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Advances in Mathematics 306 (2017) 1262–1268Contents lists available at ScienceDirectAdvances in Mathematicswww.elsevier.com/locate/aimOn the nuclear dimension of strongly purely inﬁnite C∗-algebrasGábor Szabó 1Fraser Noble Building, Institute of Mathematics, University of Aberdeen,Aberdeen AB24 3UE, Scotland, UKa r t i c l e i n f o a b s t r a c tArticle history:Received 7 October 2015Received in revised form 5 November 2016Accepted 8 November 2016Available online 17 November 2016Communicated by Dan VoiculescuMSC:46L8546L35Keywords:Nuclear dimensionPurely inﬁnite C∗-algebraToms–Winter conjectureWe show that separable, nuclear and strongly purely inﬁnite C∗-algebras have ﬁnite nuclear dimension. In fact, the value is at most three. This exploits a deep structural result of Kirchberg and Rørdam on strongly purely inﬁnite C∗-algebras that are homotopic to zero in an ideal-system preserving way.© 2016 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).0. IntroductionOver the last decade, our understanding of simple, nuclear C∗-algebras has advanced enormously. This has largely been spurred by the interplay between certain topologi-cal and algebraic regularity properties, such as ﬁnite topological dimension, tensorial absorption of suitable strongly self-absorbing C∗-algebras and order completeness of E-mail address: gabor.szabo@abdn.ac.uk.1 Supported by: DFG SFB 878 Groups, Geometry and Actions and EPSRC grant EP/N00874X/1.http://dx.doi.org/10.1016/j.aim.2016.11.0090001-8708/© 2016 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).G. Szabó / Advances in Mathematics 306 (2017) 1262–1268 1263homological invariants, see [6] for an overview. Indeed, this reﬂects the Toms–Winter regularity conjecture [6,20,21], which has spawned an investigation of simple C∗-algebras that has lead to deep results at a fast and furious pace. We refer to [2] and its excellent introduction for an overview of the current state-of-the-art concerning the Toms–Winter regularity conjecture. The most recent highlight of the Elliott classiﬁcation program is certainly the combination of papers [7,5,4,16] (by many hands), which have completed the classiﬁcation of separable, unital, simple C∗-algebras with ﬁnite nuclear dimension and satisfying the UCT. See in particular [16, Section 6] for a comprehensive summary.Apart from applications in classiﬁcation, it now stands to reason (see [17,13,1]) that the interplay between ﬁnite nuclear dimension and Z-stability goes beyond the simple case and should indeed extend to the realm of non-simple, nuclear C∗-algebras that are suﬃciently non-type I. In particular, it is tempting to conjecture that all separable, nuclear and Z-stable C∗-algebras have ﬁnite nuclear dimension. Given that Kirchberg has established a deep classiﬁcation result [8] for separable, nuclear, strongly purely inﬁnite C∗-algebras long ago, a natural step toward this conjecture is to handle the O∞-absorbing case ﬁrst. Note that a separable, nuclear C∗-algebra A is strongly purely inﬁnite if and only if A ∼= A ⊗ O∞ (see [18, 3.2] and [11, 5.11(iii), 8.6]), and if and only if A ∼= A ⊗ Z and A is traceless (see [9, 3.12]).In this note, we take this ﬁrst step towards the above conjecture and verify that all separable, nuclear and O∞-stable C∗-algebras have ﬁnite nuclear dimension, and in fact the nuclear dimension is at most three. In [1], it was shown that a nuclear dimension estimatedim+1nuc(A ⊗ O∞) ≤ 2 dim+1nuc(A ⊗ O2)holds for all C∗-algebras A. As an application, one could see that separable, nuclear, O∞-absorbing C(X)-algebras with simple ﬁbres have ﬁnite nuclear dimension. Extending this technique in the ﬁrst section, we show that in fact, a more general nuclear dimension estimatedim+1nuc(A ⊗ O∞) ≤ 2 dim+1nuc(A ⊗ B)holds for all C∗-algebras A and all non-zero, separable C∗-algebras B with B ∼= B ⊗O∞. This will be an application of Voiculescu’s observation [19] that cones over C∗-algebras are quasidiagonal, combined with a trick from [1] involving positive elements with full spectrum in simple, purely inﬁnite C∗-algebras. In particular, the above estimate implies that O∞ has a dimension-reducing eﬀect on separable, nuclear C∗-algebras upon tensorial stabilization if and only if there is some non-zero, separable C∗-algebra with this property. The deep structural results by Kirchberg and Rørdam [12] on C∗-algebras homotopic to zero in an ideal-system preserving way then show that a certain purely inﬁnite AH algebra constructed by Rørdam in [15] has such a dimension-reducing eﬀect. This yields the main result.1264 G. Szabó / Advances in Mathematics 306 (2017) 1262–12681. 2-Coloured embeddings into purely inﬁnite ultrapowersThe following is contained in the proof of [1, 3.3] and was originally observed by Winter:Lemma 1.1. Let A be a unital, simple and purely inﬁnite C∗-algebra. Let h ∈ A be a positive contraction with full spectrum [0, 1]. For every ε > 0, there is some n ∈ N, numbers 0 ≤ λ0 ≤ λ1 ≤ · · · ≤ λn ≤ 1 and pairwise orthogonal projections p0, . . . , pn ∈ Asuch that h =ε∑nj=0 λjpj. Moreover, all the projections pj may be chosen to represent the trivial class in K0(A).Proof. Since A has real rank zero by [14, 4.1.1], the only thing left to show is that the projections may be chosen to be trivial in K-theory. Let ε > 0. Choose n ∈ N, numbers 0 ≤ λ0 ≤ · · · ≤ λn ≤ 1 and pairwise orthogonal projections q0, . . . , qn ∈ A as above without the K-theory assumption.By [3, Section 1], we can ﬁnd a non-trivial projection q˜n ≤ qn with [q˜n]0 = 0 ∈ K0(A), and set pn = q˜n. Now assume that j ∈ {1, . . . , n} is a number for which we have already constructed projections q˜n, . . . , q˜n−j+1 and pn, . . . , pn−j+1. Apply [3, Section 1] again to ﬁnd a non-trivial projection q˜n−j ≤ qn−j such that [q˜n−j ]0 = −[qn−j+1 − pn−j+1]0 ∈K0(A), and set pn−j = q˜n−j + qn−j+1 − pn−j+1. After n steps, we have constructed pairwise orthogonal projections p0, . . . , pn, all being trivial in K0(A), such thatn∑j=0λn(qn − pn) = λ0(q0 − q˜0) +n∑j=1(λj − λj−1)(qj − q˜j).Since h has full spectrum, we necessarily have λ0 ≤ ε and λj − λj−1 ≤ ε for all j ∈ {1, . . . , n}. It follows that the above sum has norm at most ε. Hence h =2ε∑nj=0 λnpn. As a consequence, we get:Lemma 1.2. Let A be a unital, simple and purely inﬁnite C∗-algebra. Let h0, h1 ∈ A be two positive contractions with full spectrum [0, 1]. For every ε > 0, there is a unitary u ∈ A with h1 =ε uh0u∗.Proof. Let ε > 0. By 1.1, we can ﬁnd numbers k, n ∈ N, numbers 0 ≤ λ0 ≤ · · · ≤ λk ≤ 1and 0 ≤ μ0 ≤ · · · ≤ μn ≤ 1, pairwise orthogonal projections p0, . . . , pk and q0, . . . , qnin A that are all trivial in K0(A), such thatk∑λipi =ε h0 andn∑μjqj =ε h1.i=0 j=0G. Szabó / Advances in Mathematics 306 (2017) 1262–1268 1265By breaking up projections, if necessary, we may assume that k = n and that |λi−μi| ≤ εfor all i = 0, . . . , n. By breaking up the projections p0, q0, if necessary, we may moreover assume that ∑ni=0 pi = 1 =∑ni=0 qi.It follows from [3, Section 1] that there exist partial isometries v0, . . . , vn, v ∈ A withv∗v = 1A −n∑j=0pj , vv∗ = 1A −n∑j=0qjandv∗j vj = pj , vjv∗j = qj for all j = 0, . . . , n.The element u = v + v0 + · · · + vn ∈ A then deﬁnes a unitary. We haveuh0u∗ =ε u( n∑j=0λjpj)u∗ =n∑j=0λjupju∗=n∑j=0λjvj · v∗j vj · v∗j =n∑j=0λjqj=εn∑j=0μjqj =ε h1.This ﬁnishes the proof. Lemma 1.3. Let A be a unital, simple and purely inﬁnite C∗-algebra. Let ω ∈ βN \ N be a free ultraﬁlter. Let h0, h1 ∈ Aω be two positive contractions with full spectrum [0, 1]. Then h0 and h1 are unitarily equivalent.Proof. Since Aω is again simple and purely inﬁnite, this follows directly from 1.2 and a standard reindexation argument. The main technical observation of this section is the following:Proposition 1.4. Let A be a separable C∗-algebra with a positive element e ∈ A of norm 1. Let ω ∈ βN \ N be a free ultraﬁlter. Then there exist two c.p.c. order zero maps ϕ0, ϕ1 :A → (O∞)ω with ϕ0(e) + ϕ1(e) = 1.Proof. By a result of Voiculescu [19], the cone over A is quasidiagonal. Hence we can ﬁnd a ∗-monomorphism κ : C0((0, 1], A) → Qω, where Q is the universal UHF algebra. By choosing some (necessarily non-unital) embedding μ : Q → O∞,2 the composition 2 For example, this can be a composition of some non-unital embedding O2 ↪−→ O∞ and some unital embedding Q ↪−→ O2, which exists by [10, 2.8].1266 G. Szabó / Advances in Mathematics 306 (2017) 1262–1268ψ0 = μω ◦ κ : C0((0, 1], A) → (O∞)ω is a ∗-monomorphism. The positive element h =ψ0(id[0,1] ⊗e) ∈ (O∞)ω then has full spectrum [0, 1].Applying 1.3, we ﬁnd a unitary u ∈ (O∞)ω with uhu∗ = 1 −h. Then ψ1 = Ad(u) ◦ψ0 :C0((0, 1], A) → (O∞)ω is also a ∗-monomorphism. This gives rise to two c.p.c. order zero maps ϕ0, ϕ1 : A → (O∞)ω via ϕi(x) = ψi(id[0,1] ⊗x) for all x ∈ A and i = 0, 1. We then haveϕ0(e) + ϕ1(e) = h + uhu∗ = 1,as required. As a consequence, we get a general nuclear dimension estimate for O∞-stable C∗-algebras that generalizes [1, 3.3]:Corollary 1.5. Let A be a C∗-algebra and B a non-zero, separable, O∞-stable C∗-algebra. Thendim+1nuc(A ⊗ O∞) ≤ 2 dim+1nuc(A ⊗ B).Proof. Let d = dimnuc(A ⊗ B). Since B is assumed to be O∞-absorbing, we may also assume without loss of generality that A is O∞-absorbing. We may certainly also assume that A is nuclear, as there is otherwise nothing to show. By [17, 2.5, 2.6], it suﬃces to ﬁnd an estimate for the nuclear dimension of the embedding (idA ⊗1)ω : A → (A ⊗ O∞)ω.So let F⊂⊂A and ε > 0 be arbitrary. As B is non-zero, we can ﬁnd some positive contraction e ∈ B of norm 1. Apply 1.4 to ﬁnd two c.p.c. order zero maps ϕ0, ϕ1 : B →(O∞)ω with ϕ0(e) + ϕ1(e) = 1. By the deﬁnition of d, we can ﬁnd a ﬁnite-dimensional C∗-algebra F , a c.p.c. map μ : A ⊗ B → F and c.p.c. order zero maps κ(0), . . . , κ(d) :F → A ⊗ B withx ⊗ e =εd∑j=0κ(j) ◦ μ(x ⊗ e) for all x ∈ F.Deﬁne the c.p.c. map Ψ : A → F via Ψ(x) = μ(x ⊗e). Moreover, deﬁne for all i = 0, 1 and j = 0, . . . , d the c.p.c. order zero map Φ(i,j) : F → (A ⊗O∞)ω via Φ(i,j) = (idA ⊗ϕi) ◦κ(j). Then we have∑i=0,1d∑j=0Φ(i,j) ◦ Ψ(x) =∑i=0,1(idA ⊗ϕi)⎛⎝d∑j=0κ(j) ◦ μ(x ⊗ e)⎞⎠=2ε∑x ⊗ ϕi(e) = x ⊗ 1i=0,1G. Szabó / Advances in Mathematics 306 (2017) 1262–1268 1267for all x ∈ F . This yields a 2(d +1)-coloured decomposition of the map (idA ⊗1)ω : A →(A ⊗ O∞)ω through ﬁnite-dimensional C∗-algebras and ﬁnishes the proof. 2. O∞-stable C∗-algebras have ﬁnite nuclear dimensionIn this section, we prove our main result. This will be a consequence of the estimate 1.5, together with a deep structural result of Kirchberg and Rørdam from [12]. Let us now recall some of these results:Deﬁnition 2.1 (see [15]). Let {tn}n∈N ⊂ [0, 1) be a dense sequence. For every n, deﬁne the ∗-homomorphismϕn : C0([0, 1), M2n) → C0([0, 1), M2n+1)viaϕn(f)(t) = diag(f(t), f(max(t, tn)))for all t ∈ [0, 1).Set A[0,1] = lim−→{C([0, 1), M2n), ϕn}.In [12], Kirchberg and Rørdam have shown a deep structural result for separable, nuclear, strongly purely inﬁnite C∗-algebras that are homotopic to zero in an ideal-system preserving way. An interesting consequence of this observation is that tensoring a given nuclear C∗-algebra with A[0,1] yields an AH0 algebra with low dimension:Theorem 2.2 (see [12, 5.12, 6.2]). For every non-zero, separable, nuclear C∗-algebra B, the C∗-algebra B ⊗ A[0,1] is an AH0 algebra of topological dimension one. In particular, the decomposition rank of B ⊗ A[0,1] is one.Combining this structural result with our nuclear dimension estimate from the previ-ous section, we obtain our main result:Theorem 2.3. Let A be a separable, nuclear C∗-algebra. Then the nuclear dimension of A ⊗ O∞ is at most three.Proof. The C∗-algebra A[0,1] is O∞-stable by [15, 5.3]. So combining 1.5 and 2.2, we getdim+1nuc(A ⊗ O∞) ≤ 2 dim+1nuc(A ⊗ A[0,1]) ≤ 4.This shows our claim. Remark 2.4. At ﬁrst glance, it might appear that 2.2 (and therefore 2.3) depends on Kirchberg’s classiﬁcation [8] of non-simple, strongly purely inﬁnite C∗-algebras. However, 1268 G. Szabó / Advances in Mathematics 306 (2017) 1262–1268Kirchberg’s classiﬁcation theorem is only used in [12] to deduce that A[0,1] (or more gen-erally any separable, nuclear, strongly purely inﬁnite C∗-algebras homotopic to zero in an ideal system-preserving way) is in fact O2-absorbing. While this is an important ingre-dient in verifying the AH0-structure of the C∗-algebras in question, one can refrain from appealing to classiﬁcation by weakening the statement in 2.2 to dr(B ⊗ A[0,1] ⊗ O2) = 1for all non-zero, separable, nuclear C∗-algebras B. In particular, this weaker statement can be proved exclusively with the methods from [12], without classiﬁcation, and is suﬃcient to deduce our main result 2.3.References[1] S. Barlak, D. Enders, H. Matui, G. Szabó, W. Winter, The Rokhlin property vs. Rokhlin dimension 1 on unital Kirchberg algebras, J. Noncommut. Geom. 9 (4) (2015) 1383–1393.[2] J. Bosa, N. Brown, Y. Sato, A. Tikuisis, S. White, W. Winter, Covering dimension of C∗-algebras and 2-coloured classiﬁcation, Mem. Amer. Math. Soc. (2016), in press, http://arxiv.org/abs/1506.03974.[3] J. Cuntz, K-theory for certain C∗-algebras, Ann. of Math. 113 (1981) 181–197.[4] G.A. Elliott, G. Gong, H. Lin, Z. Niu, On the classiﬁcation of simple C∗-algebras with ﬁnite de-composition rank, II, http://arxiv.org/abs/1507.03437, 2015.[5] G.A. Elliott, Z. Niu, On the classiﬁcation of simple unital C∗-algebras with ﬁnite decomposition rank, in: Operator Algebras and Their Applications: A Tribute to Richard V. Kadison, in: Contem-porary Mathematics, Amer. Math. Soc., Providence, RI, 2016.[6] G.A. Elliott, A.S. Toms, Regularity properties in the classiﬁcation program for separable amenable C∗-algebras, Bull. Amer. Math. Soc. 45 (2008) 229–245.[7] G. Gong, H. Lin, Z. Niu, Classiﬁcation of simple amenable Z-stable C∗-algebras, http://arxiv.org/abs/1501.00135, 2015.[8] E. 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Rørdam, A purely inﬁnite AH-algebra and an application to AF-embeddability, Israel J. Math. 141 (2004) 61–82.[16] A. Tikuisis, S. White, W. Winter, Quasidiagonality of nuclear C∗-algebras, Ann. of Math. (2016), in press, http://arxiv.org/abs/1509.08318.[17] A. Tikuisis, W. Winter, Decomposition rank of Z-stable C∗-algebras, Anal. PDE 7 (2014) 673–700.[18] A.S. Toms, W. Winter, Strongly self-absorbing C∗-algebras, Trans. Amer. Math. Soc. 359 (8) (2007) 3999–4029.[19] D. Voiculescu, A note on quasi-diagonal C∗-algebras and homotopy, Duke Math. J. 62 (2) (1991) 267–271.[20] W. Winter, Decomposition rank and Z-stability, Invent. Math. 179 (2) (2010) 229–301.[21] W. Winter, J. Zacharias, The nuclear dimension of C∗-algebras, Adv. Math. 224 (2) (2010) 461–498.

On the nuclear dimension of strongly purely infinite C*-algebras

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On the nuclear dimension of strongly purely infinite C*-algebras

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