Trivialization of <span class="mathjax-formula">$\mathcal {C}(X)$</span>-algebras with strongly self-absorbing fibres

Bulletin

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Publié avec le concours du Centre national de la recherche scienti�que

de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Tome 136 Fascicule 4

2008

pages 575-606

TRIVIALIZATION OF ^C(X ) -ALGEBRAS WITH STRONGLY

SELF-ABSORBING FIBRES

Marius Dadarlat & Wilhelm Winter

TRIVIALIZATION OFC(X)-ALGEBRAS WITH STRONGLY SELF-ABSORBING FIBRES

by Marius Dadarlat &Wilhelm Winter

Abstract. — Suppose Ais a separable unital C(X)-algebra each fibre of which is isomorphic to the same strongly self-absorbing andK1-injective C^∗-algebraD. We show that Aand C(X)⊗ D are isomorphic asC(X)-algebras provided the compact Hausdorff spaceXis finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

Résumé(Trivialisation deC(X)-algèbres à fibres fortement auto-absorbantes) SoitAuneC(X)-algèbre séparable unital dont chaque fibre est isomorphe à une même C^∗-algèbreD K1-injective et fortement auto-absorbante. Nous montrons que si l’espace compact et HausdorffX est de dimension finie, alorsAetC(X)⊗ Dsont isomorphes en tant queC(X)-algèbres. Ce resultat est connu pour ne pas s’étendre au cas des espaces de dimension infinie.

Texte reçu le 2 août 2007, accepté le 4 avril 2008

Marius Dadarlat, Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA • E-mail :[email protected]

Wilhelm Winter, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom • E-mail : wil- [email protected]

2000 Mathematics Subject Classification. — 46L05, 47L40.

Key words and phrases. — Strongly self-absorbingC^∗-algebra, asymptotic unitary equiva- lence, continuous field ofC^∗-algebras.

The first named author was partially supported by NSF grant #DMS-0500693. The second named author was supported by the DFG (SFB 478), and by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280).

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1. Introduction

A classical problem in the theory of C^∗-algebras is to classify C^∗-algebras with primitive spectrum a given compact Hausdorff spaceX. These algebras are known to correspond to continuous fields of simple C*-algebras; in general they are very far from being locally trivial even if one assumes that all their fibres are mutually isomorphic. In the case of continuous trace C^∗-algebras, a most satisfactory solution was provided by Dixmier and Douady in [6]; there, all fibres are isomorphic to the compact operatorsK(H)on a separable Hilbert space, and the classifying invariant is a certain class in the cohomology group H³(X;Z).

It seems natural to look for similar results in the case of continuous fields with more complicated fibre algebras. In view of recent progress in Elliott’s program to classify nuclearC^∗-algebras byK-theory data (cf. [14] and [13]), a most obvious choice of possible fibre algebras would be the so-called strongly self-absorbing C^∗-algebras. A crucial property of these algebras is that their unital endomorphism semigroup is weakly homotopy equivalent to their auto- morphism group. This property is analogous to but still substantially weaker than the property of K(H) that all its endomorphisms inducing the identity map in K-theory are inner automorphisms.

A unital and separableC^∗-algebraD 6=Cis strongly self-absorbing if there is an isomorphismD→ D ⊗ D^∼= which is approximately unitarily equivalent to the inclusion map D → D ⊗ D, d7→d⊗1D, cf. [12]. Strongly self-absorbing C^∗-algebras are known to be simple and nuclear; moreover, they are either purely infinite or stably finite. The only known examples are UHF algebras of infinite type (i.e., every prime number that occurs in the respective super- natural number occurs with infinite multiplicity), the Cuntz algebras O₂ and O_∞, the Jiang–Su algebra Z and tensor products ofO_∞ with UHF algebras of infinite type, see [12]. All these examples are known to beK1-injective, i.e., the canonical mapU(D)/U₀(D)→K₁(D)is injective.

We shall state our results in terms of C(X)-algebras; this is a concept gen- eralizing continuous fields ofC^∗-algebras, cf. [9]. The main result of our paper is the following:

Theorem 1.1. — Let A be a separable unital C(X)-algebra over a finite di- mensional compact metrizable spaceX. Suppose that all the fibres ofAare iso- morphic to the same strongly self-absorbing K1-injective C^∗-algebra D. Then, A andC(X)⊗ D are isomorphic asC(X)-algebras.

In the case where D = O_∞ or D = O2, the preceding result was already obtained by the first named author in [3]; it is new for UHF algebras of infinite type and for the Jiang–Su algebra. While it would already be quite satisfactory

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to have this trivialization result for the known strongly self-absorbing examples, it is remarkable that it can be proven without any of the special properties of the concrete algebras. In this sense, the theorem further illustrates the importance of strongly self-absorbing algebras for the Elliott program. (In fact, Theorem1.1may clearly be regarded as a classification result forC(X)-algebras with strongly self-absorbing fibres, and as such it contributes to the non-simple case of the Elliott conjecture.) This point of view is one of the reasons why we think that not only the theorem, but also the methods developed for its proof are of independent interest. In the subsequent sections we shall therefore present two rather different proofs of Theorem1.1. The first approach follows the strategy of [3], using the main results of [7] and [5]. The second one follows ideas from Section 4 of [7]. We outline both approaches in Sections 3and4.

In [7, Example 4.7], Hirshberg, Rørdam and the second named author con- structed examples ofC(X)-algebras with X =Q

NS² and fibres isomorphic to any prescribed UHF algebra of infinite type, which do not absorb this UHF al- gebra (and hence cannot be trivial). In [4], the first named author modified this example to construct a separable, unital C(X)-algebra over the Hilbert cube with each fibre isomorphic to O2, but which does not have trivial K-theory (and hence cannot be trivial either). Therefore, the dimension condition on X in Theorem 1.1 cannot be removed. However, at the present stage, it is not known whether the theorem also fails for infinite dimensional spaces X if D=O∞ orD=Z.

Theorem1.1remains valid if one replacesK1-injectivity ofDby the (appar- ently weaker) condition that the commutator subgroup U(D)^c of the unitary group ofDis contained inU₀(D), the connected component of1. Indeed, the only role ofK1-injectivity in the proofs from this paper as well as the proofs of the results from [12], [7] and [5] that we use here is to ensure that approximate unitary equivalence for any two unital ∗-homomorphisms D → D ⊗A is in- duced by unitaries(un)n inU0(D). But as noted in [10] and [5] one can always choose the unitaries(un)n inU(D)^c. Conversely, it was observed by Kirchberg [10] that if the continuous field consisting of all continuous f : [0,1]→ D ⊗ D such that f(0) ∈ D ⊗1 and f(1) ∈ 1⊗ D is trivial (or just D-stable) then U(D)^c⊆U0(D).

We wish to point out that, in view of Elliott’s program, it is not so surprising that a dimension type condition is necessary to obtain a satisfactory classifi- cation result. In fact, all the known counterexamples to the Elliott conjecture exhibit high-dimensional behavior, whereas the conjecture has been success- fully confirmed for large classes of topologically low-dimensional C^∗-algebras, see [11] for an overview.

2. C(X)-Algebras

We recall some facts and notation aboutC(X)-algebras (introduced in [9]). For our purposes, it will be enough to restrict to compact spaces.

Definition 2.1. — LetAbe aC^∗-algebra andX a compact Hausdorff space.

Ais aC(X)-algebra, if there is a unital∗-homomorphismµ:C(X)→ Z(M(A)) from C(X)to the center of the multiplier algebra ofA.

The mapµis called thestructure map. We will not always write it explicitly.

IfAis as above andY ⊂X is a closed subset, then J_Y :=C₀(X\Y)·A

is a (closed) two-sided ideal of A; we denote the quotient map byπ_Y and set A(Y) =A_Y :=A/J_Y.

A_Y may be regarded as a C(X)-algebra or as a C(Y)-algebra in the obvious way.

If a∈A, we sometimes write aY for πY(a). If Y consists of just one point x, we will slightly abuse notation and write A_x (or A(x)), J_x, π_x and a_x (or a(x)) in place ofA_{x},J_{x},π_{x}anda_{x}, respectively. We say thatAxis the fibre of A at x. Ifϕ: A→ B is a morphism of C(X)-algebras, we denote by ϕ_Y the corresponding restriction mapA_Y →B_Y.

For anya∈A we have

kak= sup{kaxk : x∈X}.

Moreover, the function x 7→ kaxk from X to R is upper semicontinuous. If the map is continuous for anya∈A, then Ais said to be a continuousC(X)- algebra. By [3, Lemma 2.3], any unital C(X)-algebra with simple nonzero fibres is automatically continuous. In this case the structure map is injective and henceX is metrizable ifAis separable.

3. Proving the main result: The first approach

In this section we give a proof of Theorem1.1which follows ideas of [3] and relies on the main absorption result Theorem 4.6 of [7] and on [5, Theorem 2.2]:

Theorem 3.1. — Let A and D be unital C^∗-algebras, with D separa- ble, strongly self-absorbing and K1-injective. Then, any two unital ∗- homomorphisms σ, γ : D → A ⊗ D are strongly asymptotically unitarily equivalent, i.e. there is a unitary-valued continuous mapu: (0,1]→ U(A⊗ D), t 7→ ut, with u1 =1A⊗D and such that limt→0kutσ(d)u^∗_t −γ(d)k = 0 for all d∈D.

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Let us give an outline of the strategy. While a unital, separable, strongly self-absorbing C^∗-algebraD is not in general weakly semiprojective, we prove nevertheless thatD satisfies a property analogous to weak semiprojectivity in the category of unital and D-stableC^∗-algebras, see Proposition3.7. IfAis a unital, separable, C(X)-algebra over a finite dimensional compact space with fibres isomorphic toD, thenAisD-stable by [7]. This enables us to use the rel- ative weak semiprojectivity ofDdescribed above and approximateAby certain pullback C(X)-subalgebras with fibres isomorphic to D, see Proposition 3.10.

Using Theorem 3.1, we can deform continuously families of endomorphisms to families of automorphisms of D, and thus show that these pullback C(X)- subalgebras are actually trivial, see Proposition 3.16. We then complete the proof by applying Elliott’s intertwining argument.

Letη:B →Aandψ:E→Abe∗-homomorphisms. The pullback of these maps is

B⊕_η,ψE={(b, e)∈B⊕E: η(b) =ψ(e)}.

We are going to use pullbacks in the context ofC(X)-algebras.

We need the following three lemmas from [3], reproduced here for the con- venience of the reader.

Lemma 3.2. — Let Abe aC(X)-algebra and letB⊂Abe aC(X)-subalgebra.

Let a∈A and letY be a closed subset of X.

(i) The map x7→ ka(x)k is upper semi-continuous.

(ii) kπ_Y(a)k= max{kπ_x(a)k:x∈Y}

(iii) If a(x)∈πx(B)for all x∈X, thena∈B.

(iv) If δ >0 anda(x)∈δ πx(B)for allx∈X, thena∈δB.

(v) The restriction of π_x :A→A(x) toB induces an isomorphism B(x)∼= πx(B)for all x∈X.

Let B ⊂ A(Y) and E ⊂ A(Z) be C(X)-subalgebras such that X = Y ∪ Z and π_Y^Z_∩Z(E) ⊆ π_Y^Y_∩Z(B). By a basic property of C(X)-algebras, see [3, Lemma 2.4], the pullbackB⊕_πZ

Y∩Z,π^Y_Y∩ZEis isomorphic to theC(X)-subalgebra B⊕Y∩ZE ofAdefined as

B⊕Y∩ZE={a∈A:πY(a)∈B, πZ(a)∈E}.

Lemma 3.3. — The fibres of B⊕Y∩ZE are given by πx(B⊕Y∩ZE) =

(π_x(B),ifx∈X\Z, πx(E),ifx∈Z, and there is an exact sequence of C^∗-algebras

(1) 0 //{b∈B:πY∩Z(b) = 0} //B⊕Y∩ZE π_Z //E //0

Let X, Y, Z and A be as above. Let η : B ,→ A(Y) be a C(Y)-linear

∗-monomorphism and let ψ : E ,→ A(Z) be a C(Z)-linear∗-monomorphism.

Assume that

(2) π^Z_Y∩Z(ψ(E))⊆π^Y_Y∩Z(η(B)).

This gives a mapγ=η_Y⁻¹_∩ZψY∩Z:E(Y∩Z)→B(Y∩Z). To simplify notation we letπstand for bothπ_Y^Y_∩Z andπ_Y^Z_∩Z in the following lemma.

Lemma 3.4. — (a) There are isomorphisms ofC(X)-algebras B⊕π,γπE∼=B⊕πη,πψE∼=η(B)⊕Y∩Zψ(E),

where the second isomorphism is given by the mapχ:B⊕_πη,πψE→Ainduced by the pair (η, ψ). Its componentsχx can be identified with ψx forx∈Z and with ηx forx∈X\Z.

(b) Condition (2)is equivalent toψ(E)⊂π_Z A⊕_Y η(B) .

(c)IfFis a finite subset ofAsuch thatπY(F)⊂εη(B)andπZ(F)⊂εψ(E), then F ⊂εη(B)⊕Y∩Zψ(E) =χ B⊕πη,πψE

.

Notation. — Let ϕ : A → B be a completely positive contractive (c.p.c.) map between C^∗-algebras; let F be a subset of A and δ > 0. We say ϕ is δ-multiplicative on F, or (F, δ)-multiplicative, if kϕ(ab)−ϕ(a)ϕ(b)k < δ for a, b∈ F. We sayϕisδ-bimultiplicative onF, ifkϕ(abc)−ϕ(a)ϕ(b)ϕ(c)k< δ fora, b, c∈ F.

Proposition 3.5. — Let D be a separable unital strongly self-absorbing C^∗- algebra. For any finite set F ⊂ D and ε > 0 there are a finite set G ⊂ D and δ >0 with the following property. For any unital C^∗-algebra A∼=A⊗ D and any two (G, δ)-multiplicative u.c.p. maps ϕ, ψ:D →A there is a unitary u∈U(A)such thatkϕ(d)−uψ(d)u^∗k< εfor all d∈ F. If D is additionally assumed to be K₁-injective, we may chooseu∈U₀(A).

Proof. — Seeking a contradiction let us assume that for some given F and ε there are no G and δ satisfying the conclusion of the Proposition. Let (Gn) be a sequence of increasing finite subsets of Dwhose union is dense inD and let δn = 1/n . Then there exist a sequence of D-stable unital C^∗-algebras (A_n) and two sequences consisting of(G_n, δ_n)-multiplicative u.c.p. maps(ϕ_n) and (ψn) with ϕn, ψn : D → An such that for any n and any un ∈ U(An), kϕn(d)−unψn(d)u^∗_nk ≥εfor some d∈ Fn.

SetB_n=Q

k≥nA_k and letν_n :B_n →B_n+1 be the natural projection. Let us define Φn,Ψn:D →Bn by

Φ_n(d) = (ϕ_n(d), ϕ_n+1(d), . . .) and

Ψ_n(d) = (ψ_n(d), ψ_n+1(d), . . .),

o

for alldinD. One verifies immediately that if we setΦ = lim

−→Φn,Ψ = lim

−→Ψn

andB = lim

−→(Bn, νn), thenΦandΨare unital∗-homomorphisms fromDtoB.

SinceDis nuclear,B⊗ D= lim

−→(Bn⊗ D, νn⊗id_D). By applying [12, Cor. 1.12]

to Φ⊗1_D,Ψ⊗1_D : D → B ⊗ D, we find a unitary V ∈ B ⊗ D such that kΦ(d)⊗1_D−V(Ψ(d)⊗1_D)V^∗k< ε/4for alld∈ F. One then finds a sequence of unitariesVn ∈Bn⊗ Dsuch thatkΦn(d)⊗1_D−Vn(Ψn(d)⊗1_D)V_n^∗k< ε/2 for all d ∈ F and sufficiently large n. Projecting to A_n ⊗ D, we find that if vn∈U(An⊗D)denotes the component ofVn inAn⊗ D, then

(3) kϕn(d)⊗1_D−vn(ψn(d)⊗1_D)v^∗_nk< ε/2

for alld∈ F. By [12, Prop. 1.9] there is a sequence of unital∗-homomorphisms θn : D ⊗ D → D such that lim_n→∞kθn(d⊗1_D)−dk = 0 for all d ∈ D.

Then γ_n : A⊗ D ⊗ D → A⊗ D, γ_n = id_A⊗θ_n is a sequence of unital ∗- homomorphisms such that lim_n→∞kγ_n(a⊗1_D)−ak = 0 for all a ∈ A⊗ D, where A is any unital C*-algebra. Therefore, since An isD-stable, there is a unital ∗-homomorphismγn:An⊗ D →An such thatkγn(a⊗1D)−ak< ε/4 for all a of the form ϕ_n(d) and ψ_n(d) with d ∈ F. From (3) we see that kγn(ϕn(d)⊗1_D)−γn(vn)γn(ψn(d)⊗1_D)γn(vn)^∗k< ε/2and hencekϕn(d)− γn(vn)ψn(d)γn(vn)^∗k< εfor alld∈ F. This contradicts our initial assumption.

In the K1-injective case, to reach a contradiction we assume that the vn

above are inU0(An). By Theorem3.1, we may then assume that V ∈U0(B⊗ D); it is straightforward to show that theV_nmay also be chosen inU₀(B_n⊗D).

But then,vn andγ(vn)are connected to the respective identities as well.

Lemma 3.6. — Let Dbe a separable unital nuclearC^∗-algebra. For any finite set G0 ⊂ D and δ0 > 0 there are a finite set G ⊂ D and δ > 0 with the following property. For any pair of unital C^∗-algebras 1_B ∈A ⊂B, and any (G, δ)-multiplicative u.c.p. map ϕ : D →B satisfying ϕ(G)⊂δ A, there is a (G0, δ0)-multiplicative u.c.p. mapµ:D →A such thatkϕ(d)−µ(d)k< δ0 for any d∈ G₀.

Proof. — This is a known consequence of the Choi-Effros lifting theorem. One proves this by contradiction along the lines of the proof of the previous propo- sition.

Proposition 3.7. — Let D be a separable unital strongly self-absorbing C^∗- algebra. For any finite set F ⊂ D andε >0 there are a finite set G ⊂ D and δ > 0 with the following property. For any pair of unital C^∗-algebras 1B ∈ A⊂B with A∼=A⊗ D, and any (G, δ)-multiplicative u.c.p. mapϕ:D →B satisfying ϕ(G)⊂δ A, there is a unital∗-homomorphism ψ:D →A such that kϕ(d)−ψ(d)k< εfor anyd∈ F.

Proof. — LetG0andδ0 be given by Proposition3.5applied toD,F andε/2.

We may assume that F ⊂ G0 andδ0< ε/2. LetG andδbe given by applying Lemma 3.6 forD, G₀ andδ₀. Suppose now that ϕis as in the statement and let µ : D → A be the perturbation of ϕ yielded by Lemma 3.6. Thus µ is (G0, δ0)-multiplicative andkϕ(d)−µ(d)k< δ0< ε/2for alld∈ F ⊂ G0. Since A ∼= A⊗ D, there is a unital ∗-homomorphism η : D → A. Moreover, by Proposition 3.5, there is a unitary u∈A such that kµ(d)−u η(d)u^∗k < ε/2 for alld∈ F. Let us setψ=u η u^∗:D →A.Then

kϕ(d)−ψ(d)k ≤ kϕ(d)−µ(d)k+kµ(d)−u η(d)u^∗k< ε/2 +ε/2 =ε for alld∈ F.

Proposition 3.7 shows that separable unital strongly self-absorbing C^∗- algebras satisfy a relative perturbation property similar to weak semiprojec- tivity. This is one of the key tools used in the proof of Theorem1.1.

Lemma 3.8. — Let Dbe a separable unital strongly self-absorbing C^∗-algebra.

LetXbe a compact metrizable space and letAbe a unitalD-stableC(X)-algebra with all fibres isomorphic toD. LetF ⊂Abe a finite set and letε >0. For any x∈X there exist a closed neighborhoodU ofxand a unital∗-homomorphism ϕ:D →A(U)such thatπU(F)⊂εϕ(D).

Proof. — Since A is unital and D-stable there is a unital ∗-homomorphism ψ : D → A. By [12, Cor. 1.12], for each x∈ X, the map ψx : D →A(x) is approximately unitarily equivalent to some ∗-isomorphism D →A(x). There- fore there is a unitary v ∈ A(x) such that π_x(F) ⊂_ε v ψ_x(D)v^∗. Using the semiprojectivity of C(T) we lift v to a unitary in u ∈ A(V) for some closed neighborhood V of xand so ψx(F) ⊂ε πx(u)πxψ(D)πx(u)^∗. Using also the upper continuity of the norm inC(X)-algebras, we conclude that there is closed neighborhoodU ⊂V ofxsuch that πU(F)⊂επU(u)πUψ(D)πU(u)^∗.

The following lemma is useful for constructing fibered morphisms, see3.

Lemma 3.9. — LetDbe a separable unital strongly self-absorbingK₁-injective C^∗-algebra. Let(Dj)_j∈J be a finite family ofC^∗-algebras isomorphic toD. Let ε >0 and for eachj∈J letHj ⊂Dj be a finite set. LetGj ⊂Dj and δj >0 be given by Proposition 3.7 applied to D_j, H_j and ε/2. Let X be a compact metrizable space, let(Zj)_j∈J be disjoint nonempty closed subsets ofX and letY be a closed nonempty subset ofX such thatX=Y∪ ∪jZj

. LetA=A(X)be a unital separableD-stableC(X)-algebra and letB(Y)be a be a unital separable D-stableC(Y)-algebra. LetF be a finite subset ofA. Let η:B(Y)→A(Y) be aC(Y)-linear unital∗-monomorphism. Suppose thatϕj :Dj →A(Zj),j ∈J, are unital∗-homomorphisms which satisfy the following conditions:

o

(i) πZ_j(F)⊂_ε/2ϕj(Hj), for allj ∈J, (ii) πY(F)⊂εη(B),

(iii) π^Z_Y^j_∩Z

jϕj(Gj)⊂δ_j π_Y^Y_∩Z

jη(B), for allj∈J.

Then, there are C(Zj)-linear unital ∗-monomorphisms ψj : C(Zj)⊗Dj → A(Zj), satisfying

(4) kϕj(c)−ψ_j(c)k< ε/2, for allc∈ Hj, andj∈J, and such that if we set E=L

jC(Z_j)⊗D_j,Z =∪_jZ_j, andψ:E→A(Z) = L

jA(Zj), ψ=⊕jψj, then π^Z_Y∩Z(ψ(E))⊆π_Y^Y_∩Z(η(B)), πZ(F)⊂εψ(E) and hence

F ⊂εη(B)⊕Y∩Zψ(E) =χ(B⊕πη,πψE),

where χ is the isomorphism induced by the pair (η, ψ). Moreover B⊕πη,πψE isD-stable.

Proof. — Let F ={a₁, . . . , a_r} ⊂ Abe as in the statement. By (i), for each j ∈ J we find {c^(j)₁ , . . . , c^(j)r } ⊆ Hj such that kϕj(c^(j)_i )−πZ_j(ai)k < ε/2 for alli. The C(X)-algebra A⊕Y η(B)⊂A is an extension of separableD-stable C^∗-algebras by (1), and hence it is D-stable by [12, Thm. 4.3], since D is a separable unital strongly self-absorbing K₁-injective C^∗-algebra. In particular πZ_j(A⊕Y η(B))isD-stable for eachj∈J.

By (iii) and Lemmas3.2,3.3we obtain

ϕj(Gj)⊂δ_j πZ_j(A⊕Y η(B)).

Applying Proposition 3.7 we perturb ϕj to a ∗-homomorphism ψj : Dj → πZ_j(A⊕Yη(B))satisfying (4), and hence such thatkϕj(c^(j)_i )−ψj(c^(j)_i )k< ε/2, for alli, j. Therefore

kψj(c^(j)_i )−πZ_j(ai)k ≤ kψj(c^(j)_i )−ϕj(c^(j)_i )k+kϕj(c^(j)_i )−πZ_j(ai)k< ε.

This shows that πZj(F) ⊂ε ψj(Dj). We extend ψj to a C(Zj)-linear

∗-monomorphism ψ_j : C(Z_j) ⊗ D_j → π_Zj(A ⊕_Y η(B)) and then we define E, ψ and Z as in the statement. In this way we obtain that ψ:E→(A⊕Y η(B))(Z)⊂A(Z)satisfies

(5) π_Z(F)⊂εψ(E).

The property ψ(E) ⊂ (A ⊕_Y η(B))(Z) is equivalent to π_Y^Z_∩Z(ψ(E)) ⊂ π^Y_Y∩Z(η(B)) by Lemma 3.4(b). Finally, from (ii), (5) and Lemma 3.4(c) we get F ⊂ε η(B)⊕Y∩Z ψ(E). One verifies the D-stability of B ⊕πη,πψ E by repeating the argument that showed the D-stability ofA⊕_Y η(B).

LetDbe a separable unital strongly self-absorbingK1-injectiveC^∗-algebra.

LetAbe aC(X)-algebra, letF ⊂Abe a finite set and letε >0. An(F, ε,D)- approximation of Ais a finite collection

(6) α={F, ε,{Ui, ϕ_i:D_i→A(U_i),Hi,Gi, δ_i}i∈I},

with the following properties: (Ui)_i∈I is a finite family of closed subsets ofX whose interiors cover X; (D_i)_i∈I are C^∗-algebras isomorphic to D; for each i ∈ I, ϕi : Di → A(Ui) is a unital∗-homomorphism; Hi ⊂ Di is a finite set such thatπU_i(F)⊂_ε/2ϕi(Hi); the finite set Gi ⊂Di and δi >0 are given by Proposition 3.7applied toD_i for the input dataHi andε/2.

It is useful to consider the following operation of restriction. Suppose that Y is a closed subspace ofX and let(Vj)_j∈J be a finite family of closed subsets ofY which refines the family(Y∩Ui)i∈I and such that the interiors of theVj0s form a cover of Y. Letι:J →I be a map such thatV_j⊆Y ∩U_ι(j). Define

ι^∗(α) ={πY(F), ε,{Vj, πVjϕι(j):Dι(j)→A(Vj),Hι(j),Gι(j), δι(j)}j∈J}.

It is obvious that ι^∗(α) is a (πY(F), ε,D)-approximation of A(Y). The op- eration α 7→ ι^∗(α) is useful even in the case Y = X. Indeed, by applying this procedure we can refine the cover of X that appears in a given(F, ε,D)- approximation ofA.

An(F, ε,D)-approximationαis subordinate to an(F⁰, ε⁰,D)-approximation, α⁰={F⁰, ε⁰,{U_i⁰, ϕ_i⁰ :D_i⁰ →A(U_i⁰),H_i⁰,G_i⁰, δ_i⁰}_i⁰_∈I⁰},writtenα≺α⁰, if

(i) F ⊆ F⁰,

(ii) ϕi(Gi)⊆πU_i(F⁰)for alli∈I, and (iii) ε⁰ <min {ε} ∪ {δi, i∈I}

.

Let us note that, with notation as above, we have ι^∗(α) ≺ ι^∗(α⁰) whenever α≺α⁰.

Let us recall some terminology from [3]. A C(Z)-algebra E is called D- elementary if E ∼= C(Z)⊗ D. Let A be a unital C(X)-algebra. A unital n-fibered D-monomorphism (ψ₀, . . . , ψ_n) into A consists of (n+ 1) unital ∗- monomorphisms ofC(X)-algebrasψi:Ei→A(Yi), whereY0, . . . , Ynis a closed cover ofX, eachEi is a D-elementaryC(Yi)-algebra and

π_Y^Yi

i∩Yjψi(Ei)⊆π_Y^Yj

i∩Yjψj(Ej), for all0≤i≤j≤n.

Given an n-fibered morphism intoA we have an associatedcontinuous C(X)- algebra defined as the fibered product (pullback) of the∗-monomorphisms ψi: A(ψ₀, . . . , ψ_n) ={(d₀, . . . d_n) : d_i∈E_i, π_Y^Yi

i∩Yjψ_i(d_i) =π_Y^Yj

i∩Yjψ_j(d_j)for alli, j}

and an induced C(X)-linear monomorphism

η=η_{(ψ0,...,ψn)}:A(ψ₀, . . . , ψ_n)→A⊂

n

M

i=0

A(Y_i),

o

η(d0, . . . dn) = ψ0(d0), . . . , ψn(dn) .

Let us set X_k = Y_k ∪ · · · ∪Y_n. Then (ψ_k, . . . ψ_n) is an (n−k)-fibered D-monomorphism into A(Xk). Let ηk : A(Xk)(ψk, . . . ψn) → A(Xk) be the induced map and set Bk = A(Xk)(ψk, . . . ψn). Let us note that B₀=A(ψ₀, . . . , ψ_n)and that there are naturalC(Xk−1)-isomorphisms

(7) B_k−1∼=B_k⊕πηk,πψk−1E_k−1∼=B_k⊕π,γkπE_k−1,

where π stands for π_Xk∩Yk−1 and γ_k :E_k−1(X_k∩Y_k−1)→ B_k(X_k∩Y_k−1) is defined by (γk)x= (ηk)⁻¹_x (ψ_k−1)x, for allx∈Xk∩Y_k−1.

We say that a unital n-fibered D-monomorphism (ψ0, . . . , ψn) into A is locally extendable if there is another unital n-fibered D-monomorphism (ψ⁰₀, ..., ψ_n⁰)into Asuch thatψ_k⁰ :E⁰_k→A(Y_k⁰),Y_k⁰ is a closed neighborhood of Yk,E⁰_k(Yk) =Ek andπY_kψ_k⁰ =ψk,k= 0, ..., n.

The following proposition is a crucial technical result in this section.

Proposition 3.10. — LetDbe a separable unital strongly self-absorbingK₁- injective C^∗-algebra. Let X be a finite dimensional compact metrizable space and let A be a unital separable continuous C(X)-algebra the fibres of which are isomorphic to D. For any finite set F ⊂ A and any ε > 0 there exist n ≤ dim(X) and an n-fibered unital D-monomorphism (ψ0, . . . , ψn) into A which induces a unital ∗-monomorphismη:A(ψ0, ..., ψn)→A such thatF ⊂ε

η(A(ψ₀, ..., ψ_n)).

Proof. — TheC^∗-algebraAisD-stable by [7]. By Lemma3.8, Proposition3.7 and the compactness ofX, for any finite setF ⊂Aand anyε >0there is an (F, ε,D)-approximation of A. Moreover, for any finite setF ⊂ A, any ε >0 and anyn, there is a sequence {αk: 0≤k≤n}of(Fk, ε_k,D)-approximations ofA such that(F0, ε0) = (F, ε)andαk is subordinated to αk+1:

α0≺α1≺ · · · ≺αn.

Indeed, assume thatαk was constructed. Let us choose a finite setFk+1which containsFk and liftings toAof all the elements inS

i_k∈Ikϕi_k(Gi_k).This choice takes care of the above conditions (i) and (ii). Next we chooseε_k+1 sufficiently small such that (iii) is satisfied. Letαk+1 be an(Fk+1, εk+1,D)-approximation ofAgiven by Lemma3.8and Proposition3.7. Then obviouslyαk≺αk+1. Fix a tower of approximations ofAas above wheren= dim(X).

We may assume that the set X is infinite. By [2, Lemma 3.2], for every open cover V of X there is a finite open cover U which refines V and such that the set U can be partitioned into n+ 1 nonempty subsets consisting of elements with pairwise disjoint closures. Since we can refine simultaneously the covers that appear in a finite family{αk: 0≤k≤n} of approximations while preserving subordination, we may arrange not only that all α_k share the same

cover(Ui)_∈I,but moreover, that the cover(Ui)_i∈I can be partitioned inton+ 1 subsets U0, . . . ,Un consisting of mutually disjoint elements. For definiteness, let us writeU_k ={U_ik:i_k∈I_k}. Now for eachkwe consider the closed subset ofX

Y_k = [

i_k∈Ik

U_ik,

the map ιk:Ik→Iand the(πY_k(Fk), εk,D)-approximation ofA(Yk)induced byαk, which is of the form

ι^∗_k(α_k) ={πYk(Fk), ε,{Uik, ϕ_ik:D_ik →A(U_ik),Hik,Gik, δ_ik}ik∈Ik}, where eachU_ik is nonempty. We have

(8) π_Uik(F_k)⊂_εk/2ϕ_ik(H_ik), by construction. Since α_k ≺α_k+1we also have

(9) Fk⊆ Fk+1,

(10) ϕ_ik(G_ik)⊆π_Uik(F_k+1), for alli_k ∈I_k, (11) ε_k+1<min {ε_k} ∪ {δ_ik, i_k ∈I_k}

.

Set X_k = Y_k ∪ · · · ∪Y_n and E_k = ⊕_ikC(U_ik)⊗D_ik for 0 ≤ k ≤ n. We shall construct a sequence of unitalC(Yk)-linear∗-monomorphisms,ψk :Ek → A(Yk),k=n, ...,0, such that(ψk, . . . , ψn)is an(n−k)-fibered monomorphism into A(X_k). Each map

ψk=⊕i_kψi_k :Ek →A(Yk) =⊕i_kA(Ui_k)

will have components ψi_k : C(Ui_k)⊗Di_k →A(Ui_k) whose restrictions to Di_k

will be perturbations of ϕi_k :Di_k →A(Ui_k), ik ∈Ik. We shall construct the maps ψ_k by induction on decreasingk such that if B_k =A(X_k)(ψ_k, . . . , ψ_n) andηk:Bk→A(Xk)is the map induced by the(n−k)-fibered monomorphism (ψk, . . . , ψn), then

(12) πX_k+1∩U_ik ψi_k(Di_k)

⊂πX_k+1∩U_ik ηk+1(Bk+1)

,∀ik∈Ik, and

(13) πX_k(Fk)⊂ε_k ηk(Bk).

Note that (12) is equivalent to (14) π_Xk+1∩Yk ψ_k(E_k)

⊂π_Xk+1∩Yk η_k+1(B_k+1) .

Let us note thatB_k isD-stable by [7], since each of its fibres areD-stable and X is finite dimensional.

For the first step of induction, k=n, we chooseψn =⊕i_nϕei_n where ϕei_n : C(U_in)⊗D_in→A(U_in)areC(U_in)-linear extensions of the originalϕ_in. Then

o

we setBn=Enandηn=ψn. Assume now thatψn, . . . , ψk+1were constructed.

Next we shall constructψk. Condition (13) formulated fork+ 1becomes (15) πX_k+1(Fk+1)⊂ε_k+1ηk+1(Bk+1).

Sinceεk+1< δi_k, by using (10) and (15) we obtain (16) πX_k+1∩U_ik ϕi_k(Gi_k)

⊂δ_ik πX_k+1∩U_ik ηk+1(Bk+1)

, for allik∈Ik. SinceFk ⊆ Fk+1 andεk+1< εk, condition (15) gives

(17) πX_k+1(Fk)⊂ε_kηk+1(Bk+1).

Conditions (8), (16) and (17) enable us to apply Lemma3.9 and perturbϕi_k

to unital ∗-homomorphisms Di_k →A(Ui_k)whoseC(Ui_k)-linear extensionψi_k: C(U_ik)⊗D_ik →A(U_ik) satisfy (12) and (13). We set ψ_k =⊕_ikψ_ik and this completes the construction ofΨk and hence of(ψ0, . . . , ψn). Condition (13) for k = 0 givesF ⊂ε η0(B0) =η(A(ψ0, . . . , ψn)). Thus (ψ0, . . . , ψn) satisfies the conclusion of the theorem. Finally let us note that it can happen thatX_k=X for some k >0. In this caseF ⊂ε A(ψk, ..., ψn)and for this reason we write n≤dim(X)in the statement of the theorem.

Remark 3.11. — We can arrange that the n-fibered morphism(ψ0, . . . , ψn) from the conclusion of Proposition 3.10 is locally extendable. Fix a metric d for the topology of X. Then we may arrange that there is a closed cover {Y₀⁰, ..., Y_n⁰} of X and a number ` > 0 such that {x : d(x, Y<sub>i</sub><sup>0</sup>) ≤ `} ⊂ Yi for i = 0, ..., n. Indeed, when we choose the finite closed coverU = (Ui)_i∈I of X in the proof of Proposition 3.10 which can be partitioned into n+ 1 subsets U0, . . . ,Unconsisting of mutually disjoints elements, as given by [2, Lemma 3.2], and which refines all the coversU(α0), ...,U(αn)corresponding toα0, ..., αn, we may assume thatU also refines the covers given by the interiors of the elements ofU(α0), ...,U(αn). Since eachUiis compact andIis finite, there is` >0such that if Vi ={x: d(x, Ui)≤ `}, then the coverV = (Vi)_i∈I still refines all of U(α₀), ...,U(α_n)and for eachk= 0, ..., n, the elements ofVk={Vi :U_i∈ Uk}, are still mutually disjoint. We shall use the coverV rather than U in the proof of Proposition3.10and observe that Y_k^{0 def}= S

i_k∈IkUi_k ⊂S

i_k∈IkVi_k =Yk has the desired property. Finally let us note that if we defineψ_i⁰:E_i(Y_i⁰)→A(Y_i⁰) by ψ_i⁰ = π_Y⁰

iψi, then (ψ₀⁰, . . . , ψ⁰_n) is a locally extendable n-fibered unital D- monomorphism into Awhich satisfies the conclusion of Proposition3.10since π_Y⁰

i(F)⊂_εψ_i⁰(E_i(Y_i⁰))for alli= 0, . . . , nandX =Sn i=1Y_i⁰.

For a unital C^∗-algebra Dwe let End(D) denote the space of all unital∗- endomorphisms ofDendowed with the point-norm topology. Throughout the remainder of this section we assume that X is a compact metrizable space

and that Dis a unital, separable,K1-injective and strongly self-absorbingC^∗- algebra.

Lemma 3.12. — Any continuous map α : X× {0} → End(D) extends to a continuous map Φ :X×[0,1]→End(D) such thatΦ_(x,1)= id_D and Φ_(x,t) ∈ Aut (D)for allx∈X andt∈(0,1].

Proof. — Let us identifyαwith a unital∗-homomorphismα:D → C(X)⊗ D.

By Theorem3.1there is a unitary-valued continuous map(0,1]→U(C(X)⊗D), t7→u_t, withu₁= 1_C(X)⊗Dand such that

limt→0kutdu^∗_t−α(d)k= 0, for all d∈D.

Therefore the equation Φ_(x,t)(d) =

(α_x(d), ift= 0, ut(x)d ut(x)^∗,ift∈(0,1],

defines a continuous map Φ :X ×[0,1]→End(D)which extends α, Φ_(x,1)= id_D, and such thatΦ(X×(0,1])⊂Aut(D).

Proposition 3.13. — Let Y be a closed subset of X and let V be a closed neighborhood of Y. Suppose that a map γ:Y →End(D)extends to a contin- uous map α : V → End(D). Then there is a continuous extension η : X → End(D)of γ such thatη(X\Y)⊂Aut(D).

Proof. — First we prove the proposition in the case when V = X. By Lemma 3.12, there exists a continuous map Φ : X×[0,1] → End(D) which extends α, i.e. Φ_(x,0)=αx forx∈X, and such that Φ(X×(0,1])⊂Aut(D).

Letdbe a metric for the topology ofX such thatdiam(X)≤1. The equation η_x = Φ_(x,d(x,Y)) defines a map on X that satisfies the conclusion of the proposition.

Suppose now thatV is a closed neighborhood ofY. By Lemma3.12, there is a continuous map (homotopy)Φ :V ×[0,1]→End (D)such thatΦ_(x,0)= id_D andΦ_(x,1)=αxfor allx∈V. LetΦbbe the extension ofΦtoV×[0,1]∪X×{0}

obtained by setting Φb_(x,0) = id_D for x∈X\V. Let us define λ: X → [0,1]

by λ(x) = d(x, X\V) d(x, X \V) +d(x, Y)⁻¹

and αb : X → End (D) by αbx=Φb_(x,λ(x))and observe thatαbextendsγtoX. The conclusion follows now from the first part of the proof.

Ifϕ:D → C(X)⊗ Dis a∗-homomorphism, we denote by ϕe:C(X)⊗ D → C(X)⊗ DitsC(X)-linear extension.

o

Lemma 3.14. — Let Y, Z be closed subsets of X such that X =Y ∪Z. Let γ : D → C(Y ∩Z)⊗ D be a unital ∗-homomorphism. Assume that there is a closed neighborhood V of Y ∩Z in Y and a unital ∗-homomorphism α : D → C(V)⊗ D such that αx = γx for all x ∈ Y ∩Z. Then the pullback C(Y)⊗ D ⊕_π

Y∩Z,

e^γπY∩Z

C(Z)⊗ D is isomorphic toC(X)⊗ D.

Proof. — By Prop. 3.13there is a unital ∗-homomorphismη:D → C(Y)⊗ D such that η_x = γ_x for x ∈ Y ∩Z and such that η_x ∈ Aut(D) for x ∈ Y \ Z. One checks immediately that the pair(η,e id_C(Z)⊗D) defines a C(X)-linear isomorphism:

C(X)⊗ D=C(Y)⊗ D ⊕πY∩Z,πY∩ZC(Z)⊗ D → C(Y)⊗ D ⊕_π

Y∩Z,

eγπ_Y∩ZC(Z)⊗ D.

Indeed if P denotes the later pullback, there is commutative diagram 0 //C0(Y \Z)⊗ D //

ηe_Y\Z

C(X)⊗ D //

(η,e id)

C(Z)⊗ D //0

0 //C₀(Y \Z)⊗ D //P //C(Z)⊗ D //0 and hence(eη,id_C(Z)⊗D)must be bijective, since ηe_Y\Z is so.

Corollary 3.15. — Let Y,Z andZ⁰ be closed subsets of a compact metriz- able space X such that Z⁰ is a neighborhood of Z and X =Y ∪Z. Let B be a C(Y)-algebra isomorphic to C(Y)⊗ D and let E =C(Z⁰)⊗ D. Suppose that α:E(Y∩Z⁰)→B(Y∩Z⁰)is a unital∗-monomorphism ofC(Y∩Z⁰)-algebras.

If γ=α_Y∩Z, thenB(Y)⊕πY∩Z,γ πY∩ZE(Z)is isomorphic to C(X)⊗ D.

Proof. — This follows from Lemma3.14sinceY∩Z⁰ is a closed neighborhood ofY ∩Z in Y.

Proposition 3.16. — LetAbe a unitalC(X)-algebra. If(ψ0, . . . , ψn)is a lo- cally extendable, unital n-fiberedD-monomorphism into A, thenA(ψ0, . . . , ψn) is isomorphic to C(X)⊗ D.

Proof. — By assumption, there exists another unital n-fibered D-monomor- phism (ψ₀⁰, ..., ψ_n⁰) into A such that ψ_k⁰ : C(Y_k⁰)⊗ D → A(Y_k⁰), Y_k⁰ is a closed neighborhood of Yk, and πY_kψ⁰_k =ψk, k= 0, ..., n. LetXk, Bk, ηk and γk be as in 3. We need to show that B0 is isomorphic to C(X)⊗ D. To this pur- pose we prove by induction on decreasing k that the C(Xk)-algebras B_k are trivial. Indeed Bn =C(Xn)⊗ Dand assuming thatBk is trivial, it follows by Corollary3.15thatBk−1is trivial, since by (7)

B_k−1∼=B_k⊕_{πηk,πψk−1}E_k−1∼=B_k⊕_π,γkπE_k−1, (π=π_Xk∩Yk−1)

and γk : E_k−1(Xk ∩Y_k−1) → Bk(Xk ∩Y_k−1), (γk)x = (ηk)⁻¹_x (ψ_k−1)x, ex- tends to a ∗-monomorphism α : Ek−1(Xk ∩Y_k−1⁰ ) → Bk(Xk ∩Y_k−1⁰ ), αx = (ηk)⁻¹_x (ψ_k−1⁰ )x.

We are now ready to complete the first proof of our main result:

Proof. — (of Theorem1.1): SetB=C(X)⊗D. By Propositions3.16, 3.10and Remark3.11there is a sequence(θk)^∞_k=1consisting of unital∗-monomorphisms ofC(X)-algebrasθ_k:B→Asuch that the sequence(θ_k(B))^∞_k=1exhaustsAin the sense that for any finite subsetGofAand anyδ >0,G ⊂δ θk(B)for some k ≥1. Using Proposition3.7, after passing to a subsequence of (θk)^∞_k=1, we construct a sequence of finite setsF_k ⊂Band a sequence of unitalC(X)-linear

∗-monomorphismsµk :B→B such that

(i) kθk+1µk(a)−θk(a)k<2^−k for alla∈ Fk and allk≥1, (ii) µk(Fk)⊂ Fk+1 for allk≥1,

(iii) S∞

j=k+1 µj−1◦ · · · ◦µk

−1

(Fj)is dense in B andS∞

j=k θj(Fj)is dense in Afor allk≥1.

Arguing as in the proof of [11, Prop. 2.3.2], one verifies that

∆k(a) = lim

j→∞θj◦ µ_j−1◦ · · · ◦µk (a)

defines a sequence of ∗-monomorphisms ∆k:B →Asuch that ∆k+1µk = ∆k

and the induced map∆ : lim−→k(B, µk)→Ais an isomorphism ofC(X)-algebras.

Let us show that lim

−→^k(B, µ_k) is isomorphic to B. To this purpose, in view of Elliott’s intertwining argument, it suffices to show that each map µk is approximately unitarily equivalent to idB. But this follows from either [12, Cor. 1.12] or Theorem 3.1.

4. Proving the main result: The second approach

This section is devoted to another proof of Theorem 1.1, which to some extent follows ideas of [7]. Before turning to the actual proof, we outline our strategy.

We will obtain the isomorphism betweenA and C(X)⊗ D by constructing c.p.c. maps ψ : A→ C(X)⊗ D and ϕ: C(X)⊗ D → A which implement an asymptotic intertwining (in the sense of [1]) ofAandC(X)⊗DasC(X)-algebras (Proposition 4.1).

Since X is finite-dimensional, we may assume that X ⊂ [0,1]ⁿ for some n∈N. We may then regardAas aC(Y)-algebra, whereY = pr₁(X)⊂[0,1]is the image ofXunder the first coordinate projection. By an induction argument it will then be enough to prove that A ∼= C(X)⊗ D under the additional assumption that there is an isomorphism θ_t:A_t∼=C(X_t)⊗ Dfor each t∈Y,

o

where Xt = pr⁻¹({t}) is the (n−1)-dimensional pre-image of t in X. The mapsθt,t∈Y,induce an embeddingθ:A→Q

t∈Y C(Xt)⊗ D.

The first problem then is that we do not know a priori how to choose these isomorphisms in a locally trivial manner. In other words, it is not clear whether we may assume that for any t ∈ Y there are a neighborhood Y^(t) ⊂ Y of t and isomorphisms θs : As ∼= C(Xs)⊗ D, s ∈ Y^(t), such that the induced map θ^(t) : A_Y(t) → Q

s∈Y^(t)C(Xs)⊗ D actually maps A_Y(t) to C(X_Y(t))⊗ D ⊂ Q

s∈Y^(t)C(Xs)⊗ D. However, this problem can be circumvented by the concept of ‘approximate local trivializations’ (θs:As

∼=

−→ C(Xs)⊗ D)_s∈Y(t) of A (introduced in Definition4.3). The crucial property of such an approximate local trivialization is that it maps A_Y(t) to somethingclose toC(X_Y(t))⊗ D ⊂ Q

s∈Y^(t)C(Xs)⊗ D. (This and the other technical properties of approximate local trivializations will be controlled by the c.p.c. maps ζ^(t) : C(X_t)⊗ D → C(X_Y(t))⊗ Dandσ^(t):C(X_Y(t))⊗ D →A_Y(t) of4.3.) Using thatDis strongly self-absorbing and K1-injective, and involving an argument somewhat similar to Lemma 3.8, the existence of such local approximations will be established in Lemma4.5.

Since Y is compact and one-dimensional, we may then pick0≤y₀<· · ·<

yM ≤1and coverY by closed setsY^(y0), . . . , Y^(yM)such that the intersections Y^(yi)∩Y^(yi0)are empty unless|i−i⁰| ≤1. We setI:={i|Y^(yi)∩Y^(yi+1)6=∅};

we may assume that for eachi∈Ithe setY^(yi)∩Y^(yi0)is a closed neighborhood of some ti∈Y.

The next problem is that the local trivializationsθ^(yi)overY^(yi)andθ^(yi+1) over Y^(yi+1) do not necessarily agree over Y^(yi)∩Y^(yi+1) (they need not even be close to each other in any way). However, we may apply Lemma4.5 again to obtain a small interval‹Y^(ti)⊂Y^(yi)∩Y^(yi+1)around eachti (for eachi∈I) and approximate local trivializations θe^(ti) over ‹Y^(ti). Again using that D is strongly self-absorbing andK₁-injective, this will allow us to modify theeθ^(ti)to obtain new approximate local trivializations (overY‹^(ti)) which (approximately) intertwine θ^(yi) and θ^(yi+1) ‘along’ ‹Y^(ti). Here, ‘along’ means that the fibre mapsθ^(ysⁱ⁾are continuously deformed into maps close to the fibre mapsθs^(yi+1), where the parameter sruns through the interval‹Y^(ti).

By superposing theθ^(yi)and the (modified)θe^(ti)we then obtain an approx- imate trivialization of A, which yields the desired approximate intertwining betweenAandC(X)⊗ D.

We start by adapting Elliott’s approximate intertwining argument toC(X)- algebras as follows:

Proposition 4.1. — Let X be a compact metrizable space and let A be a separable unital C(X)-algebra with structure map µ; let D be a strongly self- absorbing C^∗-algebra. Suppose that, for any ε >0 and for arbitrary compact subsets F ⊂A,G1⊂ C(X)andG2⊂ D, there are c.p.c. maps

ψ:A→ C(X)⊗ D and

ϕ:C(X)⊗ D →A satisfying

(i) kϕψ(a)−ak< εfora∈ F

(ii) kϕ(f ⊗1_D)−µ(f)k< εforf ∈ G1

(iii) kψµ(f)−f⊗1_Dk< εforf ∈ G₁ (iv) ϕis ε-multiplicative on (1X⊗id_D)(G2)

(v) ψ isε-multiplicative on F.

Then, AandC(X)⊗ D are isomorphic asC(X)-algebras.

Proof. — From conditions (i)–(v) above in connection with Proposition 3.5 it is straightforward to construct unitaries u_i ∈ C(X)⊗ D and c.p.c. maps ϕi : C(X)⊗ D → A and ψi : A → C(X)⊗ D, i = 1,2, . . ., such that the following diagram is an asymptotic intertwining in the sense of [1, 2.4]:

A

ψ1

$$I

II II II

II = //A

ψ2

$$I

II II II

II = //. . .

A

AA AA AA AA A C(X)⊗ D

ϕ1

::u

uu uu uu

uu ad (1X⊗u1)//C(X)⊗ D ϕ2

::u

uu uu uu uu

u ad (1X⊗u2) //. . .

Note that the inductive limit of the first row is just A, whereas the inductive limit of the second row is isomorphic to C(X)⊗ D. Theϕiandψithen induce

∗-isomorphismsψ¯:A→ C(X)⊗ D andϕ¯:C(X)⊗ D →A which are mutual inverses (cf. the remark after Definition 2.4.1 of [1]). Using (ii), (iii) and the fact that theuicommute withC(X)⊗1_D, one can even assume thatψ◦µ¯ = id_C(X)⊗ 1_D andϕ¯◦(id_C(X)⊗1_D) =µ, which means thatϕ¯andψ¯are isomorphisms of C(X)-algebras.

It will be convenient to note the following easy consequence of Proposition 3.5explicitly.

Lemma 4.2. — Let W be a compact metrizable space and D a strongly self- absorbing K1-injective C^∗-algebra. Then, for any compact subset 1 ∈ F ⊂ C(W)⊗ D and γ > 0 there are a compact subset E(F, γ) ⊂ C(W)⊗ D and 0< δ(F, γ)< γ/2 such that the following holds:

o