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-stable C*-algebras, where Z 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-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 Szabo to the case of free and aperiodic integer (in fact, Z^d) 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, [15,8] and references therein for actions of Z, and [4,9,10,20], 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 [23].
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 II 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). Our main result in this direction is 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.
Covering dimension for topological spaces was generalized to the context of C *algebras in [14] and [34], 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 [29].
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 the tracial sense (see [19,25,1,6,22]).
We also wish to point out that finite Rokhlin dimension can be interpreted as a dynamic version of Kirchberg's covering number, introduced in [12] (also cf. [13]).
Hiroki Matui and Yasuhiko Sato have informed us that they have a proof that all strongly outer automorphisms of separable, simple, unital, monotracial and Zstable C * -algebras have Rokhlin dimension at most 1, thus partially generalizing our Theorem 3.4. Their argument is based on the techniques developed in [18].
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 [17].
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 Section 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 Section 2 we introduce various versions of Rokhlin dimension for Z-actions and study their interplay. In Section 3 we show that finite Rohklin dimension is generic in the Zstable case. A similar argument shows that the Rokhlin property is generic in the UHF-stable case. In Section 4 we show that finite nuclear dimension is preserved under forming crossed products by automorphisms with finite Rokhlin dimension, and in Section 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.
The remainder of the paper is devoted to showing that minimal Z-actions on finite dimensional compact Hausdorff spaces always have finite Rokhlin dimension. As the proof is quite technical, we give a (different) short and direct argument for irrational rotations of the circle in Section 6. The proof of the general result is carried out in Sections 7 through 12. The argument closely follows that of [31], computing the decomposition rank for orbit breaking large subalgebras of transformation group C *algebras while at the same time carefully keeping track of the underlying dynamics.
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 will refer to the family (f (l) g ) l=0,...,d; g∈G as a multiple tower and to the d + 1 families (f 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 [9, 10]).
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 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: 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 set 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 (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.
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. [15], [8] 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.
Remarks 2.2. (i) It follows immediately from the definition that the Rokhlin projections 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 (A, α) = ∞. b) If, moreover, for all towers we can arrange [f r,j ] < ε for all q, i, l, r, j, m, then we say that α has Rokhlin dimension d with commuting towers, dim c Rok (A, α) = d. c) If one can obtain this property such that for all l one of the towers (f 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. (ii) Note that in the single tower version defined in c) above, property (4) of 2.3a) forces property (3) to hold cyclically, i.e., for j ∈ {0, 1, . . . , p − 1 + r}.
(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 (l) 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)-(v) 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,ar(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: such that (1) ζ (l) (e)ζ (l) (e ) ≤ 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, A combination of b), c) and d) yields the respective statements for dim 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, and dim s Rok (C(T ), α) = dim s Rok (C(T ), α). Proof. We clearly have dim Rok (C(T ), α) ≤ dim Rok (C(T ), α). For the reverse inequality, assume dim Rok (C(T ), α) ≤ d and suppose we are given p ∈ N, 0 <ε ≤ 1 and F ⊂ C(T ) finite (in fact, the subset By approximating f uniformly by polynomials we can find 0 < δ ≤ε 4 such that, for any positive a with a < d + 1, if α 1 (a) − a < δ then α 1 (f (a)) − f (a) <ε 2 . Set 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) we see that By the choice of δ and by (2) this entails Define mapsζ (l) : 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 ζ (l) (e)ζ (l) (e ) ≤ 4ε e e ≤ε d + 1 e e 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 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 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 = 0, 1, . . . , p and for l = 1, . . . , d One now readily checks that g (l) j satisfy the requirements. If p is not sufficiently large, we choose a k such that ε > 1 kp−3/2 , choose Rokhlin towers (f (l) r,j ) for j = 1, ..., kp − 1 + r and l = 0, . . . , d with tolerance ε/k instead of ε, and repeat the previous procedure to obtain Rokhlin towers (g j+np for j = 0, 1, . . . , p − 1 and l = 0, . . . , 2d gives us Rokhlin towers of height p as required. 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.
Lemma 3.1. There exists a unitary u ∈ Z p,p+1 and positive elements (3) uf 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: 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.
)·1⊗b x gives us elements as required.
Lemma 3.2. Let A be a separable C * -algebra. Let a 1 , . . . , a n , b 1 , . . . , b n ∈ A, set c = n i=1 a 1 ⊗ max b i ∈ A ⊗ max A, 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 ,ψ(y)] = 0 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: 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, 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,Fn, 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 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 : 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, Φ, Ψ) 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 Define decay factors Notice that where a ∈ A and |m| ≤ q. In fact these estimates hold for all bounded operator Next letF 1 :=F ∪ α(F ) ∪ . . . ∪ α p (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 . Consider Let now B F be the closed norm compact unit ball in F 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 ) < cpε 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) Thus, given η > 0 we could first choose p large enough and then ε > 0 small enough so that ρ With a, m and η as above, we have that To summarize, we constructed the following maps.
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 One obtains that is a 2(d+1)(N +1)−1-decomposable approximation for F to within tolerance 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 PropositionA.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 Lemma 5.2. Let 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.
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.
It is easy to construct a unital homomorphism π : The following is a simple modification of Lemma 2.4 from [7]. While this lemma can be generalized to actions of non-discrete groups in a straightfoward way (as in the said lemma), we state it here just for actions of discrete groups to avoid notation which we do not need in this paper. 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 [27].
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 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 [7], 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.
. 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 ϕ = γ • θ •ψ 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 [7].
The previous lemma, together with the characterization of prime dimension drop algebras from [24] immediately gives the following.
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 θ(1) + η(1) = 1 and such that ᾱ g (θ(x)) − θ(x) < ε and ᾱ g (η(x)) − η(x) < ε 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 Γ then it is sufficient to consider finite subsets G 0 ⊆ Γ rather than all finite subsets in G. In particular, for actions of Z we will consider a single generator.
Lemma 5.7. Let A be a unital separable C * -algebra. Let G be a discrete countable group with an action α : G → Aut(A). Let d be a non-negative integer. Suppose that for any positive integer n, any ε > 0 and any finite subset G 0 ⊆ G there are order zero maps θ 0 , . . . , θ d : commuting ranges such that for all g ∈ G 0 , k = 0, 1, . . . , d and all x in the unit ball of M n or M n+1 respectively we have that  ). We repeat the same procedure for D (d+1) n+1 , denoting the resulting maps, sets and elements by ζ j , β , h , Y , µ and F .
Choose ε as in Lemma 5.1 with respect to the compact set of generators Y and the compact set F and with respect to the compact set of generators Y and the compact set F (take the least of the two). Let be the unique homomorphism which satisfies that π • ζ k • β = θ k for k = 0, 1, . . . d.
Notice that (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 and that for any x ∈ M 1 n and any g ∈ G 0 we have We similarly obtain a homomorphism π : D (d+1) n+1 → A ∞ ∩ A whose range commutes with that of π, such that and such that π • µ satisfies the analogous properties to that of π • µ.
The main theorem of this section is a partial generalization of Theorem 4.4 from [7] (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.
Theorem 5.8. Let A be a separable unital Z-stable C * -algebra, and let α be an automorphism of A with dim c Rok (α) = d < ∞, then the crossed product A α Z is Z-stable as well.
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 [7]). 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 as α 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 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 Similarly, we choose unitaries 1 = v Let {f 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 Corollary 5.5.
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 , 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 g (ι i (Z)) for all g ∈ G and i < k. We define µ 0 , . . . , µ d in a similar way, such that the image of µ k commutes withᾱ g (µ i (Z)) for all g ∈ G 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. Let f (l) g l=0,...,d; g∈G ⊆ A ∞ ∩ A be Rokhlin elements, and as in the statement of the theorem we assume that they all commute with each other, and which are furthermore chosen to commute with B 0 , B 1 , . . . , B d . Define 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.

Irrational rotations
In this section we first give a direct proof of the fact that irrational rotations have Rokhlin dimension 1. In the subsequent sections we consider much more generally minimal actions on finite dimensional compact spaces and show that these must always have finite Rokhlin dimension.
Remark 6.2. 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 6.1. Given a prime number p and ε > 0, we will exhibit positive {f i ,g i } i=0,1,...,p−1 such that Thef j 's are pairwise orthogonal and theg j 's are pairwise orthogonal.
We first recall some basic facts about continued fractions (see [26], 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 np ≤ x ≤ 1 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.
Notice furthermore that each f j and g j are Lipschitz with Lipschitz constant 2np. For j = 0, 1, . . . , p − 1, setf Notice that again, thef j 's are pairwise orthogonal, theg j 's are pairwise orthogonal, we havef Now, we note that for all x we have The remainder of the paper is devoted to our general version of 6.1, Theorem 12.1 below. The argument closely follows that of [31], computing the decomposition rank for (approximately subhomogeneous) orbit breaking subalgebras of transformation group C * -algebras. In this computation we need to carefully keep track of the underlying dynamics, which causes a considerable amount of technical difficulty. The role of the orbit breaking subalgebras is that of a book-keeping device, which keeps track of first return times for points of closed subsets with nonempty interiors.

Cyclic vs. non-cyclic shifts
Below we fix notation and some elementary facts on diagonal subalgebras of matrix algebras. Proposition 7.2 may be thought of as a splicing principle, which allows to compare a truncated shift to two cyclic shifts. We have already seen a very similar phenomenon in the proof of Proposition 2.8.
Notation 7.1. For r ∈ N we denote the set of diagonal elements of M r by D r ; we call diagonal elements with vanishing (r, r)-entry shiftable and denote the shiftable diagonal elements of M r by Dσ r r . We denote byσ r the truncated shift σ r : D r → D r , given byσ and note that (6)σ r (e i,i ) = S r e i,i S * r and thatσ r is injective on Dσ r r . Define a partial inversē σ − r : D r → D r ofσ r byσ − r (e) := S * r eS r . When there is no ambiguity, we will omit the subscript and just write D, Dσ,σ, σ − and S. If is a finite dimensional C * -algebra, we will usually denote by Note that this notation relies on the particular identification (7) of the finite dimensional C * -algebra F with a sum of matrix algebras.
Proposition 7.2. Given k ∈ N and δ > 0 there is s ∈ N such that the following holds: For any natural number r ≥ 4s there is a c.p.c. order zero map µ : C k → Dσ ⊂ M r such that (8) σ(µ(e)) − µ(σ(e)) ≤ δ e for all 0 ≤ e ∈ C k , and such that whereσ is the truncated shift on Dσ,σ − is its partial inverse andσ is the cyclic shift on C k , cf. 7.1.
Finally, we have

Minimal dynamics, first return times and compatible approximations
In this section we fix notation on transformation group C * -algebras and certain recursive subhomogeneous subalgebras of these. We then define what it means for approximations of those subalgebras to be compatible with the underlying dynamics.
Notation 8.1. (Cf. [16].) Let (T, h) be a minimal dynamical system with T compact and metrizable. We let be the crossed product C * -algebra, where u is the unitary which implements the action by If Z ⊂ T is closed with nonempty interior, consider Let m 1 < m 2 < . . . < m L ∈ N denote the first return times for Z with associated pairwise disjoint locally compact subsets Z 1 , . . . , Z L ⊂ Z.

Note that
Define * -homomorphisms for f ∈ C(T ) and for f ∈ C 0 (T \ Z) (these are easily checked to be well-defined). Define (17) and set where the horizontal maps are the canonical projections. Note that the map is an isomorphism.
Letσ l : E Z → C(Z l \ Z l ) ⊗ M m l denote the composition of σ l with the restriction map. If t ∈ Z l+1 \ Z l+1 , then it is straightforward to check that the kernel of ev t •σ l+1 contains the kernel of ρ l ; from this one concludes thatσ l+1 factorizes through σ l , i.e., there is a * -homomorphism (19) π l : B l → C(Z l \ Z l ) ⊗ M m l such thatσ l+1 = π l • ρ l . This in turn implies that B l+1 can be regarded as a pullback, i.e., Next, let Dσ m l m l ⊂ D m l ⊂ M m l denote the (shiftable) diagonal elements of M m l . We claim that  [ϕ(1 Mr j ), ϕ(M r i )] = 0 for i ≺ j, we say ϕ is piecewise commuting (p.c.) with respect to ≺. be c.p.c., p.c. and n-decomposable; let η ≥ 0. We say ϕ is (ρ, η)-compatible, if there is a c.p.c. map We call ϕ an η-compatible approximate c.p.c. lift for ϕ.
Proposition 8.4. For every n ∈ N and η > 0 there isη > 0 such that the following holds: Let (T, h) be a minimal dynamical system with T compact and metrizable; suppose that dim T ≤ n < ∞ and that Z ⊂ T is a closed subset with nonempty interior. Let Proof. Let n ∈ N and η > 0 be given; we may clearly assume that η ≤ 1. Choose Now suppose T , h, Z, ρ L , F and ϕ are as in the proposition. Suppose ϕ is decomposable with respect to denote the respective components of ϕ by ϕ (i) , similar forσ (i) , S (i) , D (i) and (D (i) )σ. For i ∈ {0, . . . , n} and e ∈ (D (i) )σ + we estimate Define a function f β ∈ C([0, 1]) by We apply functional calculus for the order zero maps ρ −1 L ϕ (i) (as introduced in [32]) to obtain for all a, b ∈ F (i) .

Relative barycentric subdivision
Proof. By construction, the relative barycentric subdivision Sd J K of K is obtained by inductively subdividing certain faces of K with the help of [31, Proposition 5.5], also cf. [31, 5.6]. At each step one obtains a linear homeomorphism, calledβ in [31, Proposition 5.5], between the geometric realizations. The composition of all these is our map S J . If we keep subdividing Sd J K inductively at the remaining original faces of K, eventually we arrive at the (full) barycentric subdivision SdK. This yields the factorization

The map
h × : C Σ + → C Γ is then just given by h × := ⊕ γ∈Γ ev γ (S −1 J (h)), where ev γ : C(|Sd J K|) → C denotes evaluation at the vertex γ (regarded as a point in the geometric realization of Sd J K).
Similarly, one defines With these maps it is straightforward to check that the diagram (31) indeed commutes, since the maps between the geometric realizations are linear homeomorphisms (given explicitly in the proof of [31, Proposition 5.5]).
Proposition 9.2. Given L ∈ N and θ > 0, there isθ > 0 such that the following holds: Suppose K is a simplicial complex with at most L vertices; denote by Σ its vertex set and byh : C Σ → C(|K|) the u.c.p. coordinate map.
Denote by SdK the barycentric subdivision of K, ∆ its vertex set, the u.c.p. coordinate map and the canonical isomorphism. Let D be a C * -algebra and leth 1 ,h 2 : C(|K|) → D be two * -homomorphisms satisfying Then, the c.p.c. maps the u.c.p. coordinate map, and let SdK • , ∆ • , • and S • denote the respective data for the barycentric subdivision. Approximate each coordinate function uniformly by a polynomial in L commuting variables, for each δ ∈ ∆ • ; this is possible since C(|K • |) ∼ = C(|SdK • |) is generated by the image ofh • .
Chooseθ > 0 such that, whenever a 1 , . . . , a L , b 1 , . . . , b L are positive elements of norm at most 1 in some C * -algebra satisfying be the induced surjection. Note that we have a commutative diagram We then estimate for each σ ∈ {1, . . . , L} by (32). From this it follows immediately that Before we proceed, let us fix notation for some continuous functions as follows; this will be useful for the present and the subsequent sections.
Proposition 9.4. Let n, R ∈ N be given. Then, for any θ > 0 there isθ > 0 such that the following holds: As in [31, 5.3], let K be a simplicial complex with vertex set let J be the subcomplex generated by V (J) = Σ (1) . Let B be an R-subhomogeneous C * -algebra and let be a u.c.p. map (with C commutative) as in [31, 5.3], in particular with h| C Σ ndecomposable and such that h factorizes through C(|K|) (see (50)). Suppose that is another u.c.p. map with commutative range satisfying Then, there is a u.c.p. map k ‡ : C Γ → C ‡ such that k ‡ | C Γ\{ * } is n-decomposable and such that where Γ and k : C Γ → C are as in [31, 5.3]. We may choose k ‡ | C Γ\{ * } to be n-decomposable with respect to the same decomposition as k| C Γ\{ * } .
Upon extending the map 5.4]) to a map (also denoted) we may furthermore assume that Proof. Given θ, n and R, set L := 2R(n + 2); we may clearly assume θ ≤ 1. Obtain Now let B, K, J, C, C ‡ , h, h ‡ as in the proposition be given; note that h factorizes as Define a u.c.p. map for σ ∈ Σ + ; we will check below that the inverse exists. Letπ : B → M r , r ≤ R, be an irreducible representation. Sinceh| C Σ is n-decomposable, K is at most (n+1)dimensional. Nowπh is a sum of at most R characters of C(|K|), and each of the corresponding points of |K| sits in some face with at most n + 2 vertices. Therefore, there is a subset Σ ⊂ Σ + with at most R(n + 2) elements such that and we estimate Sinceπ was arbitrary and A similar reasoning shows that ) < R(n + 2)θ +θ for any normalized (λ σ ) σ∈Σ + ∈ C Σ + . We then obtain Next, suppose that for some σ, σ ∈ Σ + we have Then, From this and (50) it follows in particular that h • also factorizes through C(|K|), say via a u.c.p. maph • , and that h • | C Σ is n-decomposable with respect to the same decomposition as h. Let ∆ denote the vertex set of the barycentric subdivision SdK of K, and consider the commutative diagrams To show that Both πh and π ‡h• can be written as sums of at most R characters of C(|K|). Again each of the corresponding points in |K| sits in some face with at most n + 2 vertices. Let (61)Σ ⊂ Σ + denote the set of these at most 2R(n + 2) vertices, and letǨ denote the subcomplex of K generated byΣ. Then, πh and π ‡h• both factorize through C(|Ǩ|), i.e. we have commuting diagrams where the first vertical maps are compatible with (61). It now follows that ȟ −ȟ • (59),(60),(62),(63) π| C ∆\∆ = π ‡• | C ∆\∆ = 0 and that we have commuting diagrams As a consequence, we have sinceπ was an arbitrary irreducible representation of B, it follows that in fact so (58) holds. Note that the vertical maps of (56) and (57) factorize through the relative barycentric subdivision, cf. Proposition 9.1, so that we have commuting diagrams We then clearly have The decomposition for k| C Γ\{ * } only depends on a coloring for the covering of |Sd J K| by open stars around vertices, cf. [31, 5.7]; therefore, k ‡ | C Γ\{ * } can be decomposed in the same way.
Proof. The subdivision Sd J K, and hence the map come from a repeated application of [31, Proposition 5.5]. Let K − denote the subcomplex of K generated by Σ = Σ (1) ∪ Σ (2) , then K − is at most n-dimensional.

Elementary polynomials
The notion of elementary polynomials introduced below is used frequently in Section 11. where each y (k,l) is either x j or 1 − x j for some j ∈ {1, . . . , r}.
Remark 10.2. Given r ∈ N, there are no more than (2r) r 2 elementary polynomials in r commuting variables x 1 , . . . , x r .
Proof. It will suffice to show that, for any projection e which is minimal in C * (q j | j = 1, . . . , s), we have e = h 1 . . . h r , where each h i is either q j i or 1 − q j i for some j i ∈ {1, . . . , s}. The result will then follow since each projection in C * (q j | j = 1, . . . , s) is a sum of at most r minimal ones (and since 0 can be written in the form r l=1 y (l) with y (l) either q j or 1 − q j ). So let e ∈ C * (q j | j = 1, . . . , s) be minimal, then e ≤ q j 1 for some j 1 ∈ {1, . . . , s}. Set In the first case, set d 2 := d 1 q j 2 ; in the second case, set d 2 := d 1 (1 − q j 2 ). If d 2 is minimal in C * (q j | j = 1, . . . , s), then e = d 2 = (q j 1 ) r−2 d 2 is of the desired form. Otherwise, keep repeating the construction inductively to obtain e ≤ d l . . . d 1 . Since d 1 has rank (as an element of M r ) at most r and the ranks of the d j are reduced at each step, the construction will terminate for somel ≤ r, in which case dl is minimal and e = dl = (q j 1 ) r−l dl is of the desired form.

Existence of compatible approximations
The purpose of this section is it to show that compatible approximations indeed exist. To this end, we carefully revisit the proof of [31, Theorem 6.1] in the case of recursive subhomogeneous subalgebras of transformation group C * -algebras.
Lemma 11.1. Let (T, h) be a minimal dynamical system with T compact and metrizable and such that dim T ≤ n < ∞. Let Z ⊂ T be closed with nonempty interior and let and be as in 8.1. Then, for each l ∈ {1, . . . , L} and η > 0 there is a system of c.p.c., p.c., ndecomposable and (ρ l , η)-compatible approximations for B l ; the approximating finite dimensional C * -algebras may be chosen so that all their irreducible representations have rank at least m 1 .
Proof. Let us first prove the assertion of the lemma for l = 1. Note that Z 1 ⊂ Z is closed with dim Z 1 ≤ n, and that It follows from (15), (21), (16) and (22) that and that be a system of c.p.c., n-decomposable approximations for C(Z 1 ), then is a system of c.p.c., p.c., n-decomposable approximations for C(Z 1 ) ⊗ M m 1 . Upon identifying C k λ ⊗ M m 1 with k λ M m 1 , we writeψ λ andφ λ for ψ λ ⊗ id Mm 1 and ϕ λ ⊗ id Mm 1 , respectively. We check that eachφ λ is (ρ 1 , η)-compatible. Let denote the (shiftable) diagonal elements, and observe that It is straightforward to construct a c.p.c. lift so that we have properties 8.3(i) and (ii). Property 8.3(iii) follows directly from (16), so that indeed eachφ λ is (ρ 1 , η)-compatible. Note that in fact our argument shows each thatφ λ is (ρ 1 , 0)-compatible, since ϕ λ is an exact lift ofφ λ | D λ ; for higher values of l we will only be able to produce approximate lifts, which will complicate matters significantly. Now suppose the lemma has been established for some l ∈ {1, . . . , L − 1}. To verify the statement for l + 1, it will suffice to show that, for any 0 < ε, η < 1 and positive contractions a 1 , . . . , a k ∈ B l+1 (where we may assume a 1 = 1 B l+1 ), there is a c.p.c., p.c., n-decomposable approximation (F, ψ, ϕ) (for B l+1 ) of {a 1 , . . . , a k } within ε such that ϕ is (ρ l+1 , η)-compatible. (We also need to make sure the ranks of the irreducible representations of F are at least m 1 -but this will turn out automatically, since only m 1 , . . . , m l+1 occur as possible ranks.) Our construction will be a variation of the proof of [31, Theorem 6.1]. Let us first adjust our notation. Set B := B l , A := B l+1 , r := m l+1 , Ω := Z l+1 and X := Z l+1 \ Z l+1 .
Note that X ⊂ Ω is closed and that is the map π l from (19). Note that A is r-subhomogeneous of topological dimension at most n, cf. [21], hence satisfies the hypotheses of [31, Theorem 6.1]. We will run the proof of [31, Theorem 6.1] almost verbatim to obtain a c.p.c., p.c., n-decomposable approximation for A. In a few places along the way we will have to be more careful about certain choices to make the approximations (ρ l+1 , η)compatible.
We claim that, upon making Y smaller if necessary, we may in addition assume thatφ is (ρ Y , 3η )-compatible, where is the natural surjection.
To verify this, note that by (6.1) of [31] we have ≤ η x for x ∈ F + , where (as in [31, (6.1)]) we use β to denote both the projection maps and B ⊕ π,X (C(Y ) ⊗ M r ) → B (this will cause no ambiguity). Let (D )σ ⊂ D ⊂ F denote the (shiftable) diagonal elements of F ; now any (βρ Y , η )-compatible c.p.c. approximate lift D → C(T ) ⊂ E Z for ϕ will be a (βρ Y , 2η )-compatible c.p.c. approximate lift for β •φ by (96); upon making Y smaller if necessary, it will then also be a 3η -compatible c.p.c. approximate lift forφ.
We now run Step 2 of the proof of [31, Theorem 6.1] to obtain It is straightforward to show that, upon making the U λ smaller if necessary, we may in addition assume that the following are satisfied: (vii) there is a map (cf. property (vi) and (77)), such that and such that is a projection for each λ ∈ Λ and i ∈ {1, . . . , s}, cf. property (vi); (viii) for each λ ∈ Λ and each t ∈ U λ , the map ev t : C * (q(λ, j) | j = 1, . . . , s) → C * (q(λ, j)(t) | j = 1, . . . , s) is an isomorphism of finite dimensional commutative C * -algebras with dimension at most r. Just as in the proof of [31, 6.1], we complete Step 2 by choosing functions we set We note that in the proof of [31, 6.1] each C 0 (U λ ) was identified with (where Z(A) denotes the center of A); it follows from (17) that in our setting and with this identification we have From here on, the construction runs exactly as in the proof of [31, 6.1].
In the remainder of the proof we will show that ϕ indeed is (ρ l+1 , η)-compatible.

Compatible approximations and Rokhlin dimension
We are finally prepared to show that actions coming from minimal homeomorphisms on finite dimensional spaces indeed have finite Rokhlin dimension.
Theorem 12.1. 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 Rok (C(T ), α) ≤ 2n + 1.
Suppose ϕ is n-decomposable with respect to the decomposition for j = 0, . . . , n let ϕ (j) : F (j) → B L denote the order zero components of ϕ and let µ (j) : C k → (D (j) )σ ⊂ F (j) denote the respective components of µ.

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 xy = 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. [14] and [34].) Definition A.2. Let A be a C * -algebra, F a finite-dimensional C * -algebra and n ∈ N.
(1) A c.p. map ϕ : F → A is n-decomposable if there is a decomposition such that the restriction ϕ (i) of ϕ to F (i) has order zero for each i ∈ {0, . . . , 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 finitedimensional 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 xy = 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 ([14, 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 [11] and [28] 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 [24] one can give the following useful alternative formulation, cf. [32].