A Characterization of Semiprojectivity

for Subhomogeneous C^{∗}-Algebras

Dominic Enders

Received: October 1, 2015 Revised: April 22, 2016

Communicated by Joachim Cuntz

Abstract. We study semiprojective, subhomogeneous C^{∗}-algebras and give a detailed description of their structure. In particular, we find two characterizations of semiprojectivity for subhomogeneous C^{∗}-algebras: one in terms of their primitive ideal spaces and one by means of special direct limit structures over one-dimensional NCCW complexes. These results are obtained by working out several new permanence results for semiprojectivity, including a complete description of its behavior with respect to extensions by homogeneous C^{∗}-algebras.

2010 Mathematics Subject Classification: Primary 46L05; Secondary 46L80, 46L85, 54C55, 54F50 Keywords and Phrases: C^{∗}-algebras, semiprojectivity, subhomogeneous, quantum permutation algebras

1 Introduction

The concept of semiprojectivity is a type of perturbation theory for C^{∗}-algebras which has become a frequently used tool in many different aspects of C^{∗}- algebra theory. Due to a certain kind of rigidity, semiprojective C^{∗}-algebras are technically important in various situations. In particular, the existence and comparison of limit structures via approximate interwinings, which is an integral part of the Elliott classification program, often relies on perturbation properties of this type. This is one of the reasons why direct limits over semiprojective C^{∗}-algebras, e.g., AF- or A𝕋-algebras, are particularly tractable and one therefore constructs models preferably from semiprojective building blocks. The most popular of those are without doubt the non-commutative CW-complexes (NCCWs) introduced by Eilers, Loring and Pedersen. These are in fact semiprojective in dimension one ([ELP98], but see also [End14]). In this paper, we study semiprojectivity for general subhomogeneous C^{∗}-algebras and see whether there exist more interesting examples, i.e., besides the onedimensional NCCW complexes (1-NCCWs), that could possibly serve as useful building blocks in the construction of ASH-algebras. In Theorem 5.1.2, we give two characterizations of semiprojectivity for subhomogenous C^{∗}-algebras: an abstract one in terms of primitive ideal spaces and a concrete one by means of certain limit structures. These show that it is quite a restriction for a subhomogeneous C^{∗}-algebra to be semiprojective, though many examples beyond the class of 1-NCCWs exist. On the other hand, a detailed study of the structure of these algebras further reveals that they can always be approximated by 1-NCCWs in a very strong sense, see Corollary 5.2.1, and hence essentially share the same properties. The work of this paper is based on the characterization of semiprojectivity for commutative C^{∗}-algebras, which was recently obtained by Sørensen and Thiel in [ST12]. They showed that a commutative C^{∗}-algebra C(X) is semiprojective if and only if X is an absolute neighborhood retract of dimension at most 1 (a 1-ANR), thereby confirming a conjecture of Blackadar and generalizing earlier work of Chigogidze and Dranishnikov on the projective case ([CD10]). Their characterization further applies to trivially homogeneous C^{∗}-algebras, i.e. to algebras of the form C(X, 𝕄_{n}). In a first step, we generalize their result to general homogeneous C^{∗}-algebras. The main difficulty, however, is to understand which ways of ’gluing together’ several homogeneous C^{∗}-algebras preserve semiprojectivity, or more precisely: Which extensions of semiprojective, homogeneous C^{∗}-algebras are again semiprojective? Conversely, is semiprojectivity preserved when passing to a homogeneous subquotient? These questions essentially ask for the permanence behavior of semiprojectivity along extensions of the form 0 → C_{0}(X, 𝕄_{n}) → A → B → 0. While it is known that the permanence properties of semiprojectivity with respect to extensions are rather bad in general, we are able to work out a complete description of its behavior in the special case of extensions by homogeneous ideals, see Theorem 4.3.2. With this permanence result at hand, it is then straightforward to characterize semiprojectivity for subhomogeneous C^{∗}-algebras in terms of their primitive ideal spaces. In particular, it is a necessary condition that the subspaces corresponding to a fixed dimension are all 1-ANRs. Combining this with the structure result for one-dimensional ANR-spaces from [ST12], we further obtain a more concrete description of semiprojective, subhomogeneous C^{∗}-algebras by identifying them with certain special direct limits of 1-NCCWs.

This work was supported by the SFB 878 Groups, Geometry and Actions and the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

Documenta Mathematica 21 (2016) 987–1049

Semiprojectivity for Subhomogeneous C^{∗}-Algebras 989

This paper is organized as follows. In section 2, we briefly recall some topological definitions and results that will be used troughout the paper. We further remind the reader of some facts about semiprojectivity, subhomogeneous C^{∗}-algebras and their primitive ideal spaces. We then start by constructing a lifting problem which is unsolvable for strongly quasidiagonal C^{∗}-algebras. This lifting problem then allows us to extend the results of [ST12] from the commutative to the homogeneous case. Section 3 contains a number of new contructions for semiprojective C^{∗}-algebras. We first introduce a technique to extend lifting problems, a method that can be used to show that in certain situations semiprojectivity passes to ideals. After that, we introduce a class of maps which give rise to direct limits that preserve semiprojectivity. Important examples of such maps are given and discussed. Section 4 is devoted to the study of extensions by homogeneous C^{∗}-algebras, i.e. extensions of the form 0 → C_{0}(X, 𝕄_{n}) → A → B → 0. In 4.1, we define and study a certain set-valued retract map R: Prim(A) → 2^{Prim(B)} associated to such an extension. We discuss regularity concepts for R, i.e. continuity and finiteness conditions, and show how regularity of R relates to lifting properties of the corresponding Busby map and, by that, to splitting properties of the extension itself. In particular, we identify conditions under which regularity of R implies the existence of a splitting map s: B → A with good multiplicative properties. After that, we verify the required regularity properties for R in the case of a semiprojective extension A. In section 4.2 it is shown how certain limit structures for the space X give rise to limit structures for the extension A, again provided that the associated retract map R is sufficiently regular. Putting all these results together in 4.3, we find a ’2 out of 3’-type statement, Theorem 4.3.2, which gives a complete description for the behavior of semiprojectivity along extensions of the considered type. In section 5.1, we use this permanence result to work out two characterizations of semiprojectivity for subhomogeneous C^{∗}-algebras. These are presented in Theorem 5.1.2, the main result of this paper. Based on this, we find a number of consequences for the structure of these algebras, e.g. information about their K-theory and dimension. Further applications, such as closure and approximation properties, are discussed in 5.2. We finish by illustrating how this also gives a simple method to exclude semiprojectivity and show that the higher quantum permutation algebras are not semiprojective.

2 Preliminaries

2.1 The structure of 1-dimensional ANR-spaces

We are particularly interested in ANR-spaces of dimension at most one. The structure of these spaces has been studied and described in detail in [ST12, section 4]. Here we recall the most important notions and results. More information about ANR-spaces can be found in [Bor67]. For proofs and further reading on the theory of continua, we refer the reader to Nadler’s book [Nad92].

Definition 2.1.1. A compact, metric space X is an absolute retract (abbreviated AR-space) if every map f: Z → X from a closed subspace Z of a compact, metric space Y extends to a map g: Y → X, i.e. g ∘ ι = f with ι: Z → Y the inclusion map:

\xymatrix{ & Y \ar@{-->}[dl]_{g} \\ X & Z \ar[l]^{f} \ar[u]_{ι}}

If every map f: Z → X from a closed subspace Z of a compact, metric space Y extends to a map g: V → X on a closed neighborhood V of Z

\xymatrix{ & Y \\  & V \ar@{-->}[dl]_{g} \ar[u] \\ X & Z \ar[l]^{f} \ar[u]_{ι}}

then X is called an absolute neighborhood retract (abbreviated ANR-space).

A compact, locally connected, metric space is called a Peano space. A connected Peano space is called a Peano continuum. By the Hahn-Mazurkiewicz Theorem, these continua can be characterized as the continuous images of the unit interval. In particular, every Peano continuum is path-connected, while every Peano space is locally path-connected. Now given an ANR-space X, we can embed it into the Hilbert cube Q and obtain a retract from a neighborhood of X in Q onto X. Hence an ANRspace inherits all local properties of the Hilbert cube which are preserved under retracts. These properties include local connectedness, so that all ANR-spaces are Peano spaces. The converse, however, is not true in general. But as we will see, it is possible to identify the ANR-spaces among all Peano spaces, at least in the one-dimensional case. A closed subspace Y of a space X is a retract of X if there exists a continuous map r: X → Y such that r_{|Y} = id_{Y}. If the retract map r: X → Y regarded as a map to X is homotopic to the identity, then Y is called a deformation retract of X. It is a strong deformation retract if in addition the homotopy can be chosen to fix the subspace Y. The following concept of a core continuum is due to Meilstrup. It is crucial for understanding the structure of one-dimensional ANR-spaces.

Definition + Lemma 2.1.2 ([Mei05]). Let X be a non-contractible one-dimen-sional Peano continuum. Then there exists a unique strong deformation retract which contains no further proper deformation retract. We call it the core of X and denote it by core(X).

As in [ST12], we define the core of a contractible, one-dimensional Peano continuum to be any fixed point. Many questions about one-dimensional Peano continua can be reduced to questions about their cores. This reduction step uses a special retract map, the so-called first point map. Recall that an arc between two points x_{0}, x_{1} ∈ X is a path [0, 1] → X from x_{0} to x_{1} which is a homeomorphism onto its image.

Definition + Lemma 2.1.3 ([ST12, 4.14-16]). Let X be a one-dimensional Peano continuum and Y a subcontinuum with core(X) ⊂ Y. For each x ∈ X\Y there is a unique point r(x) ∈ Y such that r(x) is a point of an arc in X from x to any point of Y. Setting r(x) = x for all x ∈ Y, we obtain a map r: X → Y. This map is called the first point map, it is continuous and a strong deformation retract from X onto Y.

The following follows directly from the proof of [ST12, Lemma 4.14].

Lemma 2.1.4. Let X be a one-dimensional Peano continuum, Y ⊆ X a subcontinuum containing core(X) and r: X → Y the first point map onto Y. Then the following is true:

(i) For every point x ∈ X\Y there exists an arc from x to r(x) ∈ Y which is unique up to reparametrization.

(ii) If α is a path from x ∈ X\Y to y ∈ Y, then r(im(α)) ⊆ im(α).

The simplest example of a one-dimensional Peano space is a graph, i.e. a finite, one-dimensional CW-complex. The order of a point x in a graph X is defined as the smallest number n ∈ ℕ such that for every neighborhood V of x there exists an open neighborhood U ⊆ V of x with |∂U| = |\overline{U}\U| ≤ n. We denote the order of x in X by order(x, X). Given a one-dimensional Peano continuum X, one can reconstruct the space X from its core by ’adding’ the arcs which connect points of X\ core(X) with the core as described in 2.1.4. This procedure yields a limit structure for onedimensional Peano spaces which first appeared as Theorem 4.17 of [ST12]. In the case of one-dimensional ANR-spaces, the core is a finite graph and hence the limit structure entirely consists of finite graphs.

Theorem 2.1.5 ([ST12, Theorem 4.17]). Let X be a one-dimensional Peano continuum. Then there exists a sequence {Y_{k}}_{k=1}^{∞} such that

(i) each Y_{k} is a subcontinuum of X.

(ii) Y_{k} ⊂ Y_{k+1}.

(iii) lim_{k} Y_{k} = X.

(iv) Y_{1} = core(X) and for each k, Y_{k+1} is obtained from Y_{k} by attaching a line segment at a single point, i.e., \overline{Y_{k+1}\Y_{k}} is an arc with end point p_{k} such that \overline{Y_{k+1}\Y_{k}} ∩ Y_{k} = \{p_{k}\}.

(v) letting r_{k}: X → Y_{k} be the first point map for Y_{k} we have that {r_{k}}_{k=1}^{∞} converges uniformly to the identity map on X.

If X is also an ANR, then all Y_{k} are finite graphs. If X is even contractible (i.e. an AR), then core(X) is just some point and all Y_{k} are finite trees.

We will need a local criterion for identifying one-dimensional ANR-spaces among general Peano spaces. It was observed by Ward how to get such a characterization in terms of embeddings of circles.

Definition 2.1.6. Let X be a compact, metric space, then X does not contain small circles if there is an ϵ > 0 such that diam(ι(S^{1})) ≥ ϵ for every embedding ι: S^{1} → X.

Note that the property of containing arbitrarily small circles does not depend on the particular choice of metric.

Theorem 2.1.7 ([War60]). For a Peano space X the following are equivalent:

(i) X does not contain small circles.

(ii) X is an ANR-space of dimension at most one.

This statement can also be interpreted as follows. Non-embeddability of circles into X is the same as uniqueness of arcs in X, i.e. an arc between to two given points is unique up to reparametrization. More precisely, a Peano continuum is a one-dimensional AR-space if and only if there is no embedding S^{1} ↪ X if and only if X has unique arcs. Similarly, Theorem 2.1.7 can be read as: A Peano continuum X is a one-dimensional ANR-space if and only if it has locally unique arcs, meaning that every point has a neighbouhood in which any two points can be joined by a unique arc.

2.2 Subhomogeneous C^{∗}-algebras

In this section we collect some well known results on subhomogeneous C^{∗}- algebras. In particular, we recall some facts on their primitive ideal spaces. More detailed information can be found in [Dix77, Chapter 3] and [Bla06, Section IV.1.4].

Definition 2.2.1. Let N ∈ ℕ. A C^{∗}-algebra A is N-homogeneous if all its irreducible representations are of dimension N. A is N-subhomogeneous if every irreducible representation of A has dimension at most N.

The standard example of a N-homogeneous C^{∗}-algebra is C_{0}(X, 𝕄_{N}) for some locally compact space X. As the next proposition shows, subhomogeneous C^{∗}- algebras can be characterized as subalgebras of such. A proof of this fact can be found in [Bla06, IV.1.4.3-4].

Proposition 2.2.2. A C^{∗}-algebra A is N-subhomogeneous if and only if it is isomorphic to a subalgebra of some N-homogeneous C^{∗}-algebra C(X, 𝕄_{N}). If A is separable, we may choose X to be the Cantor set K.

Example 2.2.3 (1-NCCWs). One of the most important examples of subhomogeneous C^{∗}-algebras is the class of non-commutative CW-complexes (NCCWs) defined by Eilers, Loring and Pedersen in [ELP98]. The one-dimensional NCCWs, which we will abbreviate by 1-NCCWs, are defined as pullbacks of the form

\xymatrix{1-NCCW \ar@{-->}[r] \ar@{-->}[d] & G \ar[d] \\ C([0, 1], F) \ar[r]^{ev_{0} ⊕ ev_{1}} & F ⊕ F}

with F and G finite-dimensional C^{∗}-algebras. These are particularly interesting since they are semiprojective by [ELP98, Theorem 6.2.2].

For a subhomogeneous C^{∗}-algebra A, the primitive ideal space Prim(A), i.e. the set of kernels of irreducible representations endowed with the Jacobson topology, contains a lot of information. Another useful decription of the topology on Prim(A) is given by the folllowing lemma which we will make use of regularly. For an ideal J in a C^{∗}-algebra A we write ∥x∥_{J} to denote the norm of the image of the element x ∈ A in the quotient A/J.

Lemma 2.2.4 ([Bla06, II.6.5.6]). Let A be a C^{∗}-algebra.

1. If x ∈ A, define x̌: Prim(A) → ℝ_{≥0} by x̌(J) = ∥x∥_{J}. Then x̌ is lower semicontinuous.

2. If {x_{i}} is a dense set in the unit ball of A, and U_{i} = {J ∈ Prim(A) : x̌_{i}(J) > 1/2}, then {U_{i}} forms a base for the topology of Prim(A).

3. If x ∈ A and λ > 0, then {J ∈ Prim(A): x̌(J) ≥ λ} is compact (but not necessarily closed) in Prim(A).

Since we will mostly be interested in finite-dimensional representations, we consider the subspaces

Prim_{n}(A) = {ker(π) ∈ Prim(A): dim(π) = n}

for each finite n. Similarly, we write

Prim_{≤n}(A) = {ker(π) ∈ Prim(A): dim(π) ≤ n} = ⋃_{k≤n} Prim_{k}(A).

The following theorem describes the structure of these subspaces of Prim(A) and the relations between them.

Theorem 2.2.5 ([Dix77, 3.6.3-4]). Let A be a C^{∗}-algebra. The following holds for each n ∈ ℕ:

(i) Prim_{≤n}(A) is closed in Prim(A).

(ii) Prim_{n}(A) is open in Prim_{≤n}(A).

(iii) Prim_{n}(A) is locally compact and Hausdorff.

Now assume that A is a N-subhomogeneous C^{∗}-algebra. In this case Theorem 2.2.5 gives a set-theoretical (but in general not a topological) decomposition of its primitive spectrum

Prim(A) = ⨆_{n=1}^{N} Prim_{n}(A).

While each subspace in this decomposition is nice, in the sense that it is Hausdorff, Prim(A) itself is typically non-Hausdorff. In the subhomogeneous setting it is at least a T_{1}-space, i.e. points are closed. If we further assume A to be separable and unital, the space Prim(A) will also be separable and quasi-compact. Given a general C^{∗}-algebra A, there is a one-to-one correspondence between (closed) ideals J of A and closed subsets of Prim(A). More precisely, one can identify Prim(A/J) with the closed subset {K ∈ Prim(A): J ⊆ K}. In particular, we can consider the quotient A_{≤n} corresponding to the closed subset Prim_{≤n}(A) ⊆ Prim(A). This quotient is the maximal n-subhomogeneous quotient of A and has the following universal property: Any ^{∗}-homomorphism φ: A → B to some n-subhomogeneous C^{∗}-algebra B factors uniquely through A_{≤n}:

\xymatrix{A \ar[rr]^{φ} \ar@{->>}[dr] &  & B \\  & A_{≤n} \ar@{-->}[ur]}

2.3 Semiprojective C^{∗}-algebras

We recall the definition of semiprojectivity for C^{∗}-algebras, the main property of study in this paper. More detailed information about lifting properties for C^{∗}-algebras can be found in Loring’s book [Lor97].

Definition 2.3.1 ([Bla85, Definition 2.10]). A separable C^{∗}-algebra A is semiprojective if for every C^{∗}-algebra B and every increasing chain of ideals J_{n} in B with J_{∞} = \overline{⋃_{n} J_{n}}, and for every ^{∗}-homomorphism φ: A → B/J_{∞} there exist n ∈ ℕ and a ^{∗}-homomorphism \overline{φ}: A → B/J_{n} making the following diagram commute:

\xymatrix{ & B \ar@{->>}[d]^{π_{0}^{n}} \\  & B/J_{n} \ar@{->>}[d]^{π_{n}^{∞}} \\ A \ar[r]^{φ} \ar@{-->}[ur]^{\overline{φ}} & B/J_{∞}}

In this situation, the map \overline{φ} is called a partial lift of φ. The C^{∗}-algebra A is projective if, in the situation above, we can always find a lift \overline{φ}: A → B for φ. Let C be a class of C^{∗}-algebras. A C^{∗}-algebra A is (semi)projective with respect to C if it satisfies the definitions above with the restriction that the C^{∗}-algebras B, B/J_{n} and B/J_{∞} all belong to the class C.

Remark 2.3.2. One may also define semiprojectivity as a lifting property for maps to certain direct limits: an increasing sequence of ideals J_{n} in B gives an inductive system (B/J_{n})_{n} with surjective connecting maps π_{n}^{n+1} : B/J_{n} → B/J_{n+1} and limit (isomorphic to) B/J_{∞}. On the other hand, it is easily seen that every such system gives an increasing chain of ideals (ker(π_{0}^{n}))_{n}. Hence, semiprojectivity is equivalent to being able to lift maps to \underrightarrow{lim} D_{n} to a finite stage D_{n} provided that all connecting maps of the system are surjective. It is sometimes more convenient to work in this picture.

2.3.1 An unsolvable lifting problem

In order to show that a C^{∗}-algebra does not have a certain lifting property, we need to construct unsolvable lifting problems. One such construction by Loring ([Lor97, Proposition 10.1.8]) uses the fact that normal elements in quotient C^{∗}- algebras do not admit normal preimages in general, e.g. Fredholm operators of non-zero index. Here, we generalize Loring’s construction and obtain a version which also works for almost normal elements. Combining this with Lin’s theorem on almost normal matrices, we are able to construct unsolvable lifting problems not only for commutative C^{∗}-algebras, as in Loring’s case, but for the much larger class of strongly quasidiagonal C^{∗}-algebras. First we observe that almost normal elements in quotient C^{∗}-algebras always admit (almost as) almost normal preimages. Given an element x of some C^{∗}- algebra and ϵ > 0, we say that x is ϵ-normal if ∥x^{∗}x − xx^{∗}∥ ≤ ϵ∥x∥ holds.

Lemma 2.3.3. Let A, B be C^{∗}-algebras and π: A → B a surjective ^{∗}-homomor-phism. Then for every ϵ-normal element y ∈ B there exists a (2ϵ)-normal element x ∈ A with π(x) = y and ∥x∥ = ∥y∥.

Proof. Let (u_{λ})_{λ∈Λ} denote an approximate unit for ker(π) which is quasicentral for A. Pick any preimage x_{0} of x with ∥x_{0}∥ = ∥x∥ and set x := (1 − u_{λ_{0}})x_{0} for a suitable λ_{0} ∈ Λ.

The next lemma is due to Halmos. A short proof using the Fredholm alternative can be found in [BH74, Lemma 2].

Lemma 2.3.4 (Halmos). Let S ∈ B(H) be a proper isometry, then

dist (S, {N + K | N, K ∈ B(H), N normal, K compact}) = 1.

It is a famous result by H. Lin that in matrix algebras almost normal elements are uniformly close to normal ones ([Lin97]). A short, alternative proof involving semiprojectivity arguments can be found in [FR01].

Theorem 2.3.5 (Lin). For every ϵ > 0, there is a δ > 0 so that, for any d and any X in 𝕄_{d} satisfying

∥XX^{∗} − X^{∗}X∥ ≤ δ and ∥X∥ ≤ 1

there is a normal Y in 𝕄_{d} such that

∥X − Y ∥ ≤ ϵ.

The following is the basis for most of our unsolvable lifting problems appearing in this paper. Recall that a C^{∗}-algebra A is strongly quasidiagonal if every representation of A is quasidiagonal. See [Bla06, Section V.4.2] or [Bro00] for more information on quasidiagonality. In the following, let T denote the Toeplitz algebra C^{∗}(S|S^{∗}S = 1) and ϱ: T → C(S^{1}) the quotient map given by mapping S to the canonical generator z of C(S^{1}).

Proposition 2.3.6. There exists δ > 0 such that the following holds for all n ∈ ℕ: If A is strongly quasidiagonal and φ: A → C(S^{1}) ⊗ 𝕄_{n} is any ∗-homomorphism with dist(z ⊗ 1_{n}, im(φ)) < δ, then φ does not lift to a ∗-homomorphism from A to T ⊗ 𝕄_{n}:

\xymatrix{ & T ⊗ 𝕄_{n} \ar@{->>}[d]^{ϱ⊗id} \\ A \ar[r]^{φ} \ar@{-->}[ur]^{∄} & C(S^{1}) ⊗ 𝕄_{n}}

Proof. Choose δ′ > 0 corresponding to ϵ = 1/6 as in Theorem 2.3.5 and set δ = δ′/14. Let a′ ∈ A be such that ∥φ(a′)−z⊗1_{n}∥ < δ, then ∥[φ(a′), φ(a′)^{∗}]∥ ≤ 2δ(∥φ(a′)∥+1) < 5δ∥φ(a′)∥. Hence by Lemma 2.3.3 there exists a (10δ)-normal element a ∈ A with φ(a) = φ(a′) and 5/6 < ∥a∥ = ∥φ(a′)∥ < 6/5. Now if ψ is a ^{∗}-homomorphism with (ϱ ⊗ id) ∘ ψ = φ as indicated, we regard ψ as a representation on H^{⊕n} with T generated by the unilateral shift S on H. By assumption, ψ is then a quasidiagonal representation. In particular, ψ(a) can be approximated arbitrarily well by block-diagonal operators ([Bro00, Theorem 5.2]). We may therefore choose a (11δ)-normal block-diagonal operator B with 5/6 ≤ ∥B∥ ≤ 6/5 within distance at most 1/3 from ψ(a). Applying Lin’s Theorem to the normalized, (14δ)-normal block-diagonal operator ∥B∥^{−1}B shows the existence of a normal element N ∈ H^{⊕n} with ∥ψ(a) − N∥ ≤ 2/3. But then we find

∥(N − S ⊗ 1_{n}) + K(H^{⊕n})∥
≤ ∥N − ψ(a)∥ + ∥(ϱ ⊗ id)(ψ(a) − S ⊗ 1_{n})∥
≤ \frac{2}{3} + ∥φ(a′) − z ⊗ 1_{n}∥
≤ \frac{2}{3} + δ < 1

in contradiction to Lemma 2.3.4.

2.3.2 The homogeneous case

In [ST12], A. Sørensen and H. Thiel characterized semiprojectivity for commutative C^{∗}-algebras. Moreover, they gave a description of semiprojectivity for homogeneous trivial fields, i.e. C^{∗}-algebras of the form C_{0}(X, 𝕄_{N}). Note that the projective case was settled earlier by A. Chigogidze and A. Dranishnikov in [CD10]. Their result is as follows.

Theorem 2.3.7 ([ST12]). Let X be a locally compact, metric space and N ∈ ℕ. Then the following are equivalent:

1. C_{0}(X, 𝕄_{N}) is (semi)projective.

2. The one-point compactification αX is an A(N)R-space and dim(X) ≤ 1.

The work of Sørensen and Thiel will be the starting point for our analysis of semiprojectivity for subhomogeneous C^{∗}-algebras. In this section, we reduce the general N-homogeneous case to their result by showing that semiprojectivity for homogeneous, locally trivial fields implies global triviality. We further obtain some information about parts of the primitive ideal space for general semiprojective C^{∗}-algebras.

Lemma 2.3.8. Let I be a N-homogeneous ideal in a C^{∗}-algebra A. If A is semiprojective with respect to N-subhomogeneous C^{∗}-algebras, then the onepoint compactification α Prim(I) is a Peano space. If A is semiprojective, we further have dim(α Prim(I)) ≤ 1.

Proof. Let A_{≤N} be the maximal N-subhomogeneous quotient of A, then I is also an ideal in A_{≤N}. Being N-homogeneous, the ideal I is isomorphic to the section algebra Γ_{0}(E) of a locally trivial 𝕄_{N}-bundle E over the locally compact, second countable, metrizable Hausdorff space Prim(I) by [Fel61, Theorem 3.2]. Since A_{≤N} is separable and N-subhomogenous, we can embed it into C(K, 𝕄_{N}) with K the Cantor set by Proposition 2.2.2. Using the well known middle-third construction of K = \underleftarrow{lim}_{k}(⨆^{2^{k}}[0, 1]), we can write C(K) as

a direct limit \underrightarrow{lim}_{k} C([0, 1])^{⊕2^{k}} with surjective connecting maps. After tensoring with 𝕄_{N}, we obtain a lifting problem for A and hence can apply semiprojectivity of A with respect to N-subhomogenous C^{∗}-algebras. As the solution to this lifting problem factors through A_{≤N}, we obtain an embedding of A_{≤N} into C([0, 1], 𝕄_{N})^{⊕2^{k}} for some k.

\xymatrix{ & A \ar@{-->}[r] \ar@{->>}[d] & C([0, 1], 𝕄_{N})^{⊕2^{k}} \ar@{->>}[d] \\ I \ar[ur] \ar[r] & A_{≤N} \ar[r] \ar@{.>}[ur] & C(K, 𝕄_{N})}

The restriction of this embedding to I induces a continuous surjection π of ⨆^{2^{k}}[0, 1] onto α Prim(I). By the Hahn-Mazurkiewicz Theorem ([Nad92, Theorem 8.18]), this shows that α Prim(I) is a Peano space. Furthermore, we find a basis of compact neighborhoods consisting of Peano continua for any point x of α Prim(I) by [Nad92, Theorem 8.10]. Now let A be semiprojective and assume that dim(Prim(I)) = dim(α Prim(I)) > 1. Arguing precisely as in [ST12, Proposition 3.1], we use our basis of neighborhoods for points of Prim(I) to find arbitrarily small circles around a point x ∈ Prim(I). Using triviality of E around x, we obtain a lifting problem for A:

[figure: A; ((⨁_{ℕ} T )^{+} / (⨁_{1}^{n} 𝕂)) ⊗ 𝕄_{N}; I; ⊆; (⨁_{ℕ} C(S^{1}))^{+} ⊗ 𝕄_{N}; ((⨁_{ℕ} T )^{+} / (⨁_{ℕ} 𝕂)) ⊗ 𝕄_{N}]

Semiprojectivity of A allows us to solve this lifting problem. Now restrict a partial lift to the ideal I and consider its coordinates to obtain a commutative diagram

\xymatrix{ & T ⊗ 𝕄_{N} \ar@{->>}[d] \\ I \ar@{->>}[r] \ar@{-->}[ur] & C(S^{1}) ⊗ 𝕄_{N}.}

The map on the bottom is surjective since it is induced by the inclusion of one of the circles around x. But a diagram like this does not exist by Proposition 2.3.6 because I is homogeneous and by that strongly quasidiagonal.

Corollary 2.3.9. Let A be a semiprojective C^{∗}-algebra, then α Prim_{n}(A) is a Peano space for every n ∈ ℕ.

Proof. If A is semiprojective, each A_{≤n} is semiprojective with respect to n-subhomogeneous C^{∗}-algebras. Hence we can apply Lemma 2.3.8 to the n-homogeneous ideal ker(A_{≤n} → A_{≤n−1}) in A_{≤n} whose primitive ideal space is homeomorphic to Prim_{n}(A).

It is known to the experts that there are no non-trivial 𝕄_{n}-valued fields over one-dimensional spaces and we are indebted to L. Robert for pointing this fact out to us. Since we couldn’t find a proof in the literature, we include one here.

Lemma 2.3.10. Let E be a locally trivial field of C^{∗}-algebras over a separable, metrizable, locally compact Hausdorff space X with fiber 𝕄_{N} and Γ_{0}(E) the corresponding section algebra. If dim(X) ≤ 1, then Γ_{0}(E) is C_{0}(X)-isomorphic to C_{0}(X, 𝕄_{N}).

Proof. First assume that X is compact. One-dimensionality of X implies that that the Dixmier-Douady invariant δ ∈ Ȟ^{3}(X, ℤ) corresponding to Γ_{0}(E) vanishes. Therefore Γ_{0}(E) is stably C(X)-isomorphic to C(X, 𝕄_{N}) by Dixmier-Douady classification (see e.g. [RW98, Corollary 5.56]). Let ψ: Γ(E) ⊗ K → C(X, 𝕄_{N})⊗K be such an isomorphism and note that Γ(E) ≅ her(ψ(1_{Γ(E)} ⊗e)) via ψ with e a minimal projection in K. Equivalence of projections over onedimensional spaces is completely determined by their rank by [Phi07, Proposition 4.2]. Since ψ(1_{Γ(E)} ⊗ e) and 1_{C(X,𝕄_{N})} ⊗ e share the same rank N everywhere we therefore find v ∈ C(X, 𝕄_{N}) ⊗ K with v^{∗}v = ψ(1_{Γ(E)} ⊗ e) and vv^{∗} = 1_{C(X,𝕄_{N})} ⊗ e. But then Ad(v) gives a C(X)-isomorphism from her(ψ(1_{Γ(E)} ⊗ e)) onto her(1_{C(X,𝕄_{N})} ⊗ e) = C(X, 𝕄_{N}). Now consider the case of non-compact X. Since X is σ-compact, it clearly suffices to prove the following: Given compact subsets X_{1} ⊆ X_{2} of X and a C(X_{1})- isomorphism φ_{1} : Γ(E_{|X_{1}}) → C(X_{1}, 𝕄_{N}) there exists a C(X_{2})-isomorphism φ_{2} : Γ(E_{|X_{2}}) → C(X_{2}, 𝕄_{N}) extending φ_{1}. By the first part of the proof there is a C(X_{2})-isomorphism ψ_{2} : Γ(E_{|X_{2}}) → C(X_{2}, 𝕄_{N}). One-dimensionality of X_{1} implies Ȟ^{2}(X_{1}, ℤ) = 0, which means that every C(X_{1})-automorphism of C(X_{1}, 𝕄_{N}) is inner by [RW98, Theorem 5.42]. In particular, φ_{1} ∘ (ψ_{2}^{−1})_{|X_{1}} is of the form Ad(u) for some unitary u ∈ C(X_{1}, 𝕄_{N}). It remains to extend u to a unitary in C(X_{2}, 𝕄_{N}). This, however, follows from one-dimensionality of X and [HW48, Theorem VI.4].

We are now able to extend the results of [ST12] to the case of general Nhomogeneous C^{∗}-algebras:

Theorem 2.3.11. Let A be a N-homogeneous C^{∗}-algebra. The following are equivalent:

1. A is (semi)projective.

2. A ≅ C_{0}(Prim(A), 𝕄_{N}) and α Prim(A) is an A(N)R-space of dimension at most 1.

Proof. We know that (1) implies A ≅ C_{0}(Prim(A), 𝕄_{N}) by Lemma 2.3.8 and Lemma 2.3.10. The remaining implications are given by Theorem 2.3.7.

3 Constructions for semiprojective C^{∗}-algebras

Unfortunately, the class of semiprojective C^{∗}-algebras lacks good permanence properties. In fact, semiprojectivity is not preserved by most C^{∗}-algebraic standard constructions and the list of positive permanence results, most of which can be found in [Lor97], is surprisingly short. Here, we extend this list by a few new results.

3.1 Extending lifting problems

In this section, we introduce a technique to extend lifting problems from ideals to larger C^{∗}-algebras. This technique can be used to show that in many situations lifting properties of a C^{∗}-algebra pass to its ideals.

Lemma 3.1.1. Given a surjective inductive system of short exact sequences

\xymatrix{0 \ar[r] & C_{n} \ar[r]^{ι_{n}} \ar@{->>}[d]_{π_{n}^{n+1}} & D_{n} \ar[r]^{ϱ_{n}} \ar@{->>}[d]_{_{\overline{π}{n+1 \atop n}}} & E_{n} \ar[r] \ar@{->>}[d]_{_{\overline{\overline{π}}{n+1 \atop n}}} & 0 \\ 0 \ar[r] & C_{n+1} \ar[r]^{ι_{n+1}} & D_{n+1} \ar[r]^{ϱ_{n+1}} & E_{n+1} \ar[r] & 0}

and a commutative diagram of extensions

\xymatrix{0 \ar[r] & \underrightarrow{lim} C_{n} \ar[r]^{ι_{∞}} & \underrightarrow{lim} D_{n} \ar[r]^{ϱ_{∞}} & \underrightarrow{lim} E_{n} \ar[r] & 0 \\ 0 \ar[r] & I \ar[r]^{i} \ar[u]^{φ} & A \ar[r]^{p} \ar[u]^{\overline{φ}} & B \ar[r] \ar[u]^{\overline{\overline{φ}}} & 0}

the following holds: If both A and B are semiprojective, then φ lifts to C_{n} for some n. If both A and B are projective, then φ lifts to C_{1}.

Proof. First observe that we may assume the^{∗}-homomorphism \overline{\overline{φ}} to be injective since otherwise we simply pass to the system of extensions

\xymatrix{0 \ar[r] & C_{n} \ar[r]^{ι_{n}} & D_{n} ⊕ B \ar[r]^{ϱ_{n}⊕id_{B}} & E_{n} ⊕ B \ar[r] & 0}

and replace \overline{φ} by \overline{φ} ⊕ p and \overline{\overline{φ}} by \overline{\overline{φ}} ⊕ id_{B}. Using semiprojectivity of B, we can find a partial lift ψ: B → E_{n_{0}} of \overline{\overline{φ}} for some n_{0}, i.e. \overline{\overline{π}}_{n_{0}}^{∞} ∘ ψ = \overline{\overline{φ}}. Now consider the C^{∗}-subalgebras

D′_{n} := ϱ_{n}^{−1}((\overline{\overline{π}}_{n_{0}}^{n} ∘ ψ)(B)) ⊆ D_{n}

and observe that the restriction of \overline{π}_{n}^{n+1} to D′_{n} surjects onto D′_{n+1}. We also find that the direct limit \underrightarrow{lim} D′_{n} = \overline{π}_{n_{0}}^{∞}(D′_{n_{0}}) of this new system contains \overline{φ}(A). Hence semiprojectivity of A allows us to lift \overline{φ} (regarded as a map to \underrightarrow{lim} D′_{n}) to D′_{n} for some n ≥ n_{0}. Let σ: A → D′_{n} be a suitable partial lift, i.e. \overline{π}_{n}^{∞} ∘ σ = \overline{φ}, then the restriction of σ to the ideal I will be a solution to the original lifing problem for φ: The only thing we need to check is that the image of I under σ is in fact contained in C_{n}. But we know that \overline{\overline{π}}_{n}^{∞} is injective on (ϱ_{n} ∘ σ)(A) ⊆ (\overline{\overline{π}}_{n_{0}}^{n} ∘ ψ)(B) since \overline{\overline{φ}} = \overline{\overline{π}}_{n}^{∞} ∘ (\overline{\overline{π}}_{n_{0}}^{n} ∘ ψ) was assumed to be injective. Hence the identity

(\overline{\overline{π}}_{n}^{∞} ∘ϱ_{n} ∘σ)(i(I)) = (ϱ_{∞} ∘\overline{π}_{n}^{∞} ∘σ)(i(I)) = (ϱ_{∞} ∘\overline{φ})(i(I)) = (ϱ_{∞} ∘ι_{∞})(φ(I)) = 0

confirms that σ(i(I)) ⊆ i_{n}(C_{n}) holds.

Now assume that we are given an inductive system

\xymatrix{⋯ \ar@{->>}[r] & C_{n} \ar@{->>}[r]^{π_{n}^{n+1}} & C_{n+1} \ar@{->>}[r] & ⋯}

of separable C^{∗}-algebras with surjective connecting homomorphisms. Then each connecting map π_{n}^{n+1} canonically extends to a surjective^{∗}-homomorphism \overline{π}_{n}^{n+1} on the level of multiplier C^{∗}-algebras ([WO93, Theorem 2.3.9]), i.e., we automatically obtain a surjective inductive system of extensions

\xymatrix{0 \ar[r] & C_{n} \ar[r] \ar@{->>}[d]_{π_{n}^{n+1}} & M(C_{n}) \ar[r] \ar@{->>}[d]_{_{\overline{π}{n+1 \atop n}}} & Q(C_{n}) \ar[r] \ar@{->>}[d]_{_{\overline{\overline{π}}{n+1 \atop n}}} & 0 \\ 0 \ar[r] & C_{n+1} \ar[r] & M(C_{n+1}) \ar[r] & Q(C_{n+1}) \ar[r] & 0}.

We would like to apply Lemma 3.1.1 to such a system of extensions. However, the reader should be really careful when working with multipliers and direct limits at the same time since these constructions are not completely compatible: Each π_{n}^{∞}: C_{n} → \underrightarrow{lim} C_{n} extends to a ^{∗}-homomorphism M(C_{n}) → M(\underrightarrow{lim} C_{n}). The collection of these maps induces a ^{∗}-homomorphism p_{M}: \underrightarrow{lim} M(C_{n}) → M(\underrightarrow{lim} C_{n}) which is always surjective but only in trivial cases injective. The same occurs for the quotients, i.e. for the system of corona algebras Q(C_{n}). The situation can be summarized in the commutative diagram with exact rows

[figure: 0; C_{n}; M(C_{n}); Q(C_{n}); 0; 0; \underrightarrow{lim} C_{n}; \underrightarrow{lim} M(C_{n}); p_{M}; \underrightarrow{lim} Q(C_{n}); p_{Q}; 0; 0; \underrightarrow{lim} C_{n}; M(\underrightarrow{lim} C_{n}); Q(\underrightarrow{lim} C_{n}); 0]

where the quotient maps p_{M} and p_{Q} are the obstacles for an application of Lemma 3.1.1. The following proposition makes these obstacles more precise.

Proposition 3.1.2. Let A and B be semiprojective C^{∗}-algebras and

\xymatrix{0 \ar[r] & I \ar[r] & A \ar[r] & B \ar[r] & 0 & [τ]}

a short exact sequence with Busby map τ: B → Q(I). Let I ⟶^{∼} \underrightarrow{lim} C_{n} be an isomorphism from I to the limit of an inductive system of separable C^{∗}- algebras C_{n} with surjective connecting maps. If the Busby map τ can be lifted as indicated

[figure: \underrightarrow{lim} Q(C_{n}); p_{Q}; B; τ; Q(I) ≅ Q(\underrightarrow{lim} C_{n})] ,

then I → \underrightarrow{lim} C_{n} lifts to C_{n} for some n. If both A and B are projective, we can obtain a lift to C_{1}.

Proof. Keeping in mind that p_{Q} is the Busby map associated to the extension 0 → \underrightarrow{lim} C_{n} → \underrightarrow{lim} M(C_{n}) → \underrightarrow{lim} Q(C_{n}) → 0, the claim follows by combining Theorem 2.2 of [ELP99] with Lemma 3.1.1.

One special case, in which the existence of a lift for the Busby map τ as in Proposition 3.1.2 is automatic, is when the quotient B is a projective C^{∗}-algebra. Hence we obtain a new proof for the permanence result below which has the advantage that it does not use so-called corona extendability (cf. [Lor97, Section 12.2]).

Corollary 3.1.3 ([LP98], Theorem 5.3). Let 0 → I → A → B → 0 be short exact. If A is (semi)projective and B is projective, then I is also (semi)projective.

Another very specific lifting problem for which Proposition 3.1.2 applies, is the following mapping telescope contruction due to Brown.

Lemma 3.1.4. Let a sequence (C_{k})_{k} of separable C^{∗}-algebras be given and consider the telescope system (T_{n}, ϱ_{n}^{n+1}) associated to ⨁_{k=0}^{∞} C_{k} = \underrightarrow{lim}_{n} ⨁_{k=0}^{n} C_{k}, i.e.

T_{n} = {f ∈ C ([n, ∞], ⨁_{k=0}^{∞} C_{k}) : t ≤ m ⇒ f(t) ∈ ⨁_{k=1}^{m} C_{k}}

with ϱ_{n}^{n+1} : T_{n} → T_{n+1} the (surjective) restriction maps, so that \underrightarrow{lim}_{n}(T_{n}, ϱ_{n}^{n+1}) ≅ ⨁_{k=1}^{∞} C_{k}. Then both canonical quotient maps in the diagram

[figure: 0; \underrightarrow{lim} T_{n}; \underrightarrow{lim} M(T_{n}); p_{M}; \underrightarrow{lim} Q(T_{n}); p_{Q}; 0; 0; \underrightarrow{lim} T_{n}; M(\underrightarrow{lim} T_{n}); Q(\underrightarrow{lim} T_{n}); 0]

split.

Proof. It suffices to produce a split for p_{M} which is the identity on \underrightarrow{lim} T_{n}. Under the identification \underrightarrow{lim} T_{n} ≅ ⨁_{k=0}^{∞} C_{k} we have M(\underrightarrow{lim} T_{n}) ≅ ∏_{k=0}^{∞} M(C_{k}). One checks that

T_{n} = ⨁_{k=0}^{n} C([n, ∞], C_{k}) ⊕ ⨁_{k>n} C_{0}((k, ∞], C_{k})

and hence

∏_{k=0}^{∞} C([max{n, k}, ∞], M(C_{k})) ⊂ M(T_{n}).

It follows that the sum of embeddings as constant functions

∏_{k=0}^{∞} M(C_{k}) → ∏_{k=0}^{∞} C([max{n, k}, ∞], M(C_{k})) ⊂ M(T_{n})

defines a split for the quotient map \underrightarrow{lim} M(T_{n}) → M(\underrightarrow{lim} T_{n}). It is easily verified that this split is the identity on ⨁_{k=1}^{∞} C_{k}.

Remark 3.1.5 (Lifting the Busby map). Given an extension 0 → I → A → B → 0 with both A and B semiprojective, the associated Busby map does in general not lift as in 3.1.2. However, there are a number of interesting situations where it does lift and we therefore can use Propostion 3.1.2 to obtain lifting properties for the ideal I. One such example is studied in [End14], where it is (implicitly) shown that the Busby map lifts if B is a finite-dimensional C^{∗}- algebra. This observation leads to the fact that semiprojectivity passes to ideals of finite codimension. Further examples will be given in section 4, where we study Busby maps associated to extensions by homogeneous ideals and identify conditions which guarantee that 3.1.2 applies.

3.2 Direct limits which preserve semiprojectivity

3.2.1 Weakly conditionally projective homomorphisms

The following definition characterizes ^{∗}-homomorphisms along which lifting solutions can be extended in an approximate manner. This type of maps is implicitly used in [CD10] and [ST12] in the special case of finitely presented, commutative C^{∗}-algebras.

Definition 3.2.1. A ^{∗}-homomorphism φ: A → B is weakly conditionally projective if the following holds: Given ϵ > 0, a finite subset F ⊂ A and a commuting square

\xymatrix{A \ar[d]_{φ} \ar[r]^{ψ} & D \ar@{->>}[d]^{π} \\ B \ar[r]^{ϱ} & D/J,}

there exists a ^{∗}-homomorphism ψ′: B → D as indicated

\xymatrix{A \ar[d]_{φ} \ar[r]^{ψ} & D \ar@{->>}[d]^{π} \\ B \ar[r]^{ϱ} \ar@{-->}[ur]^{ψ′} & D/J}

which satisfies π ∘ ψ′ = ϱ and ∥(ψ′ ∘ φ)(a) − ψ(a)∥ < ϵ for all a ∈ F.

The definition above is a weakening of the notion of conditionally projective morphisms, as introduced in section 5.3 of [ELP98], where one asks the homomorphism ψ′ in 3.2.1 to make both triangles of the lower diagram to commute exactly. While conditionally projective morphisms are extremely rare (even when working with projective C^{∗}-algebras, cf. the example below), there is a sufficient supply of weakly conditionally projective ones, as we will show in the next section.

Example 3.2.2. The inclusion map id ⊕ 0: C_{0}(0, 1] → C_{0}(0, 1] ⊕ C_{0}(0, 1] is weakly conditionally projective but not conditionally projective. This can be illustrated by considering the commuting square

[figure: C_{0}(0, 1]; ψ; id ⊕ 0; C_{0}(0, 3); π; C_{0}(0, 1] ⊕ C_{0}[2, 3); C_{0}(0, 1] ⊕ C_{0}[2, 3)]

where π is the restriction map and ψ is given by sending the canonical generator t of C_{0}(0, 1] to the function

(ψ(t))(s) = \begin{cases}s & if s ≤ 1 \\ 1 − s & if 1 < s ≤ 2 \\ 0 & if 2 ≤ s \end{cases} .

It is clear that there is no lift for the generator of C_{0}[2, 3) which is orthogonal to ψ(t). This shows that the map id ⊕ 0 is not conditionally projective. However, after replacing ψ(t) with (ψ(t)−ϵ)_{+} for any ϵ > 0, finding an orthogonal lift for the generator of the second summand is no longer a problem. Using this idea, it will be shown in Proposition 3.2.4 that id ⊕ 0 is in fact weakly conditionally projective.

If A is a (semi)projective C^{∗}-algebra and φ: A → B is weakly conditionally projective, then B is of course also (semi)projective. The next lemma shows that (semi)projectivity is even preserved along a sequence of such maps. Its proof is of an approximate nature and relies on a one-sided approximate intertwining argument (cf. section 2.3 of [Rør02]), a technique borrowed from the Elliott classification program.

Lemma 3.2.3. Suppose \xymatrix{A_{1} \ar[r]^{φ_{1}^{2}} & A_{2} \ar[r]^{φ_{2}^{3}} & A_{3} \ar[r]^{φ_{3}^{4}} & ⋯} is an inductive system of separable C^{∗}-algebras. If A_{1} is (semi)projective and all connecting maps φ_{n}^{n+1} are weakly conditionally projective, then the limit A_{∞} = \underrightarrow{lim}(A_{n}, φ_{n}^{n+1}) is also (semi)projective.

Proof. We will only consider the projective case, the statement for the semiprojective case is proven analogously with obvious modifications. Choose finite subsets F_{n} ⊂ A_{n} with φ_{n}^{n+1}(F_{n}) ⊆ F_{n+1} such that the union ⋃_{m=n}^{∞}(φ_{n}^{m})^{−1}(F_{m}) is dense in A_{n} for all n. Further let (ϵ_{n})_{n} be a sequence in ℝ_{>0} with ∑_{n=1}^{∞} ϵ_{n} < ∞. Now let ϱ: A_{∞} → D/J be a ^{∗}-homomorphism to some quotient C^{∗}-algebra D/J. By projectivity of A_{1} there is a^{∗}-homomorphism s_{1} : A_{1} → D with π ∘ s_{1} = ϱ ∘ φ_{1}^{∞}. Since the maps φ_{n}^{n+1} are weakly conditionally projective, we can inductively choose s_{n+1} : A_{n+1} → D with π ∘ s_{n+1} = ϱ ∘ φ_{n+1}^{∞} such that

∥s_{n}(a) − (s_{n+1} ∘ φ_{n}^{n+1})(a)∥ < ϵ_{n}

holds for all a ∈ F_{n}. It is now a standard computation (and therefore ommited) to check that ((s_{m} ∘ φ_{n}^{m})(x))_{m} is a Cauchy sequence in D for every x ∈ F_{n}. Furthermore, the induced map φ_{n}^{∞}(x) ↦ lim_{m}(s_{m} ∘ φ_{n}^{m})(x) extends from the dense subset ⋃_{n} φ_{n}^{∞}(F_{n}) to a ^{∗}-homomorphism s: A_{∞} → D.

\xymatrix{A_{n} \ar[r]^{s_{n}} \ar[d]_{φ_{n}^{n+1}} & D \ar@{->>}[dd]^{π} \\ A_{n+1} \ar[d]_{φ_{n+1}^{∞}} \ar@{-->}[ur]^{s_{n+1}} \\ A_{∞} \ar[r]^{ϱ} \ar@{.>}[uur]^{s} & D/J}

Since each s_{n} lifts π, the same holds for their pointwise limit, i.e. the limit map s satisfies π ∘ s = ϱ. This shows that A_{∞} is projective.

3.2.2 Adding non-commutative edges

In order to make Lemma 3.2.3 a useful tool for constructing semiprojective C^{∗}-algebras, we have to ensure the existence of weakly projective ∗-homomorphisms as defined in 3.2.1. The examples we work out in this section arise in special pullback situations where one ’adds a non-commutative edge’ to a given C^{∗}-algebra A. By this we mean that we form the pullback of A and C([0, 1]) ⊗ 𝕄_{n} over a n-dimensional representation of A and the evaluation map ev_{0}. In the special case of A = C(X) being a commutative C^{∗}-algebra and n = 1 this pullback construction already appeared in [CD10] and [ST12] where it indeed corresponds to attaching an egde [0, 1] at one point to the space X. Here we show that the map obtained by extending elements of A as constant functions onto the attached non-commutative edge gives an example of a weakly conditionally projective ^{∗}-homomorphism. As an application, we observe that the AF-telescopes studied in [LP98] arise from weakly projective ^{∗}-homomorphisms and hence projectivity of these algebras is a direct consequence of Lemma 3.2.3.

Adapting notation from [ELP98], we set

T (ℂ, G) = {f ∈ C_{0}((0, 2], G) : t ≤ 1 ⇒ f(t) ∈ ℂ · 1_{G}},
S(ℂ, G) = {f ∈ C_{0}((0, 2), G) : t ≤ 1 ⇒ f(t) ∈ ℂ · 1_{G}}

for G a unital C^{∗}-algebra. We further write

T (ℂ, G, F) = \{f ∈ C_{0}((0, 3], F) : \begin{matrix}t ≤ 2 ⇒ f(t) ∈ G \\ t ≤ 1 ⇒ f(t) ∈ ℂ · 1_{G} \end{matrix} \}

with respect to a fixed inclusion G ⊆ F. We have the diagram

\xymatrix{T (ℂ, G, F) \ar[r] \ar[d] & C([2, 3], F) \ar[d]^{_{ev_{2}}} \\ T (ℂ, G) \ar[r]^{_{ev_{2}}} & F}

which is a special case of the pullback situation considered in the next proposition. However, this example is in some sense generic and implementing it into the general situation is an essential part of proving the following.

Proposition 3.2.4. Given a (semi)projective C^{∗}-algebra Q and a ^{∗}-homomor-phism τ: Q → 𝕄_{n}, the following holds:

1. The pullback P over τ and ev_{0} : C([0, 1], 𝕄_{n}) → 𝕄_{n}, i.e.

P = {(q, f) ∈ Q ⊕ C([0, 1], 𝕄_{n}) : τ(q) = f(0)},

is (semi)projective.

2. The canonical split s: Q → P, q ↦ (q, τ(q)⊗1_{[0,1]}) is weakly conditionally projective.

Proof. (1) Semiprojectivity of the pullback P follows from [End14, Corollary 3.4]. Since P is homotopy equivalent to Q, the projective statement follows from the semiprojective one using [Bla12, Corollary 5.2].

(2) For technical reasons we identify the attached interval [0, 1] with [2, 3] and consider the pullback

\xymatrix{P \ar@{-->}[r] \ar@{-->}[d] & C([2, 3], 𝕄_{n}) \ar[d]^{_{ev_{2}}} \\ Q \ar[r]^{τ} & 𝕄_{n}}

with s: Q → P, q ↦ (q, τ(q) ⊗ 1_{[2,3]}) instead. Denote by G ⊆ 𝕄_{n} the image of τ. According to [ELP98, Theorem 2.3.3], we can find a ^{∗}-homomorphism \overline{φ}: T (ℂ, G) → Q such that

[figure: 0; ker(τ); Q; τ; G; 0; 0; S(ℂ, G); T (ℂ, G); \overline{φ}; ev_{2}; G; 0]

commutes and \overline{φ}_{|S(ℂ,G)} is a proper ^{∗}-homomorphism to ker(τ) (meaning that the hereditary subalgebra generated by its image is all of ker(τ)). Using the pullback property of P, \overline{φ} can be extended to φ: T (ℂ, G, 𝕄_{n}) → P such that

[figure: 0; C_{0}((2, 3], 𝕄_{n}); P; Q; s; 0; 0; C_{0}((2, 3], 𝕄_{n}); T (ℂ, G, 𝕄_{n}); φ; T (ℂ, G); \overline{φ}; s′; 0]

commutes. In particular we have φ ∘ s′ = s ∘ \overline{φ}, where s′ is the canonical split which simply extends functions constantly onto [2, 3]. Choose generators f_{1}, ..., f_{l} of norm 1 for C_{0}((2, 3], 𝕄_{n}) and generators g_{1}, ..., g_{k} of norm 1 for T (ℂ, G). We need the following ’softened’ versions of P: For δ > 0 we consider the universal C^{∗}-algebra

P_{δ} = C^{∗} ({f^{δ}, q^{δ} : f ∈ C_{0}((2, 3], 𝕄_{n}), q ∈ Q} |R_{C_{0}((2,3],𝕄_{n})}&R_{Q}&R_{δ})

which is generated by copies of C_{0}((2, 3], 𝕄_{n}) and Q (here R_{C_{0}((2,3],𝕄_{n})}, R_{Q} denote all the relations from C_{0}((2, 3], 𝕄_{n}) resp. from Q) and additional, finitely many relations

R_{δ} = \{∥f_{i}^{δ}(\overline{φ}(g_{j}))^{δ} − (f_{i}(g_{j}(2) ⊗ 1_{[2,3]}))^{δ}∥ ≤ δ\}\begin{matrix}1≤i≤l \\ 1≤j≤k \end{matrix} .

Note that P = \underrightarrow{lim} P_{δ} with respect to the canonical surjections p_{δ,δ′}: P_{δ} → P_{δ′} (for δ > δ′) and denote the induced maps P_{δ} → P, f^{δ} ↦ f, q^{δ} ↦ s(q) by p_{δ,0}. Since P is semiprojective by part (1) of this proposition, we can find a partial lift j_{δ}: P → P_{δ} for some δ > 0, i.e. p_{δ,0} ∘ j_{δ} = id_{P}. Now let a finite set F = {x_{1}, ..., x_{m}} ⊆ Q and ϵ > 0 and be given. Denoting the inclusions Q → P_{δ}, q ↦ q^{δ} by s_{δ}, we can (after decreasing δ if necessary) assume that ∥s_{δ}(x_{i}) − (j_{δ} ∘ s)(x_{i})∥ ≤ ϵ holds for all 1 ≤ i ≤ m. Now given any commuting square

\xymatrix{Q \ar[d]^{s} \ar[r]^{ψ} & D \ar@{->>}[d]^{π} \\ P \ar[r]^{ϱ} & D/J}

it only remains to construct a ^{∗}-homomorphism ψ_{δ}: P_{δ} → D such that in the diagram

\xymatrix{Q \ar[rr]^{ψ} \ar[dd]_{s} \ar[dr]^{s_{δ}} &  & D \ar@{->>}[dd]^{π} \\  & P_{δ} \ar@{-->}[ur]^{ψ_{δ}} \ar[dl]_{p_{δ,0}} \\ P \ar[ur]_{j_{δ}} \ar[rr]_{ϱ} &  & D/J}

Documenta Mathematica 21 (2016) 987–1049

the upper central triangle and the lower right triangle commute. We consider the following subalgebras of T (ℂ, G) and S(ℂ, G) for any η > 0:

T_{η}(ℂ, G) = {f ∈ T (ℂ, G) : f is constant on (0, η] ∪ [2 − η, 2]}
S_{η}(ℂ, G) = {f ∈ S(ℂ, G) : f is constant (=0) on (0, η] ∪ [2 − η, 2]}

Since

T (ℂ, G) = \overline{⋃_{η>0} T_{η}(ℂ, G)}

we find 0 < η < \frac{1}{2} and elements g̃_{j} ∈ T_{η}(ℂ, G) with g̃_{j}(2) = g_{j}(2) and ∥g_{j}−g̃_{j}∥ < δ for every 1 ≤ j ≤ k. Let h ∈ T (ℂ, G) be the scalar-valued function which equals 1_{G} on [η, 2 − η], satisfies h(0) = h(2) = 0 and is linear in between. Consider the hereditary C^{∗}-subalgebra D′ = \overline{(1 − (ψ ∘ \overline{φ})(h))D(1 − (ψ ∘ \overline{φ})(h))} and define

D′′ := (ψ ∘ \overline{φ})(T_{η}(ℂ, G)) + D′ ⊆ D.

Then (ψ ∘\overline{φ})(S_{η}(ℂ, G)) and D′ are orthogonal ideals in D′′ because h is central in T (ℂ, G). We further have (ϱ ∘ φ)(C_{0}((2, 3], 𝕄_{n})) ⊆ π(D′) and hence obtain a commutative diagram

\xymatrix{0 \ar[r] & (ψ ∘ \overline{φ})(S_{η}(ℂ, G)) \ar[r] \ar@{->>}[d]^{π} & D′′ \ar[r] \ar@{->>}[d]^{π} & H_{D} + D′ \ar[r] \ar@{->>}[d] & 0 \\ 0 \ar[r] & (ϱ ∘ φ ∘ s′)(S_{η}(ℂ, G)) \ar[r] & π(D′′) \ar[r] & H_{D/J} + π(D′) \ar[r] & 0 \\ 0 \ar[r] & s′(S_{η}(ℂ, G)) \ar[r] \ar[u]^{ϱ∘φ} & s′(T_{η}(ℂ, G)) + C_{0}((2, 3], 𝕄_{n}) \ar[u]^{ϱ∘φ} \ar[r] & T̂(G, 𝕄_{n}) \ar[r] \ar@{-->}[u] & 0}

where H_{D} and H_{D/J} are finite-dimensional C^{∗}-algebras given by

H_{D} = (ψ ∘ \overline{φ})(T_{η}(ℂ, G))/(ψ ∘ \overline{φ})(S_{η}(ℂ, G)),

H_{D/J} = (ϱ ∘ φ ∘ s′)(T_{η}(ℂ, G))/(ϱ ∘ φ ∘ s′)(S_{η}(ℂ, G))

and T̂(G, 𝕄_{n}) denotes what is called a crushed telescope in [ELP98]:

T̂(G, 𝕄_{n}) = {f ∈ C([2, 3], 𝕄_{n}): f(2) ∈ G}

By [ELP98, Proposition 6.1.1], the embedding G → T̂(G, 𝕄_{n}) as constant functions is a conditionally projective map (in the sense of [ELP98, Section 5.3]). It is hence possible to extend the map

\xymatrix{G \ar[r]^{∼} & T_{η}(ℂ, G)/S_{η}(ℂ, G) \ar[r] & H_{D} ⊂ H_{D} + D′} to a ^{∗}-homomorphism

ψ′: T̂(G, 𝕄_{n}) → H_{D} + D′ such that the diagram with exact rows

[figure: 0; D′; π; H_{D} + D′; H_{D}; 0; 0; π(D′); H_{D/J} + π(D′); H_{D/J}; 0; 0; C_{0}((2, 3], 𝕄_{n}); ϱ∘φ; T̂(G, 𝕄_{n}); _{ev_{2}}; ψ′; G; 0]

commutes. In particular, ψ′ restricts to a^{∗}-homomorphism C_{0}((2, 3], 𝕄_{n}) → D′ which we will also denote by ψ′. But then a diagram chase confirms that

ψ′(f_{i}) · (ψ ∘ \overline{φ})(g̃_{j}) = ψ′(f_{i} · (g̃_{j}(2) ⊗ 1_{[2,3]}))

holds for every i, j. Finally, define ψ_{δ}: P_{δ} → D by

q^{δ} ↦ ψ(q) and f_{i}^{δ} ↦ ψ′(f_{i}).

It needs to be checked that ψ_{δ} is well-defined, i.e. that the elements ψ_{δ}(f_{i}^{δ}) and ψ_{δ}(\overline{φ}(g_{j})^{δ}) satisfy the relations R_{δ}:

∥ψ_{δ}(f_{i}^{δ})ψ_{δ}(\overline{φ}(g_{j})^{δ}) − ψ_{δ}((f_{i}(g_{j}(2) ⊗ 1_{[2,3]}))^{δ})∥
= ∥ψ′(f_{i})((ψ ∘ \overline{φ})(g_{j})) − ψ′(f_{i}(g_{j}(2) ⊗ 1_{[2,3]}))∥
≤ ∥ψ′(f_{i})(ψ ∘ \overline{φ})(g̃_{j}) − ψ′(f_{i}(g̃_{j}(2) ⊗ 1_{[2,3]}))∥ + ∥f_{i}∥ · ∥g_{j} − g̃_{j}∥ < δ

Since we also have ψ_{δ} ∘ s_{δ} = ψ and π ∘ ψ_{δ} = ϱ ∘ p_{δ,0}, the proof is hereby complete.

One example, where pullbacks as in 3.2.4 show up, is the class of so-called AF-telescopes defined by Loring and Pedersen:

Definition 3.2.5 ([LP98]). Let A = \overline{⋃ A_{n}} be the inductive limit of an increasing union of finite-dimensional C^{∗}-algebras A_{n}. We define the AF-telescope associated to this AF-system as

T (A) = {f ∈ C_{0}((0, ∞], A) : t ≤ n ⇒ f(t) ∈ A_{n}}.

We have an obvious limit structure for T (A) = \underrightarrow{lim} T (A_{k}) over the finite telescopes

T (A_{k}) = {f ∈ C_{0}((0, k], A_{k})) : t ≤ n ⇒ f(t) ∈ A_{n}}.

Now the embedding of T (A_{k}) into T (A_{k+1}) is given by extending the elements of T (A_{k}) constantly onto the attached interval [k, k + 1]. This is nothing but a finite composition of maps as in part (2) of 3.2.4. Hence the connecting maps in the system of finite telescopes are weakly conditionally projective and using Lemma 3.2.3 we recover [LP98, Theorem 7.2]:

Corollary 3.2.6. All AF -telescopes are projective.

In contrast to the original proof we didn’t have to work out any description of the telescopes by generators and relations. Such a description would have to encode the structure of each A_{n} as well as the inlusions A_{n} ⊂ A_{n+1} (i.e., the Bratteli-diagram of the system). Showing that such an infinite set of generators and relations gives rise to a projective C^{∗}-algebra is possible but complicated. Instead we showed that these algebras are build up from the projective C^{∗}- algebra T (A_{0})=0 using operations which preserve projectivity.

4 Extensions by homogeneous C^{∗}-algebras

In this section we study extensions by (trivially) homogeneous C^{∗}-algebras, i.e. extensions of the form

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] & A \ar[r] & B \ar[r] & 0.}

Our final goal is to understand the behavior of semiprojectivity along such extensions, and we will eventually achieve this in Theorem 4.3.2.

4.1 Associated retract maps

Identifying X with an open subset of Prim(A), we make the following definition of an associated retract map. This map will play a key role in our study of extensions.

Definition 4.1.1. Let X be locally compact space with connected components (X_{i})_{i∈I} and

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] & A \ar[r] & B \ar[r] & 0}

a short exact sequence of C^{∗}-algebras. We define the (set-valued) retract map R associated to the extension to be the map

R: Prim(A) → 2^{Prim(B)}

given by

R(z) = \begin{cases}z & if z ∈ Prim(B), \\ ∂X_{i} = \overline{X_{i}}\X_{i} & if z ∈ X_{i} ⊆ X. \end{cases}

Note that R defined as above takes indeed values in 2^{Prim(B)} because the connected components X_{i} are always closed in X. However, in our cases of interest the components X_{i} will actually be clopen in X (e.g. if X is locally connected) so that we have a topological decomposition X = ⨆_{i} X_{i}.

4.1.1 Regularity properties for set-valued maps

Let X, Y be sets and S: X → 2^{Y} a set-valued map. We say that S has pointwise finite image if S(x) ⊆ Y is a finite set for every x ∈ X. If furthermore X and Y are topological spaces, we will use the following notion of semicontinuity for S (cf. [AF90, Section 1.4]).

Definition + Lemma 4.1.2. Let X, Y be topological spaces. A set-valued map S: X → 2^{Y} is lower semicontinuous if one of the following equivalent conditions holds:

(i) {x ∈ X: S(x) ⊆ B} is closed in X for every closed B ⊆ Y.

(ii) For every neighborhood N(\overline{y}) of \overline{y} ∈ S(\overline{x}) there exists a neighborhood N(\overline{x}) of \overline{x} with S(x) ∩ N(\overline{y}) ≠ ∅ for every x ∈ N(\overline{x}).

(iii) For every net (x_{λ})_{λ∈Λ} ⊂ X with x_{λ} → x and every y ∈ S(x) there exists a net (y_{μ})_{μ∈M} ⊂ {S(x_{λ}): λ ∈ Λ} such that y_{μ} → y.

Proof. (i) ⇒ (ii): Let N(\overline{y}) be an open neighborhood of \overline{y} ∈ S(\overline{x}). Then \{x ∈ X : S(x) ⊂ Y \N(\overline{y})\} is closed and does not contain \overline{x}. Hence we find an open neighborhood N(\overline{x}) of \overline{x} in X\\{x ∈ X : S(x) ⊂ Y \N(\overline{y})\} = \{x ∈ X : S(x) ∩ N(\overline{y}) ≠ ∅\}. (ii) ⇒ (iii): Denote by N the family of neighborhoods of y ordered by reversed inclusion. Set M = {(λ, N) ∈ Λ × N : S(x_{λ′}) ∩ N ≠ ∅ ∀ λ′ ≥ λ}, then by assumption M is nonempty and directed with respect to the partial order (λ_{1}, N_{1}) ≤ (λ_{2}, N_{2}) iff λ_{1} ≤ λ_{2} and N_{2} ⊆ N_{1}. Now pick a y_{(λ,N)} ∈ S(x_{λ}) ∩ N for each (λ, N) ∈ M, then (y_{μ})_{μ∈M} constitutes a suitable net converging to y. (iii) ⇒ (i): Let a closed set B ⊆ Y and (x_{λ})_{λ∈Λ} ⊂ {x ∈ X : S(x) ⊆ B} with x_{λ} → \overline{x} be given. Then for any \overline{y} ∈ S(\overline{x}) we find a net y_{μ} → \overline{y} with (y_{μ}) ⊂ \{S(x_{λ}) : λ ∈ Λ\} ⊂ B. Since B is closed we have \overline{y} ∈ B showing that S(\overline{x}) ⊂ B.

Remark 4.1.3. An ordinary (i.e. a single-valued) map is evidently lower semicontinuous in the sense above if and only if it is continuous. If both spaces X and Y are first countable, we may use sequences instead of nets in condition (iii).

Examples of set-valued maps that are lower semicontinuous in the sense above arise from split extensions by homogeneous C^{∗}-algebras as follows.

Example 4.1.4. Let a split-exact sequence of separable C^{∗}-algebras

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{n}) \ar[r] & A \ar[r]_{π} & B \ar[r] \ar[l]_{s} & 0}

be given and consider the set-valued map R_{s}: Prim(A) → 2^{Prim(B)} given by

R_{s}(z) = \begin{cases}z & if z ∈ Prim(B) \\ \{[π_{z,1}], ..., [π_{z,r(z)}]\} & if z ∈ X \end{cases}

where π_{z,1} ⊕ ... ⊕ π_{z,r(z)} is the decomposition of B ⟶^{s} A → C_{b}(X, 𝕄_{n}) ⟶^{ev_{z}} 𝕄_{n}

into irreducible summands. Then R_{s} is lower semicontinuous in the sense of 4.1.2.

Proof. We verify condition (ii) of 4.1.2: Let z_{n} → \overline{z} in Prim(A) and a neighborhood N(\overline{y}) of \overline{y} ∈ R_{s}(\overline{z}) in Prim(B) be given. By Lemma 2.2.4 we may assume that N(\overline{y}) is of the form \{z ∈ Prim(B): b̌(z) > 1/2\} for some b ∈ B. By definition of R_{s}, we find š(b)(z) = max_{y∈R_{s}(z)} b̌(y) for all z ∈ Prim(A). Hence N(\overline{z}) = \{z ∈ Prim(A): š(b) > 1/2\} constitutes a neighborhood of \overline{z} in Prim(A) which satisfies 4.1.2 (ii).

Note that the retract map R_{s} in 4.1.4 highly depends on the choice of splitting s while the retract map R from 4.1.1 is associated to the underlying extension in a natural way. It is the goal of section 4.1.2 to find a splitting s such that R = R_{s} holds. This is, however, not always possible. It can even happen that the underlying extension splits while R is not of the form R_{s} for any splitting s (cf. remark 4.3.3). Under suitable conditions, we will at least be able to arrange R = R_{s} outside of a compact set K ⊂ X, i.e. we can find a (not necessarily multiplicative) splitting map s such that B ⟶^{s} A → C_{b}(X, 𝕄_{n}) is multiplicative on X\K so that R_{s}(x) is still well-defined and coincides with R(x) for all x ∈ Prim(A)\K.

4.1.2 Lifting the Busby map

In this section we identify conditions on an extension

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] & A \ar[r] & B \ar[r] \ar@{.>}[l]|{?} & 0 & [τ]}

which allow us to contruct a splitting s: B → A. This is evidently the same as asking for a lift of the corresponding Busby map τ as indicated on the left of the commutative diagram

[figure: C(βX, 𝕄_{N}); ϱ; ∏_{i∈I} C(βX_{i}, 𝕄_{N}); B; τ; s; ⊕τ_{i}; C(χ(X), 𝕄_{N}); ∏_{i∈I} C(χ(X_{i}), 𝕄_{N}).]

We will produce a suitable lift of τ in two steps:

1. For every component X_{i} of X, we trivialize the map τ_{i}: B → C(χ(X_{i}), 𝕄_{N}), i.e. we conjugate it to a constant map, so that it can be lifted to C(βX_{i}, 𝕄_{N}). This step requires the associated retract map R from 4.1.1 to have pointwise finite image and the spaces χ(X_{i}) to be connected and low-dimensional.

2. We extend the collection of lifts for the τ_{i}’s to a lift for τ. Here we need the associated retract map R to be lower semicontinuous.

In many cases of interest, the spaces χ(X_{i}) will not be connected, so that we have to modify the first step of the lifting process. This results in the fact that we cannot find a (multiplicative) split s in general. Instead we will settle for a lift s of τ with slightly weaker multiplicative properties. First we give the connection between the retract map R and the Busby map τ of the extension.

Lemma 4.1.5. Let a short exact sequence

\xymatrix{0 \ar[r] & I \ar[r] & A \ar[r]^{π} & B \ar[r] & 0}

with Busby map τ: B → Q(I) be given. Identifying Prim(I) with the open subset {J|I ⊈ J} of Prim(A) and denoting by ∂ Prim(I) its boundary in Prim(A), the following statements hold:

(i) J ∈ ∂ Prim(I) ⇔ I + I^{⊥} ⊆ J for every J ∈ Prim(A),

(ii) ∂ Prim(I) = {J: ker(τ ∘ π) ⊆ J} ≅ Prim(τ(B)).

If in addition I is subhomogeneous, we further have

(iii) |∂ Prim(I)| < ∞ ⇔ dim(τ(B)) < ∞.

Proof. For (i) it suffices to check that Prim(I^{⊥}) = Prim(A)\\overline{Prim(I)} where I^{⊥} denotes the annihilator of I in A. But this follows directly from the definition of the Jacobson topology on Prim(A):

J ∉ \overline{Prim(I)} ⇔ ⋂_{K∈Prim(I)} K ⊈ J
⇔ ∃x ∈ A: x ∉ J while ∥x∥_{K} = 0 ∀ K ∈ Prim(I)
⇔ ∃x ∈ I^{⊥}: x ∉ J
⇔ I^{⊥} ⊈ J
⇔ J ∈ Prim(I^{⊥}).

As I^{⊥} = ker(A → M(I)), one finds ker(τ ∘ π) = I + I^{⊥}. Together with (i) this shows

Prim(τ(B)) = Prim((τ ∘ π)(A))
≅ {J ∈ Prim(A): ker(τ ∘ π) ⊆ J}
= {J ∈ Prim(A): I + I^{⊥} ⊆ J}
= ∂ Prim(I).

For the last statement note that if all irreducible representations of I have dimension at most n, the same holds for all irreducible representations π of A with ker(π) contained in \overline{Prim(I)}. So by the correspondence described in (ii), irreducible representations of τ(B) are also at most n-dimensional. Hence, in this case, finitenesss of ∂ Prim(I) is equivalent to finite-dimensionality of τ(B).

For technical reasons we would prefer to work with unital extensions. However, it is not clear whether unitization preserves the regularity of R, i.e. whether the retract map associated to a unitized extension 0 → C_{0}(X, 𝕄_{N}) → A^{+} → B^{+} → 0 is lower semicontinuous provided that the retract map associated to the original extension is. As the next lemma shows, this is true and holds in fact for more general extensions.

Lemma 4.1.6. Let a locally compact space X with clopen connected components and a commutative diagram

[figure: 0; 0; 0; C_{0}(X, 𝕄_{N}); A; B; 0; 0; C_{0}(X, 𝕄_{N}); C; π; D; 0; F; F; 0; 0]

of short exact sequences of separable C^{∗}-algebras be given. Let R: Prim(A) → 2^{Prim(B)} (resp. S: Prim(C) → 2^{Prim(D)}) be the set-valued retract map associated to the upper (resp. the lower) horizontal sequence as in 4.1.1. If the quotient F is a finite-dimensional C^{∗}-algebra, then the following holds:

1. If R has pointwise finite image, then so does S.

2. If R is lower semicontinuous, then so is S.

Proof. (1) This is immediate since Prim(F) is a finite set and one easily verifies S(x) ⊆ R(x) ∪ Prim(F) for all x ∈ X.

(2) We may assume that F is simple and hence π is irreducible. Note that S(J) = R(J) for all J ∈ Prim(B) ⊂ Prim(D), while for x ∈ X we have either S(x) = R(x) or S(x) = R(x) ∪ {[π]}. Given a closed subset K ⊆ Prim(D), we need to verify that {J ∈ Prim(C): S(J) ⊆ K} is closed in Prim(C). If [π] ∈ K, then {J ∈ Prim(C): S(J) ⊆ K} = {J ∈ Prim(A): R(J) ⊆ K} ∪ {[π]} is closed in Prim(C) because {J ∈ Prim(A): R(J) ⊆ K} is closed in Prim(A) by semicontinuity of R. Now if [π] ∉ K, the only relevant case to check is a sequence x_{n} ⊂ X converging to \overline{x} ∈ Prim(D) with S(x_{n}) ⊆ K for all n. We then need to show that S(\overline{x}) = \overline{x} ∈ K as well. Decompose X = ⋃_{i∈I} X_{i} into its clopen connected components and write x_{n} ∈ X_{i_{n}} for suitable i_{n} ∈ I. We may assume that i_{n} ≠ i_{m} for n ≠ m since otherwise \overline{x} ∈ ∂X_{i_{n}} = S(x_{n}) for some n. Since R is lower semicontinuous, we know that the boundary of ⋃_{n} X_{i_{n}} in Prim(A) is contained in K ∩ Prim(A) and hence ∂ (⋃_{n} X_{i_{n}}) ⊂ K ∪ {[π]} in Prim(C). Let p denote the projection of C_{0}(X, 𝕄_{N}) onto C_{0}(⋃_{n} X_{i_{n}}, 𝕄_{N}). This map canonically extends to \overline{p} and \overline{\overline{p}} making the diagram

[figure: 0; C_{0}(X, 𝕄_{k}); p; C; \overline{p}; D; \overline{\overline{p}}; 0; 0; ⨁_{n} C_{0}(X_{i_{n}}, 𝕄_{N}); ⊆; ∏_{n} C(βX_{i_{n}}, 𝕄_{N}); \frac{∏_{n} C(βX_{in} ,𝕄_{N})}{⨁_{n} C_{0}(X_{in} ,𝕄_{N})}; q; 0; 0; ∏_{n} C_{0}(X_{i_{n}}, 𝕄_{N}); ∏_{n} C(βX_{i_{n}}, 𝕄_{N}); ∏_{n} C(χ(X_{i_{n}}), 𝕄_{N}); 0]

commute. Using Lemma 4.1.5, we can indentify the boundary of ⋃_{n} X_{i_{n}} in Prim(C) with Prim (\overline{\overline{p}}(D)). We already know that \overline{\overline{p}} factors through D_{K} ⊕ F, where D_{K} denotes the quotient corresponding to the closed subset K of Prim(D), and denote the induced map by φ:

[figure: D; \overline{\overline{p}}; π_{K}⊕π; \frac{∏_{n} C(βX_{in} ,𝕄_{N})}{⨁_{n} C_{0}(X_{in} ,𝕄_{N})}; D_{K} ⊕ F; φ]

We further know that the composition q ∘ φ_{|F} : F → ∏_{n} C(χ(X_{i_{n}}), 𝕄_{N}) vanishes because [π] ∉ ∂X_{i_{n}} = R(x_{n}) ⊆ K for all n. Hence the image of F under φ is contained in ker(q) = \frac{∏_{n} C_{0}(X_{in} ,𝕄_{N})}{⨁_{n} C_{0}(X_{in} ,𝕄_{N})}. But since this C^{∗}-algebra is projectionless and F is finite-dimensional, we find φ_{|F} = 0. Consequently, \overline{\overline{p}} factors through D_{K} which means nothing but \overline{x} ∈ ∂ (⋃_{n} X_{i_{n}}) = Prim (\overline{\overline{p}}(D)) ⊆ K.

Lemma 4.1.7. Let X be a connected, compact space of dimension at most 1. For every finite-dimensional C^{∗}-algebra F ⊆ C(X, 𝕄_{n}) there exists a unitary u ∈ C(X, 𝕄_{n}) such that uF u^{∗} is contained in the constant 𝕄_{n}-valued functions on X.

Proof. Since dim(X) ≤ 1, equivalence of projections in C(X, 𝕄_{n}) is completely determined by their rank ([Phi07, Proposition 4.2]). In particular, the C^{∗}- algebra C(X, 𝕄_{n}) has cancellation. Hence [RLL00, Lemma 7.3.2] shows that the inclusion F ⊂ C(X, 𝕄_{n}) is unitarily equivalent to any constant embedding ι: F → 𝕄_{n} ⊆ C(X, 𝕄_{n}) with rank(ι(p)) = rank(p) for all minimal projections p ∈ F.

Lemma 4.1.8. Let X be a connected, locally compact, metrizable space of dimension at most 1. Then every unitary in C(χ(X), 𝕄_{n}) lifts to a unitary in C(βX, 𝕄_{n}).

Proof. By [Phi07, Proposition 4.2], we have K_{0}(C(αX, 𝕄_{n})) ≅ ℤ via [p] ↦ rank(p). Using the 6-term exact sequence in K-theory, this shows that the induced map K_{1}(C(βX, 𝕄_{n})) → K_{1}(C(χ(X), 𝕄_{n})) is surjective. Combining this with K_{1}-bijectivity of C(βX, 𝕄_{n}), which is guaranteed by dim(βX) = dim(X) ≤ 1 ([Nag70, Thm. 9.5]) and [Phi07, Theorem 4.7], the claim follows.

Proposition 4.1.9. Let a short exact sequence of separable C^{∗}-algebras

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] & A \ar[r] & B \ar[r] & 0 & [τ]}

with Busby invariant τ be given. Assume that X is at most one-dimensional, has clopen connected components (X_{i})_{i∈I} and that every corona space χ(X_{i}) has only finitely many connected components. If the associated set-valued retract map R as in 4.1.1 has pointwise finite image, then there is a unitary U ∈ C(βX, 𝕄_{N}) such that for each i ∈ I the composition

B ⟶^{τ} C(χ(X), 𝕄_{n}) ⟶^{Ad(ϱ(U))} C(χ(X), 𝕄_{N}) → C(χ(X_{i}), 𝕄_{n})

has image contained in the locally constant 𝕄_{N}-valued functions on χ(X_{i}).

Proof. By Lemma 4.1.5, the image of each τ_{i}: B ⟶^{τ} C(χ(X), 𝕄_{N}) → C(χ(X_{i}), 𝕄_{N}) is finite-dimensional. Since by [Nag70, Thm. 9.5] furthermore dim χ(X_{i}) ≤ dim βX_{i} = dim X_{i} ≤ dim X ≤ 1, we can apply Lemma 4.1.7 to obtain unitaries u_{i} ∈ C(χ(X_{i}), 𝕄_{N}) such that u_{i}τ_{i}(B)u_{i}^{∗} is contained in the locally constant functions on χ(X_{i}). These unitaries can be lifted to unitaries U_{i} ∈ C(βX_{i}, 𝕄_{N}) by Lemma 4.1.8. Now U = ⊕_{i}U_{i} ∈ ∏_{i} C(βX_{i}, 𝕄_{N}) = C(βX, 𝕄_{N}) has the desired property.

Lemma 4.1.10. Let a short exact sequence of separable C^{∗}-algebras

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] & A \ar[r] & B \ar[r] & 0 & [τ]}

with Busby map τ be given. Assume that X is at most one-dimensional and that the connected components (X_{i})_{i∈I} of X are clopen. Further assume that the image of τ is constant on each χ(X_{i}) ⊆ χ(X). Denote by ι: A → M(C_{0}(X, 𝕄_{N})) = C(βX, 𝕄_{N}) the canonical map. If the set-valued retract map R: Prim(A) → 2^{Prim(B)} as defined in 4.1.1 is lower semicontinuous, the following statement holds: For every finite set G ⊂ A, every ϵ > 0 and almost every i ∈ I there exists a unitary U_{i} ∈ C(αX_{i}, 𝕄_{N}) ⊂ C(βX_{i}, 𝕄_{N}) such that

∥(U_{i}ι(a)_{|βX_{i}}U_{i}^{∗})(x) − ι(a)(y)∥ < ϵ

holds for all a ∈ G, x ∈ βX_{i} and y ∈ χ(X_{i}).

Proof. We may assume that A is unital by Lemma 4.1.6. Let a finite set G ⊂ A and ϵ > 0 be given. For each x ∈ βX, we write F_{x} = im(ev_{x} ∘ι) ⊆ 𝕄_{N} and

T_{1}(F_{x}) = {f ∈ C([0, 1], F_{x}): f(0) ∈ ℂ · 1_{F_{x}}},
S_{1}(F_{x}) = {f ∈ C_{0}([0, 1), F_{x}): f(0) ∈ ℂ · 1_{F_{x}}}.

Further let h_{η} ∈ C_{0}[0, 1) denote the function t ↦ max{1 − t − η, 0}. Using the Urysohn-type result [ELP98, Theorem 2.3.3], we find for each x ∈ βX a commuting diagram

[figure: 0; J_{x}; A; ev_{x} ∘ι; F_{x}; 0; 0; S(ℂ, F_{x}); φ_{x}; T (ℂ, F_{x}); _{\overline{φ}_{x}}; _{ev_{1}}; F_{x}; s_{x}; 0]

such that \overline{φ}_{x} is unital and φ_{x} is proper. Let s_{x}: F_{x} → T (ℂ, F_{x}) be any map satisfying s_{x}(b)(t) = b for t ≥ 1/2, so that in particular ev_{1} ∘s_{x} = id_{F_{x}} holds. Now consider

V_{x,δ} = \{y ∈ βX: (ev_{y} ∘ι)(\overline{φ}_{x}(h_{δ})) = 0\}

which is, for δ > 0, a closed neighborhood of x in βX. Note that by assumption χ(X_{i}) ∩ V_{x,δ} ≠ ∅ implies χ(X_{i}) ⊆ V_{x,δ}. We further claim the following: For almost every i ∈ I the inclusion χ(X_{i}) ⊆ V_{x,δ} implies X_{i} ⊂ V_{x,2δ}. Assume otherwise, then we find pairwise different i_{n} ∈ I, points x_{n} ∈ X_{i_{n}} such that χ(X_{i_{n}}) ⊆ V_{x,δ} while x_{n} ∉ V_{x,2δ} for all n. We may assume that ev_{x_{n}} ∘ι converges pointwise to a representation π. Then

∥π(\overline{φ}_{x}(h_{δ}))∥ = lim_{n} ∥(ev_{x_{n}} ∘ι ∘ \overline{φ}_{x})(h_{δ})∥ ≥ δ

since x_{n} ≠ V_{x,2δ} implies that ev_{x_{n}} ∘ι ∘ \overline{φ} contains irreducible summands corresponding to evaluations at points t with t < 1 − 2δ. On the other hand, since the retract map R is lower semicontinuous, we find each irreducible summand of π to be the limit of irreducible subrepresentations ϱ_{n} of ev_{y_{n}} ∘ι where y_{n} ∈ χ(X_{i_{n}}) ⊆ V_{x,δ}. Hence

∥π(\overline{φ}_{x}(h_{δ}))∥ ≤ lim inf_{n} ∥ϱ_{n}(\overline{φ}_{x}(h_{δ}))∥ = 0

by 2.2.4, giving a contradiction and thereby proving our claim. Since φ_{x} is proper, we have J_{x} = \overline{⋃_{η>0} her(φ_{x}(h_{η}))}. Hence there exists 1/2 > δ(x) > 0 such that

inf \{∥(a − (\overline{φ}_{x} ∘ s_{x} ∘ ev_{x} ∘ι)(a)) − b∥: b ∈ her(φ_{x}(h_{2δ(x)}))\} < \frac{ϵ}{2}

for all a ∈ G. By compactness of χ(X), we find x_{1}, ..., x_{m} such that

χ(X) ⊆ ⋃_{j=1}^{m} V_{x_{j},δ(x_{j})}.

Then by the claim proved earlier, for almost every i with χ(X_{i}) ⊆ V_{x_{j},δ(x_{j})} we have a factorization as indicated

\xymatrix{A \ar[d]_{ev_{x_{j}} ∘ι} \ar@{->>}[dr]^{π_{j}} \ar[r]^{ι} & ∏_{i} C(αX_{i}, 𝕄_{N}) \ar[r] & C(αX_{i}, 𝕄_{N}), \\ F_{x_{j}} \ar[r]_{π_{j}∘\overline{φ}_{x_{j}} ∘s_{x_{j}}} & A/⟨\overline{φ}_{x_{j}}(h_{2δ(x_{j})})⟩ \ar@{-->}[ur]_{_{\overline{ι}_{i}}}}

where ⟨\overline{φ}_{x_{j}}(h_{2δ(x_{j})})⟩ denotes the ideal generated by \overline{φ}_{x_{j}}(h_{2δ(x_{j})}) and π_{j} the corresponding quotient map. By the choice of δ(x_{j}), the lower left triangle commutes up to ϵ/2 on the finite set G. Also note that the map π_{j} ∘ \overline{φ}_{x_{j}} ∘ s_{x_{j}} is multiplicative. Finally, by Lemma 4.1.7 there exists a unitary U_{i} ∈ C(αX_{i}, 𝕄_{N}) such that Ad(U_{i}) ∘ (\overline{ι}_{i} ∘ π_{j} ∘ \overline{φ}_{x_{j}} ∘ s_{x_{j}}) is a constant embedding. Of course, we may arrange U(∞) = 1. We then verify

∥(U_{i}ι(a)_{|βX_{i}}U_{i}^{∗})(x) − ι(a)(y)∥
≤ ∥(U_{i}(\overline{ι}_{i} ∘ π_{j} ∘ \overline{φ}_{x_{j}} ∘ s_{x_{j}})((ev_{x_{j}} ∘ι)(a))U_{i}^{∗})(x) − (\overline{ι}_{i} ∘ π_{j})(a)(y)∥ + \frac{ϵ}{2}
≤ ∥(U_{i}(\overline{ι}_{i} ∘ π_{j} ∘ \overline{φ}_{x_{j}} ∘ s_{x_{j}})((ev_{x_{j}} ∘ι)(a))U_{i}^{∗})(y) − (\overline{ι}_{i} ∘ π_{j})(a)(y)∥ + \frac{ϵ}{2}
= ∥(\overline{ι}_{i} ∘ (π_{j} ∘ \overline{φ}_{x_{j}} ∘ s_{x_{j}}) ∘ (ev_{x_{j}} ∘ι))(a)(y) − (\overline{ι}_{i} ∘ π_{j})(a)(y)∥ + \frac{ϵ}{2}
≤ ϵ.

Applying this procedure to each of the finitely many points x_{1}, ..., x_{m}, the statement of the lemma follows.

Using Lemma 4.1.10 we can now construct a split for our sequence of interest - at least in the case of τ(B) being constant on each χ(X_{i}).

Corollary 4.1.11. If 0 → C_{0}(X, 𝕄_{N}) → A → B → 0 is a short exact sequence of separable C^{∗}-algebras such that the assumptions of Lemma 4.1.10 hold, then this sequence splits.

Proof. Let τ: B → Q(C_{0}(X, 𝕄_{N})) = C(χ(X), 𝕄_{N}) denote the Busby map of the sequence. We have the canonical commutative diagram

[figure: 0; ⨁_{i} C_{0}(X_{i}, 𝕄_{N}); C(βX, 𝕄_{N}); ϱ; C(χ(X), 𝕄_{N}); 0; 0; C_{0}(X, 𝕄_{N}); A; π; ι; B; τ; 0.]

Choose points y_{i} ∈ χ(X_{i}) for every i ∈ I. Using separability of A and Lemma 4.1.10, we find a unitary U ∈ ∏_{i} C(αX_{i}, 𝕄_{N}) ⊂ ∏_{i} C(βX_{i}, 𝕄_{N}) = C(βX, 𝕄_{N}) with

Uι(a)U^{∗} − ∏_{i} ι(a)(y_{i}) · 1_{αX_{i}} ∈ ⨁_{i} C_{0}(X_{i}, 𝕄_{N})

for all a ∈ A (where ι(a)(y_{i}) · 1_{αX_{i}} denotes the function on αX_{i} with constant value ι(a)(y_{i})). By setting s(π(a)) = U^{∗} (∏_{i}(ι(a)(y_{i}) · 1_{αX_{i}}) U we find s: B → C(βX, 𝕄_{N}) with (ϱ ∘ s)(π(a)) = (ϱ ∘ ι)(a) = τ(π(a)) by the formula above. Identifying A with the pullback over ϱ and τ, we can regard s as a map from B to A with π ∘ s = id_{B}, i.e. we have constructed a split for the sequence.

As the example 0 → C_{0}(0, 1) → C[0, 1] → ℂ^{2} → 0 shows, we cannot expect extensions by C_{0}(X, M_{N}) to split if the corona space of X (or of one of its components) is not connected. We will now deal with these components and show that one can still obtain a split s: B → A which, though not multiplicative in general, has still good multiplicative properties.

Lemma 4.1.12. Let 0 → C_{0}(X, 𝕄_{N}) → A → B → 0 be a short exact sequence with Busby map τ. Assume that the corona space χ(X) of X has only finitely many connected components and that the image of τ is contained in the locally constant functions on χ(X). Then there exists a compact set K ⊂ X and a completely positive split s: B → C(βX, 𝕄_{N}) which is multiplicative outside of an open set U ⊂ K.

Proof. Let χ(X) = ⋃_{k=1}^{K} Y_{k} be the decomposition of the corona space into its connected components. By assumption τ decomposes as ⊕_{k=1}^{K}τ_{k} with im(τ_{k}) ⊂ 𝕄_{N} · 1_{Y_{k}} ⊆ C(χ(X), 𝕄_{N}). Lift the indicator functions 1_{Y_{1}}, ⋯, 1_{Y_{K}} to pairwise orthogonal contractions h_{1}, ⋯, h_{K} in C(βX, ℂ · 1_{M_{N}}) and let f: [0, 1] → [0, 1] be the continuous function which equals 1 on [\frac{1}{2}, 1], satisfies f(0) = 0 and is linear in between. We define a completely positive map s: B → C(βX, 𝕄_{N}) by s(b)(x) = ∑_{k=1}^{K} τ_{k}(b) · f(h_{k})(x) and check that in the diagram

\xymatrix{A \ar[r]^{ι} \ar[d] & C(βX, 𝕄_{N}) \ar[d] \\ B \ar[r]^{τ} \ar@{-->}[ur]^{s} & C(χ(X), 𝕄_{N})}

the right triangle commutes. Set K = ⋂_{k=1}^{K} h_{k}^{−1}([0, \frac{1}{2}]) ⊂ X, then s is multiplicative outside of the open set U = ⋂_{k=1}^{K} h_{k}^{−1}([0, \frac{1}{2})) ⊂ K ⊂ X.

Proposition 4.1.13. Let 0 → C_{0}(X, 𝕄_{N}) → A → B → 0 be a short exact sequence of separable C^{∗}-algebras with Busby map τ. Assume that X is at most one-dimensional and has clopen connected components (X_{i})_{i∈I}. Further assume that each corona space χ(X_{i}) has only finitely many connected components and that χ(X_{i}) is connected for almost all i ∈ I. If for each i ∈ I the image of τ_{i}: B ⟶^{τ} C(χ(X), 𝕄_{N}) → C(χ(X_{i}), 𝕄_{N}) is locally constant on χ(X_{i}) and the set-valued retract map R: Prim(A) → 2^{Prim(B)} as in 4.1.1 is lower semicontinuous, the following holds: There exists a compact set K ⊂ X and a completely positive split s: B → C(βX, 𝕄_{N}) which is multiplicative outside of an open set U ⊂ K.

Proof. Let I_{0} ⊆ I be a finite set such that χ(X_{i}) is connected for every i ∈ I_{1} := I\I_{0}. We may then study the extensions (∗ = 0 or 1)

[figure: 0; ⨁_{i∈I} C_{0}(X_{i}, 𝕄_{N}); pr_{I∗}; A; B; φ_{∗}; 0; 0; ⨁_{i∈I_{∗}} C_{0}(X_{i}, 𝕄_{N}); A; ι_{∗}; A/ ⨁_{i∈I_{∗}} C_{0}(X_{i}, 𝕄_{N}); τ_{∗}; s_{∗}; 0; 0; ⨁_{i∈I_{∗}} C_{0}(X_{i}, 𝕄_{N}); ∏_{i∈I_{∗}} C(βX_{i}, 𝕄_{N}); \frac{∏_{i∈I∗} C(βX_{i},𝕄_{N})}{⨁_{i∈I∗} C_{0}(X_{i},𝕄_{N})}; 0]

with Busby maps τ_{∗}. Denote the map B → A/ ⨁_{i∈I_{∗}} C_{0}(X_{i}, 𝕄_{N}) induced by the projection pr_{I_{∗}} by φ_{∗}. It is now easy to check that for ∗ = 1 the short exact sequence in the middle row satisfies the assumptions of Lemma 4.1.10 and hence admits a splitting s_{1} by Corollary 4.1.11. For ∗ = 0, we apply Lemma 4.1.12 to obtain a compact set K ⊂ ⨆_{i∈I_{0}} X_{i} and a completely positive split s_{0} which is multiplicative outside of an open set U ⊂ K ⊂ ⨆_{i∈I_{0}} X_{i}. Setting s = s_{0} ∘ φ_{0} ⊕ s_{1} ∘ φ_{1}, we now get a split for the original sequence. In particular, ϱ ∘ s = τ holds due to the commutative diagram

[figure: 0; C_{0}(X, 𝕄_{N}); A; ι_{0}⊕ι_{1}; B; τ_{0}⊕τ_{1}; s; τ; 0; 0; ⨁_{∗=0,1} ⨁_{i∈I_{∗}} C_{0}(X_{i}, 𝕄_{N}); ⨁_{∗=0,1} ∏_{i∈I_{∗}} C(βX_{i}, 𝕄_{N}); _{⨁_{∗=0,1} \frac{∏_{i∈I∗} C(βX_{i},𝕄_{N})}{⨁_{i∈I∗} C_{0}(X_{i},𝕄_{N})}}; 0; 0; ⨁_{i∈I} C_{0}(X_{i}, 𝕄_{N}); ∏_{i∈I} C(βX_{i}, 𝕄_{N}); ϱ; \frac{∏_{i∈I} C(βX_{i},𝕄_{N})}{⨁_{i∈I} C_{0}(X_{i},𝕄_{N})}; 0.]

Summarizing the results of this section, we obtain the following.

Theorem 4.1.14. Let a short exact sequence of separable C^{∗}-algebras

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] & A \ar[r] & B \ar[r] & 0 & [τ]}

with Busby map τ be given. Assume that X satisfies the conditions

1. dim X ≤ 1,

2. the connected components (X_{i})_{i∈I} of X are clopen,

3. each χ(X_{i}) has finitely many connected components,

4. almost all χ(X_{i}) are connected,

then the following holds: If the associated set-valued retract map R: Prim(A) → 2^{Prim(B)} given as in 4.1.1 by

R(z) = \begin{cases}z if z ∈ Prim(B) \\ ∂X_{i} = \overline{X_{i}}\X_{i} if z ∈ X_{i} ⊆ X \end{cases}

is lower semicontinuous and has pointwise finite image, then there exists a compact set K ⊂ X and a completely positive split s: B → A for the sequence such that the composition

\xymatrix{B \ar[r]^{s} & A \ar[r] & M(C_{0}(X, 𝕄_{N})) = C_{b}(X, 𝕄_{N})}

is multiplicative outside of an open set U ⊂ K.

Proof. Note that we can replace the given extension by any strongly unitarily equivalent one (in sense of [Bla06, II.8.4.12]) without changing the retract map R. Hence, by Proposition 4.1.9, we may assume that the image of τ is locally constant on each χ(X_{i}). Now Proposition 4.1.13 provides a split s with the desired properties.

4.1.3 Retract maps for semiprojective extensions

We now verify the regularity properties for the set-valued retract map R: Prim(A) → 2^{Prim(B)} associated to an extension 0 → C_{0}(X, 𝕄_{N}) → A → B → 0 in the case that both the ideal C_{0}(X, 𝕄_{N}) and the extension A are semiprojective C^{∗}-algebras.

First we need the following definition which is an adaption of 2.1.2 and 2.1.3 to the setting of pointed spaces.

Definition 4.1.15. Let (X, x_{0}) be a pointed one-dimensional Peano continuum and r: X → core(X) the first point map onto the core of X as in 2.1.3 (where we choose core(X) to be any point x ≠ x_{0} if X is contractible). Denote the unique arc from x_{0} to r(x_{0}) by [x_{0}, r(x_{0})], then we say that

core(X, x_{0}) := core(X) ∪ [x_{0}, r(x_{0})]

is the core of (X, x_{0}). It is the smallest subcontinuum of X which contains both core(X) and the point x_{0}.

Now let X be a non-compact space with the property that its one-point compactification αX = X ∪ {∞} is a one-dimensional ANR-space. We are interested in the structure of the space X at around infinitity (i.e. outside of large compact sets) which is reflected in its corona space χ(X) = βX\X. At least some information about χ(X) can be obtained by studying neighborhoods of the point ∞ in αX. The following lemma describes some special neighborhoods which relate nicely to the finite graph core(αX, ∞).

Lemma 4.1.16. Let X be a connected, non-compact space such that its onepoint compactification αX = X ∪ {∞} is a one-dimensional ANR-space. Fix a geodesic metric d on αX (which exists by [Bin49], [Moi49]), then for any compact set C ⊂ αX\{∞} there exists a closed neighborhood V of ∞ with the following properties:

(i) {x ∈ X: d(x, ∞) ≤ ϵ} ⊆ V ⊆ X\C for some ϵ > 0.

(ii) V ∩ core(αX, ∞) is homeomorphic to the space of K many intervals [0, 1] glued together at the 0-endpoints with K = order(∞, core(αX, ∞)). The gluing point corresponds to ∞ under this identification.

Let D^{(k)} ⊆ V denote the k-th copy of [0, 1] under the identification described above and let r be the first point map onto core(αX, ∞). We can further arrange:

(iii) V = ⋃_{k=1}^{K} r^{−1} (D^{(k)}) and r^{−1} (D^{(k )}) ∩ r^{−1} (D^{(k′)}) = {∞} for k ≠ k′.

(iv) The connected components of V \{∞} are given by V^{(k)} := r^{−1} (D^{(k)}\{∞}).

(v) Every path in V from x ∈ V^{(k)} to x′ ∈ V^{(k′)} with k ≠ k′ contains ∞.

Proof. We first note that r^{−1}({∞}) ∩ X is open. Assume there is x ∈ X with r(x) = ∞ and d(x, ∞) = r > 0. Then given any y ∈ X with d(x, y) < r we choose an isometric arc α: [0, d(x, y)] → αX from x to y. Now the arc from y to ∞ given by first following α in reverse direction and then going along the unique arc from x to ∞ must run through r(y) by 2.1.3. Since every point on the second arc gets mapped to ∞ by r, we find either r(y) = ∞ or there is 0 < t < d(x, y) such that α(t) = r(y) ∈ core(αX, ∞). In the second case, the arc α_{|[0,t]} must run through r(x) = ∞ which, using the fact that α was isometric, gives the contradiction d(x, ∞) < t < d(x, y) < r. Since r^{−1}({∞}) is also closed, connectedness of X implies in fact r^{−1}({∞}) = {∞}. By definition of K (see section 2.1), the closed set {x ∈ core(αX, ∞): d(x, ∞) ≤ ϵ} satisfies the description in (ii) for all sufficiently small ϵ > 0. We set

V = {x ∈ αX: d(r(x), ∞) ≤ ϵ},

then V ∩core(αX, ∞) = r(V) = {x ∈ core(αX, ∞): d(x, ∞) ≤ ϵ} so that condition (ii) is satisfied. For (i), we observe that d(x, ∞) ≤ ϵ implies d(r(x), ∞) ≤ ϵ since d is geodesic and every arc from x to ∞ runs through r(x). Since ∞ ∉ r(C), we have min{d(r(x), ∞): x ∈ C} > 0 and therefore V ∩ C = ∅ for ϵ sufficiently small. Condition (iii) follows immediately from the definition of V. The sets V^{(k)} are connected and open by construction, so that (iv) holds.

(v) follows from (iv).

We now collect some information about the corona space χ(X) in the case of connected X. These observations are mostly based on the work of Grove and Pedersen in [GP84] and the graph-like structure of one-dimensional ANRspaces.

Lemma 4.1.17. Let X be a connected, non-compact space such that its onepoint compactification αX is a one-dimensional ANR-space. Then the corona space χ(X) has covering dimension at most 1 and its number of connected components is given by K = order(∞, core(αX, ∞)) < ∞. In particular, if αX is a one-dimensional AR-space, then χ(X) is connected.

Proof. Apply Lemma 4.1.16 to (αX, ∞). It is straightforward to check that the map

C(χ(X)) = C_{b}(X)/C_{0}(X) → ⨁_{k=1}^{K} C_{b}(V^{(k)})/C_{0}(V^{(k)}) = ⨁_{k=1}^{K} C(χ(V^{(k)}))

is an isomorphism. Therefore we find χ(X) = ⨆_{k=1}^{K} χ(V^{(k)}) and it suffices to check that each χ(V^{(k)}) is connected. By Proposition 3.5 of [GP84], it is now enough to show that each V^{(k)} is connected at infinity. So let a compact set C_{1} ⊂ V^{(k)} be given and denote by r: V^{(k)} ∪ {∞} → D^{(k)} the first point map. Using the identification [0, 1] ≅ D^{(k)} where the point 0 corresponds to the point ∞, we find t > 0 such that r(C_{1}) ⊂ [t, 1]. But C_{2} := r^{−1}([t, 1]) is easily seen to be compact while V^{(k)}\C_{2} = r^{−1}((0, t)) is path-connected by definition of r. For the dimension statement we note that dim(χ(X)) ≤ dim(βX) = dim(X) ≤ 1 by [Nag70, Theorem 9.5].

Remark 4.1.18. The assumption that X is connected in 4.1.17 is necessary. If we drop it, the corona space χ(X) may no longer have finitely many connected components, but the following weaker statement holds: If αX is a onedimensional ANR-space, so will be αX_{i} for any connected component X_{i} of X. However, it follows from 2.1.7 that all but finitely many components lead to contractible spaces αX_{i}, i.e. to one-dimensional AR-spaces. Since in this case core(αX_{i}, ∞) is just an arc [x, ∞] for some x ∈ X_{i}, we see from Lemma 4.1.17 that χ(X_{i}) is connected for almost every component X_{i} of X.

We will now see that, in the situation described in the beginning of this section, the set-valued retract map R has pointwise finite image, i.e. |R(z)| < ∞ for all z ∈ Prim(A). The cardinality of these sets is in fact uniformly bounded and we give an upper bound which only depends on N and the structure of the finite graph core(αX, ∞).

Proposition 4.1.19. Let A be a semiprojective C^{∗}-algebra containing an ideal of the form C_{0}(X, 𝕄_{N}). If αX = X ∪ {∞} is a one-dimensional ANR-space, then every connected component C of X has finite boundary ∂C = \overline{C}\C in Prim(A). More precisely, we find

|∂C| ≤ N · order(∞, (αC, ∞)) < ∞.

Proof. Since X is locally connected, the connected components of X are clopen and αC is again a one-dimensional ANR-space for every component C of X. Hence we may assume that C = X. Fix a geodesic metric d on αX = X ∪ {∞} ([Bin49], [Moi49]) and let V be a neighborhood of ∞ as constructed in Lemma 4.1.16, satisfying {x ∈ αX: d(x, ∞) ≤ ϵ} ⊆ V for some ϵ > 0. We further choose sequences (x_{n}^{(k)})_{n} ⊆ D^{(k)}\{∞} converging to ∞ and write x_{∞}^{(1)} = ⋯ = x_{∞}^{(K)} = ∞. By compactness of the unit ball in 𝕄_{N} and separability of A, we may assume that the representation

π^{(k)} : A → 𝕄_{N}, a ↦ lim_{n→∞} a(x_{n}^{(k)})

exists for all 1 ≤ k ≤ K. Here, a(x) denotes the image of a ∈ A under the extension of the point evaluation ev_{x}: C_{0}(X, 𝕄_{N}) → 𝕄_{N} to A. For a sequence (x_{n})_{n} in X ⊆ Prim(A) we write Lim(x_{n}) = {z ∈ Prim(A): x_{n} → z}. Our goal is then to show that there exists a finite set S ⊂ Prim(A) such that Lim(x_{n}) ⊂ S for every sequence (x_{n})_{n} ⊂ X with x_{n} → ∞ in αX. We will show that each S^{(k)} := Lim(x_{n}^{(k)}) consists of at most N elements and that S := ⋃_{k=1}^{K} S^{(k)} has the desired property described above. First observe that

S^{(k)} = {[π_{1}^{(k)}] , …, [π_{r(k)}^{(k)}]}

holds, where π^{(k)} ≃ π_{1}^{(k)}⊕⋯⊕π_{r(k)}^{(k)} is the decomposition of π^{(k)} into irreducible summands. The ⊇-inclusion is immediate, for the other direction assume that x_{n}^{(k)} → ker(ϱ) for some irreducible representation ϱ with ϱ ≄ π_{i}^{(k)} for all i. Since all x_{n}^{(k)} correspond to N-dimensional representations, we also have dim(ϱ) ≤ N. Therefore all π_{i}^{(k)} and ϱ drop to irreducible representations of the maximal Nsubhomogeneous quotient A_{≤N} of A (cf. section 2.2). Because Prim(A_{≤N}) is a T_{1}-space, the finite set {[π_{1}^{(k)}], …, [π_{r(k)}^{(k)}]} is closed and [ϱ] can be separated from it. In terms of 2.2.4, this means that there exists a ∈ A such that ∥ϱ(a)∥ > 1 while ∥π_{i}^{(k)}(a)∥ ≤ 1 for all i. On the other hand, we find

∥ϱ(a)∥ ≤ lim inf_{n→∞} ∥a(x_{n}^{(k)})∥ = ∥π^{(k)}(a)∥ = max_{i=1...r(k)} ∥π_{i}^{(k)}(a)∥ ≤ 1,

using 2.2.4 again. Hence [ϱ] = [π_{i}^{(k)}] for some i and in particular |S^{(k)}| = r(k) ≤ N for every k. It now suffices to show that Lim(x_{n}) ⊆ S^{(k)} for sequences (x_{n}) ⊂ X with x_{n} → ∞ such that r(x_{n}) ∈ D^{(k)} for some fixed k and all n. Let such a sequence (x_{n})_{n} for some fixed k be given and pick z ∈ Lim(x_{n}). In order to show that z ∈ S^{(k)}, we consider the compact spaces

Y_{n} := \{(t, t)|0 ≤ t ≤ \frac{1}{n}\} ∪ ⋃_{m≥n} (\{\frac{1}{m}\} × [0, \frac{1}{m}]) ⊂ ℝ^{2}.

Note that Y_{n+1} ⊂ Y_{n} and ⋂_{n} Y_{n} = (0, 0). We will now ’glue’ C(Y_{n}, 𝕄_{N}) to A in the following way: As before, we may assume that x_{n} → z in Prim(A) and that π_{∞}(a) = lim_{n} a(x_{n}) exists for every a ∈ A. In particular, we find z = [π_{i,∞}] for some i where π_{∞} ≃ π_{1,∞} ⊕ ⋯ ⊕ π_{r_{∞},∞} is the decomposition of π_{∞} into irreducible summands. Let c denote the C^{∗}-algebra of convergent 𝕄_{N}-valued sequences, we can then form the pullback A_{n} := A ⊕_{c} C(Y_{n}, 𝕄_{N}) over the two ^{∗}-homomorphisms

\begin{matrix}A ⟶ c \\ a ↦ (a(x_{n}), a(x_{n+1}), a(x_{n+2}), …) \end{matrix}

and

\begin{matrix}C(Y_{n}, 𝕄_{N}) ⟶ c. \\ f ↦ f((\frac{1}{n}, 0), f(\frac{1}{n+1}, 0), f(\frac{1}{n+2}, 0), …) \end{matrix}

These pullbacks form an inductive system in the obvious way. Further note that the connecting maps A_{n} → A_{n+1} are all surjective. The limit \underrightarrow{lim} A_{n} can be identified with A via the isomorphism induced by the projections A_{n} = A ⊕_{c} C(Y_{n}, 𝕄_{N}) → A onto the left summand. Using semiprojectivity of A, we can find a partial lift to some finite stage A_{n} of this inductive system:

\xymatrix{ &  & A_{n} = A ⊕_{c} C(Y_{n}, 𝕄_{N}) \ar[r] \ar@{->>}[d] & C(Y_{n}, 𝕄_{N}) \\ C_{0}(X, 𝕄_{N}) \ar[r]^{⊆} & A \ar[r]^{≅} \ar@{-->}[ur] & \underrightarrow{lim} A_{n}}

Let φ: A → C(Y_{n}, 𝕄_{N}) be the composition of this lift with the projection A_{n} → C(Y_{n}, 𝕄_{N}) to the right summand. The restriction of φ to the ideal C_{0}(X, 𝕄_{N}) then induces a continuous map φ^{∗}: Y_{n} → αX with φ^{∗} (\frac{1}{m}, 0) = x_{m} for all m ≥ n and φ^{∗}(0, 0) = ∞. Denote by h the strictly positive element of C_{0}(X, 𝕄_{N}) given by h(x) = d(x, ∞) · 1_{𝕄_{N}}. After increasing n, we may assume that ∥φ(h)∥ < ϵ holds. For m ≥ n, we consider the paths α_{m}: [0, \frac{2}{m}] → Y_{n} given by

α_{m}(t) = \begin{cases}(\frac{1}{m}, t) & if 0 ≤ t ≤ \frac{1}{m} \\ (\frac{2}{m} − t, \frac{2}{m} − t) & if \frac{1}{m} ≤ t ≤ \frac{2}{m}. \end{cases}

Set t_{∞,m} = min\{t: φ(h)(α_{m}(t)) = 0\}, then 0 < t_{∞,m} ≤ \frac{2}{m} because of ∥φ(h)(α_{m}(0))∥ = ∥φ(h)(\frac{1}{m}, 0)∥ = ∥h(x_{m})∥ = d(x_{m}, ∞) > 0 and φ(h)(α_{m}(\frac{2}{m})) = φ(h)(0, 0) = h(∞) = 0. By setting β_{m}(t) = φ^{∗}(α_{m}(t)) we obtain paths β_{m}: [0, t_{∞,m}] → αX which have the properties

(1) β_{m}(0) = x_{m},
(2) β_{m}(t) = ∞ if and only if t = t_{∞,m},
(3) im(β_{m}) ⊆ V^{(k)} for all m,
(4) x_{l}^{(k)} ∈ im(β_{m}) for fixed m and all sufficiently large l.

The first property is clear while the second one follows directly from the definition of t_{∞,m}. In order to verify properties (3) and (4) we have to involve the structure of the neighborhood V and by that the special structure of αX as a one-dimensional ANR-space. From ∥φ(h)∥ < ϵ we obtain im(β_{m}) ⊆ im(α_{m}) ⊆ {x ∈ αX: d(x, ∞) ≤ ϵ} ⊆ V, it then follows from (1),

(2) and property (v) in Lemma 4.1.16 that im(β_{m}) ⊆ V^{(k)}. For (4), observe that im(β_{m}) contains r(im(β_{m})) by part (ii) of Lemma 2.1.4, where r is the first-point map αX → core(αX, ∞). Under the identification D^{(k)} ≅ [0, 1], the connected set r(im(β_{m})) corresponds to a proper interval containing the 0-endpoint and hence it contains x_{l}^{(k)} for almost every l. Now set π_{m} = ev_{β(t_{∞,m})} ∘φ: A → 𝕄_{N} and let π_{m} ≃ π_{1,m} ⊕ ⋯ ⊕ π_{r_{m},m} be the decomposition into irreducible summands. We claim that the identity

S^{(k)} = {[π_{1,m}] , ⋯, [π_{r_{m},m}]}

holds for all m. Involving property (4) for the path β_{m}, we find

∥π_{m}(a)∥ = lim_{t↗t_{∞,m}} ∥(ev_{β(t)} ∘φ)(a)∥ = lim_{l→∞} ∥a (x_{l}^{(k)})∥ = ∥π^{(k)}(a)∥

for every fixed m and all a ∈ A. Now the same separation argument as in the beginning of the proof shows that the finite-dimensional representations π^{(k)} and π_{m} share the same irreducible summands for every m. Since β_{m}(t_{∞,m}) → (0, 0) in Y_{n}, we find π_{m} = ev_{β(t_{∞,m})} ∘φ → ev_{(0,0)} ∘φ = π_{∞} pointwise. Hence by the above identity, π_{∞} and π^{(k)} also share the same irreducible summands. In particular, we find z ∈ S^{(k)} which finishes the proof.

Next, we show that in our situation the set-valued retract map R is also lower semicontinuous in the sense of 4.1.2.

Proposition 4.1.20. Let 0 → C_{0}(X, 𝕄_{N}) → A → B → 0 be a short exact sequence of separable C^{∗}-algebras. If αX is a one-dimensional ANR-space and A is semiprojective, then the associated retract map R as in 4.1.1 is lower semicontinuous.

Proof. Let X = ⨆_{i∈I} X_{i} denote the decomposition of X into its connected components. By separability of A it suffices to verify condition (iii) of Lemma 4.1.2 for a given sequence x_{n} → z in Prim(A). The case z ∈ X is trivial since X is locally connected and therefore has open connected components. The critical case is when x_{n} ∈ X for all n but z ∈ Prim(B). In this case, we write x_{n} ∈ X_{i_{n}} and we may assume that π_{∞}(a) = lim_{n} a(x_{n}) is well defined for all a ∈ A. In particular, z corresponds to the kernel of an irreducible summand π_{j,∞} of π_{∞} ≃ π_{1,∞} ⊕ ⋯ ⊕ π_{r,∞}, as we have already seen in the beginning of the proof of Proposition 4.1.19. Using exactly the same construction of ’gluing the space Y to A along the sequence (x_{n})’ as in the proof of 4.1.19, one now shows that

\{[π_{1,∞}], ⋯, [π_{r,∞}]\} ⊆ \overline{⋃_{n} ∂X_{i_{n}}}.

Hence we find y_{n} ∈ ∂X_{i_{n}} = R(x_{n}) with y_{n} → [π_{j,∞}] = z showing that the retract map R is in fact lower semicontinuous.

4.2 Existence of limit structures

Consider an extension of separable C^{∗}-algebras

0 → C_{0}(X, 𝕄_{N}) → A → B → 0

where the one-point compactification of X is assumed to be a one-dimensional ANR-space. We know from Theorem 2.1.5 that in this case αX comes as a inverse limit of finite graphs over a surprisingly simple system of connecting maps. Here we show that under the right assumptions on the set-valued retract map R: Prim(A) → 2^{Prim(B)} associated to the sequence above, this limit structure for αX is compatible with the extension of B by C_{0}(X, 𝕄_{N}) in the following sense: We prove the existence of a direct limit structure for A which describes it as the C^{∗}-algebra B with a sequence of non-commutative finite graphs (1-NCCW’s) attached. The connecting maps of this direct system are obtained from the limit structure for αX and hence can be described in full detail.

Lemma 4.2.1. Let a short exact sequence of separable C^{∗}-algebras 0 → C_{0}(X, 𝕄_{N}) → A → B → 0 with Busby map τ be given. Assume that αX is a one-dimensional ANR-space and that the associated set-valued retract map R: Prim(A) → 2^{Prim(B)} as in 4.1.1 is lower semicontinuous and has pointwise finite image. Then A is isomorphic to the direct limit B_{∞} = \underrightarrow{lim} (B_{i}, s_{i}^{i+1}) of an inductive system

\xymatrix{B_{0} \ar[r]_{s_{0}^{1}} & B_{1} \ar[r]_{s_{1}^{2}} \ar@{->>}[l]_{r_{1}^{0}} & B_{2} \ar[r] \ar@{->>}[l]_{r_{2}^{1}} & ⋯ \ar[r]_{s_{i−1}^{i}} \ar@{->>}[l] & B_{i} \ar[r] \ar[rr]_{s_{i}^{∞}} \ar@{->>}[l]_{r_{i}^{i−1}} & ⋯ \ar[r] & B_{∞} \ar@{->>}[ll]_{r_{∞}^{i}}}

where

• B_{0} is given as a pullback B ⊕_{F} D with D a 1-NCCW and dim(F) < ∞. Furthermore, if αX is contractible, we may even arrange B_{0} ≅ B.

and

• for every i ∈ ℕ there is a representation π_{i}: B_{i} → 𝕄_{N} such that B_{i+1} is

defined as the pullback \xymatrix{B_{i+1} \ar@{->>}[d]^{r_{i+1}^{i}} \ar@{.>}[r] & C([0, 1], 𝕄_{N}) \ar@{->>}[d]^{ev_{0}} \\ B_{i} \ar[u]^{s_{i}^{i+1}} \ar[r]^{π_{i}} & 𝕄_{N}.}

The map s_{i}^{i+1} : B_{i} → B_{i+1} is given by a ↦ (a, π_{i}(a) ⊗ 1_{[0,1]}) and hence satisfies r_{i+1}^{i} ∘ s_{i}^{i+1} = id_{B_{i}}.

Proof. Let X = ⨆_{j∈J} C_{j} be the decomposition of X into its clopen connected components. Denote by J_{1} ⊆ J the subset of those indices for which the corona space χ(C_{j}) is connected and note that J_{0} := J\J_{1} is finite by Remark 4.1.18. We have the canonical commutative diagram

[figure: 0; C_{0}(X, 𝕄_{N}); A; ι_{0}⊕ι_{1}; B; τ_{0}⊕τ_{1}; 0; 0; ⨁_{{∗=0,1 \atop j∈J∗}} C_{0}(C_{j}, 𝕄_{N}); ⨁_{∗=0,1} ∏_{j∈J_{∗}} C(βC_{j}, 𝕄_{N}); q_{0}⊕q_{1}; _{⨁_{∗=0,1} \frac{∏_{j∈J∗} C(βC_{j},𝕄_{N})}{⨁_{j∈J∗} C_{0}(C_{j},𝕄_{N})}}; 0]

where τ_{0} ⊕ τ_{1} is the Busby map τ and the right square is a pullback diagram. Since we can pass to any strongly unitarily equivalent extension (in the sense of [Bla06, II.8.4.12]) without changing the retract map R, we can, by Proposition 4.1.9 and the finiteness condition on R, assume that for every j the image of

τ_{j}: B ⟶^{τ} \frac{∏_{j′} C(βC_{j′}, 𝕄_{N})}{⨁_{j′} C_{0}(C_{j′}, 𝕄_{N})} → \frac{C(βC_{j}, 𝕄_{N})}{C_{0}(C_{j}, 𝕄_{N})} = C(χ(C_{j}), 𝕄_{N})

is locally constant on χ(C_{j}), and even constant if j ∈ J_{1}. Furthermore, using lower semicontinuity of R and arguing as in the proof of Corollary 4.1.11, we may assume that

ι_{1}(A) ⊆ ∏_{j∈J_{1}} 𝕄_{N} · 1_{βC_{j}} + ⨁_{j∈J_{1}} C_{0}(C_{j}, 𝕄_{N}).

Next, we write αX = X ∪ {∞} as a limit of finite graphs. By Theorem 2.1.5 we can find a sequence of finite graphs X_{i} ⊂ X_{i+1} ⊂ αX such that X_{0} = core(αX, ∞) (in the sense of 4.1.15) and each X_{i+1} is obtained from X_{i} by attaching a line segment [0, 1] at the 0-endpoint to a single point y_{i} of X_{i}. Furthermore we have \underleftarrow{lim} X_{i} = αX along the sequence of first point maps ϱ_{∞}^{i}: αX → X_{i}. We need to fix some notation: Denote the inclusion of X_{i} into X_{i+1} by ι_{i}^{i+1} and the retract from X_{i+1} to X_{i} by collapsing the attached interval to the attaching point y_{i} by ϱ_{i+1}^{i}. An analogous notation is used for the inclusion X_{i} ⊆ αX:

[figure: X_{i}; ι_{i}^{i+1}; X_{i+1}; ϱ_{i+1}^{i}; ι_{i+1}^{∞}; αX; ϱ_{∞}^{i+1}]

Now for every pair of indices i, j we have X_{i} ∩ C_{j} sitting inside C_{j}. Note that X_{i+1}\X_{i} ∩ C_{j(i)} ≠ ∅ for a unique j(i) ∈ J since ∞ ∈ X_{0}. We define suitable compactifications α_{j}(X_{i} ∩ C_{j}) of X_{i} ∩ C_{j} as follows: if X_{0} ∩ C_{j} = ∅, we let α_{j}(X_{i} ∩ C_{j}) = α(X_{i} ∩ C_{j}) be the usual one-point compactification for any i ∈ ℕ. In the case X_{0} ∩ C_{j} ≠ ∅, which will occur only finitely many times, we have an inclusion C_{b}(X_{i} ∩ C_{j}) ⊆ C_{b}(C_{j}) induced by the surjective retract ϱ_{∞|C_{j}}^{i}: C_{j} → X_{i} ∩ C_{j} and we define α_{j}(X_{i} ∩ C_{j}) via

C(α_{j}(X_{i} ∩ C_{j})) = \{f ∈ C_{b}(X_{i} ∩ C_{j}) ⊆ C_{b}(C_{j}) = C(βC_{j}): \begin{matrix}f is locally \\ constant \\ on χ(C_{j}) \end{matrix} \} .

Since the corona space χ(C_{j}) has only finitely many connected components by Lemma 4.1.17, α_{j}(X_{i} ∩ C_{j}) will be a finite-point compactification of X_{i} ∩ C_{j} (meaning that α_{j}(X_{i} ∩ C_{j})\(X_{i} ∩ C_{j}) is a finite set). In particular, α_{j}(X_{i} ∩ C_{j}) is a finite graph for any pair of indices i and j. We are now ready to iteratively define the C^{∗}-algebras B_{i} as the pullbacks over

[figure: B_{i}; B; τ; ∏_{j} C(α_{j}(X_{i} ∩ C_{j}), 𝕄_{N}); q; \frac{∏_{j} C(βC_{j},𝕄_{N})}{⨁_{j} C_{0}(C_{j},𝕄_{N})}]

with respect to the inclusions (ϱ_{∞|C_{j}}^{i})^{∗} ⊗ id_{𝕄_{N}} : C(α_{j}(X_{i} ∩ C_{j}), 𝕄_{N}) ⊆ C(βC_{j}, 𝕄_{N}). Let us first simplify the description of B_{i}. For every fixed i, the set X_{i}∩C_{j} is empty for almost every j ∈ J so that C(α_{j}(X_{i}∩C_{j}), 𝕄_{N}) = 𝕄_{N} ·1_{βC_{j}} for almost every j. Given ((f_{j})_{j}, b) ∈ B_{i}, this implies f_{j} = τ_{j}(b)·1_{βC_{j}} for almost every j. Hence B_{i} is isomorphic to the pullback

[figure: B_{i}; B; ⊕_{j∈J(i)} τ_{j}; ⨁_{j∈J(i)} C(α_{j}(X_{i} ∩ C_{j}), 𝕄_{N}); q; _{⨁_{j∈J(i)} \frac{C(βC_{j},𝕄_{N})}{C_{0}(C_{j},𝕄_{N})}}]

for the finite set J(i) = {j ∈ J: X_{i} ∩ C_{j} ≠ ∅} ⊆ J. Since every α(X_{i} ∩ C_{j}) is a finite graph, the C^{∗}-algebra on the lower left side is a 1-NCCW. One also checks that the pullbacks are taken over finite-dimensional C^{∗}-algebras because (⊕_{j∈J(i)}τ_{j})(B) consists of locally constant functions on the space ⨆_{j∈J(i)} χ(C_{j}) which has only finitely many connected components by Lemma 4.1.17. Next, we specify the inductive structure, i.e. the connecting maps s_{i}^{i+1} : B_{i} → B_{i+1} and retracts r_{i+1}^{i}: B_{i+1} → B_{i}. By definition, we find B_{i} ⊆ B_{i+1} ⊆ A with the inclusions coming from (ϱ_{i+1}^{i})^{∗} ⊗ id_{𝕄_{N}} resp. by (ϱ_{∞}^{i+1})^{∗} ⊗ id_{𝕄_{N}}. We denote them by s_{i}^{i+1} resp. by s_{i}^{∞}. Since \overline{⋃_{i} C(α_{j}(X_{i} ∩ C_{j}), 𝕄_{N})} ⊇ \overline{⋃_{i} C_{0}(X_{i} ∩ C_{j}, 𝕄_{N})} = C_{0}(X ∩ C_{j}, 𝕄_{N}) for every j ∈ J, we find C_{0}(X, 𝕄_{N}) ⊆ \overline{⋃_{i} B_{i}}. One further checks that ⨁_{j∈J_{0}} C(α_{j}(X_{0} ∩ C_{j}), 𝕄_{N}) surjects via q onto the locally constant functions on ⨆_{j∈J_{0}} χ(C_{j}). Together with τ_{1}(B) ⊆ q_{1}(∏_{j∈J_{1}} 𝕄_{N} · 1_{βC_{j}}) ⊆ q_{1}(∏_{j∈J_{1}} C(α_{j}(X_{0} ∩ C_{j}), 𝕄_{N})) it follows that \overline{⋃_{i} B_{i}} is the pullback over τ and q, and hence \overline{⋃_{i} B_{i}} = A.

It remains to verify the description of the connecting maps s_{i}^{i+1}. We have X_{i} ∩ C_{j} = X_{i+1} ∩ C_{j} if j ≠ j(i) and α_{j}(X_{i} ∩ C_{j(i)}) ⊆ α_{j}(X_{i+1} ∩ C_{j(i)}) ≅ α_{j}(X_{i} ∩ C_{j(i)}) ∪_{{y_{i}}={0}} [0, 1]. This means there is a pullback diagram

\xymatrix{C(α_{j}(X_{i+1} ∩ C_{j(i)}), 𝕄_{N}) \ar@{.>}[r] \ar@{->>}[d] & C([0, 1], 𝕄_{N}) \ar@{->>}[d]^{_{ev_{0}}} \\ C(α_{j}(X_{i} ∩ C_{j(i)}), 𝕄_{N}) \ar[r]^{_{ev_{y_{i}}}} \ar[u]^{(ϱ_{i+1}^{i})^{∗}⊗id_{𝕄_{N}}} & 𝕄_{N}}

where (ϱ_{i+1}^{i})^{∗} ⊗ id_{𝕄_{N}} corresponds to f ↦ (f, f(y_{i}) ⊗ 1_{[0,1]}) in the pullback picture and the downward arrow on the left side comes from the inclusion α_{j}(X_{i} ∩ C_{j(i)}) ⊆ α_{j}(X_{i+1} ∩ C_{j(i)}). This map induces a surjection B_{i+1} → B_{i} which will be denoted by r_{i+1}^{i} and gives the claimed pullback diagram. Finally, if αX is an AR-space, the core X_{0} = core(αX, ∞) = [x_{0}, ∞] is nothing but an arc from some point x_{0} ∈ X to ∞. In this case the finite set J(0) consists of a single element j(0), namely the index corresponding to the component containing x_{0}. By definition, B_{0} comes as a pullback

\xymatrix{B_{0} \ar@{.>}[r] \ar@{.>}[d] & B \ar[d]^{τ_{j(0)}} \\ C([x_{0}, ∞], 𝕄_{N}) \ar[r]^{_{ev_{∞}}} & 𝕄_{N} · 1_{χ(C_{j(0)})}}

and hence an index shift allows us to start with B_{0} ≅ B.

The procedure of forming extensions by C^{∗}-algebras of the form C_{0}(X, 𝕄_{N}) can of course be iterated. The next proposition shows that, if all the attached spaces X are one-dimensional ANRs up to compactification, the limit structures which we get from Lemma 4.2.1 for each step can be combined into a single one.

Proposition 4.2.2. Let a short exact sequence of separable C^{∗}-algebras 0 → C_{0}(X, 𝕄_{N}) → A → B → 0 be given. Assume that αX is a one-dimensional ANR-space and that the associated set-valued retract map R: Prim(A) → 2^{Prim(B)} as in 4.1.1 is lower semicontinuous and has pointwise finite image. Further assume that there exists a direct limit structure for B

\xymatrix{B_{0} \ar[r]_{s_{0}^{1}} & B_{1} \ar[r]_{s_{1}^{2}} \ar@{->>}[l]_{r_{1}^{0}} & B_{2} \ar[r] \ar@{->>}[l]_{r_{2}^{1}} & ⋯ \ar[r]_{s_{i−1}^{i}} \ar@{->>}[l] & B_{i} \ar[r] \ar[rr]_{s_{i}^{∞}} \ar@{->>}[l]_{r_{i}^{i−1}} & ⋯ \ar[r] & B \ar@{->>}[ll]_{r_{∞}^{i}}}

such that all B_{i} are 1-NCCWs and at each stage there is a representation p_{i}: B_{i} → 𝕄_{n_{i}} such that B_{i+1} is defined as the pullback

\xymatrix{B_{i+1} \ar@{->>}[d]^{r_{i+1}^{i}} \ar@{.>}[r]^{t_{i+1}} & C([0, 1], 𝕄_{n_{i}}) \ar@{->>}[d]^{_{ev_{0}}} \\ B_{i} \ar[u]^{s_{i}^{i+1}} \ar[r]^{p_{i}} & 𝕄_{n_{i}}}

and s_{i}^{i+1} : B_{i} → B_{i+1} is given by a ↦ (a, p_{i}(a) ⊗ 1_{[0,1]}). Then A is isomorphic to the limit A_{∞} of an inductive system

\xymatrix{A_{0} \ar[r]_{σ_{0}^{1}} & A_{1} \ar[r]_{σ_{1}^{2}} \ar@{->>}[l]_{ϱ_{1}^{0}} & A_{2} \ar[r] \ar@{->>}[l]_{ϱ_{2}^{1}} & ⋯ \ar[r]_{σ_{i−1}^{i}} \ar@{->>}[l] & A_{i} \ar[r] \ar[rr]_{σ_{i}^{∞}} \ar@{->>}[l]_{ϱ_{i}^{i−1}} & ⋯ \ar[r] & A_{∞} \ar@{->>}[ll]_{ϱ_{∞}^{i}}}

where all A_{i} are 1-NCCWs and at each stage there is a representation π_{i}: A_{i} → 𝕄_{m_{i}} such that A_{i+1} is defined as the pullback

\xymatrix{A_{i+1} \ar@{->>}[d]^{ϱ_{i+1}^{i}} \ar@{.>}[r] & C([0, 1], 𝕄_{m_{i}}) \ar@{->>}[d]^{_{ev_{0}}} \\ A_{i} \ar[u]^{σ_{i}^{i+1}} \ar[r]^{π_{i}} & 𝕄_{m_{i}}}

and σ_{i}^{i+1} : A_{i} → A_{i+1} is given by a ↦ (a, π_{i}(a) ⊗ 1_{[0,1]}). Furthermore, if αX is an AR-space we may even arrange A_{0} ≅ B_{0}.

Proof. By Lemma 4.2.1, we know that A can be written as an inductive limit

[figure: \overline{A}_{0}; _{\overline{s}{1 \atop 0}}; \overline{A}_{1}; _{\overline{s}{2 \atop 1}}; ^{\overline{r}{0 \atop 1}}; \overline{A}_{2}; ^{\overline{r}{1 \atop 2}}; ⋯; _{\overline{s}{i \atop i−1}}; \overline{A}_{i}; _{\overline{s}{∞ \atop i}}; ^{\overline{r}{i−1 \atop i}}; ⋯; A; ^{\overline{r}{i \atop ∞}}]

with a pullback structure

\xymatrix{\overline{A}_{i+1} \ar@{->>}[d]^{_{\overline{r}{i \atop i+1}}} \ar@{.>}[r] & C([0, 1], 𝕄_{N}) \ar@{->>}[d]^{_{ev_{0}}} \\ _{\overline{A}_{i}} \ar[u]^{_{\overline{s}{i+1 \atop i}}} \ar[r]^{_{\overline{p}_{i}}} & 𝕄_{N}}

at every stage and \overline{s}_{i}^{i+1} : \overline{A}_{i} → \overline{A}_{i+1} given by a ↦ (a, \overline{p}_{i}(a) ⊗ 1_{[0,1]}). The starting algebra \overline{A}_{0} comes as a pullback

\xymatrix{\overline{A}_{0} \ar[r] \ar[d] & D \ar[d]^{φ} \\ B \ar[r]^{ψ} & F}

with D a 1-NCCW and dim(F) < ∞. In the case of αX being an AR-space, we may choose \overline{A}_{0} = B, i.e. D = 0. For j ∈ ℕ we now define A_{0,j} to be the pullback

\xymatrix{A_{0,j} \ar@{.>}[r] \ar@{.>}[d]_{ϱ_{0,j}} & D \ar[d]^{φ} \\ B_{j} \ar[r]^{ψ∘s_{j}^{∞}} & F.}

The maps s_{j}^{j+1},s_{j}^{∞} homomorphisms σ_{0,j}^{0,j+1} : A_{0,j} → A_{0,j+1} and σ_{0,j}^{0,∞}: A_{0,j} → \overline{A}_{0}, leading to an inductive limit structure with \underrightarrow{lim}_{j}(A_{0,j}, σ_{0,j}^{0,j+1}) = \overline{A}_{0}. We proceed iteratively, defining A_{i+1,j} to be the pullback

\xymatrix{A_{i+1,j} \ar@{.>}[r] \ar@{.>}[d]^{ϱ_{i+1,j}^{i,j}} & C([0, 1], 𝕄_{N}) \ar[d]^{_{ev_{0}}} \\ A_{i,j} \ar[r]_{\overline{p}_{i}∘σ_{i,j}^{i,∞}} \ar@{.>}[u]^{σ_{i,j}^{i+1,j}} & 𝕄_{N}}

with σ_{i,j}^{i+1,j}: A_{i,j} → A_{i+1,j} given by a ↦ (a, (\overline{p}_{i} ∘ σ_{i,j}^{i,∞})(a) ⊗ 1_{[0,1]}). It is then checked that σ_{i,j}^{i,j+1} and σ_{i,j}^{i,∞} induce compatible homomorphisms σ_{i+1,j}^{i+1,j+1} : A_{i+1,j} → A_{i+1,j+1} and σ_{i+1,j}^{i+1,∞}: A_{i+1,j} → \overline{A}_{i+1} with \underrightarrow{lim}_{j}(A_{i+1,j}, σ_{i+1,j}^{i+1,j+1}) = \overline{A}_{i+1}. Observing that for every i and j

\xymatrix{A_{i,j+1} \ar[r]^{t_{j+1}∘ϱ_{0,j+1}∘ϱ_{i,j+1}^{0,j+1}} \ar[d]^{ϱ_{i,j+1}^{i,j}} & C([0, 1], 𝕄_{n_{j}}) \ar[d]^{_{ev_{0}}} \\ A_{i,j} \ar[r]_{p_{j}∘ϱ_{0,j}∘ϱ_{i,j}^{0,j}} \ar[u]^{σ_{i,j}^{i,j+1}} & 𝕄_{n_{j}}}

is indeed a pullback diagram, we get the desired limit structure for A by following the diagonal in the commutative diagram

Documenta Mathematica 21 (2016) 987–1049

\xymatrix{A_{00} \ar[r] \ar@{.>}[d] & A_{01} \ar[r] \ar[d] & A_{02} \ar[r] \ar[d] & A_{03} \ar[r] \ar[d] & ⋯ \ar[r] & \overline{A}_{0} \ar[d] \\ A_{10} \ar@{.>}[r] \ar[d] & A_{11} \ar[r] \ar@{.>}[d] & A_{12} \ar[r] \ar[d] & A_{13} \ar[r] \ar[d] & ⋯ \ar[r] & _{\overline{A}_{1}} \ar[d] \\ A_{20} \ar[r] \ar[d] & A_{21} \ar@{.>}[r] \ar[d] & A_{22} \ar[r] \ar@{.>}[d] & A_{23} \ar[r] \ar[d] & ⋯ \ar[r] & _{\overline{A}_{2}} \ar[d] \\ A_{30} \ar[r] \ar[d] & A_{31} \ar[r] \ar[d] & A_{32} \ar@{.>}[r] \ar[d] & A_{33} \ar[r] \ar@{.>}[d] & ⋯ \ar[r] & _{\overline{A}_{3}} \ar[d] \\ ⋮ & ⋮ & ⋮ & ⋮ & _{...} & ⋮}

as indicated. Note that, since all connecting maps are injective, the limit over the diagonal equals \underrightarrow{lim} \overline{A}_{n} = A.

4.3 Keeping track of semiprojectivity

We now reap the fruit of our labor in the previous sections and work out a ’2 out of 3’-type statement describing the behavior of semiprojectivity with respect to extensions by homogeneous C^{∗}-algebras. While for general extensions the behavior of semiprojectivity is either not at all understood or known to be rather bad, Theorem 4.3.2 gives a complete and satisfying answer in the case of homogeneous ideals. It is the very first result of this type and allows to understand semiprojectivity for C^{∗}-algebras which are built up from homogeneous pieces, see chapter 5.1.

Proposition 4.3.1. Let 0 → C_{0}(X, 𝕄_{N}) → A → B → 0 be a short exact sequence of C^{∗}-algebras. If both A and B are (semi)projective, then the onepoint compactification αX is a one-dimensional A(N)R-space.

Proof. The projective case follows directly from Corollary 3.1.3 and Theorem 2.3.7 while the semiprojective case requires some more work. By Lemma 2.3.8 we know that αX is a Peano space of dimension at most 1. The proof of 2.3.8 further shows that there are no small circles accumulating in X. However, in order to show that αX is an ANR-space we have to exclude the possibility of smaller and smaller circles accumulating at ∞ ∈ αX, see Theorem 2.1.7. Assume that we find a sequence of circles with diameters converging to 0 (with respect to some fixed geodesic metric ([Bin49], [Moi49])) at around ∞ ∈ αX. After passing to a subsequence, there are two possible situations: either each circle contains the point ∞ or none of them does. Both cases are treated exactly the same, for the sake of brevity we only consider the situation where ∞ is contained in all circles. In this case have a ^{∗}-homomorphism φ: C_{0}(X, 𝕄_{N}) → ⨁_{n=1}^{∞} C_{0}((0, 1)_{n}, 𝕄_{N}) where (0, 1)_{n} ≅ (0, 1) is the part of the n-th circle contained in X. Note that each coordinate projection gives a surjection φ_{n}: C_{0}(X, 𝕄_{N}) → C_{0}((0, 1), 𝕄_{N}) while φ itself is not necessarily surjective (because the circles might intersect in X). We make use of Brown’s mapping telescope associated to ⨁_{n=1}^{∞} C_{0}((0, 1)_{n}, 𝕄_{N}), i.e.

T_{k} = {f ∈ C([k, ∞], ⨁_{n=1}^{∞} C_{0}((0, 1)_{n}, 𝕄_{N})): t ≤ l ⇒ f(t) ∈ ⨁_{n=1}^{l} C_{0}((0, 1)_{n}, 𝕄_{N})}

with the obvious (surjective) restrictions as connecting maps giving \underrightarrow{lim} T_{k} = ⨁_{n=1}^{∞} C_{0}((0, 1)_{n}, 𝕄_{N}). Using Lemma 3.1.4, we find a commutative diagram

\xymatrix{0 \ar[r] & \underrightarrow{lim} T_{k} \ar[r] & \underrightarrow{lim} M(T_{k}) \ar[r] & \underrightarrow{lim} Q(T_{k}) \ar[r] & 0 \\ 0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] \ar[u]^{φ} & A \ar[r] \ar[u]^{\overline{φ}} & B \ar[r] \ar[u]^{\overline{\overline{φ}}} & 0}.

It now follows from the semiprojectivity assumptions and Lemma 3.1.1 that φ lifts to T_{k} for some k, which is equivalent to a solution of the original lifting problem

\xymatrix{ & ⨁_{n=1}^{k} C_{0}((0, 1)_{n}, 𝕄_{N}) \ar[d]^{⊆} \\ C_{0}(X, 𝕄_{N}) \ar[r]^{φ} \ar@{.>}[ur] & ⨁_{n=1}^{∞} C_{0}((0, 1)_{n}, 𝕄_{N})}

up to homotopy. This, however, implies

\begin{matrix}im(K_{1}(φ)) ⊆ & K_{1} ( ⨁_{n=1}^{k} C_{0}((0, 1)_{n}, 𝕄_{N})) \\ = & ∑_{n=1}^{k} ℤ \\ ⊂ & ∑_{n=1}^{∞} ℤ = K_{1} ( ⨁_{n=1}^{∞} C_{0}((0, 1)_{n}, 𝕄_{N})) \end{matrix}

which gives a contradiction as follows. Because φ_{k+1} is surjective and dim(αX) ≤ 1, we can extend the canonical unitary function from α((0, 1)_{n}) to a unitary u on all of αX by [HW48, Theorem VI.4]. This unitary then satisfies u − 1 ∈ C_{0}(X) and K_{1}(φ)([u ⊗ 1_{𝕄_{N}}]) = N ∈ ℤ = K_{1}(C_{0}((0, 1)_{k+1}, 𝕄_{N})). This shows that there are no small circles at around ∞ in αX and hence that αX is a one-dimensional ANR-space by Theorem 2.1.7.

Theorem 4.3.2. Let a short exact sequence of C^{∗}-algebras 0 → I → A → B → 0 be given and assume that the ideal I is a N-homogeneous C^{∗}-algebra with Prim(I) = X. Denote by (X_{i})_{i∈I} the connected components of X and consider the following statements:

(I) I is (semi)projective.

(II) A is (semi)projective.

(III) B is (semi)projective and the set-valued retract map R: Prim(A) → 2^{Prim(B)} given as in 4.1.1 by

R(z) = \begin{cases}z & if z ∈ Prim(B), \\ ∂X_{i} = \overline{X_{i}}\X_{i} & if z ∈ X_{i} ⊆ X = Prim(I) \end{cases}

is lower semicontinuous and has pointwise finite image.

If any two of these statements are true, then the third one also holds.

Proof. (I)+(II)⇒(III): By Theorem 2.3.11, we know that the sequence is isomorphic to an extension

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] & A \ar[r]^{π} & B \ar[r] & 0}

with the one-point compactification of X a one-dimensional A(N)R-space. The set-valued retract map R is then lower semicontinuous by Proposition 4.1.20 and has pointwise finite image by Proposition 4.1.19. But now Theorem 4.1.14 applies and shows that there is a completely positive split s for the quotient map π such that the composition B ⟶^{s} A ⟶^{ι} C_{b}(X, 𝕄_{N}) is multiplicative outside of an open set U ⊂ K ⊂ X where K is compact. Let a lifting problem φ: B ⟶^{∼} D/J = \underrightarrow{lim} D/J_{n} be given. Since A is semiprojective, we can solve the resulting lifting problem for A, meaning we find ψ: A → D/J_{n} for some n with π_{n} ∘ ψ = φ ∘ π. Restricting to her_{D/J_{n}}(ψ(C_{0}(X, 𝕄_{N}))) + ψ(A) ⊆ D/J_{n}, we may assume that ψ_{|C_{0}(X,𝕄_{N})} is proper as a ^{∗}-homomorphism to J/J_{n} and hence induces a map M(ψ) between multiplier algebras. Since the restriction of π_{n} ∘ ψ to the ideal C_{0}(X, 𝕄_{N}) vanishes, we may use compactness of K to assume that ψ maps C_{0}(U, 𝕄_{N}) to 0 (after increasing n if necessary). This further implies that M(ψ) factors through r: C_{b}(X, 𝕄_{N}) → C_{b}(X\U, 𝕄_{N}). We then find s′ := r ∘ ι ∘ s to be multiplicative and hence a ^{∗}-homomorphism:

[figure: A; ι; π; ψ; D/J_{n}; ι_{n}; π_{n}; C_{b}(X, 𝕄_{N}); r; M(ψ); M(J/J_{n}); ϱ_{n}; C_{b}(X\U, 𝕄_{N}); M(ψ)′; B; φ; s; s′; D/J; τ_{n}; Q(J/J_{n})]

The inclusion of J/J_{n} as an ideal in D/J_{n} gives canonical ^{∗}-homomorphisms ι_{n} and τ_{n} as in the diagram above. One now checks that ϱ_{n} ∘(M(ψ)′ ∘s′) = τ_{n} ∘φ holds. Combining this with the fact that the trapezoid on the right is a pullback diagram, we see that the pair (φ, (M(ψ)′ ∘ s′)) defines a lift B → D/J_{n} for φ. This shows that the quotient B is semiprojective. For the projective version of the statement, one uses Corollary 4.1.11 to see that the sequence admits a multiplicative split s: B → A rather than just a completely positive one.

(I)+(III)⇒(II): We know that I ≅ C_{0}(X, 𝕄_{N}) with αX a one-dimensional A(N)R-space by Theorem 2.3.11. Now Lemma 4.2.1 applies and we obtain a limit structure for A

\xymatrix{B_{0} \ar[r]_{s_{0}^{1}} & B_{1} \ar[r]_{s_{1}^{2}} \ar@{->>}[l]_{r_{1}^{0}} & B_{2} \ar[r] \ar@{->>}[l]_{r_{2}^{1}} & ⋯ \ar[r]_{s_{i−1}^{i}} \ar@{->>}[l] & B_{i} \ar[r] \ar[rr]_{s_{i}^{∞}} \ar@{->>}[l]_{r_{i}^{i−1}} & ⋯ \ar[r] & \underrightarrow{lim} (B_{i}, s_{i}^{i+1}) ≅ A \ar@{->>}[ll]_{r_{∞}^{i}}}

with B_{0} given as a pullback of B and a 1-NCCW D over a finite-dimensional C^{∗}-algebra. In particular, B_{0} is semiprojective by [End14, Corollary 3.4]. In the projective case, we can take B_{0} = B to be projective. In both cases, the connecting maps in the system above arise from pullback diagrams

\xymatrix{B_{i+1} \ar@{->>}[d]^{r_{i+1}^{i}} \ar@{.>}[r] & C([0, 1], 𝕄_{N}) \ar@{->>}[d]^{_{ev_{0}}} \\ B_{i} \ar[u]^{s_{i}^{i+1}} \ar[r]^{π_{i}} & 𝕄_{N}}

with s_{i}^{i+1}(a) = (a, π_{i}(a) ⊗ 1_{[0,1]}). Since these maps are weakly conditionally projective by Proposition 3.2.4, we obtain (semi)projectivity of A from Lemma 3.2.3.

(II)+(III)⇒(I): This implication holds under even weaker hypothesis. More precisely, we show that (semi)projectivity of both A and B implies I to be (semi)projective. The assumption on the retract map R is not needed here. First we apply Lemma 2.3.8 to find the one-point compactification of Prim(I) to be a Peano space of dimension at most 1, and hence I is trivially homogeneous by Lemma 2.3.10. Now Proposition 4.3.1 shows that αX is in fact an ANRspace which, together with Theorem 2.3.11, means that I is semiprojective. The projective version is Corollary 3.1.3.

Remark 4.3.3. Theorem 4.3.2 shows that regularity properties of the retract map R: Prim(A) → 2^{Prim(B)} are crucial for semiprojectivity to behave nicely with respect to extensions by homogeneous C^{∗}-algebras. This can already be observed and illustrated in the commutative case. Given an extension of commutative C^{∗}-algebras

0 → C_{0}(X) → C_{0}(Y) → C_{0}(Y \X) → 0,

the following holds: If both the ideal C_{0}(X) and the quotient C_{0}(Y \X) are (semi)projective, then the extension C_{0}(Y) is (semi)projective if and only if the retract map R: Y → 2^{Y \X} is lower semicontinuous and has pointwise finite image. The following examples show that both properties for R are not automatic:

(a) An examples with R not having pointwise finite image is contained as example 5.5 in [LP98], we include it here for completeness. Let X = {(x, sin x^{−1}) : 0 < x ≤ 1} ⊂ ℝ^{2} and Y = X ∪ {(0, y): − 1 ≤ y < 1}, then we get an extension isomorphic to

0 → C_{0}(0, 1] → C_{0}(Y) → C_{0}(0, 1] → 0.

Here both the ideal and the quotient are projective, but the extension C_{0}(Y) is not (because αY is not locally connected and hence not an AR-space). In this example, we find R(x) = {(0, y): − 1 ≤ y < 1} to be infinite for all x ∈ X.

(b) The following is an example where R fails to be lower semicontinuous. Consider Y = {(x, 0): 0 ≤ x < 1} ∪ ⋃_{n} C_{n} ⊂ ℝ^{2} with C_{n} = {(t, (1 − t)/n): 0 ≤ t < 1} the straight line from (0, 1/n) to (1, 0) with the endpoint (1, 0) removed. With X = ⋃_{n} C_{n} ⊂ Y we obtain an extension isomorphic to

0 → ⨁_{n} C_{0}(0, 1] → C_{0}(Y) → C_{0}(0, 1] → 0.

Here both the ideal and the quotient are projective while the extension C_{0}(Y) is not (again because αY is not locally connected). We also find (0, 1/n) → (0, 0) in Y but R((0, 1/n)) = ∅ for all n, which shows that R is not lower semicontinuous. The descriptive reason for C_{0}(Y) not being projective in this case is that the length of the attached intervals C_{n} does not tend to 0 as n goes to infinity.

5 The structure of semiprojective subhomogeneous C^{∗}-algebras

5.1 The main result

With Theorem 4.3.2 at hand, we are now able to keep track of semiprojectivity when decomposing a subhomogeneous C^{∗}-algebra into its homogeneous subquotients. On the other hand, Theorem 4.3.2 also tells us in which manner homogeneous, semiprojective C^{∗}-algebras may be combined in order to give subhomogeneous C^{∗}-algebras which are again semiprojective. This leads to the main result of this chapter, Theorem 5.1.2, which gives two characterizations of projectivity and semiprojectivity for subhomogeneous C^{∗}-algebras.

Lemma 5.1.1. Let A be a N-subhomogeneous C^{∗}-algebra. If A is semiprojective, then the maximal N-homogeneous ideal of A is also semiprojective.

Proof. By Lemma 2.3.8 we know that the one-point compactification of X = Prim_{N}(A) is a one-dimensional Peano space. Since any locally trivial 𝕄_{N}-bundle over X is globally trivial by Lemma 2.3.10, we are concerned with an extension of the form

\xymatrix{0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] & A \ar[r]^{π} & A_{≤N−1} \ar[r] & 0}

where A_{≤N−1} denotes the maximal (N-1)-subhomogeneous quotient of A. Since A is semiprojective, A_{≤N−1} will be semiprojective with respect to (N-1)-subhomogeneous C^{∗}-algebras. In order to show that C_{0}(X, 𝕄_{N}) is semiprojective, it remains to show that αX = X ∪ {∞} does not contain small circles at around ∞, cf. Theorem 2.1.7. The proof for this is similar to the one of 4.3.1. We use notations from 2.3.8 and follow the proof there to arrive at a commutative diagram

\xymatrix{0 \ar[r] & \underrightarrow{lim} T_{k} \ar[r] & \underrightarrow{lim} M(T_{k}) \ar[r] & \underrightarrow{lim} Q(T_{k}) \ar[r] & 0 \\ 0 \ar[r] & C_{0}(X, 𝕄_{N}) \ar[r] \ar[u]^{φ} & A \ar[r]^{π} \ar[u]^{\overline{φ}} & A_{≤N−1} \ar[r] \ar[u]^{\overline{\overline{φ}}} & 0}.

We may not solve the lifting problem for A_{≤N−1} directly since the algebras Q(T_{k}) are not (N-1)-subhomogeneous. Instead we will replace the Q(T_{k}) by suitable (N-1)-subhomogeneous subalgebras which will then lead to a solvable lifting problem for A_{≤N−1}. Let ι_{n} denote the n-th coordinate of the map A → C_{b}(X, 𝕄_{N}) → ∏_{n} C_{b}((0, 1)_{n}, 𝕄_{N}). We then have a lift of \overline{φ} given by

\begin{matrix}A & → & C([k, ∞], ∏_{n} C_{b}((0, 1), 𝕄_{N})) & → & M(T_{k}) \\ a & ↦ & 1_{[k,∞]} ⊗ (ι_{n}(a))_{n=1}^{∞} \end{matrix}

where the map on the right is induced by the inclusion of T_{k} as an ideal in C([k, ∞], ∏_{n} C_{b}((0, 1), 𝕄_{N})). Consider in there the central element f = (f_{n})_{n=1}^{∞} with f_{n} the scalar function that equals 0 on [k, n], 1 on [n + 1, ∞] and which is linear in between. Then

\begin{matrix}ψ: A & → & C([k, ∞], ∏_{n} C_{b}((0, 1), 𝕄_{N})) & → & M(T_{k}) \\ a & ↦ & (f_{n} ⊗ ι_{n}(a))_{n=1}^{∞} \end{matrix}

is a completely positive lift of \overline{φ} which sends C_{0}(X, 𝕄_{N}) to T_{k}. Hence ψ induces a completely positive lift ψ′: A_{≤N−1} → Q(T_{k}) of \overline{\overline{φ}}. We claim that C^{∗}(ψ′(A_{≤N−1})) is in fact (N-1)-subhomogeneous. To see this, we use the algebraic characterization of subhomogenity as described in [Bla06, IV.1.4.6]. It suffices to check that γ(C^{∗}(ψ′(A_{≤N−1}))) satisfies the polynomial relations p_{r(N−1)} for every irreducible representation γ of Q(T_{k}). By definition of ψ, we find γ ∘ ψ′(π(a)) = t · γ′(ι(a)) for some representation γ′ of ι(A), some t ∈ [0, 1] and every a ∈ A. Moreover, since ψ′ maps C_{0}(X, 𝕄_{N}) to 0, we obtain γ ∘ ψ′(π(a)) = t · γ′′(π(a)) for some representation γ′′ of A_{≤N−1}. Using (N-1)-subhomogeneity of A_{≤N−1}, it now follows easily that the elements of γ(C^{∗}(ψ′(A_{≤N−1}))) satisfy the polynomial relations p_{r(N−1)} from [Bla06, IV.1.4.6]. Knowing that the image of \overline{\overline{φ}} has a (N-1)-subhomogenous preimage in Q(T_{k}), we may now solve the lifting problem for A_{≤N−1}. It then follows from Lemma 3.1.1 (and its proof) that φ lifts to T_{k} for some k. The remainder of the proof is exactly the same as the one of Proposition 4.3.1.

We now present two characterizations of projectivity and semiprojectivity for subhomogeneous C^{∗}-algebras. The first one describes semiprojectivity of these algebras in terms of their primitive ideal spaces. The second description characterizes them as those C^{∗}-algebras which arise from 1-NCCWs by adding a sequence of non-commutative edges (of bounded dimension), cf. section 3.2.2.

Theorem 5.1.2. Let A be a N-subhomogeneous C^{∗}-algebra, then the following are equivalent:

(1) A is semiprojective (resp. projective).

(2) For every n = 1, ..., N the following holds:

• The one-point compactification of Prim_{n}(A) is an ANR-space (resp. an AR-space) of dimension at most 1.

• If (X_{i})_{i∈I} denotes the family of connected components of Prim_{n}(A), then the set-valued retract map

R_{n}: Prim_{≤n}(A) → 2^{Prim_{≤n−1}(A)}

given by

z ↦ \begin{cases}z & if z ∈ Prim_{≤n−1}(A) \\ ∂X_{i} & if z ∈ X_{i} ⊂ Prim_{n}(A) \end{cases}

is lower semicontinuous and has pointwise finite image.

(3) A is isomorphic to the direct limit \underrightarrow{lim}_{k}(A_{k}, s_{k}^{k+1}) of a sequence of 1-NCCWs

[figure: ⋯; A_{k}; s_{k}^{k+1}; A_{k+1}; r_{k+1}^{k}; ⋯]

(with A_{0} = 0) such that for each stage there is a pullback diagram

\xymatrix{A_{k+1} \ar@{.>}[r] \ar@{->>}[d]^{r_{k+1}^{k}} & C([0, 1], 𝕄_{n}) \ar[d]^{_{ev_{0}}} \\ A_{k} \ar[r]^{π_{k}} \ar[u]^{s_{k}^{k+1}} & 𝕄_{n}}

with n ≤ N and s_{k}^{k+1} given by a ↦ (a, π_{k}(a) ⊗ 1_{[0,1]}).

Proof. (1) ⇒ (2): We prove the implication by induction over N. The base case N = 1 is given by Theorem 2.3.7. Now given a N-subhomogeneous, (semi)projective C^{∗}-algebra A, we know by Lemma 5.1.1 that the maximal N-homogeneous ideal A_{N} of A is (semi)projective as well. This forces α Prim_{N}(A) to be a one-dimensional A(N)R-space by Theorem 2.3.11. Applying Theorem 4.3.2 to the sequence

0 → A_{N} → A → A_{≤N−1} → 0

now shows that the retract map R_{N}: Prim_{N}(A) → 2^{Prim_{≤N−1}(A)} is lower semicontinuous, has pointwise finite image and that the maximal (N-1)- subhomogeneous quotient A_{≤N−1} is (semi)projective. The remaining statements follow from the induction hypothesis applied to A_{≤N−1}.

(2) ⇒ (3): By Lemma 2.3.10, we know that the maximal N-homogeneous ideal A_{N} of A is of the form C_{0}(Prim_{N}(A), 𝕄_{N}). Using induction over N, the statement then follows from Proposition 4.2.2 applied to the sequence

0 → C_{0}(Prim_{N}(A), 𝕄_{N}) → A → A_{≤N−1} → 0.

The base case N = 1 is given by Theorem 2.1.5.

(3) ⇒ (1): Note that the connecting maps are weakly conditionally projective by Proposition 3.2.4, then apply Lemma 3.2.3.

Remark 5.1.3. The most prominent examples of subhomogeneous, semiprojective C^{∗}-algebras are the one-dimensional non-commutative CW-complexes (1-NCCWs, see Example 2.2.3). The structure theorem 5.1.2 shows that these indeed play a special role in the class of all subhomogeneous, semiprojective C^{∗}-algebras. By part (2) of 5.1.2, they are precisely those subhomogeneous, semiprojective C^{∗}-algebras for which the spaces α Prim_{n} are all finite graphs rather than general one-dimensional ANR-spaces. Hence 1-NCCWs should be thought of as the elements of ’finite type’ in the class of subhomogeneous, semiprojective C^{∗}-algebras. Moreover, part (3) of 5.1.2 shows that every subhomogeneous, semiprojective C^{∗}-algebra can be constructed from 1-NCCWs in a very controlled manner. Therefore these algebras share many properties with 1-NCCWs, as we will see in section 5.2.1 in more detail.

5.2 Applications

Now we discuss some consequences of Theorem 5.1.2. First we collect some properties of semiprojective, subhomogeneous C^{∗}-algebras which follow from the descriptions in 5.1.2. This includes information about their dimension and K-theory as well as details about their relation to 1-NCCWs and some further closure properties. At least in principle one can use the structure theorem 5.1.2 to test any given subhomogeneous C^{∗}-algebra A for (semi)projectivity. Since this would require a complete computation of the primitive ideal space of A, it is not recommended though. Instead one might use 5.1.2 as a tool to disprove semiprojectivity for a candidate A. In fact, showing directly that a C^{∗}-algebra A is not semiprojective can be surprisingly difficult. One might therefore take one of the conditions from 5.1.2 which are easier to verify and test A for those instead. We illustrate this strategy in section 5.2.2 by proving the quantum permutation algebras to be not semiprojective. This corrects a claim in [Bla04] on semiprojectivity of universal C^{∗}-algebras generated by finitely many projections with order and orthogonality relations.

5.2.1 Further structural properties

By part (3) of Theorem 5.1.2, we know that any semiprojective, subhomogeneous C^{∗}-algebra comes as a direct limit of 1-NCCWs. Since the connecting maps are explicitly given and of a very special nature, it is possible to show that these limits are approximated by 1-NCCWs in a very strong sense. The following corollary makes this approximation precise.

Corollary 5.2.1 (Approximation by 1-NCCWs). Let A be a subhomogeneous C^{∗}-algebra. If A is semiprojective, then for every finite set G ⊂ A and every ϵ > 0 there exist a 1-NCCW B ⊆ A and a ^{∗}-homomorphism r: A → B such that G ⊂_{ϵ} B and r is a strong deformation retract for B, meaning that there exists a homotopy H_{t} from H_{0} = id_{A} to H_{1} = r with H_{t|B} = id_{B} for all t. In particular, A is homotopy equivalent to a one-dimensional non-commutative CW-complex.

Proof. Use part (3) of Theorem 5.1.2 to write A = \underrightarrow{lim} A_{n} and find a suitable 1-NCCW B = A_{n_{0}} which almost contains the given finite set G. It is straightforward to check that the strong deformation retracts r_{n}^{n_{0}} : A_{n} → A_{n_{0}} give rise to a strong deformation retract r: \underrightarrow{lim} A_{n} → A_{n_{0}}.

In particular, 1-NCCWs and semiprojective, subhomogeneous C^{∗}-algebras share the same homotopy invariant properties. For example, we obtain the following restrictions on the K-theory of these algebras:

Corollary 5.2.2. Let A be a subhomogeneous C^{∗}-algebra. If A is semiprojective, then its K-theory is finitely generated and K_{1}(A) is torsion free.

Another typical phenomenon of (nuclear) semiprojective C^{∗}-algebras is that they appear to be one-dimensional in some sense. In the context of subhomogeneous C^{∗}-algebras, we can now make this precise using the notion of topological dimension which, for subhomogeneous A, is given by topdim(A) = max_{n} dim(α Prim_{n}(A)).

Corollary 5.2.3. Let A be a subhomogeneous C^{∗}-algebra. If A is semiprojective, then A has stable rank 1 and topdim(A) ≤ 1.

Proof. The statement on the stable rank of A follows from Corollary 5.2.1, while the topological dimension can be estimated using part (2) of Theorem 5.1.2.

Our structure theorem can also be used to study permanence properties of semiprojectivity when restricted to the class of subhomogeneous C^{∗}-algebras. In fact, these turn out to be way better then in the general situation. This can be illustrated by the following longstanding question by Blackadar and Loring: Given a short exact sequence of C^{∗}-algebras

\xymatrix{0 \ar[r] & I \ar[r] & A \ar[r] & F \ar[r] & 0}

with finite-dimensional F, does the following hold?

I semiprojective ⇔ A semiprojective

While we showed the ’⇐’-implication to hold in general in [End14], S. Eilers and T. Katsura proved the ’⇒’-implication to be wrong ([EK]), even in the case of split extensions by ℂ. We refer the reader to [Sør12] for counterexamples which involve infinite C^{∗}-algebras. However, when one restricts to the class of subhomogeneous C^{∗}-algebras, this implication holds:

Corollary 5.2.4. Let a short exact sequence of C^{∗}-algebras

\xymatrix{0 \ar[r] & I \ar[r] & A \ar[r]^{π} & F \ar[r] & 0}

with finite-dimensional F be given. If I is subhomogeneous and semiprojective, then A is also semiprojective.

Proof. We verify condition (2) in Theorem 5.1.2 for A. By assumption, each Prim_{k}(I) is a one-dimensional ANR-space after compactification and the same holds for any space obtained from Prim_{k}(I) by adding finitely many points ([ST12, Theorem 6.1]). Hence the one-point compactifications of Prim_{k}(A) are 1-dimensional ANRs for all k. If we assume F = 𝕄_{n}, then the set-valued retract maps R_{k} are unchanged for k < n. For k = n, regularity of R_{k} follows from regularity of the retract map for I and the fact that {[π]} is closed in Prim_{≤k}(A) = Prim_{≤k}(I) ∪ {[π]}. For k > n, we apply Lemma 4.1.6 to

[figure: 0; 0; 0; C_{0}(Prim_{k}(I), 𝕄_{k}); I_{≤k}; I_{≤k−1}; 0; 0; C_{0}(Prim_{k}(A), 𝕄_{k}); A_{≤k}; π; A_{≤k−1}; 0; F; F; 0; 0]

and see that R_{k}: Prim_{≤k}(A) → 2^{Prim_{≤k−1}(A)} is again lower semicontinuous and has pointwise finite image.

5.2.2 Quantum permutation algebras

We are now going to demonstrate how the structure theorem 5.1.2 can be used to show that certain C^{∗}-algebras fail to be semiprojective. We would like to thank T. Katsura for pointing out to us the quantum permutation algebras ([Wan98], [BC08]) as a testcase:

Definition 5.2.5 ([BC08]). For n ∈ ℕ, the quantum permutation algebra A_{s}(n) is the universal C^{∗}-algebra generated by n^{2} elements u_{ij}, 1 ≤ i, j ≤ n, with relations

u_{ij} = u_{ij}^{∗} = u_{ij}^{2} & ∑_{j} u_{ij} = ∑_{i} u_{ij} = 1.

It is not clear from the definition whether the C^{∗}-algebras A_{s}(n) are semiprojective or not. For n ∈ {1, 2, 3} one easily finds A_{s}(n) ≅ ℂ^{n!} so that we have semiprojectivity in that cases. For higher n one might expect semiprojectivity of A_{s}(n) because of the formal similarity to graph C^{∗}-algebras. In fact, their definition only involves finitely many projections and orthogonality resp. order relations between them. Since graph C^{∗}-algebras associated to finite graphs are easily seen to be semiprojective, one might think that we also have semiprojectivity for the quantum permutation algebras. This was even erroneously claimed to be true in [Bla04, example 2.8(vi)]. In this section we will show that the C^{∗}-algebras A_{s}(n) are in fact not semiprojective for all n ≥ 4. One can reduce the question for semiprojectivity of these algebras to the case n = 4. The following result of Banica and Collins shows that the algebra A_{s}(4) is 4-subhomogeneous, so that our machinery applies. The idea is to get enough information about the primitive spectrum of A_{s}(4) to show that it contains closed subsets of dimension strictly greater than 1. This will then contradict part (2) of 5.1.2, so that A_{s}(4) cannot be semiprojective.

We follow notations from [BC08] and denote the Pauli matrices by

c_{1} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , c_{2} = (\begin{matrix}i & 0 \\ 0 & −i \end{matrix}) , c_{3} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , c_{4} = (\begin{matrix}0 & i \\ i & 0 \end{matrix}) .

Set ξ_{ij}^{x} = c_{i}xc_{j} and regard 𝕄_{2} as a Hilbert space with respect to the scalar product < a|b >= tr(b^{∗}a). Then for any x ∈ SU(2) we find {ξ_{ij}^{x}}_{j=1..4} and {ξ_{ij}^{x}}_{i=1..4} to be a basis for 𝕄_{2}. Under the identification 𝕄_{4} ≅ B(𝕄_{2}), Banica and Collins studied the following representation of A_{s}(4):

Proposition 5.2.6 (Theorem 4.1 of [BC08]). The ^{∗}-homomorphism given by

π: A_{s}(4) ⟶ C(SU(2), 𝕄_{4})
u_{ij} ↦ (x ↦ rank one projection onto ℂ · ξ_{ij}^{x})

is faithful. It is called the Pauli representation of A_{s}(4).

For the remainder of this section let S denote the following subset of SU(2):

S := \{(\begin{matrix}λ & −\overline{μ} \\ μ & \overline{λ} \end{matrix}) ∈ SU(2): min \{ \begin{matrix}| Re(λμ)|, | Im(λμ)|, | Re(\overline{λ}μ)|, \\ | Im(\overline{λ}μ)|, ||λ| − |μ|| \end{matrix} \} = 0\}

We will now study the representations of A_{s}(4) obtained by composing the Pauli representation with a point evaluation. As we will see, most points of SU(2) lead to irreducible representations which are furthermore locally pairwise inequivalent.

Lemma 5.2.7. The representation π_{x} = ev_{x} ∘π: A_{s}(4) → 𝕄_{4} is irreducible for every x ∈ SU(2)\S.

Proof. Let x = (\begin{matrix}λ & −\overline{μ} \\ μ & \overline{λ} \end{matrix}) ∈ SU(2)\S be given, we show that the com-

mutant of π_{x}(A_{s}(4)) equals the scalars. Therefore we will check the matrix entries of the elements π_{x}(u_{ij}) with respect to the orthonormal basis \{\frac{1}{\sqrt{2}}ξ_{11}^{x}, \frac{1}{\sqrt{2}}ξ_{12}^{x}, \frac{1}{\sqrt{2}}ξ_{13}^{x}, \frac{1}{\sqrt{2}}ξ_{14}^{x}\} of M_{2} ≅ ℂ^{4}. Since in this picture π_{x}(u_{1i}) equals

the elementary matrix e_{ii}, every element in (π_{x}(A_{s}(4)))′ is diagonal. But we also find

(π_{x}(u_{23}))_{12} = \frac{1}{2} < π_{x}(u_{23})ξ_{12}^{x}|ξ_{11}^{x} >
= \frac{1}{4} < ξ_{12}^{x}|ξ_{23}^{x} >< ξ_{23}^{x}|ξ_{11}^{x} >= 4 · Re(λμ)Im(λμ) ≠ 0,
(π_{x}(u_{22}))_{13} = \frac{1}{4} < ξ_{13}^{x}|ξ_{22}^{x} >< ξ_{22}^{x}|ξ_{11}^{x} >= 2 · Re(λμ)(|λ|^{2} − |μ|^{2}) ≠ 0,
(π_{x}(u_{22}))_{14} = \frac{1}{4} < ξ_{14}^{x}|ξ_{22}^{x} >< ξ_{22}^{x}|ξ_{11}^{x} >= −2 · Im(λμ)(|λ|^{2} − |μ|^{2}) ≠ 0.

So the only elements of 𝕄_{4} commuting with all of π_{x}(A_{s}(4)) are the scalars.

Proposition 5.2.8. Every x ∈ SU(2)\S admits a small neighborhood V ⊆ SU(2)\S such that for all distinct y, y′ ∈ V the representations π_{y} and π_{y′} are not unitarily equivalent.

Proof. Let x = (\begin{matrix}λ_{0} & −\overline{μ_{0}} \\ μ_{0} & \overline{λ_{0}} \end{matrix}) ∈ SU(2)\S be given, then

ϵ := min \{ \begin{matrix}| Re(λ_{0}μ_{0})|, | Im(λ_{0}μ_{0})|, | Re(\overline{λ_{0}}μ_{0})|, \\ | Im(\overline{λ_{0}}μ_{0})|, ||λ_{0}| − |μ_{0}|| , |λ_{0}| \end{matrix} \} > 0.

Define a neighborhood V ⊆ SU(2)\S of x by

V = \{(\begin{matrix}λ & −\overline{μ} \\ μ & \overline{λ} \end{matrix}) ∈ SU(2)\S : |λ − λ_{0}| < \frac{ϵ}{3}, |μ − μ_{0}| < \frac{ϵ}{3}\} .

Now let y, y′ ∈ V with unitarily equivalent representations π_{y} and π_{y′} be given. We compute the value

∥π_{y}(u_{11}u_{22})∥ = \frac{1}{4}∥(< − |ξ_{11}^{y} > ξ_{11}^{y}) ∘ (< − |ξ_{22}^{y} > ξ_{22}^{y})∥
= \frac{1}{4}| < ξ_{22}^{y}|ξ_{11}^{y} > | · ∥(< − |ξ_{22}^{y} > ξ_{11}^{y})∥
= \frac{1}{4}| < ξ_{22}^{y}|ξ_{11}^{y} > | · ∥ξ_{22}^{y}∥∥ξ_{11}^{y}∥
= ||λ|^{2} − |μ|^{2}|

which is invariant under unitary equivalence. So we find ||λ|^{2} − |μ|^{2}|| = ||λ′|^{2} − |μ′|^{2}|. This implies

(|λ| = |λ′| ∧ |μ| = |μ′|) ∨ (|λ| = |μ′| ∧ |μ| = |λ′|)

because of |λ|^{2} + |μ|^{2} = 1 = |λ′|^{2} + |μ′|^{2}. By definition of V we have

||λ| − |μ′|| ≥ ||λ_{0}| − |μ_{0}|| − ||λ| − |λ_{0}|| − ||μ′| − |μ_{0}|| > \frac{ϵ}{3} > 0,

so that we can exclude the second case. Analogously, computing the invariants ∥π_{y}(u_{13}u_{22})∥ and ∥π_{y}(u_{14}u_{22})∥ gives

| Re(λμ)| = | Re(λ′μ′)| and | Im(λμ)| = | Im(λ′μ′)|

and checking ∥π_{y}(u_{11}u_{42})∥ and ∥π_{y}(u_{11}u_{32})∥ shows

| Re(\overline{λ}μ)| = | Re(\overline{λ′}μ′)| and | Im(\overline{λ}μ)| = | Im(\overline{λ′}μ′)| .

The last four equalities imply λμ = λ′μ′ and \overline{λ}μ = \overline{λ′}μ′ by the choice of V. Together with |λ| = |λ′| and |μ| = |μ′| we find (λ, μ) = (λ′, μ′) or (λ, μ) = (−λ′, −μ′). In the second case we get |λ − λ′| = 2|λ| ≥ 2|λ_{0}| − 2|λ − λ_{0}| ≥ \frac{4ϵ}{3} contradicting |λ − λ′| ≤ |λ − λ_{0}| + |λ′ − λ_{0}| < \frac{2ϵ}{3} by the choice of V. It follows that y = y′.

By now we have obtained enough information about Prim(A_{s}(4)) to show that it does not satisfy condition (2) of Theorem 5.1.2. Hence we find:

Theorem 5.2.9. The C^{∗}-algebra A_{s}(4) is not semiprojective.

Proof. Choose a point x_{0} ∈ SU(2)\S and a neighborhood V of x_{0} as in Proposition 5.2.8. Since SU(2) is a real 3-manifold, there is a neighborhood of x_{0} contained in V which is homeomorphic to 𝔻^{3} = {x ∈ ℝ: ∥x∥ ≤ 1}. The restriction of the Pauli representation π to this neighborhood gives a^{∗}-homomorphism φ: A_{s}(4) → C(𝔻^{3}, 𝕄_{4}) with the property that ev_{x} ∘φ and ev_{y} ∘φ are irreducible but not unitarily equivalent for all distinct x, y ∈ 𝔻^{3}. The pointwise surjectivity of φ given by Lemma 5.2.7 and a Stone-Weierstraß argument ([Kap51, Theorem 3.1]) show that φ is in fact surjective. This implies that Prim_{4}(A_{s}(4)) contains a closed 3-dimensional subset and hence dim(Prim_{4}(A_{s}(4))) ≥ 3. As a consequence, A_{s}(4) cannot be semiprojective because it is subhomogeneous by Proposition 5.2.6 but fails to satisfy condition (2) of Theorem 5.1.2.

It is not hard to show that semiprojectivity of A_{s}(n) for some n > 4 would force A_{s}(4) to be semiprojective. Since we have just shown that this is not the case, we obtain:

Corollary 5.2.10. The C^{∗}-algebras A_{s}(n) are not semiprojective for n ≥ 4.

Proof. For n ≥ 4 there is a canonical surjection ϱ_{n}: A_{s}(n) → A_{s}(4) given by

u_{ij}^{(n)} ↦ \begin{cases}u_{ij}^{(4)} & if 1 ≤ i, j ≤ 4 \\ 1 & if i = j > 4 \\ 0 & otherwise \end{cases} .

The kernel of ϱ_{n} is generated by the finite set of projections {u_{ij}^{(n)} : ϱ_{n} (u_{ij}^{(n)}) = 0}. It follows from [Sør12, Proposition 3], which extends the idea of [Neu00, Proposition 5.19], that semiprojectivity of A_{s}(n) would imply semiprojectivity of ϱ_{n}(A_{s}(n)) = A_{s}(4). Since this is not the case by Theorem 5.2.9, A_{s}(n) cannot be semiprojective for all n ≥ 4.

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Dominic Enders University of Copenhagen Department of Mathematical Sciences Universitetsparken 5 2100 Copenhagen, Denmark d.enders@math.ku.dk 1050

Documenta Mathematica 21 (2016)