Rokhlin Dimension and C*-Dynamics

We develop the concept of Rokhlin dimension for integer and for finite group actions on C*-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{Z}$$\end{document}-stable C*-algebras, where Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{Z}$$\end{document} denotes the Jiang–Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{Z}}$$\end{document} -stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}^{d}}$$\end{document}-actions on compact metrizable and finite dimensional spaces.


Introduction
An action of a countable discrete group G on a (possibly noncommutative) space X has a Rokhlin property if there are systems of subsets of X , indexed by large subsets of G, (i) which are (approximately) pairwise disjoint, (ii) which (approximately) cover X and (iii) on which the group action is (approximately) compatible with translation on the index set.
In a sense, Rokhlin properties create a 'reflection' of the acting group in the underlying space which often allows for a dynamic viewpoint on properties of the latter. The concrete interpretation of conditions (i), (ii) and (iii) above allows for a certain amount of freedom, and may vary with the applications one has in mind. For a general group it is usually not so clear how to synchronize indexing by large subsets and compatibility of the group action with translation on these subsets. The situation is much less ambiguous if G is finitely generated, in particular for Z (or Z d ) or for finite groups.
In this paper we will be mostly interested in the noncommutative case, more specifically in actions on C * -algebras. The study of such actions, and their associated crossed products, has always been a central theme in operator algebras, beginning with Murray and von Neumann's group measure space construction which associates a von Neumann algebra to an action of a group on a measure space. Since then the crossed product construction has been generalized to actions on noncommutative von Neumann and C *algebras, providing an inexhaustible source of highly nontrivial examples. It combines in an intricate way dynamical properties of the action and properties of the algebra of coefficients.
A basic, yet important early result in the dynamics of group actions on measure spaces is the Rokhlin Lemma saying that a measure preserving aperiodic (i.e., almost everywhere non-periodic) action of Z can be approximated by cyclic shifts, in the sense that there exists a finite partition of the space (Rokhlin tower) which is almost cyclically permuted. It can be reformulated as a result about strongly outer automorphisms of commutative von Neumann algebras involving partitions of unity by orthogonal projections (towers of projections) and this formulation has led to the various versions of C * -algebraic Rokhlin properties. (We recall the version which is now most commonly used in Definition 2.1.) As it turns out, there are many important examples of actions of finite groups and Z possessing Rokhlin properties. See, for instance, [9,16] and references therein for actions of Z, and [4,10,11,23], for the finite group case. In fact, Rokhlin properties, especially for the single automorphism case, are quite prevalent, and indeed as a byproduct we establish in this paper that they are generic for automorphisms of unital C * -algebras which absorb a UHF algebra of infinite type. For related work establishing genericity of Rokhlin type conditions, see [25].
Most currently used C * -algebraic Rokhlin properties involve towers consisting of projections. However, unlike in the case of von Neumann algebras, requiring the existence of projections poses severe restrictions on the coefficient algebra. In particular, by lack of projections, automorphisms of the Jiang-Su algebra Z (the smallest possible C * -algebraic analogue of the hyperfinite I I 1 factor R) or automorphisms arising from homeomorphisms of connected spaces will not satisfy most of the current definitions of the Rokhlin property. Moreover, even if there are sufficiently many projections (e.g. in the real rank zero case) Rokhlin properties do not always hold. In the C * -setting they become regularity properties of actions which may be regarded as a strong form of outerness (producing mostly simple C * -algebras).
The main purpose of this paper is to develop generalizations of Rokhlin properties motivated by the idea of covering dimension. Roughly speaking, instead of requiring towers consisting of projections, we allow for partitions of unity involving several towers of positive elements, where elements from different towers are no longer required to be orthogonal, but the number of different towers is restricted. (A further requirement, which we need for some results and which is automatic in many important cases, is that these towers commute.) This is analogous to the definition of covering dimension using partitions of unity with functions allowing for controlled overlaps. We like to think of the number of towers as the coloring number, or dimension, of the cover of the dynamical system, the usual Rokhlin property corresponding to the zero dimensional case. It turns out that this generalization provides a much more flexible concept, which applies to a much wider range of examples than the currently used Rokhlin properties. We can even show that in the Z-stable case finite Rokhlin dimension is topologically generic. A very nice and explicit example for which we can show finiteness of our Rokhlin dimension and where the usual Rohklin property clearly fails are irrational rotation automorphisms of C(T). In the initial version of this paper we showed that that automorphisms corresponding to minimal homeomorphisms of compact spaces of finite covering dimension always have finite Rokhlin dimension. This result might be regarded as a topological version of the classical Rokhlin Lemma. After this result appeared on the arXiv, Szabó in [29] generalized it to the case of free and aperiodic Z d -actions. Szabó's argument is based on results by Gutman and by Lindenstrauss; it is technically easier than our original approach, which we do not include in the present version (it is still available in arXiv:1209.1618v1).
Covering dimension for topological spaces was generalized to the context of C *algebras in [15,35], in which the related concepts of decomposition rank and nuclear dimension were introduced. Those are based on placing a uniform bound on the decomposability of c.p. approximations of the identity map (cf. [5]). Simple C * -algebras with finite decomposition rank or finite nuclear dimension have been shown to have significant regularity properties, in particular they are Z-stable [32,33]. Whilst these classes have good permanence properties, in particular those with finite nuclear dimension, there are still no general results known involving crossed products. Our original motivation was to find bounds on the decomposition rank and/or nuclear dimension of crossed products, which is indeed possible for Rokhlin actions of finite groups. Moreover it turns out that similar bounds can be established for automorphisms satisfying higher dimensional Rokhlin properties, at least for the nuclear dimension. Combining these results with the finiteness of Rokhlin dimension for minimal systems on compact spaces of finite covering dimension we obtain finiteness of the nuclear dimension of crossed products of such systems. This provides a different proof of the nuclear dimension result in [31]. Moreover, the generalization obtained by Szabó (see [29]) together with the results of [17] have now led to the classification of crossed products associated to Cantor free minimal Z d -actions with compact space of ergodic measures, see [34] (also see [21] for the uniquely ergodic case). In fact, the respective result holds for actions on compact metrizable and finite dimensional spaces, provided that traces are separated by projections.
In this paper we focus our attention on unital C * -algebras. The non-unital case, as well as further properties of finite Rokhlin dimension for finite group actions, will be explored in a forthcoming paper of the first named author and Phillips, [7].
We note that other generalizations of the Rokhlin property have been considered which are based on the idea of leaving out a remainder which is small in a tracial sense (see [1,6,22,24,27]).
We also wish to point out that finite Rokhlin dimension can be interpreted as a dynamic version of Kirchberg's covering number, introduced in [13] (also cf. [14]).
Hiroki Matui and Yasuhiko Sato have informed us that they have a proof that all strongly outer automorphisms of separable, simple, unital, monotracial and Z-stable C *algebras have Rokhlin dimension at most 1, thus partially generalizing our Theorem 3.4. Their argument is based on the techniques developed in [20].
As a further variation of our Rokhlin dimension one might in addition ask the Rokhlin systems to approximate the underlying algebra; one might also compare the necessary number of colors with the possible lengths of the Rokhlin towers of such approximations (instead of asking for a global bound on the number of colors). This idea leads to a version of 'slow dimension growth' which in turn is closely related to the notion of mean dimension introduced by Lindenstrauss and Weiss in [18].
Finally, we mention that Bartels, Lück and Reich have studied a concept closely related to ours in [2]; this idea plays a key role in their proof of the Farrell-Jones conjecture for hyperbolic groups in [3].
These connections will be studied in subsequent work. The organization of the paper is as follows. We first consider the case of finite group actions in Sect. 1, which in some respects is simpler than the case of actions of Z and serves as a good motivation displaying some essential ideas. In Sect. 2 we introduce various versions of Rokhlin dimension for Z-actions and study their interplay. In Sect. 3 we show that finite Rohklin dimension is generic in the Z-stable case. A similar argument shows that the Rokhlin property is generic in the UHF-stable case. In Sect. 4 we show that finite nuclear dimension is preserved under forming crossed products by automorphisms with finite Rokhlin dimension, and in Sect. 5 we show that Z-stability is preserved if we assume a further commutativity condition in the definition of Rokhlin dimension (which holds in all cases we are aware of); the latter result holds for actions of finite groups and of Z.
In Sect. 6 we consider minimal Z-actions on finite dimensional compact Hausdorff spaces, which always turn out to have finite Rokhlin dimension. We give a direct argument of this fact for irrational rotations and compare it to Szabó's generalization to free and aperiodic Z d -actions.
In an appendix we recall the crossed product construction as well as the notions of nuclear dimension and of Z-stability for the reader's convenience.
We would like to thank the referee for carefully reading the manuscript and for a number of helpful comments.

Rokhlin Dimension for Actions of Finite Groups
In this section we define the concept of Rokhlin dimension for finite group actions on C * -algebras and show how this notion can be used to prove permanence of finite nuclear dimension under taking crossed products. Definition 1.1. Let G be a finite group, A a unital C * -algebra and α : G → Aut(A) an action of G on A. We say that α has Rokhlin dimension d if d is the least integer such that the following holds: for any ε > 0 and every finite subset F ⊂ A there are positive contractions d l=0 hg < ε for all l ∈ {0, . . . , d} and g ∈ G, g , a < ε for all l ∈ {0, . . . , d}, g ∈ G and a ∈ F.
We will refer to the family ( f (l) g ) l=0,...,d; g∈G as a multiple tower and to the d +1 families g ) g∈G as towers of color 0, . . . , d, respectively. Notice that if d = 0 then we obtain the usual Rokhlin property for actions of finite groups (see [10,11]).
A possible variant of the above definition would be to weaken (2) to If the towers can be chosen to commute or approximately commute it's not hard to see that if (2 ) is true then, upon replacing f (l) g f −1/2 and ε by a sufficiently small fraction of ε, we can also obtain (2), so that (2) and (2 ) are equivalent in this case. Remark 1.2. By approximating the function t → t 1/2 by polynomials on [0, 1] we may assume in Definition 1.1 above that for h, g ∈ G and a ∈ F we have for all l ∈ {0, . . . , d}. Also, we may replace the finite set F in the definition by a norm compact set.

Theorem 1.3. Let G be a finite group,
A a unital C * -algebra with finite decomposition rank and α : G → Aut(A) be an action with Rokhlin dimension d. Then the crossed product A α G has finite decomposition rank as well-in fact, The respective statement is true for nuclear dimension.
Proof. We will only give a proof for the decomposition rank; the proof for nuclear dimension is similar. Set n = |G| and recall that dr(M n (A)) = dr(A) and denote this number by N . Let F ⊂ A α G ⊂ M n (A) finite and ε > 0 be given; we may assume that F consists of contractions. Choose an N -decomposable c.p.c. approximation (for M n (A)) consisting of F = F (0) ⊕ · · · ⊕ F (N ) , ψ : M n (A) → F, ϕ : F → M n (A), ϕ = ϕ (0) + · · · + ϕ (N ) for F to within ε/6. We will construct c.p. approximations (for A α G) as below, and then use stability of order zero maps to replace the maps ρϕ by ones that are decomposable into order zero maps: I  I  I  I  I  I  I  I  I where B F is the closed unit ball in F, and let K ⊆ A be the collection of all matrix entries of elements in K 0 . Notice that K 0 , K are compact subsets of M n (A) and A respectively. Using stability of order zero maps, we choose η > 0 such that η < ε/6 and such that if θ : F → A α G is a c.p.c. map which satisfies θ(a)θ (b) ≤ (n + 4)n 3 η for all positive orthogonal contractions a, b ∈ F, then there exists a c.p.c. order zero map θ : F → A α G such that θ − θ < ε/(6(d + 1)(N + 1)). Finally we also require η to be such that (2(d + 1)n + 1)nη < ε/6 which will be used later.
Every element x ∈ F ⊆ A α G can be written in the form x = g∈G x(g)u g , where x(g) ∈ A are uniquely determined contractions. LetF be the finite setF For the particular η > 0 we have chosen and the compact set K ∪F, choose a multiple tower f (l) g l=0,...,d; g∈G such that (1)-(4) of Definition 1.1 hold true for ε replaced by η and a ∈ K ∪F and also f (l) (as a row vector whose entries are indexed by the elements of G). Thus the sum ρ = d l=0 ρ (l) is also c.p. For a ∈ K ∪F, g ∈ G, l ∈ {0, . . . , d} we have Thus ρ(au g ) − au g < (2(d + 1)n + 1)η by assumption (note that a ≤ 1) and so g , therefore ρ < 1 + ε/6 and so after possible normalization (dividing by its norm) we have ρ = 1 and ρ(x . . , d}. One checks that for all a, b ∈ K ∪F, and g 1 , Consequently, for any j = 0, . . . , d, we have that for all a, b ∈ B F . In particular, if a, b ∈ B F are positive and orthogonal, then ρ (l) (ϕ ( j) (a))ρ (l) (ϕ ( j) (b)) ≤ (n + 4)n 3 η, and ρ (l) • ϕ ( j) is a c.p.c. map. By the choice of η, there are c.p.c. order zero maps If σ > 1, then we normalize σ ; since σ < 1+ε/6, normalizing will again introduce an error of at most ε/6.
Remark 1.4. We will see in Theorem 5.9 that Z-stability also passes to crossed products by finite group actions with finite Rokhlin dimension, provided one in addition assumes that the Rokhlin towers commute.

Rokhlin Dimension for Actions of Z
In this section we introduce the concept of Rokhlin dimension for Z-actions on C *algebras. There is a certain amount of freedom in the definition, just as for the original Rokhlin property. For the latter, several versions have been studied (cf. [9,16] and the references therein). Correspondingly, we introduce several versions of Rokhlin dimension and compare these. As it turns out, in general one such version is finite if and only if the other is-roughly, they bound one another by a factor 2. Since we are mostly interested in when the Rokhlin dimension is finite (as opposed to the precise value), for our purposes it will not be too important which variant we choose. The different versions of Rokhlin dimension are designed so that they are just the zero dimensional instances of the various Rokhlin properties; in this regard, our concept bridges the gaps between the latter. For our exposition we will give slight preference to the version generalizing the definition below.
Let us first recall the Rokhlin property which is now most commonly used.
Definition 2.1. Let A be a unital C * -algebra, and let α : Z → Aut(A) be an action of the integers. We say that α has the Rokhlin property if for any finite set F ⊆ A, any ε > 0 and any p ∈ N there are projections [e r, j , a] < ε for all r, j and all a ∈ F.
We shall refer to the projections in the definition as Rokhlin projections and to the set of those projections as Rokhlin towers.

Remark 2.2. (i) It follows immediately from the definition that the Rokhlin projec-
tions are pairwise orthogonal, and that α 1 (e 0, p−1 + e 1, p ) − (e 0,0 + e 1,0 ) < ε. In the generalization we consider, we will need to add such assumptions explicitly. (ii) It could be that the e 1, j 's (or the e 0, j 's) are all zero. If this can always be arranged then we say that α has the single tower Rokhlin property. The single tower Rokhlin property is stronger than the Rokhlin property-it implies immediately that the identity can be decomposed into a sum of projections which are equivalent via an automorphism, which amounts to a nontrivial divisibility requirement. For instance, there can be no automorphism on the UHF algebra of type 2 ∞ which has the single tower Rokhlin property as stated here, whereas we shall see that the Rokhlin property is generic for automorphisms of this algebra (for this C * -algebra, there are many automorphisms with single Rokhlin towers if one restricts to towers of a height which is a power of 2-in fact, basically the same proof shows that such automorphisms are generic).
We now turn to our main definition.

Definition 2.3. Let
A be a unital C * -algebra and α : Z → Aut(A) an action of the integers.
(a) We say α has Rokhlin dimension d, if d is the least natural number such that the following holds: For any finite subset F ⊂ A, any p ∈ N and any ε > 0, there are positive elements r, j , a] < ε for all r, j, l and a ∈ F. If there is no such d then we say that α has infinite Rokhlin dimension and write dim Rok (c) If one can obtain this property such that for all l one of the towers ( f 1, j ) j vanishes, then we shall say that α has Rokhlin dimension d with single towers, if α satisfes the conditions above except that (2) is replaced by the weaker condition We shall refer to each sequence ( f (l) r, j ) j∈{0,..., p−1+r } as a tower, to the length of the sequence as the height of the tower, and to the pair of towers ( f (l) r, j ) j∈{0,..., p−1+r } , r ∈ {0, 1}, as a double tower. The superscript l will sometimes be referred to as the color of the tower. (iii) It is clear that in general dim Rok (A, α) ≤ dim c Rok (A, α). However in all cases in which we establish that a Z-action has Rokhlin dimension d the towers are commuting, and we do not know if there are any examples for which there is a strict inequality. The additional assumption concerning commuting towers is needed as a hypothesis for some results but not others. (iv) Similarly, dim Rok (A, α) ≤ dim Rok (A, α) in general and it is straightforward to verify that dim Rok (A, α) = 0 if and only if α has the Rokhlin property (if and only if dim c Rok (A, α) = 0). (v) As is the case for the regular Rokhlin property, one could have defined a notion of a Rokhlin dimension using more than two towers for each color. That is, require that for any given N one could find numbers R N , p 0 , . . . , p R N such that p r > N and positive elements f r, j , l = 0, 1, . . . , d, r = 0, 1, . . . , R N , for all r and j = 0, 1, . . . , p r such that the analogous definition holds (of course, if one allows for arbitrary tower lengths and numbers of towers, then ε in conditions (1)-(5) has to be replaced by something like ε/( p 0 + · · · + p R N )). However, this can be reduced to the case of two towers in a standard way, which we quickly outline. For a given p, we find an N such that any n > N can be written as a( p − 1) + bp for some given non-negative integers a, b. Using this N , one finds f (l) r, j as above. We fix a r , b r such that a r ( p − 1) + b r p = p r + 1, and then definẽ r,a r ( p−1)+mp+ j and this reduces the case of an arbitrary number of towers of each color, each in an arbitrary length of at least N to our notion of Rokhlin dimension. (vi) The definition of Rokhlin dimension (and Rokhlin dimension with commuting towers) is equivalent to the following apriori weaker formulation: instead of asking for double towers of arbitrary height p, one can require that there are such towers for arbitrarily large p (independently of ε) but not necessarily for any p. This can be shown by using a rearrangement of tower terms as explained above, and is useful in certain examples (e.g. if it is easier to produce towers which are of prime height, or of height which is a power of 2). We note, however, that in the single tower case this indeed gives a different formulation. For example [cf. Remark 2.2(ii)], the infinite tensor shift on the CAR algebra has the Rokhlin property and one can obtain single towers of height 2 n , but if one requires heights which are not powers of 2 then one needs two towers (this follows from K -theoretic considerations).

Notation 2.5.
We like to think of finite Rokhlin dimension as a property of a group action rather than of an automorphism, even if the group is cyclic. However, when it is understood that the group is Z we will sometimes slightly abuse notation and write dim Rok (A, α) even when α ∈ Aut(A) denotes the automorphism inducing the Z-action rather than the action itself. In this situation we also say that the automorphism has finite Rokhlin dimension.
There is a useful and straightforward reformulation of the definition of Rokhlin dimension which will be used later on: Proposition 2.6. Let A be a unital C * -algebra and α : Z → Aut(A) a Z-action.

(a) Then dim Rok (A, α) ≤ d if and only if the following holds:
For every p ∈ N, F ⊂ A finite and ε > 0 there are c.p.c. maps ≤ ε e e whenever e, e ∈ C p ⊕ C p+1 are orthogonal positive elements, is the cyclic shift on each summand and e j , f j denote the canonical generators of C p and C p+1 , respectively,  When the underlying C * -algebra is commutative, then clearly each version of Rokhlin dimension agrees with its commuting tower counterpart -but in this case, we even have that dim Rok agrees with dim Rok : Proposition 2.7. Let (T, h) be a dynamical system with T compact and metrizable; let α be the induced action on C(T ). Then, By approximating f uniformly by polynomials we can find satisfying 2.6(1), (2 ), (3), (4) and (5). Note that by 2.6(2 ) and since the ζ (l) are c.p.c. we have From 2.6(3), (4) we see that By the choice of δ and by (2) this entails Define mapsζ for l = 0, . . . , d; it is clear that theζ (l) are c.p.c. and that d l=0ζ whence 2.6(2) holds for theζ (l) (in fact, with 0 in place of ε). By 2.6(1), (2 ) and since C(T ) is commutative, we have whenever e, e ∈ C p ⊕ C p+1 are positive orthogonal elements; it follows that 2.6(1) holds for theζ (l) andε in place of ζ (l) and ε. 2.6(5) holds automatically for theζ (l) (again with 0 in place of ε) since F and the range of theζ (l) are in C(T ).
Finally, we estimate for each l ∈ {0, . . . , d} and 0 ≤ e ∈ C p ⊕ C p+1 We have now shown (1). Literally the same proof also yields the single tower version.
Rokhlin dimension with single towers in general does not coincide with Rokhlin dimension. That is already true in the case of the regular Rokhlin property: there are automorphisms with the Rokhlin property but in which requiring double-towers is essential. One way to see this is via K -theoretic considerations [cf. Remarks 2.2(ii) and 2.4(vi)]: for example, if α is an approximately inner automorphism which has a Rokhlin tower then all Rokhlin projections are equivalent in K 0 , and therefore if α has the Rokhlin property with single towers then K 0 (A) must be divisible-and it is well known that there are Rokhlin automorphisms on C * -algebras which do not have divisible K 0 , e.g. the CAR algebra. However, if we allow for higher Rokhlin dimensions, then we see that the distinction is relatively mild: Proposition 2.8. Let A be a unital C * -algebra and α : Z → Aut(A) an action. Then and Proof. As in 2.7, we only show (4); the same argument will also yield (5). The left hand inequality is trivial. As for the right hand one, we may assume that α has finite Rokhlin dimension. Let f (l) r, j , l = 0, 1, . . . , d, r = 0, 1, j = 0, 1, . . . , p −1+r be Rokhlin tower elements for α, with respect to a given ε > 0, height p and finite set F ⊆ A. Let us first assume that p is large enough so that ε > 1 p−1/2 . We shall construct single towers (g (l) j ) for l = 0, 1, . . . , 2d, the same finite set F and with 2ε instead of ε. Define decay factors μ r : {0, 1, . . . , p − 1 + r } → R by where we denote μ 0 ( p) = 0. We use this decay factor to merge each double tower into two separate (cyclic) towers. Schematically, we use the decay factor μ r which is large in the middle and small in the ends to bunch up the tower terms as follows and we use 1 − μ r , which is small at the center, to bunch up the tower terms like this (treat p as even for the purpose of the diagram): For l = 0, 1, . . . , d, and j = 0, 1, . . . , p, define tower elements as follows, where in the formulas below f (l) r, j for j > p − 1 + r is meant to be read modulo p + r : 1, j j = p/2 + 1, . . . , p.
One now readily checks that g (l) j satisfy the requirements. If p is not sufficiently large, we choose a k such that ε > Remark 2.9. It follows from the preceding proof that in fact all the single towers can be chosen to be of the same height (rather than allowing for some to be of height p and others of height p + 1).

Genericity in the Z-Stable Case
We recall that the automorphism group of a C * -algebra A can be endowed with the topology of pointwise convergence generated by the sets where α runs over all α ∈ Aut(A) and a runs over all elements of A. If A is separable, then this topology is Polish.
In this section we want to show that the property of having finite Rokhlin dimension is generic, more precisely the automorphisms a Z-stable C * -algebra which have Rokhlin dimension at most 1 form a dense G δ set in the topology of pointwise norm convergence.
Recall that for p ∈ N the dimension drop interval Z p, p+1 is given as and that the Jiang-Su algebra Z is an inductive limit of such dimension drop intervals. (2) f 0 + · · · + f p−1 + g 0 + · · · + g p = 1.
(3) u f j u * = f j+1 and ug j u * = g j+1 for all j, where addition is modulo p or p + 1, respectively.
Proof. Let h ∈ C([0, 1]) be defined by: -6 r r r r r r Let v x ∈ M p be a continuous path of unitary matrices such that v x is a fixed cyclic permutation matrix of order p for all x ∈ [0, 2/3], and v 1 = 1. Similarly, let w x ∈ M p+1 be a continuous path of unitaries, which is a fixed cyclic permutation matrix of order p + 1 for all x ∈ [1/3, 1] and w 0 = 1. Let u gives us elements as required. A be a separable C  *  -algebra. Let a 1 , . . . , a n , b 1 , . . . , b n

Lemma 3.2. Let
and let p(x, x * ) be a noncommutative polynomial. For any ε > p(c, c * ) there are a δ > 0 and a finite subset W ⊆ A such that the following holds.
For any C * -algebra B and any two * -homomorphisms ϕ, ψ : Proof. Suppose not. Let W m be an increasing sequence of finite sets with dense union, then for any m > 0 we can find a C * -algebra B m and a pair of homomorphisms ϕ m , ψ m : for any x, y, and therefore we have a well-defined homomorphism But then it follows that We recall the following simple application of functional calculus: Lemma 3.3. For any ε > 0 there is a δ > 0 such that for any unital C * -algebra A, if a ∈ A satisfies max{ 1 − a * a , 1 − aa * } < δ then there is a unitary u ∈ A such that u − a < ε.

Theorem 3.4. Let A be a unital separable Z-stable C * -algebra, then a dense G δ set of automorphisms of A has Rokhlin dimension ≤ 1 with commuting towers (and in fact with single Rokhlin towers). In particular, Rokhlin dimension at most 1 is generic.
If furthermore A ∼ = A ⊗ D for D a UHF algebra of infinite type then the Rokhlin property (i.e. Rokhlin dimension 0) is generic.
Proof. We shall give a proof for the Z-stable case. The UHF-stable case is similar and will be omitted. Given p ∈ N, a finite set F ⊂ A and ε > 0, we shall say that an automorphism α ∈ Aut(A) has the ( p, F, ε)-approximate 1-dimensional Rokhlin property if there are positive elements f 0 , . . . , f p−1 , g 0 , . . . , g p ∈ A such that (1) f i f j < ε for all i = j, and g i g j < ε for all i = j, (2) f 0 + · · · + f p−1 + g 0 + · · · + g p − 1 < ε, (3) α( f j ) − f j+1 < ε and α(g j ) − g j+1 < ε for all j, where addition is modulo p or p + 1, respectively, (4) [ f j , a] < ε and [g j , a] < ε for all j and all a ∈ F, We denote by V p,F,ε the set of all automorphisms α ∈ Aut(A) which satisfy the ( p, F, ε)-approximate 1-dimensional Rokhlin property. It is clear that V p,F,ε is open. Now, if we choose an increasing sequence of finite sets F n ⊂ A with dense union, then any α ∈ p∈N n∈N V p,F n , 1 n has Rokhlin dimension at most 1. It thus suffices to prove that V p,F,ε is dense for any p, F, ε. We thus fix an automorphism α and a triple ( p, F, ε). For any finite set F 0 ⊂ A and γ > 0 we need to find β ∈ V p,F,ε such that for all a ∈ F 0 . Since enlarging F simply imposes additional conditions, we may assume without loss of generality that F ⊇ F 0 , for notational convenience, and we furthermore assume that all elements of F have norm at most 1. We may furthermore assume that γ < ε.
Fix γ /10 > η > 0 as in Lemma 3.3 such that for any unital C * -algebra A, if a ∈ A satisfies max{ 1 − a * a , 1 − aa * } < η then there is a unitary u ∈ A such that u − a < γ /10.
Choose a unitary u ∈ Z p, p+1 ⊂ Z and elements f 0 , . . . , contractions a 1 , . . . , a n , b 1 and furthermore n i, j=1 Choose W and δ as in Lemma 3.2 for the four expressions above and with 2η, 3η instead of ε. We may assume without loss of generality that δ < η/3n and that W  contains a 1 , . . . , a n , b 1 , . . . , b n . Choose a unital embedding ϕ : Now, for any x ∈ F we have that Notice that for all a ∈ F ∪ α(F), we have that [a, v i ] ≤ 2nδ < η, i = 1, 2.
Let z = u * 1 u 2 . We note that if x ∈ G then Note furthermore that if a ∈ F ∪ α(F) then [z, a] < 2η + 4γ /10. We now define So, for all a ∈ F, we have that It remains to check that β ∈ V p,F,ε . We claim that the elements ϕ( f i ), ϕ(g i ) have the required properties. The facts that they are orthogonal to each other, add up to 1, ε-commute with the elements of F and with each other follow immediately from the requirements we imposed on ϕ. It thus remains to check that they are almost permuted by β. Indeed, Remark 3.5. A modification of this argument can be used to show the following. Let D be a strongly self-absorbing C * -algebra, and let A be a unital D-stable C * -algebra, then for a dense G δ set of automorphisms of A we have that A α Z is D-stable as well. This can be done by showing that for any given ε > 0 and finite sets F ⊂ A, G ⊂ D, the set of automorphisms α such that there exists unital homomorphism ϕ : D → A such that [ϕ(x), a] < ε for all x ∈ G, a ∈ F and furthermore α(ϕ(x)) − ϕ(x) < ε for all x ∈ G is a dense open set. We omit the proof.

Permanence of Finite Nuclear Dimension
In this section we show that forming a crossed product by an automorphism with finite Rokhlin dimension preserves finiteness of nuclear dimension. Theorem 4.1. Let A be a separable unital C * -algebra of finite nuclear dimension and α ∈ Aut(A) an automorphism with finite Rokhlin dimension. Then A α Z has finite nuclear dimension with Proof. Denote the nuclear dimension of A by N and denote d = 2 dim Rok (α) + 1. By Proposition 2.8 and the subsequent remark, we have that dim s Rok (α) ≤ d and furthermore the towers can all be chosen to be of the same height. (The fact that the towers can be chosen to be of the same height is not important for the proof, but simplifies notation a bit.) Let F ⊆ A α Z be a given finite set. We need to construct a piecewise contractive c.p. approximation (F, , ) which is [2(d + 1)(N + 1) − 1]-decomposable and of tolerance ε on F.
Recall that A α Z → B( 2 (Z, H )) is generated by a copy of A and a unitary u acting on 2 (Z, H ), cf. the appendix. We may and shall assume that F consists of contractions all lying in the algebraic crossed product, i.e. there exists q ∈ N such that all elements of F are of the form x = q i=−q x(i)u i , where x(i) ∈ A are coefficients uniquely determined by x. LetF ⊆ A be the finite set of all such coefficients of the elements in F. Let p ∈ N be a positive integer (much) larger than q to be specified later. We shall furthermore require that p is even to slightly simplify notation.
Let Q be the projection onto the subspace 2 Define decay factors to be the c.p.c. map given by Notice that where a ∈ A and |m| ≤ q. In fact these estimates hold for all bounded operator matrices x = [x i, j ] of width q (i.e. x i, j = 0 implies |i − j| ≤ q). It follows that for |m| ≤ q (F) and let (F, ψ, ϕ) be a piecewise contractive N -decomposable c.p. approximation, where ϕ (i) = ϕ|F (i) is an order zero contraction for every i and ϕ(ψ(x)) − x < ε for all x ∈F 1 . Let ψ (i) denote the ith component of ψ and consider Let now B F be the closed unit ball in F (which is norm compact) and define the norm .., p−1 be single Rokhlin towers with respect to the given ε and the compact set K (we note that one may obviously replace the finite set in the definition by a compact set). For j > p − 1 we understand f (l) j to mean f (l) j (mod p) . Define ρ (l) 0 , ρ (l) 1 : for a constant c > 0 which depends only on ε and for all a, b ∈ K . The same applies for ρ (l) 1 . This is easily verified by approximating the square root function by a polynomial vanishing at 0, and we omit the calculation. Similarly, we see that there is a constant c such that for all a ∈F 1 , and a similar estimate holds for ρ (l) Let x = au m for |m| ≤ q, a ∈F. For m ≥ 0 denote y = Qx Q = p−1 i=m e i,i−m ⊗ α −i (a); the case m < 0 is handled analogously and we skip the details. Then we have It follows that ρ (l) Thus, given η > 0 we could first choose p large enough and then ε > 0 small enough so that for all a ∈ F and all |m| ≤ q. Consider the maps With a, m and η as above, we have that To summarize, we constructed the following maps.
We notice that ρ is approximately order zero, i.e. for any a, b ∈ M p (F (i) ) of norm at most 1, we have that By stability of order zero maps, given η > 0 one can choose ε > 0 small enough so that for this choice of ε, there exists a contractive order zero map ζ : Putting together such approximating order zero maps ζ for the various maps ρ is a (2(d + 1)(N + 1) − 1)-decomposable approximation for F to within Cη for some constant C which depends only on d and N . This shows that dim nuc (A α Z) ≤ 2(d + 1)(N + 1) − 1, as required.

Permanence of Z-Stability
The purpose of this section is to show that if A is a unital separable Z-stable C *algebra, then so is any crossed product by an automorphism which has finite Rokhlin dimension with commuting towers. As shown above, this property is generic and we do not know whether there is an automorphism which has finite Rokhlin dimension but without commuting towers. Our prefered criterion for Z-stability will be Proposition A.5 of the appendix. We begin with a simple preliminary lemma, whose proof is straightforward and left to the reader.
The following lemma is also easy to verify and we leave its proof to the reader. We denote by A + the unitization of A such that A is a codimension 1 ideal in A + (that is, if A is unital then A + ∼ = A ⊕ C). A, B be two C * -algebras. Let D be the kernel of the canonical map A + ⊗ max B + → C, and let us denote by ι A , ι B the following homomorphisms.

Lemma 5.2. Let
Then D has the following universal property. For any C * -algebra E and any two homomorphisms γ A : A → E, γ B : B → E with commuting images there is a unique homomorphism θ : D → E such that the following diagram commutes.
We observe that It is easy to construct a unital homomorphism π : M n → M(D (k) n ), and it is readily seen that θ(x)( t) = h( t) · θ(x)( t) satisfies the desired properties.
The following is a simple modification of Lemma 2.4 from [8]. While that lemma can be generalized to actions of non-discrete groups in a straightforward way (as was done in [8]), we state it here just for actions of discrete groups to avoid notation which we do not need in this paper. It should be pointed out that our argument is essentially the same as that of [19,Theorem 4.9]. We reproduce the full proof here for the convenience of the reader.
We denote A ∞ = ∞ (N, A)/c 0 (N, A), with A embedded as the subalgebra of constant sequences in A ∞ . We use a similar idea to one which was used in Proposition 2.2 from [30].
If α : G → Aut(A) is an action then we have naturally induced actions of G on A ∞ and A ∞ ∩ A . We denote those actions byᾱ.
Lemma 5.4. Let A, B be a unital separable C * -algebras. Let G be a discrete countable group with an action α : G → Aut(A). Suppose that B n is a sequence of unital nuclear subalgebras (with a common unit) of B with dense union. Suppose that for any n, any finite subset F ⊆ B n , any ε > 0 and any finite set G 0 ⊆ G there is a unital for all a ∈ F and all g ∈ G 0 , then there is a unital homomorphism from B into the fixed point subalgebra of A ∞ ∩ A . If B n is finitely generated, then it suffices to check this for a generating set F.
If B is furthermore strongly-self absorbing then it follows that the maximal (hence any) crossed product absorbs B tensorially.
Proof. To verify the properties of Lemma 2.4 from [8], it suffices to show that for any finite set F ⊂ B, any finite subset G 0 ⊆ G and any ε > 0 there is a unital c.p. map Doing a small perturbation of the elements of F if need be, we may assume without loss of generality that F ⊆ B n for a sufficiently large n, and that all the elements of F have norm at most 1, and that 1 ∈ F (so that F · F ⊇ F-this is just for notational convenience). Fix such an n. Since B n is nuclear, one can find a finite dimensional algebra E and unital c.p. maps for all x, y ∈ F. By the Arveson extension theorem, we can extend ψ to a unital c.p. mapψ : B → E. Choose a homomorphism γ as in the statement of the lemma, with ε/2 instead of ε. Define and ϕ satisfies the required conditions. It is straightforward to check that if one can find such a ϕ for a generating set then one could find it for any other finite set. If B is strongly self-absorbing then the fact that A α G absorbs B tensorially now follows from the results in [8].
The previous lemma, together with the characterization of prime dimension drop algebras from [26] immediately gives the following; cf. [19,Theorem 4.9].
Corollary 5.5. Let A be a unital separable C * -algebra. Let G be a discrete countable group with an action α : G → Aut(A). Suppose that for any positive integer n, any ε > 0 and any finite subset G 0 ⊆ G there are order zero maps θ : M n → A ∞ ∩ A , η : M n+1 → A ∞ ∩ A with commuting ranges such that for any x in the unit ball of M n or M n+1 , respectively, and any g ∈ G 0 , then A α G is Z-stable.
Remark 5.6. In the previous lemma and corollary, if G is generated by a subset then it is sufficient to consider finite subsets G 0 ⊆ rather than all finite subsets in G. In particular, for actions of Z we will consider a single generator.  Let (1) [and that π(h) in fact is a contraction]. We have then that π • μ : M n → A ∞ ∩ A is an order zero map, that (1) and that for any x ∈ M 1 n and any g ∈ G 0 we have We similarly obtain a homomorphism whose range commutes with that of π , such that (1) and such that π • μ satisfies the analogous properties to that of π • μ. Thus, π • μ(1) + π • μ (1) = 1, and those two order zero maps satisfy the conditions of Corollary 5.5. Therefore A α G is Z-stable, as required.
The main theorem of this section is a partial generalization of Theorem 4.4 from [8] (that theorem is for the Rokhlin property, but it applies to absorption of general strongly self-absorbing C * -algebras and not just the Jiang-Su algebra).
For the proof, we shall implicitly use the following immediate observation: by choosing a sequence of Rokhlin tower elements, one can view them as sitting in the central sequence algebra A ∞ ∩ A , in which case the approximate properties in the definition of Rokhlin dimension hold exactly.
The reason why we have to assume the Rokhlin property with commuting towers is that we want to use Lemma 5.7, which in turn relies on Corollary 5.5. The latter required us to produce almost central and almost invariant copies of prime dimension drop intervals, which in turn are given by pairs of commuting cones over M n and M n+1 , respectively. Lemma 5.7 then showed that instead of finding pairs of such matrix cones it is in fact enough to find larger ensembles of mutually commuting matrix cones, as long as they add up to the identity in a suitable way.
Theorem 5.8. Let A be a separable unital Z-stable C * -algebra, and let α be an auto- Proof. We establish that the conditions of Lemma 5.7 hold.
Let r be a given positive integer. Fix two order zero maps θ : M r → Z, η : M r +1 → Z with commuting ranges such that θ(1) + η(1) = 1. Let K be the union of the images of the unit balls of M r , M r +1 under those maps.
We define unital homomorphisms ι 0 , . . . , ι d , μ 0 , . . . , μ d : Z → A ∞ ∩A as follows. First fix ι 0 : Z → A ∞ ∩ A . Proceeding inductively, we choose ι k : Z → A ∞ ∩ A such that its image furthermore commutes withᾱ j (ι i (Z)) for all j ∈ Z and i < k (this can be done by Lemma 4.5 of [8]). We define μ 0 , . . . , μ d in a similar way, such that the image of μ k commutes withᾱ j (μ i (Z)) for all j ∈ Z and i < k as well asᾱ j (ι i (Z)) for all j ∈ Z and all i = 0, 1, . . . , d. Let note that B k ∼ = B k ⊗ Z, and that the elements of B k commute with those of B m for k = m. Choose a unitary w ∈ U (Z ⊗ Z) such that for all x ∈ K . Notice that the unitary group of the Jiang-Su algebra is connected. Thus, w can be connected to 1 via a rectifiable path. Let L be the length of such a path. Choose n such that L x /n < ε/8 for all x ∈ F. Define homomorphisms Pick unitaries 1 = w 0 , w 1 , . . . , w n = w ∈ U 0 (Z ⊗ Z) such that w j − w j+1 ≤ L/n for j = 0, . . . , n − 1. Now, for k = 0, . . . , d, let and We can check that for all x in the unit balls of M r , M r +1 , respectively. Furthermore, Therefore those maps satisfy the conditions of Lemma 5.7.
We can also obtain in a similar way the analogous result for actions of finite groups with finite Rokhlin dimension and commuting towers. Theorem 5.9. Let A be a separable unital Z-stable C * -algebra, let G be a finite group and let α : G → Aut(A) be an action with Rokhlin dimension d < ∞, such that furthermore the Rokhlin elements from Definition 1.1 can be chosen to commute with each other. Then the crossed product A α G is Z-stable as well.
Proof. The proof is a simpler version of the proof of Theorem 5.8-here we do not need the choice of correcting unitaries. We again establish that the conditions of Lemma 5.7 hold, and we start in a similar way.
Let r be a given positive integer. Fix two order zero maps θ : M r → Z, η : M r +1 → Z with commuting ranges such that θ(1) + η(1) = 1. We define unital homomorphisms ι 0 , . . . , ι d : Z → A ∞ ∩ A as follows. First fix ι 0 : Z → A ∞ ∩ A . Proceeding inductively, we choose ι k : Z → A ∞ ∩ A such that its image furthermore commutes withᾱ g (ι i (Z)) for all g ∈ G and i < k. Let One checks that the images of those maps are fixed by the action of G on A ∞ ∩ A , and satisfy the conditions of Corollary 5.5.

Topological Dynamics
In the initial version of this paper we showed that minimal homeomorphisms on finite dimensional compact metrizable spaces always have finite Rokhlin dimension: Theorem 6.1. Let (T, h) be a minimal dynamical system with T compact and metrizable; suppose that dim T ≤ n < ∞ and let α be the induced action on C(T ). Then, dim s Rok (C(T ), α) ≤ 2n + 1.
The proof relied on a detailed analysis of the representation theory of certain large orbit breaking subalgebras of the transformation group C * -algebra. After the paper appeared on the arXiv, Szabó in [29] generalized this to the case of aperiodic homeomorphisms. In fact, with essentially the same methods, he could push the result even to free and aperiodic Z d -actions.
Szabó's proof-although by no means trivial-is somewhat easier and also more conceptual than our original argument, which we therefore have removed from this version (even though at least it has the slight advantage over Szabó's proof that it provides particularly nice and fairly explicit approximations of large orbit breaking subalgebras of the crossed product). However, the reader might find the following short and direct direct proof of the fact that irrational rotations have Rokhlin dimension 1 instructive. Theorem 6.2. Let α be the irrational rotation by θ on C(T), then dim s Rok (α) = 1. Remark 6.3. It follows immediately that if (X, h) is a dynamical system which has an irrational rotation as a factor, that is there is a commuting diagram , α) ≤ 1 as well. Proof of Theorem 6.2. Given a prime number p and ε > 0, we will exhibit positive Thef j 's are pairwise orthogonal and theg j 's are pairwise orthogonal.
We first recall some basic facts about continued fractions (see [28], Chapter 1). If t is an irrational number and if m j /n j is the sequence of approximants for t, then |t − m j n j | < 1 n 2 j for all j. Furthermore, the m j 's satisfy a recursive formula of the form m j+1 = a j m j + m j−1 . It follows by induction that gcd(m j , m j+1 ) = gcd(m 0 , m 1 ) = 1, and thus any two successive m j 's are coprime. In particular, given a prime number p and an irrational number t, there are infinitely many integers m, n such that p | m and |t − m n | < 1 n 2 . Fix a prime number p. We have infinitely many coprime numbers m, n such that | pθ − m n | < 1 n 2 , and p | m, i.e. |θ − m pn | < 1 pn 2 and (m, pn) = 1. Identifying the circle with the reals mod 1, we set and we set g 0 (x) = f 0 (x − 1/2np). Let γ ( f ) = f (x − m/np) (rational rotation by m/np). Notice that γ is np-periodic (and does not have a smaller period). Set f j (x) = γ j ( f 0 ), g j (x) = γ j (g 0 ) for j = 1, 2, . . . , np − 1. One easily verifies that f 0 + f 1 + · · · + f np−1 + g 0 + g 1 + · · · + g np−1 = 1 and that f j f k = g j g k = 0 for j = k.

Appendix: Crossed Products, Nuclear Dimension and Z-Stability
Notation A.1. To fix notation we recall the construction of the reduced crossed product. Let G be a discrete group, A → B(H ) a C * -algebra acting faithfully on the Hilbert space H and α : G → Aut(A) an action of G on A. Define π : A → B( 2 (G) ⊗ H ) by π(a)(e g ⊗ ξ) = e g ⊗ α g −1 (a)ξ and λ h (e g ⊗ ξ) = e hg ⊗ ξ, where g, h ∈ G, a ∈ A and ξ ∈ H . Then λ g π(a) = π(α g (a))λ g i.e. (π, λ) is a covariant representation. The reduced crossed product A α,r G is the C * -subalgebra of B( 2 (G) ⊗ H ) generated by {π(a)λ g | a ∈ A, g ∈ G}. This algebra does not depend on the choice of the faithful representation of A. The groups we consider in this paper are either finite or equal to Z, hence amenable. For amenable groups the reduced crossed product A α,r G coincides with the universal crossed product A α G. Using matrix units, π(a) can also be written as π(a) = g∈G e g,g ⊗ α g −1 (a) and π(a)λ h = g∈G e g,h −1 g ⊗ α g −1 (a).
Thus if G is finite with n elements then we may regard A α G as a subalgebra of M n (A) in a natural way and if G = Z we will regard A α Z as a subalgebra of B( 2 (Z) ⊗ H ). We will usually drop π from the notation and denote λ g by u g .
We recall that a c.p. contraction ϕ : A → B is said to be an order zero map if whenever x, y are positive elements in A such that x y = 0 then ϕ(x)ϕ(y) = 0.
Order zero maps play a central role in the definition of decomposition rank and nuclear dimension which we recall for the reader's convenience. (Cf. [15,35].) Definition A.2. Let A be a C * -algebra, F a finite-dimensional C * -algebra and n ∈ N.
(2) A has decomposition rank n, drA = n, if n is the least integer such that the following holds: For any finite subset F ⊂ A and ε > 0, there is a finite-dimensional c.p.c. approximation (F, ψ, ϕ) for F with tolerance ε (i.e., F is finite-dimensional, ψ : A → F and ϕ : F → A are c.p.c. and ϕψ(b) − b < ε ∀ b ∈ F) such that ϕ is n-decomposable. If no such n exists, we write drA = ∞. (3) A has nuclear dimension n, dim nuc A = n, if n is the least integer such that the following holds: For any finite subset F ⊂ A and ε > 0, there is a finite-dimensional c.p. approximation (F, ψ, ϕ) for F to within ε (i.e., F is finite-dimensional, ψ : A → F and ϕ : F → A are c.p. and ϕψ(b) − b < ε ∀ b ∈ F) such that ψ is c.p.c., and ϕ is n-decomposable with c.p.c. order zero components ϕ (i) . If no such n exists, we write dim nuc A = ∞.
Definition A.3. Let F be a finite dimensional C * -algebra. Let A be a C * -algebra, and let δ > 0. A c.p. contraction ϕ : F → A is a δ-order zero map if for any positive contractions x, y ∈ F such that x y = 0 we have that ϕ(x)ϕ(y) ≤ δ.
We recall that order zero maps from finite dimensional C * -algebras have the following stability property [15,Proposition 2.5]. Let F be a finite dimensional C * -algebra, then for any ε > 0 there is a δ > 0 (depending on F and ε) such that if A is any C * -algebra and ϕ : F → A is a δ-order zero map then there is an order zero map ϕ : F → A such that ϕ − ϕ < ε. Using this, we easily obtain the following technical Lemma.
Lemma A.4. Let A be a C * -algebra. A has nuclear dimension at most n if and only if for any finite set F ⊆ A and any ε > 0 there exists a finite dimensional C * -algebra F = F (0) ⊕ · · · ⊕ F (n) such that for any δ > 0 there exist c.p. maps ψ : A → F, ϕ : F → A such that ψ is contractive, ϕ • ψ(a) − a < ε for all a ∈ F and ϕ| F ( j) is a c.p.c. δ-order zero map for any j = 0, 1, . . . , n.
A has decomposition rank at most n if the same holds where furthermore ϕ is assumed to be a contraction.
Recall from [12,30] that the Jiang-Su algebra Z can be written as an inductive limit of prime dimension drop intervals of the form and that a separable unital C * -algebra A is Z-stable if and only if it admits almost central unital embeddings of Z p, p+1 for some p ≥ 2. With the aid of [26] one can give the following useful alternative formulation, cf. [32].