes {\mathbb {C}}^{I_l^p} \hookrightarrow E^p\) of the second map are given by

where u is a permutation matrix.

Let \(E_t^p\) be the image of \(\bigoplus _l M_{n_l} \otimes {\mathbb {C}}^{I_l^p}\) in E, and set \(E_t \mathrel {:=}\bigoplus _p E_t^p \subseteq E\), for \(t=0,1\). Let \({\mathcal {R}}_t^p \subseteq {\mathcal {R}}^p\) be subgroupoids such that the identification \(C^*({\mathcal {R}}^p) \cong E^p\) restricts to \(C^*({\mathcal {R}}_t^p) \cong E_t^p\). Write \({\mathcal {R}}_t \mathrel {:=}\coprod _p {\mathcal {R}}_t^p\), so that \(C^*({\mathcal {R}}) \cong E\) restricts to \(C^*({\mathcal {R}}_t) \cong E_t\). Let \(\sigma _t^p\) be the groupoid isomorphism \(\coprod _l {\mathcal {R}}_l \times I_l^p \cong {\mathcal {R}}_t^p\), given by a bijection of the finite unit space, corresponding to conjugation by the unitary u in (27). Let \(V_{i,a}\) and \(v_{i,a}\) be as above (introduced after Lemma 6.1). We now define a map \(\varvec{b}_t: \, {\mathbb {T}}\times (\left\{ t \right\} \times {\mathcal {R}}_t) \rightarrow \Sigma \) as follows: Given an index l and \((j,i) \in {\mathcal {R}}_l\), choose indices a and c such that \(\theta ^l \in V_{j,c} \cap V_{i,a}\). Then define

Here, we are using (25). If \(\theta ^l\) is not the distinguished point \(\theta _n^{\varvec{i}}\), then we set \(z_{j,i} = 1\). For \(z \in {\mathbb {T}}\) and \(h \in I_l^p\), set

Define

where \(\sim \) is the equivalence relation on \({\check{\Sigma }}\) generated by \((z,t,\gamma ) \sim \varvec{b}_t(z,t,\gamma )\) for all \(z \in {\mathbb {T}}\), \(t=0,1\) and \(\gamma \in {\mathcal {R}}_t\). \({\check{\Sigma }}\) and \(\Sigma \) are principal \({\mathbb {T}}\)-bundles belonging to twisted groupoids, and we denote the underlying groupoids by \(\check{G}\) and G.

By construction, the canonical projection and inclusion \(\Sigma \twoheadleftarrow {\check{\Sigma }} \hookrightarrow {\mathcal {C}}\amalg {\mathcal {F}}\) induce on the level of C*-algebras

In particular, \((G,\Sigma )\) is the desired groupoid model for our building block.

In what follows, it will be necessary to keep track of the index n, so that we will consider, for all n, twisted groupoids \(({\dot{{\mathcal {C}}}}_n \amalg {\dot{{\mathcal {F}}}}_n, {\mathcal {C}}_n \amalg {\mathcal {F}}_n)\), \((\check{G}_n,{\check{\Sigma }}_n)\), \((G_n,\Sigma _n)\) describing the C*-algebras \(C_n \oplus F_n\), \(\check{A}_n\) and \(A_n\) as explained above. Moreover, for all n, let \(B_n \subseteq A_n\) be the subalgebra corresponding to \(C_0(G_n^{(0)})\) under the isomorphism \(C^*_r(G_n,\Sigma _n) \cong A_n\).

6.2 The connecting maps

Let us now describe the connecting maps \(\varphi _n: \, A_n \rightarrow A_{n+1}\) in the groupoid picture above. Let \(P_{n+1}^{\varvec{a}}\), \(\varphi _n^{\varvec{a}}\) be as in Sect. 4.2, so that \(\varphi _n = \bigoplus _{\varvec{a}} \varphi _n^{\varvec{a}}\) and \(\mathrm{im\,}(\varphi _n^{\varvec{a}}) \subseteq P_{n+1}^{\varvec{a}} A_{n+1} P_{n+1}^{\varvec{a}}\). Also, let \(\Phi _n: \, C_n \oplus F_n \rightarrow C_{n+1} \oplus F_{n+1}\) be the extension of \(\varphi _n\) as in Remark 4.1. Set \(\Phi _n^{\varvec{a}}: \, C_n \oplus F_n \rightarrow P_{n+1}^{\varvec{a}}(C_{n+1} \oplus F_{n+1}) P_{n+1}^{\varvec{a}}, \, x \mapsto P_{n+1}^{\varvec{a}} \varphi _n^{\varvec{a}}(x) P_{n+1}^{\varvec{a}}\). We obtain \({\check{\varphi }}_n: \, \check{A}_n \rightarrow \check{A}_{n+1}\) and \({\check{\varphi }}_n^{\varvec{a}}: \, \check{A}_n \rightarrow P_{n+1}^{\varvec{a}} \check{A}_{n+1} P_{n+1}^{\varvec{a}}\) by restricting \(\Phi _n\) and \(\Phi _n^{\varvec{a}}\). Set

Note that \(\check{A}[{\check{\varphi }}_n^{\varvec{a}}] = P_{n+1}^{\varvec{a}} F_{n+1} P_{n+1}^{\varvec{a}}\) if \(P_{n+1}^{\varvec{a}} \in F_{n+1}\) and \(\check{A}[{\check{\varphi }}_n^{\varvec{a}}] = \left\{ x \in P_{n+1}^{\varvec{a}} \check{A}_{n+1} P_{n+1}^{\varvec{a}} \text {: }x(t) \in \mathrm{im\,}({\text {ev}}_t \circ {\check{\varphi }}_n^{\varvec{a}}) \ \mathrm{for} \ t = 0,1 \right\} \) if \(P_{n+1}^{\varvec{a}} \in C_{n+1}\).

Let \({\mathcal {T}}_n\) be the open subgroupoid of \({\mathcal {C}}_{n+1} \amalg {\mathcal {F}}_{n+1}\), with \({\dot{{\mathcal {T}}}}_n \subseteq {\dot{{\mathcal {C}}}}_{n+1} \amalg {\dot{{\mathcal {F}}}}_{n+1}\) correspondingly, such that \(C^*_r({\dot{{\mathcal {C}}}}_{n+1} \amalg {\dot{{\mathcal {F}}}}_{n+1}, {\mathcal {C}}_{n+1} \amalg {\mathcal {F}}_{n+1}) \cong C_{n+1} \oplus F_{n+1}\) restricts to \(C^*_r({\dot{{\mathcal {T}}}}_n,{\mathcal {T}}_n) \cong (C \oplus F)[\Phi _n]\). Similarly, let \(\check{T}_n\) be the open subgroupoid of \({\check{\Sigma }}_{n+1}\), with \(\check{H}_n \subseteq \check{G}_{n+1} \) correspondingly, such that \(C^*_r(\check{G}_{n+1}, {\check{\Sigma }}_{n+1}) \cong \check{A}_{n+1}\) restricts to \(C^*_r(\check{H}_n,\check{T}_n) \cong \check{A}[{\check{\varphi }}_n]\). For \(\eta \in \check{T}_n\) and \(\eta ' \in {\check{\Sigma }}_{n+1}\), \(\eta \sim \eta '\) implies that \(\eta '\) lies in \(\check{T}_n\). It follows that \(T_n = \check{T}_n / { }_{\sim }\) is an open subgroupoid of \(\Sigma _{n+1}\). Define \(H_n = \check{H}_n / { }_{\sim }\) in a similar way. By construction, the commutative diagram at the groupoid level

induces at the C*-level

Let \(\check{T}_n = \coprod _{\varvec{a}} \check{T}_n^{\varvec{a}}\) and \(\check{H}_n = \coprod _{\varvec{a}} \check{H}_n^{\varvec{a}}\) be the decompositions into subgroupoids such that the identification \(C^*_r(\check{H}_n, \check{T}_n) \cong \check{A}[{\check{\varphi }}_n] \subseteq \check{A}_{n+1}\) restricts to \(C^*_r(\check{H}_n^{\varvec{a}}, \check{T}_n^{\varvec{a}}) \cong \check{A}[{\check{\varphi }}_n^{\varvec{a}}]\). For fixed n and every \(\varphi ^{\varvec{a}} = \varphi ^{\varvec{a}}_n\) from our list in Sect. 4.2, we now construct a map \(p^{\varvec{a}}: \, \check{T}_n^{\varvec{a}} \rightarrow \Sigma _n\) such that

commutes.

Recall that \({\check{\Sigma }}_n \subseteq {\mathcal {C}}_n \amalg {\mathcal {F}}_n = ({\mathbb {T}}\times [0,1] \times {\mathcal {R}}_n) \amalg {\mathcal {F}}_n\). Also, we denote the canonical projection \({\mathcal {F}}_n \twoheadrightarrow {\dot{{\mathcal {F}}}}_n\) by \(\sigma \mapsto {\dot{\sigma }}\).

• For \(\varphi ^{\varvec{a}}\) as in (16), let \(p^{\varvec{a}}\) be the composite

  $$\begin{aligned} \check{T}_n^{\varvec{a}} \cong {\mathbb {T}}\times [0,1] \times {\mathcal {R}}_n&\rightarrow {\check{\Sigma }}_n \overset{q}{\longrightarrow } \Sigma _n,\\ (z,t,\gamma )&\mapsto (z,\lambda (t),\gamma ) \end{aligned}$$

  where we note that the first map has image in \({\check{\Sigma }}_n\), so that we an apply the quotient map \(q: \, {\check{\Sigma }}_n \twoheadrightarrow \Sigma _n\).

• For \(\varphi ^{\varvec{a}}\) as in (17), let \(p^{\varvec{a}}\) be the composite

  $$\begin{aligned} \check{T}_n^{\varvec{a}} \cong {\mathbb {T}}\times [0,1] \times {\mathcal {R}}_n&\rightarrow {\mathcal {F}}_n \overset{q}{\longrightarrow } \Sigma _n,\nonumber \\ (z,t,\gamma )&\mapsto z \cdot \sigma (t,\gamma ) \end{aligned}$$

  (29)

  where \(\sigma \) is a continuous groupoid homomorphism such that \({\dot{\sigma }}(t,\gamma ) = (x(t),\gamma )\). For \(x(t) = \theta ^l \in \left\{ \theta _n^i \right\} \) and \(\gamma = (j,i)\), write

  $$\begin{aligned} \sigma (t,\gamma ) = [z_{j,i},\theta ^l], \end{aligned}$$

  (30)

  which has to match up with (28).

• For \(\varphi ^{\varvec{a}}\) as in (18), let \(p^{\varvec{a}}\) be the composite

  $$\begin{aligned} \check{T}_n^{\varvec{a}} \cong {\mathbb {T}}\times \{ \theta _{n+1}^j \} \times {\mathcal {R}}_n&\rightarrow {\check{\Sigma }}_n \overset{q}{\longrightarrow } \Sigma _n,\\ (z,\theta _{n+1}^j,\gamma )&\mapsto (z,\varvec{t},\gamma ). \end{aligned}$$

• For \(\varphi ^{\varvec{a}}\) as in (19), let \(p^{\varvec{a}}\) be the composite

  $$\begin{aligned} \check{T}_n^{\varvec{a}} \cong {\mathbb {T}}\times \{ \theta _{n+1}^j \} \times ({\dot{{\mathcal {F}}}}_n)^x_x&\rightarrow {\mathcal {F}}_n \overset{q}{\longrightarrow } \Sigma _n,\\ (z,\theta _{n+1}^j,\gamma )&\mapsto z \cdot \sigma (\gamma ), \end{aligned}$$

  where \(\sigma : \, ({\dot{{\mathcal {F}}}}_n)^x_x \rightarrow {\mathcal {F}}_n\) is a groupoid homomorphism with \({\dot{\sigma }}(\gamma ) = (x,\gamma )\) matching up with \(\sigma \) in (29).

• For \(\varphi ^{\varvec{a}}\) as in (20), let \(p^{\varvec{a}}\) be the composite

  $$\begin{aligned} \check{T}_n^{\varvec{a}} \cong {\mathbb {T}}\times \{ \theta _{n+1}^j \} \times ({\dot{{\mathcal {F}}}}_n)^{\theta _n^i}_{\theta _n^i}&\rightarrow {\mathcal {F}}_n \overset{q}{\longrightarrow } \Sigma _n,\\ (z,\theta _{n+1}^j,\gamma )&\mapsto (z,\theta _n^i,\gamma ). \end{aligned}$$

• For \(\varphi ^{\varvec{a}}\) as in (21), let \(p^{\varvec{a}}\) be the composite

  $$\begin{aligned} \check{T}_n^{\varvec{a}} \cong {\mathbb {T}}\times Z_{n+1} \times ({\dot{{\mathcal {F}}}}_n)^{\theta _n^i}_{\theta _n^i} \twoheadrightarrow {\mathbb {T}}\times ({\dot{{\mathcal {F}}}}_n)^{\theta _n^i}_{\theta _n^i}&\rightarrow {\mathcal {F}}_n \overset{q}{\longrightarrow } \Sigma _n,\\ (z,\gamma )&\mapsto z \cdot \sigma (\gamma ), \end{aligned}$$

  where \(\sigma : \, ({\dot{{\mathcal {F}}}}_n)^{\theta _n^i}_{\theta _n^i} \rightarrow {\mathcal {F}}_n\) is a groupoid homomorphism with \({\dot{\sigma }}(\gamma ) = (\theta _n^i,\gamma )\) matching up with (28), just as (30).

• For \(\varphi ^{\varvec{a}}\) as in (22), we have \(C^*_r(\check{H}_n^{\varvec{a}}, \check{T}_n^{\varvec{a}}) \cong \left( \sum _i \lambda ^*(p_i) \right) \cdot F_{n+1} \cdot \left( \sum _i \lambda ^*(p_i) \right) \), where \(p_i\) are the line bundles such that \(P_n = \sum _i p_i\) (see Sect. 3 and Sect. 6.1), and \(p^{\varvec{a}}\) is the composite

  $$\begin{aligned} \check{T}_n^{\varvec{a}}&\rightarrow {\mathcal {F}}_n \overset{q}{\longrightarrow } \Sigma _n,\\ [z,x]&\mapsto [z,\lambda (x)], \end{aligned}$$

  with \((z,x) \in {\mathbb {T}}\times \lambda ^{-1}(V_{i,a})\) and \((z,\lambda (x)) \in {\mathbb {T}}\times V_{i,a}\), where for a given open cover \(V_{i,a}\) and trivialization \(v_{i,a}\) for \({\mathcal {F}}_n\), we choose the open cover \(\lambda ^{-1}(V_{i,a})\) and trivialization \(v_{i,a} \circ \lambda \) for \(\check{T}_n^{\varvec{a}}\) (see Sect. 6.1).

The homomorphism

must descend to \(p_n: \, T_n \rightarrow \Sigma _n\) because \(C^*_r(\coprod _{\varvec{a}} p^{\varvec{a}}): \, C^*_r(G_n, \Sigma _n) \rightarrow C^*_r(\check{H}_n, \check{T}_n), \, f \mapsto f \circ (\coprod _{\varvec{a}} p^{\varvec{a}})\) lands in \(C^*_r(H_n, T_n)\). Moreover, the homomorphisms \(\Phi _n\) and \({\check{\varphi }}_n\) admit similar groupoid models (say \({\mathcal {P}}_n\) and \(\check{p}_n\)) as \(\varphi _n\), so that we obtain a commutative diagram

Proofs of Theorems 1.2 and 1.3

All we have to do is to check the conditions in Theorem 1.10, using Proposition 5.4 and the groupoid models in Sect. 6. We treat the unital and stably projectionless cases simultaneously. Given a prescribed Elliott invariant, let \(A_n\) and \(\varphi _n\) be as in Sect. 4.2. Consider the groupoid models for \(A_n\) and \(\varphi _n\) in Sect. 6. First of all, by construction, \((H_n,T_n)\) is a subgroupoid of \((G_{n+1},\Sigma _{n+1})\) and \(H_n \subseteq G_{n+1}\) is open. Let \((i_n, \imath _n)\) be the canonical inclusion. Secondly, \(p_n\) is proper because all the \(p^{\varvec{a}}\) in Sect. 6.2 are proper (they are closed, and pre-images of points are compact). Thirdly, \(p_n\) is fibrewise bijective because this is true for \(\check{p}_n\) and the canonical projections \({\check{\Sigma }}_n \twoheadrightarrow \Sigma _n\), \(\check{T}_n \twoheadrightarrow T_n\). By construction, all the connecting maps \(\varphi _n\) in Sect. 4.2 are of the form \(\varphi _n = (\imath _n)_* \circ (p_n)^*\). Thus, by Proposition 5.4, the conditions in Theorem 1.10 are satisfied. Hence \(\varinjlim \left\{ B_n; \varphi _n \right\} \), with \(B_n\) as in Sect. 6.1, is a Cartan subalgebra of \(\varinjlim \left\{ A_n; \varphi _n \right\} \), and actually even a C*-diagonal by Remark 5.6 because all \(G_n\) are principal.

Remark 7.1

By Remark 5.6, the twisted groupoids \((G,\Sigma )\) we obtain in the proofs of Theorems 1.2 and 1.3 are given by the Weyl twists described by (23) and (24). Moreover, it is easy to see that for the groupoids in Sect. 6, we have \(({\mathcal {C}}_n \amalg {\mathcal {F}}_n)^{(0)} = \check{G}_n^{(0)}\), and since \(\Phi _n\), \({\check{\varphi }}_n\) and \(\varphi _n\) send full elements to full elements, \({\dot{{\mathcal {T}}}}_n^{(0)} = ({\mathcal {C}}_n \amalg {\mathcal {F}}_n)^{(0)}\), \(\check{H}_n^{(0)} = \check{G}_{n+1}^{(0)}\) and \(H_n^{(0)} = G_{n+1}^{(0)}\), for all n (by Remark 5.5). So Remark 5.6 tells us that \(G^{(0)} \cong \varprojlim \{ G_n^{(0)}; {\dot{p}}_n \}\).

We now turn to the additional statements in Sect. 1. In order to prove Corollaries 1.6 and 1.7 , we need to show the following statement. In both the unital and stably projectionless cases, let \(A = C^*_r(G,\Sigma )\), \(D = C_0(G^{(0)})\), and \(\gamma = \tilde{\gamma } \vert _T\) be as in Corollaries 1.6 and 1.7 . Let \({\mathcal {C}}\) be the canonical diagonal subalgebra of the algebra of compact operators \({\mathcal {K}}\).

Proposition 7.2

There exists a positive element \(a \in D \otimes {\mathcal {C}}\subseteq A \otimes {\mathcal {K}}\) such that \(d_{\bullet }(a) = \gamma \).

Here \(d_{\bullet }(a)\) denotes the function \(T \ni \tau \mapsto d_{\tau }(a)\). For the proof, we need the following

Lemma 7.3

Given a continuous affine map \(g: \, T \rightarrow (0,\infty )\) and \(\varepsilon > 0\), there exists \(z \in D \otimes D_k \subseteq A \otimes M_k \subseteq A \otimes {\mathcal {K}}\) with \(\left\| z\right\| = 1\), \(z \ge 0\), \(z \in \mathrm{Ped}(A \otimes {\mathcal {K}})\) such that \(g - \varepsilon< d_{\bullet }(z) < g + \varepsilon \).

Here \(D_k\) is the canonical diagonal subalgebra of \(M_k\).

Proof

We treat the unital and stably projectionless cases simultaneously. Let \({\hat{F}}_n\) be as in Sect. 2.1 and \({\hat{F}} \mathrel {:=}\varinjlim {\hat{F}}_n\). Choose \(a \in {\hat{F}} \otimes {\mathcal {K}}\) with \(a \ge 0\) and \(d_{\bullet }(a) = g\). Then we can choose \(b \in {\hat{F}}_n \otimes M_k\) (for n big enough) with \(b \ge 0\), \(d_{\bullet }(b)\) continuous and

Using [1, Theorem 3.1] just as in [37, Proof of (6.2) and (6.3)], choose \(c \in D({\hat{F}}_n) \otimes D_k\) with \(c \ge 0\) such that c and b are Cuntz equivalent, where \(D({\hat{F}}_n)\) is the canonical diagonal subalgebra of \({\hat{F}}_n\). Choose \(d \in \mathrm{Ped}(A_n \otimes M_k)\) with \(d \in D(A_n) \otimes D_k\) such that \((\pi \otimes \mathrm{id})(d) = c \), where \(\pi : \, A_n \twoheadrightarrow F_n \twoheadrightarrow {\hat{F}}_n\) is the canonical projection. Let z denote the image of d under the canonical map \(A_n \otimes M_k \rightarrow A \otimes M_k\). Then \(z \in D \otimes D_k\). It is now straightforward to check, using the isomorphism \(T(A) \cong T({\hat{F}})\) from [20, § 13], that z has the desired properties.

Proof of Proposition 7.2

There is a sequence \((\gamma _i)\) of continuous affine maps \(T \rightarrow [0,\infty )\) with \(\gamma _i \nearrow \gamma - \min (\gamma )\). Choose \(\varepsilon _i > 0\) such that \(\sum _i \varepsilon _i = \min (\gamma )\). Define \(f_i \mathrel {:=}\gamma _i + \sum _{h=1}^{i-1} \varepsilon _h\). Then \(f_i \nearrow \gamma \) and \(f_i > 0\). Moreover,

Using Lemma 7.3, proceed inductively on i to find \(z_i \in D \otimes D_{k(i)}\) such that

Note that \(f_{i+1} - \sum _{h=1}^i d_{\bullet }(z_h) > 0\) since \(\sum _{h=1}^i d_{\bullet }(z_h) < f_i + \varepsilon _i \le f_{i+1}\).

By construction, we have

Now set

In this way, we obtain an element \(a \in D \otimes {\mathcal {C}}\subseteq A \otimes {\mathcal {K}}\) with \(d_{\bullet }(a) = \gamma \).

Proof of Corollaries 1.6 and 1.7

Given \(\tilde{\gamma }\) as in Corollaries 1.6 and 1.7, let \(\gamma = \tilde{\gamma } \vert _T\). Using Proposition 7.2, choose a positive element \(a \in D \otimes {\mathcal {C}}\) with \(d_{\bullet }(a) = \gamma \). In the unital case, it is straightforward to check that we can always arrange a to be purely positive. Then it is straightforward to check that \((\overline{a (A \otimes {\mathcal {K}}) a}, \overline{a (D \otimes {\mathcal {C}}) a})\) is a Cartan pair. Hence, by [36, Theorem 5.9], there is a twisted groupoid \((\tilde{G},\tilde{\Sigma })\) such that \((C^*_r(\tilde{G},\tilde{\Sigma }), C_0(\tilde{G}^{(0)})) \cong (\overline{a (A \otimes {\mathcal {K}}) a}, \overline{a (D \otimes {\mathcal {C}}) a})\). It is now easy to see (compare also [19, Corollary 6.12]) that \((\tilde{G},\tilde{\Sigma })\) has all the desired properties.

Proofs of Corollaries 1.8 and 1.9

(i) follows from the observation that we only need the twist if \(G_0\) has torsion. The claims in (ii)–(iv) about subhomogeneous building blocks and their spectra follow immediately from our constructions (see also Remark 3.9). Moreover, the inverse limit description of the unit space in Remark 7.1 and the dimension formula for inverse limits (see for instance [17, Chapter 3, § 5.3, Theorem 22]) imply that \(\mathrm{dim}\,(G^{(0)}) \le 3\) in (ii), \(\mathrm{dim}\,(G^{(0)}) \le 2\) in (iii) and \(\mathrm{dim}\,(G^{(0)}) \le 1\) in (iv) and (v). Since \(C_0(G^{(0)})\) is projectionless in Theorem 1.3, we obtain \(\mathrm{dim}\,(G^{(0)}) \ne 0\), which forces \(\mathrm{dim}\,(G^{(0)}) = 1\) in (v), in the situation of Theorem 1.3. In particular, this shows that \({\mathcal {W}}\) and \({\mathcal {Z}}_0\) have C*-diagonals with one-dimensional spectra. Similarly, given a groupoid G with \({\mathcal {Z}}\cong C^*_r(G)\), the only projections in \(C(G^{(0)})\) are 0 and 1, so that \(\mathrm{dim}\,(G^{(0)}) \ne 0\) and hence \(\mathrm{dim}\,(G^{(0)}) = 1\). It remains to prove that \(\mathrm{dim}\,(G^{(0)}) \ge 3\) in (ii), \(\mathrm{dim}\,(G^{(0)}) \ge 2\) in (iii) and \(\mathrm{dim}\,(G^{(0)}) \ge 1\) in (iv).

To do so, let us use the same notation as in Sect. 6, and write \(X_n \mathrel {:=}G_n^{(0)}\), \(Q_n \mathrel {:=}{\dot{{\mathcal {C}}}}_n^{(0)}\), and \(W_n \mathrel {:=}{\dot{{\mathcal {F}}}}_n^{(0)}\). Clearly, \(Q_n\) is homotopy equivalent to a finite set of points, so that for any cohomology theory \(H^{\bullet }\) (satisfying the Eilenberg–Steenrod axioms, see [40, Chapter 17]), we have

Let \(P_n \mathrel {:=}\left\{ (t,x) \in Q_n \text {: }t \in \left\{ 0,1 \right\} , \, x \in {\mathcal {R}}_t^{(0)} \right\} \). Then we have a pushout diagram

where \(P_n \rightarrow W_n\) is induced by \(\varvec{b}_t\) and the left vertical arrow is the canonical inclusion. The long exact (Mayer-Vietoris type) sequence attached to the pushout reads

Since \(H^{\bullet }(P_n) \cong \left\{ 0 \right\} \) and \(H^{\bullet }(Q_n) \cong \left\{ 0 \right\} \) (see (31)), we deduce that the canonical map \(W_n \rightarrow X_n\) induces a surjection \(H^{\bullet }(X_n) \rightarrow H^{\bullet }(W_n)\) for \(\bullet \ge 1\). Moreover, the map

induces for \(\bullet \ge 1\) a homomorphism \(H^{\bullet }(\check{p}_n): \, H^{\bullet }(W_n) \rightarrow H^{\bullet }(W_{n+1})\) which fits into the commutative diagram

Thus the canonical maps \(W_n \rightarrow X_n\) induce for all \(\bullet \ge 1\) surjections

Here \(\check{H}^{\bullet }\) is Čech cohomology, and the first identification follows from the inverse limit description of \(G^{(0)}\) in Remark 7.1 and continuity of Čech cohomology. By construction, \(W_n = Z_n \times I_n\) for some finite set \(I_n\) and \(Z_n\) is as in Sect. 4.2. Now it is an immediate consequence of our construction that \(\varinjlim \left\{ H^{\bullet }(W_n); H^{\bullet }(\check{p}_n) \right\} \) surjects onto \(\mathrm{Tor}(G_1)\) in case (ii) for \(\bullet = 3\), \(\mathrm{Tor}(G_0)\) in case (iii) for \(\bullet = 2\), and \(G_1\) in case (iv) for \(\bullet = 1\). Hence it follows that \(\check{H}^3(G^{(0)}) \ncong \left\{ 0 \right\} \) in case (ii), \(\check{H}^2(G^{(0)}) \ncong \left\{ 0 \right\} \) in case (iii), and \(\check{H}^1(G^{(0)}) \ncong \left\{ 0 \right\} \) in case (iv). As cohomological dimension is always a lower bound for covering dimension, this implies \(\mathrm{dim}\,(G^{(0)}) \ge 3\) in case (ii), \(\mathrm{dim}\,(G^{(0)}) \ge 2\) in case (iii), and \(\mathrm{dim}\,(G^{(0)}) \ge 1\) in case (iv), as desired.

Let us describe concrete groupoid models for the Jiang–Su algebra \({\mathcal {Z}}\), the Razak–Jacelon algebra \({\mathcal {W}}\) and the stably projectionless version \({\mathcal {Z}}_0\) of the Jiang–Su algebra as in [19, Definition 7.1]. These C*-algebras can be constructed in a way which fits into the framework of Sect. 4.2, so that our general machinery in Sect. 5 produces groupoid models as in Sect. 6. In the following, we focus on \({\mathcal {Z}}\).

First we recall the original construction of \({\mathcal {Z}}\) in [23]. For every \(n \in {\mathbb {N}}\), choose natural numbers \(p_n\) and \(q_n\) such that they are relatively prime, with \(p_n \mid p_{n+1}\) and \(q_n \mid q_{n+1}\), such that \(\frac{p_{n+1}}{p_n} > 2 q_n\) and \(\frac{q_{n+1}}{q_n} > 2 p_n\). Then \({\mathcal {Z}}= \varinjlim \left\{ A_n; \varphi _n \right\} \), where \(A_n = \left\{ (f,a) \in C([0,1], E_n) \oplus F_n \text {: }f(t) = \beta _t(a) \ \mathrm{for} \ t = 0,1 \right\} \), \(E_n = M_{p_n} \otimes M_{q_n}\), \(F_n = M_{p_n} \oplus M_{q_n}\), \(\beta _0: \, M_{p_n} \oplus M_{q_n} \rightarrow M_{p_n} \otimes M_{q_n}, \, (x,y) \mapsto x \otimes 1_{q_n}\), \(\beta _1: \, M_{p_n} \oplus M_{q_n} \rightarrow M_{p_n} \otimes M_{q_n}, \, (x,y) \mapsto 1_{p_n} \otimes y\).

To describe \(\varphi _n\) for fixed n, let \(d_0 \mathrel {:=}\frac{p_{n+1}}{p_n}\), \(d_1 \mathrel {:=}\frac{q_{n+1}}{q_n}\), \(d \mathrel {:=}d_0 \cdot d_1\), and write \(d = l_0 q_{n+1} + r_0\) with \(0 \le r_0 < q_{n+1}\), \(d = l_1 p_{n+1} + r_1\) with \(0 \le r_1 < p_{n+1}\). Note that we must have \(d_1 \mid r_0\) and \(d_0 \mid r_1\). Then

Here we think of \(A_n\) as a subalgebra of \(C([0,1],E_n)\) via the embedding \(A_n \hookrightarrow C([0,1],E_n), \, (f,a) \mapsto f\).

To construct groupoid models for building blocks and connecting maps, start with a set \({\mathcal {X}}(1)\) with \(p_1 \cdot q_1\) elements, and define recursively \({\mathcal {X}}(n+1) \mathrel {:=}{\mathcal {X}}(n) \times {\mathcal {Y}}(n)\). Let \({\mathcal {R}}(n)\) be the full equivalence relation on \({\mathcal {X}}(n)\). Let \({\mathcal {R}}(n,p)\) and \({\mathcal {R}}(n,q)\) be the full equivalence relations on finite sets \({\mathcal {X}}(n,p)\) and \({\mathcal {X}}(n,q)\) with \(p_n\) and \(q_n\) elements. For \(t=0,1\), let \(\rho _{n+1,t}\) be the bijections corresponding to conjugation by \(u_{n+1}(t)\), which induce \(\sigma _{n,t}: \, {\mathcal {R}}(n,p) \times {\mathcal {R}}(n,q) \cong {\mathcal {R}}(n)\) corresponding to conjugation by \(v_n(t)\) introduced in Remark 4.1. Now set

Define \(\check{p}_n: \, \check{H}_n \rightarrow \check{G}_n\) as the restriction of \({\mathcal {P}}_n: \, {\dot{{\mathcal {T}}}}_n \mathrel {:=}[0,1] \times {\mathcal {R}}(n) \times {\mathcal {Y}}(n) \rightarrow [0,1] \times {\mathcal {R}}(n), \, (t,\gamma ,y) \mapsto (\lambda _y(t),\gamma )\) to \(\check{H}_n \mathrel {:=}{\mathcal {P}}_n^{-1}(\check{G}_n)\). Set \(H_n \mathrel {:=}\check{H}_n / { }_{\sim }\) where \(\sim \) is the equivalence relation defining \(G_{n+1} = \check{G}_{n+1} / { }_{\sim }\). The map \(\check{p}_n\) descends to \(p_n: \, H_n \rightarrow G_n\). The groupoid G with \({\mathcal {Z}}\cong C^*_r(G)\) is now given by (23) and (24). As explained in Remark 7.1, its unit space \(X \mathrel {:=}G^{(0)}\) is given by \(X \cong \varinjlim \left\{ X_n; p_n \right\} \), where \(X_n = G_n^{(0)}\).

To further describe X, let \(\varvec{p}_n\) be the set-valued function on [0, 1] defined by \(\varvec{p}_n(s) \mathrel {:=}\left\{ \lambda _y(s) \text {: }y \in {\mathcal {Y}}(n) \right\} \). We can form the inverse limit

as in [21, § 2.2]. It is easy to see that \(X_n \mapsto [0,1], \, [(t,x)] \mapsto t\) gives rise to a continuous surjection \(X \twoheadrightarrow \varvec{X}\) whose fibres are all homeomorphic to the Cantor space. Moreover, \(\varvec{X}\) is connected and locally path connected. The space X itself is also connected. This follows easily from the construction itself (basically from \(\gcd (p_n,q_n) = 1\)) and also from abstract reasons because \({\mathcal {Z}}\) is unital projectionless. In addition, it is straightforward to check that for particular choices for \(\rho _{n,t}\) and hence \(\sigma _{n,t}\), our space X becomes locally path connected as well. In that case, it is a one-dimensional Peano continuum.

Every \(X_n\) is homotopy equivalent to a finite bouquet of circles. It is then easy to compute K-theory and Čech (co)homology:

It follows that for choices of \(\rho _{n,t}\) and \(\sigma _{n,t}\) such that X is locally path connected, X must be shape equivalent to the Hawaiian earring by [7]. In particular, its first Čech homotopy group is isomorphic to the one of the Hawaiian earring, which is the canonical projective limit of non-abelian free groups of finite rank. Moreover, by [9], the singular homology \(H_1(X)\) coincides with the singular homology of the Hawaiian earring, which is described in [10]. We refer the reader to [33] for more information about shape theory, which is the natural framework to study our space since it is constructed as an inverse limit.

Now we turn to \({\mathcal {W}}\). Recall the construction in [22]. For every \(n \in {\mathbb {N}}\), choose integers \(a_n, b_n \ge 1\) with \(a_{n+1} = 2 a_n + 1\), \(b_{n+1} = a_{n+1} \cdot b_n\). Then \({\mathcal {W}}= \varinjlim \left\{ A_n; \varphi _n \right\} \), where \(A_n = \left\{ (f,a) \in C([0,1], E_n) \oplus F_n \text {: }f \right\} (t) = \beta _t(a) \ \mathrm{for} t = 0,1\), \(E_n = M_{(a_n + 1) \cdot b_n}\), \(F_n = M_{b_n}\), with

where we put \(a_n\) copies of x on the diagonal for \(\beta _0\), and \(a_n + 1\) copies of x on the diagonal for \(\beta _1\). To describe \(\varphi _n\) for fixed n, let \(d \mathrel {:=}2 a_{n+1}\). Then

Here we think of \(A_n\) as a subalgebra of \(C([0,1],E_n)\) via the embedding \(A_n \hookrightarrow C([0,1],E_n), \, (f,a) \mapsto f\).

To construct groupoid models, start with a set \({\mathcal {X}}(1)\) with \((a_1 + 1) \cdot b_1\) elements, and define recursively \({\mathcal {X}}(n+1) \mathrel {:=}{\mathcal {X}}(n) \times {\mathcal {Y}}(n)\). Let \({\mathcal {R}}(n)\) be the full equivalence relation on \({\mathcal {X}}(n)\). Let \({\mathcal {R}}(n,a)\) and \({\mathcal {R}}(n,b)\) be the full equivalence relations on finite sets \({\mathcal {X}}(n,a)\) and \({\mathcal {X}}(n,b)\) with \(a_n + 1\) and \(b_n\) elements, and let \({\mathcal {X}}'(n,a) \subseteq {\mathcal {X}}(n,a)\) be a subset with \(a_n\) elements (corresponding to the multiplicity of \(\beta _0\)). For \(t=0,1\), let \(\rho _{n+1,t}\) be the bijections corresponding to conjugation by \(u_{n+1}(t)\), which induce \(\sigma _{n,t}: \, {\mathcal {R}}(n,a) \times {\mathcal {R}}(n,b) \cong {\mathcal {R}}(n)\) corresponding to conjugation by \(v_n(t)\) introduced in Remark 4.1. Set

Now define \(\check{p}_n: \, \check{H}_n \rightarrow \check{G}_n\) as the restriction of \({\mathcal {P}}_n: \, {\dot{{\mathcal {T}}}}_n \mathrel {:=}[0,1] \times {\mathcal {R}}(n) \times {\mathcal {Y}}(n) \rightarrow [0,1] \times {\mathcal {R}}(n), \, (t,\gamma ,y) \mapsto (\lambda _y(t),\gamma )\) to \(\check{H}_n \mathrel {:=}{\mathcal {P}}_n^{-1}(\check{G}_n)\). Set \(H_n \mathrel {:=}\check{H}_n / { }_{\sim }\) where \(\sim \) is the equivalence relation defining \(G_{n+1} = \check{G}_{n+1} / { }_{\sim }\). The map \(\check{p}_n\) descends to \(p_n: \, H_n \rightarrow G_n\). The groupoid G with \({\mathcal {W}}\cong C^*_r(G)\) is now given by (23) and (24). As explained in Remark 7.1, its unit space \(X \mathrel {:=}G^{(0)}\) is given by \(X \cong \varprojlim \left\{ X_n; p_n \right\} \), where \(X_n = G_n^{(0)}\). As in the case of \({\mathcal {Z}}\), X surjects continuously onto \(\varprojlim \left\{ {\mathbb {T}}; \varvec{p}_n \right\} \) with Cantor space fibres, where \({\mathbb {T}}= [0,1] / { }_{0 \sim 1}\) and \(\varvec{p}_n([s]) = \left\{ [\lambda _y(s)] \text {: }y \in {\mathcal {Y}}(n) \right\} \). However, it is easy to see that (at least for some choices of \(\rho _{n,t}\) and \(\sigma _{n,t}\)), X will not be connected, though its connected components all have to be non-compact.

Now let us treat \({\mathcal {Z}}_0\). For each \(m \in {\mathbb {N}}\), choose integers \(a_n, b_n, h_n \ge 1\) with \(a_{n+1} = ((2a_n + 2)h_n + 1) \cdot a_n\), \(b_{n+1} = ((2a_n + 2)h_n + 1) \cdot b_n\). Let \(A_n = \left\{ (f,a) \in C([0,1], E_n) \oplus F_n \text {: }f(t) = \beta _t(a) \ \mathrm{for} \ t = 0,1 \right\} \), with \(E_n = M_{(2a_n + 2) \cdot b_n}\), \(F_n = M_{b_n} \oplus M_{b_n}\),

where we put \(a_n\) copies of x and y on the diagonal for \(\beta _0\), and \(a_n + 1\) copies of x and y on the diagonal for \(\beta _1\). To describe the connecting maps \(\varphi _n: \, A_n \rightarrow A_{n+1}\), fix n and let \(d \mathrel {:=}(2 a_{n+1} + 2)h_n + (2 a_n h_n + 1)\). Then \((2a_{n+1} + 2) \cdot b_{n+1} = d \cdot (2a_n + 2) \cdot b_n\). It is now easy to see that for suitable choices of unitaries \(u_{n+1}\), whose values at 0 and 1 are permutation matrices, we obtain a homomorphism \(\varphi _n: \, A_n \rightarrow A_{n+1}\) by setting

As above, we think of \(A_n\) as a subalgebra of \(C([0,1],E_n)\) via \(A_n \hookrightarrow C([0,1],E_n), \, (f,a) \mapsto f\). Now arguments similar to those in [22, 23] show that \(\varinjlim \left\{ A_n; \varphi _n \right\} \) has the same Elliott invariant as \({\mathcal {Z}}_0\), so that \({\mathcal {Z}}_0 \cong \varinjlim \left\{ A_n; \varphi _n \right\} \) by [37, Corollary 6.2.4] (see also [19, Theorem 12.2]).

To construct groupoid models, start with a set \({\mathcal {X}}(1)\) with \((2a_1 + 2) \cdot b_1\) elements, and define recursively \({\mathcal {X}}(n+1) \mathrel {:=}{\mathcal {X}}(n) \times {\mathcal {Y}}(n)\). Let \({\mathcal {R}}(n)\) be the full equivalence relation on \({\mathcal {X}}(n)\). Let \({\mathcal {R}}(n,a,1)\), \({\mathcal {R}}(n,a,2)\), \({\mathcal {R}}(n,b,1)\) and \({\mathcal {R}}(n,b,2)\) be full equivalence relations on finite sets \({\mathcal {X}}(n,a,1)\), \({\mathcal {X}}(n,a,2)\), \({\mathcal {X}}(n,b,1)\) and \({\mathcal {X}}(n,b,2)\) with \(a_n + 1\), \(a_n + 1\), \(b_n\) and \(b_n\) elements, respectively. Let \({\mathcal {X}}_0(n,a,1) \subseteq {\mathcal {X}}(n,a,1)\) and \({\mathcal {X}}_0(n,a,2) \subseteq {\mathcal {X}}(n,a,2)\) be subsets with \(a_n\) elements (corresponding to the multiplicities of \(\beta _0\)), and set \({\mathcal {X}}_1(n,a,\bullet ) \mathrel {:=}{\mathcal {X}}(n,a,\bullet )\). For \(t=0,1\), let \(\rho _{n+1,t}\) be the bijections corresponding to conjugation by \(u_{n+1}(t)\), which induce \(\sigma _{n,t}: \, {\mathcal {R}}(n,a,1) \times {\mathcal {R}}(n,b,1) \amalg {\mathcal {R}}(n,a,2) \times {\mathcal {R}}(n,b,2) \cong {\mathcal {R}}(n)\) corresponding to conjugation by \(v_n(t)\) introduced in Remark 4.1. Set

Now define \({\check{p}_n}: \, \check{H}_n \rightarrow \check{G}_n\) as the restriction of \({{\mathcal {P}}_n}: \, {\dot{{\mathcal {T}}}}_n \mathrel {:=}[0,1] \times {\mathcal {R}}(n) \times {\mathcal {Y}}(n) \rightarrow [0,1] \times {\mathcal {R}}(n), \, (t,\gamma ,y) \mapsto (\lambda _y(t),\gamma )\) to \(\check{H}_n \mathrel {:=}{\mathcal {P}}_n^{-1}(\check{G}_n)\). Set \(H_n \mathrel {:=}\check{H}_n / { }_{\sim }\) where \(\sim \) is the equivalence relation defining \(G_{n+1} = \check{G}_{n+1} / { }_{\sim }\). The map \(\check{p}_n\) descends to \(p_n: \, H_n \rightarrow G_n\). The groupoid G with \({\mathcal {Z}}_0 \cong C^*_r(G)\) is now given by (23) and (24). As explained in Remark 7.1, its unit space \(X \mathrel {:=}G^{(0)}\) is given by \(X \cong \varprojlim \left\{ X_n; p_n \right\} \), where \(X_n = G_n^{(0)}\). As for \({\mathcal {W}}\), X surjects continuously onto \(\varprojlim \left\{ {\mathbb {T}}; \varvec{p}_n \right\} \) with Cantor space fibres, where \({\mathbb {T}}= [0,1] / { }_{0 \sim 1}\) and \(\varvec{p}_n([s]) = \left\{ [\lambda _y(s)] \text {: }y \in {\mathcal {Y}}(n) \right\} \). However, it is easy to see that (at least for some choices of \(\rho _{n,t}\) and \(\sigma _{n,t}\)), X will not be connected, though its connected components all have to be non-compact.