The Cuntz-Toeplitz algebras have nuclear dimension one

We prove that unital extensions of Kirchberg algebras by separable stable AF algebras have nuclear dimension one. The title follows.


Introduction
A compact space X has covering dimension at most n if every open cover can be refined and then (n + 1)-coloured so that sets with the same colour are disjoint ( [28]). Nuclear dimension is a generalisation of covering dimension to C * -algebras, introduced by Winter and the fifteenth-named author ( [40]), based on coloured finite-dimensional approximations, and has played a prominent role in the classification programme for amenable C * -algebras over the last decade (see the expository articles [14,38]).
For simple C * -algebras, the theory of the nuclear dimension, and its relationship with classification is essentially complete. The only possible values of the nuclear dimension are 0, 1 and ∞ and these values can be seen in the internal structure of the algebra ( [40,26,27,33,2,7,6]). Moreover, (in the presence of the universal coefficient theorem), the simple separable unital and nuclear C * -algebras accessible to classification by K-theory and traces are precisely those of nuclear dimension 0 and 1 ([19, 29, 18, 13, 34]). Accordingly, attention is now turning to the non-simple setting. How do we compute the nuclear dimension of non-simple C * -algebras, and is there a connection between finiteness of the nuclear dimension and classification?
In this paper, we consider C * -algebras arising from extensions 0 → J → A → B → 0. A key advantage of nuclear dimension over its predecessor (the decomposition rank, defined in [23]) is its compatibility Research partially supported by EPSRC grant EP/R025061/1. CB was partially supported by the Alexander von Humboldt-Foundation. MF was supported by GAČR project 19-05271Y, RVO: 67985840, and a scheme 2 grant of the London Mathematical Society. MW was partially supported by EPSRC grant EP/R013691/1. 1 with extensions: one can directly obtain an (n + m)-coloured approximation for A by combining an n-coloured approximation for J with an m-coloured approximation for B ( [40,Proposition 2.9]). If the extension is a direct sum (and more generally for quasidiagonal extensions), one can do better because B and J can be approximated in an orthogonal fashion ( [23,40,32]). This results in an approximation for A with only max(n, m) colours, which is also the fewest colours possible. For general extensions, we do not know where in the range between max(n, m) and n + m the correct number of colours for A falls.
For a decade, the nuclear dimension of the Toeplitz algebra T was an open problem because the extension 0 → K → T → C(S 1 ) → 0 is not quasidiagonal and moreover T is the primordial example of an extension with infinite decomposition rank, despite the ideal and quotient having finite decomposition rank. It was known that two colours were necessary in a nuclear dimension approximation of T and three suffice because two colours are needed to approximate C(S 1 ) and one for K ( [40]). Recently, Brake and Winter found an elegant way to re-use one colour to get a 2-coloured approximation to T , thereby proving that the Toeplitz algebra has nuclear dimension one ( [4]).
Motivated by the work of Brake and Winter, we achieve a similar colour reduction for a large class of unital extensions of Kirchberg algebras (simple, separable, amenable, purely infinite C * -algebras). Theorem 1. Let 0 → J → A → B → 0 be a unital extension where J is a separable stable AF algebra and B is a Kirchberg algebra. Then A has nuclear dimension one.
The class of C * -algebras covered by Theorem 1 includes the Cuntz-Toeplitz algebras T n (for n ≥ 2), defined as the universal C * -algebras generated by isometries s 1 , . . . , s n with pairwise orthogonal ranges. Indeed, the ideal of T n generated by 1 − n i=1 s i s * i is isomorphic to the compacts K and the quotient by this ideal is the Cuntz algebra O n ( [9]). The Toeplitz algebra can be viewed as the Cuntz-Toeplitz algebra T 1 , but in this case the quotient is the commutative C * -algebra C(S 1 ) rather than a Kirchberg algebra. It is therefore not surprising that our proof requires a different set of tools from Brake and Winter while sharing some structural similarities.
Although not needed in our proof, Gabe and Ruiz have shown that the C * -algebras covered by Theorem 1 (and that satisfy the universal coefficient theorem) are classified by their 6-term exact sequences in K-theory together with order and the position of the unit in the K 0groups ( [17]). Together with our result, this is further evidence that the relationship between nuclear dimension at most one and K-theoretic classification persists outside the simple setting. This connection has long been evident for AF algebras, which have nuclear dimension zero and are classified by K 0 ( [11]), and is also present for the class of non-simple algebras classified in [12], which are built as limits of suitable one-dimensional building blocks and so have nuclear dimension at most 1. Furthermore, amenable O ∞ -stable algebras have recently been shown to have nuclear dimension one ( [3]). This last class was classified by ideal related KK-theory by Kirchberg ([20]; see [15] for a new approach to this result).
We now discuss the proof of our theorem. The basic approximation has the form where π : A → B is the quotient map, µ : B → A is a c.p.c. splitting and (h i ) i is a quasicentral approximate unit for J relative to A such that h i+1 h i = h i . Note that the first term on the right of (1) is in the ideal, the last term factors through the quotient, and the first and last terms are orthogonal by the hypothesis on (h i ) i . If the middle term were to vanish, the extension would be quasidiagonal, and the nuclear dimension of A would just be the maximum of that of the ideal and the quotient.
In [4], Brake and Winter handle the non-vanishing middle term in the case of the Toeplitz algebra by using Lin's theorem on almost normal elements of matrix algebras ( [24]) to prove a lifting theorem for order zero maps from C(S 1 ) into ultraproducts of finite-dimensional C * -algebras. A similar strategy is not available when B is simple and infinite as there are no non-trivial order zero maps from B into finitedimensional C * -algebras.
The new idea in our proof is inspired by a technique that has been utilised in the study of Z-stable and O ∞ -stable algebras ( [2,7,3]). Breaking the last term in half, for b ∈ B, we set colour and get a 2-coloured approximation for A. 1 The classification results we need are in the non-simple setting, in the spirit of Kirchberg's classification of strongly purely infinite nuclear C * -algebras ( [19,20]). Precisely, we need a classification theorem for full * -homomorphisms from cones over Kirchberg algebras into ultrapowers, which we deduce from Gabe's classification theorem for full, nuclear, strongly O ∞ -stable * -homomorphisms ( [15]). We record this result as Proposition 6. The role of the positive contractions k i with spectrum [0, 1] is to ensure that the * -homomorphisms C 0 (0, 1] ⊗ B → A ω coming from the approximately order zero maps θ (0) i and θ (1) i are full, so that Gabe's theorem applies. A criterion for checking fullness of such maps is established in Proposition 4, and the construction of the auxiliary maps using quasidiagonality of cones and order zero lifting is isolated in Proposition 5. With the more technical aspects at hand, we can then give a streamlined proof of the main result at the end of the paper.

Nuclear Dimension of extensions of Kirchberg algebras
We begin with some preliminaries. Two positive elements a and b in a C * -algebra A are called orthogonal if ab = 0. A completely positive map ϕ : A → B between two C * -algebras is said to be order zero if it sends orthogonal elements to orthogonal elements. The oneto-one correspondence between c.p.c. order zero maps A → B and *homomorphisms C 0 (0, 1]⊗A → B established in [39, Corollary 3.1] will be used repeatedly in this paper. The * -homomorphism Φ : C 0 (0, 1] ⊗ A → B associated to a c.p.c. order zero map φ : A → B is characterised by We now recall the definition of nuclear dimension of a C * -algebra and more generally the nuclear dimension of a * -homomorphism. . A * -homomorphism of C *algebras π : A → B is said to have nuclear dimension at most n, denoted dim nuc (π) ≤ n, if for any finite subset F ⊆ A and any ε > 0, there exists a finite dimensional C * -algebra F and completely positive maps such that ψ is contractive and φ is n-decomposable, in the sense that we can write order zero for all i, and for every a ∈ F , The nuclear dimension of a C * -algebra A, denoted dim nuc (A), is then defined as the nuclear dimension of the identity homomorphism id A .
Next, we review ultraproducts and set out our notation conventions. Fix a free ultrafilter ω ∈ βN \ N for the entirety of this paper. Let (A i ) i be a sequence of C * -algebras. The bounded sequence algebra is the C * -algebra and the ultraproduct is the quotient C * -algebra When (A i ) i is a constant sequence, i.e. A i = A for all i, we denote the ultraproduct by A ω , and there is a canonical embedding A → A ω via constant sequences. As is standard, when working with ultraproducts we usually use representative sequences in the bounded sequence algebra to denote elements in the ultraproduct; this should be understood as standing for the image under the quotient. We write σ(a) for the spectrum of the element a. The following standard proposition will be used later to calculate the spectrum of elements in ultraproducts.
Proof. Without loss of generality, assume each A i is unital. We prove the contrapositive. Suppose λ ∈ σ(k). Then λ1 − k is invertible, An element a in a C * -algebra A is called full if it generates all of A as a closed, two-sided ideal. A * -homomorphism π : A → B is called full if π(a) is full in B for every non-zero a ∈ A. The next proposition allows us to test for fullness in the setting of Theorem 1.

Proposition 4. Consider a sequence of unital extensions
be the induced extension of the corresponding ultraproducts.
c. order zero map and Φ : C 0 (0, 1] ⊗ D → i→ω A i is the corresponding * -homomorphism out of the cone, then the following are equivalent: Proof. (i): The condition π ω (a) = 0 is clearly necessary for fullness of a, we prove that it is sufficient. Suppose that a ∈ i→ω A i satisfies π ω (a) = 0, and let I be the ideal generated by a. As 0 = π ω (a) * π ω (a) ∈ I and π ω (a * a) = π ω (a) * π ω (a), we may assume without loss of generality that a is positive.
Since each B i is simple and purely infinite, so is the ultraproduct which may be taken to be self-adjoint.
As the J i are stable, we can write J i = K ⊗ C i for some C * -algebras C i . Perturbing each d i in such a way that the equivalence class of (d i ) i is unaffected, we may assume that d i ∈ M n i ⊗C i for some n i ∈ N. Writing p n i for the unit in M n i , we may make a further perturbation and assume there exist positive contractions c i ∈ C i such that (p n i ⊗c i )d i = d i (these come from approximate units in each C i ). Let Therefore, I = i→ω A i , i.e. a is full.
With this in place, we now give the existence result needed for our theorem.
with η i c.p.c. and ξ i c.p.c. order zero, such that the induced map φ : B → i→ω A i is c.p.c. order zero and the corresponding * -homomorphism Φ : Proof. Since C 0 (0, 1] ⊗ B is quasidiagonal by [36], there exists a sequence of c.p.c. maps ψ i : C 0 (0, 1]⊗B → M n i (C) that is approximately multiplicative and approximately isometric.
Since the sequence of maps (ψ i ) i is approximately multiplicative, the sequence of maps (η i ) i is approximately order zero. Let k ∈ O ∞ be a positive contraction with spectrum [0, 1] and let α i : M n i (C) → O ∞ be a (non-unital) injective * -homomorphism. Define a c.p.c. order zero mapξ i : [22], and so we may identify the codomain ofξ i with B. The resulting c.p.c. order zero map M n i (C) → B can be lifted to a c.p.c. order zero map Set φ i to be the composition ξ i • η i . Since the sequence of maps (η i ) i is approximately order zero, and each map ξ i is order zero, the map φ : B → i→ω A i induced by (φ i ) i is c.p.c. order zero.
Our final ingredient is a complementary uniqueness result showing that, after amplifying the codomain by a matrix algebra, the *homomorphisms constructed in the previous proposition are unique up to unitary equivalence. To achieve this we use a special case of Gabe's new approach to O ∞ -stable classification ( [15]). For ease of notation, given a * -homomorphism φ : A → B, we write φ ⊕ 0 for the * -homomorphism A → M 2 (B) given by (13) a → φ(a) 0 0 0 . Proposition 6. Let (A i ) i be a sequence of separable unital C * -algebras and B be a Kirchberg algebra. Suppose Φ, Ψ : C 0 (0, 1] ⊗ B → i→ω A i are full * -homomorphisms. Then there exists a unitary u ∈ i→ω M 2 (A i ) such that Proof. Since B is nuclear and O ∞ -stable, it is automatic that Φ and Ψ are nuclear and strongly O ∞ -stable. Indeed, * -homomorphisms with O ∞ -stable domains are strongly O ∞ -stable by [16,Proposition 3.18] Using [16,Proposition 3.10], we deduce that Φ ⊕ 0 and Ψ ⊕ 0 are asymptotically unitarily equivalent and hence approximately unitarily equivalent. As B is separable, a standard reindexing argument or application of Kirchberg's ε-test (see [21,Lemma A.1]) enables one to pass from approximate unitary equivalence to unitary equivalence, i.e. there exists a unitary u ∈ i→ω M 2 (A i ) such that for all x ∈ C 0 (0, 1] ⊗ B, Ψ(x) ⊕ 0 = u(Φ(x) ⊕ 0)u * , as desired.
We are now in a position to prove Theorem 1.
Proof of Theorem 1. Let π : A → B denote the quotient map of the extension. Fix a u.c.p. splitting µ : B → A, which can be obtained from the Choi-Effros lifting theorem [8].
we can fix an approximately central sequence of positive contractions We will start by constructing a quasicentral approximate unit (h i ) i for the extension with some additional properties. Firstly, we shall need that each h i has finite spectrum and h i+1 h i = h i for all i ∈ N. Secondly, we need to ensure that the sequences of c.p.c. maps α i : for a ∈ A and b ∈ B, induce order zero maps α : A → A ω , θ (0) , θ (1) : B → A ω , and that the sum α + θ (0) • π + θ (1) • π coincides with the canonical embedding A → A ω as constant sequences.
We now perform the construction of (h i ) i . Write J as an increasing union of finite-dimensional C * -algebras (F n ) n , and let e n be the unit of F n . Then (e n ) n will be an approximate unit for J with e n+1 e n = e n for all n ∈ N. Therefore, there exists a quasicentral approximate unit (h i ) i in the convex hull of the sequence (e n ) n such that h i+1 h i = h i for all i ∈ N (see [1,II.4.3.2]). By construction, each h i lies in a finitedimensional subalgebra, so has finite spectrum. Since B is separable and (k i ) i has already been fixed, replacing (h i ) i with a subsequence, we may further assume that With this choice of (h i ) i we can now verify that the maps α, θ (0) and θ (1) are order zero. For α this is immediate from quasicentrality, but for θ (0) and θ (1) we shall need (16) and write µ ω : B ω → A ω for the c.p.c. map induced by µ. Let b, b ′ ∈ B be positive elements such that bb ′ = 0. Then we have Now as (h i ) i is quasicentral for A and is chosen so the properties (16) hold, all the terms in the right hand side of (17) are zero. Applying a similar argument to θ (1) shows that it too is order zero. Moreover, for all a ∈ A, we have using (16) and that k ∈ B ω ∩ B ′ and h,h ∈ A ω ∩ A ′ .
Since h i+1 h i = h i , the maps α i and θ (1) i have orthogonal ranges. To ensure that this orthogonality is respected by subsequent constructions, we introduce hereditary subalgebras By restricting the codomains, we may view α i as a map A → D i and θ (1) i as a map B → A i , and we note that A i and D i are orthogonal. We also have an induced short exact sequence 0 → J i → A i → B → 0, where the quotient map is the restriction of π. Since h i+1 has finite spectrum, A i is unital. Moreover, J i is stable by [31,Corollary 2.3(iii)].
Since A is separable and each D i is approximately finite dimensional (by virtue of being a hereditary subalgebra of J; see [10, Theorem 3.1]), there exist finite-dimensional subalgebras G i ⊆ D i and conditional expectations E i : D i → G i such that E i (α i (a)) − α i (a) → 0 for all a ∈ A. We now define completely positive approximations i (π(a)), η (1) i (π(a)), E i (α i (a))), χ i (x, y, z) = Ad(u i )(ξ i (x) ⊕ 0) + Ad(v i )(ξ (1) i (x) ⊕ 0) + z ⊕ 0. (24) It follows from (18) and (22) that χ i (ψ i (a)) → a along the ultrafilter ω. The restriction of χ i to F (1) i → A i is order zero, v i ∈ M 2 (A i ), and D i is orthogonal to A i . Therefore, id A ⊕ 0 has nuclear dimension at most 1.
As A is a hereditary subalgebra of M 2 (A), [3, Proposition 1.6] implies that id A , as the co-restriction of id A ⊕ 0 to A, has nuclear dimension at most 1. Hence, dim nuc (A) ≤ 1. On the other hand, [40,Proposition 2.9] implies that the nuclear dimension of A is bounded below by the nuclear dimension of B. As we assume that B is a Kirchberg algebra, it has nuclear dimension 1 by [2, Theorem G]. Therefore, we conclude dim nuc (A) = 1.
Corollary 7. The Cuntz-Toeplitz algebras T n have nuclear dimension one.