Radius of comparison equals half the mean dimension

Proof. Choose \eta>0 so small that

Choose a multiple L of k with L>e and e/L<\eta. Apply Corollary 3.1 with height L and error \eta, and denote its base by V. Set

Its first k translates partition the tower. Hence they are pairwise disjoint, and

The first requirement follows from Equation (3.4).

For each integer s with 0\leq s<L-e, let i(s) be the unique integer in \{0,\ldots,k-1\} satisfying s+i(s)\equiv0\pmod{k}, and put

These are pairwise disjoint open sets. For the indicated values of D, the integer s+i(s)+Dk is a multiple of k between 0 and L-1, which proves Equation (3.1). Also

We next shrink O uniformly over invariant measures. Take continuous functions \varphi_j:X\to[0,1] increasing pointwise to \mathbf 1_O, each with compact support in O. Such functions are obtained from increasing cutoffs of the distance to X\setminus O; if O=X, use the constant function one. For each invariant measure, monotone convergence gives \int\varphi_j\,d\mu\to\mu(O)>1-2\eta. The sets

are increasing open sets covering the compact space \mathcal M_h(X). One of them therefore equals that whole space. Fix the corresponding function \varphi and set U=\{\varphi>0\}. The continuous function \mu\mapsto\int\varphi\,d\mu attains its minimum, which is still strictly greater than 1-3\eta. Since \mu(U)\geq\int\varphi\,d\mu, we obtain \overline U\subset O and

Choose w:X\to[0,1] equal to one on \overline U and with \operatorname{supp}(w)\subset O.

For completeness, the uniform orbit estimate for the closed set C is

Indeed, a sequence violating this estimate has a subsequence of its orbit empirical measures converging weakly to an invariant probability measure. The closed-set inequality for weak convergence bounds the limsup of their masses on C by the mass of that limit measure, a contradiction. Thus, by Equations (3.5) and (3.6), there is m_0 such that every orbit interval of length m\geq m_0 spends less than b_0/(2e) of its times in C.

The set \{w<1\} is contained in C. For eJ\geq m_0 and any y\in X,

Taking y=h^vx proves Equation (3.3) for a suitable J_0. This argument counts all integer times before passing to e-spaced starts; it makes no assumption about invariant measures for h^e.

Finally, each \operatorname{supp}(w)\cap O_i is compact: the O_i form a finite open partition of a neighborhood of \operatorname{supp}(w). Let K be the union of their images under the finitely many maps in Equation (3.2). It is a compact subset of B by Equation (3.1), as required.□

Proof. For each \Lambda\subset E with |\Lambda|\leq q, failure of injectivity on \bigoplus_{s\in\Lambda}\mathbb C^b is a proper determinantal algebraic condition on L: it is proper because a map injective on that particular subspace plainly exists. There are only finitely many such conditions, so their union does not fill the vector space of linear maps. Any map outside their union works. The cases b=0 or q=0 are immediate.□

Proof. We retain the allocation data throughout the proof. In particular, all cutoffs and the Cuntz witness below are fixed before J is chosen.

Extension of the aligned functions. Put W_i=\operatorname{supp}(w)\cap O_i and choose open sets V_i with

an empty W_i may be assigned V_i=\varnothing. For fixed D,j, the maps x\mapsto F_j(h^{i+Dk}x) on the disjoint closed sets \overline V_i combine to a continuous cube-valued map. Coordinatewise extension gives a continuous map \widehat F_{D,j}:X\to[-1,1]^t agreeing with it on their union. For every J, set

This proves the asserted agreement, including the neighborhood assertion. The extension maps \widehat F_{D,j} are independent of J.

Positive matrix functions. We now encode the g torus bundles by positive elements of the crossed product. Their tracial rank gap will supply a single witness for all the aligned orbit blocks. Take a tubular neighborhood \mathcal V of L inside (-1,1)^t, with retraction \pi_L:\mathcal V\to L, and write \widetilde Q=Q\circ\pi_L. Choose a continuous function \psi:[-1,1]^t\to[0,1] with compact support in \mathcal V and equal to one on a neighborhood of L. The matrix function \psi\widetilde Q extends by zero outside \mathcal V. By compactness there is \delta>0 such that, for y\in L and v\in[-1,1]^t,

Consequently \widetilde Q(v) maps H_y injectively onto \operatorname{ran}\widetilde Q(v) in this situation.

Choose continuous scalar functions \rho_0,\rho_1:X\to[0,1] such that

The compact containment K\subset B gives \rho_0, and a cutoff of the distance to X\setminus B gives \rho_1; if B=X, take \rho_1=1. Define positive matrix functions P_0\in M_{2pg}(C(X))_+ and P_1\in M_R(C(X))_+ by

The summands of P_0 are understood to be zero where F_j(x) lies outside \mathcal V.

If \tau is a normalized trace on A, its restriction to C(X) is integration against some \mu\in\mathcal M_h(X). Functional calculus and bounded convergence give, for every positive matrix function P,

The unnormalized matrix trace is essential in this formula. The rank of P_0(x) is at most pg on B and is zero off B, whereas P_1 has rank R precisely on B. Therefore

Here the strict inequality is supplied by Corollary 3.2. Since r is a comparison bound, it follows that

A finite-propagation witness and simultaneous injection. Finite-support compression witnesses and bundle obstructions also enter the lower-bound arguments of Hirshberg–Phillips [7, proofs of Theorems 3.3 and 4.5 in the author preprint]. Here we retain the sparse parameter throughout and require simultaneous injectivity on the entire sum of the selected fibers. Taking the appropriate rectangular corner of a Cuntz witness for Equation (3.13), and then approximating its entries by finite Fourier sums, gives T_0\in M_{R,2pg}(A) such that

for some fixed M\geq0 and continuous matrix functions A_\ell. For example, first choose a witness with error less than 1/4; norm continuity of T\mapsto T^*P_1T permits the Fourier approximation while keeping the error below 1/2.

In the orbit representation at x, a function a\in C(X) acts by

and the implementing unitary acts by a shift, with direction determined by the crossed-product convention. Thus T_0 has propagation at most M. Let

Restricting the input to times 0,\ldots,N-1 and the output to I_J gives a continuous finite matrix

representing \pi_x(P_1)^{1/2}\pi_x(T_0) on that input window. There is no output outside I_J. If an input vector \xi in the window satisfies \pi_x(P_0)\xi=\xi, then

This holds on the whole sum of any selected genuine projection ranges, including sums involving different times and different bundle blocks.

The rank cost of separated output rows. The map W_J(x) initially has (N+2M)R output coordinates. We now bound the target rank needed to retain the injectivity in Equation (3.15), using the spacing of the rows that can be nonzero. The output row at time s vanishes unless h^sx\in B. These active times are separated by at least k, because B,hB,\ldots,h^{k-1}B are disjoint. There are at most

of them in I_J. By Corollary 3.3, choose one fixed linear map

injective on every coordinate subspace supported at at most q_J times. Hence C_J is injective on the range of W_J(x) for every x. The matrix C_J is chosen once for this finite window and does not depend on x or on the sparse parameter.

The first bundle map. Let \alpha=(q,D,j) index a slot. On \mathcal L_J, write

This describes the rank-pn bundle E in the proposition. Choose a continuous \chi:[-2,2]^t\to[0,1] with \chi(0)=1 and support contained in \{z:\|z\|_\infty<\delta\}. Set

Where \theta_\alpha>0, there is a unique i with h^{qe}x\in O_i. Put s_\alpha=qe+i+Dk. By Equations (3.8) and (3.11),

is injective. Furthermore, h^{s_\alpha}x\in K, so the indicated range is a genuine projection range of the jth block of P_0(h^{s_\alpha}x).

Define a bundle map

by sending v_\alpha\in H_{y_\alpha}, where \theta_\alpha>0, to

in the jth block of the row at time s_\alpha, and by setting that contribution to zero where \theta_\alpha=0. All such times lie in the qth interval [qe,(q+1)e). Within an interval the pairs (i+Dk,j) are distinct, and intervals are disjoint. Thus different active slots occupy different time/block pairs.

This map is continuous on the entire space \mathcal L_J. Indeed, on a region with positive weight the label is locally constant and the formula is continuous. The projection factor has norm at most one, so the norm of its weighted contribution is at most \theta_\alpha\|v_\alpha\| and tends to zero wherever the weight vanishes. This also handles a change of label, which can occur only outside the positive-weight region. More formally, one can use the matrix \widetilde Q(F_j(h^{s_\alpha}x))Q(y_\alpha) on the ambient \mathbb C^{2p} and extend its weighted entries by zero; the same bound proves their continuity. No nearby-projection map is needed at zero weight.

The image of \Xi_{(x,z)} lies in the subspace on which \pi_x(P_0) is the identity. The first bundle map is therefore

By Equation (3.15), the defining property of C_J, and the distinct time/block placements,

The exceptional summands. The first map now controls every positive-weight summand. To complete one embedding over the full parameter locus, we give its kernel a second map whose target rank is charged only to the sparse slots and the infrequent starts at which w<1. Using the given embeddings H_y\subset\mathbb C^{2p}, define

If 1-\theta_\alpha\ne0, then either z_\alpha\ne0 or w(h^{qe}x)<1: when both z_\alpha=0 and w(h^{qe}x)=1, Equation (3.16) gives \theta_\alpha=1. The sparse condition bounds the first kind by an slots. For J\geq J_0, Equation (3.3) bounds the second kind by

slots. The range of \Psi_{(x,z)} is therefore supported on at most (a+b_0)n coordinate groups, each of dimension 2p.

Put s_J=\lceil(a+b_0)n\rceil. Another application of Corollary 3.3 gives a fixed linear map

injective on all the relevant coordinate subspaces. Set \Phi_2(x,z)=D_J\Psi_{(x,z)}. The direct sum \Phi_1\oplus\Phi_2 is fiberwise injective. In fact, Equation (3.17) says that a vector killed by \Phi_1 has components only where \theta_\alpha=0. On those components 1-\theta_\alpha=1. Injectivity of D_J on the range support of \Psi shows that a vector also killed by \Phi_2 must be zero.

Both maps are continuous in (x,z), including when the support of z changes. They thus give a continuous bundle embedding into the trivial bundle of rank

One may also verify the bundle assertion locally by representing the map in a trivialization of E: its Gram matrix is positive definite, so its image is a continuous rank-pn subbundle. The base \mathcal L_J is compact.

The strict rank estimate. Let

Since n=J(H_0-1)g, Equation (3.18) gives

For all sufficiently large J\geq J_0 this is strictly less than two. Hence K_J<2pn, which proves the proposition.□

Proof. Choose finitely many such neighborhoods covering the compact set \Phi(Y), and add K\setminus\Phi(Y) to obtain a finite open cover of K. A sufficiently fine simplicial subdivision has its open vertex stars subordinate to this cover. These stars have order at most q: a collection of stars with nonempty intersection corresponds to vertices of a single simplex. Their pullbacks under \Phi, with empty sets omitted, form a finite open refinement of \mathcal V of order at most q.□

Proof. Choose a finite open refinement of \mathcal V of order m and a subordinate partition of unity. The partition defines a map to the nerve S_0 of that refinement, whose dimension is at most m. Assign each vertex of S_0 a containing member of \mathcal V.

Take S to be the barycentric subdivision of S_0, and regard the same map as a map to S under the canonical identification of their realizations. A vertex of S is a nonempty face F of S_0; give it color \dim F+1 and choose one original vertex v_F\in F. Faces in a simplex of S form a strictly increasing chain, so their colors are distinct. If the coordinate at the barycenter of F is positive, then the original coordinate at v_F is positive: the contribution of that barycenter to this coordinate is its positive coefficient divided by the number of vertices of F. Assign to the new vertex the member of \mathcal V already assigned to v_F. This proves Equation (4.1).□

Proof. For a color \ell, let

These continuous color weights sum to one. At any point of S at most one vertex of a given color has positive weight. Partition the first (t+1)g colors into packets

and retain the remaining colors individually. Let w_j(y) be the vector of the t+1 color weights in I_j, and let M_j(y) be their sum. At positive mass, the proportions w_j/M_j record each missing color by a zero coordinate, but division by M_j is undefined at zero mass. We first regularize these proportions near zero to obtain continuous slot maps. During reconstruction we discard packets in this range, so any colors introduced by regularization do not enter the reconstructed point. Choose

Define a continuous simplex-valued function by

The formulas agree at M_j=\tau. For 0<M_j\le\tau, the second is the convex combination of b and w_j/M_j with coefficients 1-M_j/\tau and M_j/\tau; at M_j=0 it equals b. Thus it takes values in \Delta^t throughout. Set \mathcal F_j=\kappa\circ q_j.

To implement this deletion, choose a continuous function \eta:[0,1]\to[0,1] that is zero on [0,\tau] and one on [2\tau,1]. For data as in the statement, give packet j the unnormalized new color vector

and give each leftover color its original weight. Denote all these new color weights by \widehat c_\ell(z), and their sum by Z(z). Then

The last inequality follows from Equation (4.5). Thus division by Z is defined and continuous, even if some packets have been discarded completely.

We first verify that a missing original color stays missing. For a leftover color this is immediate. For a packet whose factor \eta(M_j)M_j is zero it is also immediate. In the remaining case M_j>\tau, so q_j=w_j/M_j exactly. If one original color in that packet has weight zero, q_j lies on \partial\Delta^t and \mathcal F_j lies on \partial C_t. Equation (4.4) forces \kappa^{-1}(v_j)=q_j, and the new weight of that color is zero. We have proved

Assign the normalized color weights to the originally present vertices. More explicitly, for a vertex v of color \ell, put

This function is continuous at points where c_\ell=0 as well: the first ratio lies in [0,1], while the second tends to zero by Equation (4.8) and the positive lower bound for Z. The new vertex weights are nonnegative, sum to one, and have support contained in the original carrier. They therefore define a continuous map \widetilde y into S, with the first asserted property.

Finally, if v_j(z)\in\partial C_t, the vector \kappa^{-1}(v_j(z)) has a zero coordinate. Thus packet j contributes at most t positive colors in Equation (4.7); if the packet is discarded, it contributes none. The s(z) successful packets use disjoint sets of colors. Hence at most m+1-s(z) colors, and therefore at most m+1-s(z) vertices, are positive in the reconstructed point. Its carrier has dimension at most m-s(z).□

Proof. Suppose, to the contrary, that a cover has rate d>2r. By Equation (4.3), d is finite. We choose the parameters so that the bundle embedding and the covering-order decrease have simultaneous strict margins.

Choice of parameters. First choose an integer p\ge1 such that, for t=2p+1,

This is possible because p/(2p+2)\to1/2. Set c_0=d/(t+1)>0, and choose 0<a<1/16 so small that

Choose an integer H_0\ge2 so large that

Next choose b_0>0 satisfying

These choices precede the choice of k.

For m=m_k, put

As k\to\infty,

For every k with g>0, the first inequality in Equation (5.2) gives

The strict inequality follows from b_0<a/32.

The cover construction below treats each interval of length kH_0 with H_0-1 reconstructed length-k blocks and two unchanged edge blocks. Their nerve factors have total dimension at most (H_0+1)m before counting the missing colors. An alternative description uses H_0 unchanged blocks. Joining the two descriptions can add at most H_0m+1 dimensions, and this extra cost will occur on at most a b_0 fraction of the intervals. Each interval has (H_0-1)g slots. Compression puts at least an a/8 fraction of all slots on the boundary; discarding those in these exceptional intervals loses at most a b_0 fraction of all slots. The retained boundary slots therefore reduce the dimension bound by at least (a/8-b_0)(H_0-1)g per interval on average. These three contributions give the prospective cover-rate bound

Its limiting excess over d is

Here we used H_0\ge2 and Equations (5.2) and (5.3). We may therefore fix k large enough that g\ge1 and

In particular R>pg. All of p,t,a,H_0,b_0,k,m,g,R are now fixed. Put e=kH_0.

The subordinate functions and their compression. Apply Corollary 4.2 to \mathcal U^{[0,k)}, and write \phi:X\to S for the resulting map. For every vertex v of S, fix members U(v,s)\in\mathcal U, 0\le s<k, such that

These members come from the assigned element of the orbit join. Use Corollary 4.3 to define

The estimates in Equations (5.5) and (5.7) allow us to use Corollaries 3.2 and 3.4. Keep the open labeled regions O_i, 0\le i<k, and the continuous weight w:X\to[0,1] supplied there. For some J_0, their uniform frequency property is

Choose and fix one J\ge J_0 sufficiently large for Corollary 3.4, and put

The proposition gives a continuous map f:X\to C_t^n whose coordinates are indexed by

Whenever w(h^{qe}y)>0 and h^{qe}y\in O_i, these coordinates satisfy

Moreover the rank-pn torus bundle over the sparse test locus for f embeds in a trivial bundle of rank strictly less than 2pn.

By Corollary 2.1, for arbitrarily large integers u there is a continuous map

with at least (a/8)un boundary slots at every point, agreeing with each original slot of f^{\times u} whenever that slot is on its boundary. Pull it back along the map of actual orbit starts

Denote the resulting slot functions by \widetilde F_{b,q,D,j}(x), with 0\le b<u and the other indices as above. They retain the boundary agreement and the pointwise count

Aligned and fallback descriptions. The orbit pullback has provided the boundary slots. We next encode their reconstructed nerve points in a fixed finite polyhedron, including the points where the alignment label is unavailable. Fix one of these integers u. The uJ successive macro-intervals have starting times

On the open set where w(h^{T_{b,q}}x)>0, let i be the unique label with h^{T_{b,q}}x\in O_i. The label is locally constant there. For each 0\le D<H_0-1, use Corollary 4.3 at the original point

with replacements \widetilde F_{b,q,D,j}(x), 1\le j\le g. Equation (4.4) holds by Equation (5.10) and the boundary agreement of G. Let the resulting continuous map on this open set be \psi_{b,q,D}(x)\in S. It has exact vertex subordination to the displayed original point. If s_{b,q,D}(x) of these g replacement slots are on their boundaries, then

Use the finite polyhedra

The aligned point in the copy of \mathcal A labeled i is

Its first and last factors retain the two edge blocks. The fallback point, defined on all of X, is

In the join, give the aligned branch weight w(h^{T_{b,q}}x) and the fallback branch weight 1-w(h^{T_{b,q}}x). Although the aligned coordinates are needed only where their weight is positive, this defines a continuous map to \mathcal J. To see this directly, embed the finitely many copies of the product polyhedra in Euclidean space. Every aligned coordinate, including the label indicator, is bounded and continuous where its weight is positive; multiplying it by that weight and setting it to zero at weight zero is continuous. These are the usual coordinates for the join. Taking all the macro-intervals gives a continuous map

The descriptions retain the entire orbit cover. We give a finite open cover of \mathcal K_u whose pullback under \Phi_u refines \mathcal U^{[0,uN)}. In one join factor let \omega be the weight of its aligned branch. For an aligned label i, a factor index 0\le\ell\le H_0, and a vertex v of S, the function

is continuous on the join when extended by zero where the aligned branch is collapsed. Similarly, for 0\le\ell<H_0,

is continuous, with value zero where the fallback branch is collapsed. Here a_\ell and b_\ell denote the simplicial factors in the two descriptions.

Form open sets of two kinds: choose a label i and one vertex in each aligned factor, requiring all the corresponding \alpha’s to be positive; or choose one vertex in each fallback factor, requiring all the corresponding \beta’s to be positive. These finitely many sets cover \mathcal J, because at every point some branch has positive weight and every simplicial factor has a positive vertex coordinate. Their products form a finite open cover of \mathcal K_u.

Fix one member of this product cover. In each macro-interval it fixes a branch, its label when aligned, and one positive vertex per simplicial factor. Under \Phi_u, an updated factor with positive vertex v has that same vertex positive in its original point, by Corollary 4.3. Thus Equation (5.8) applies to every chosen factor, updated or unchanged, and gives fixed members of \mathcal U along its length-k orbit block.

For a fallback description the block starts relative to the macro-interval are 0,k,\ldots,(H_0-1)k, so these blocks cover [0,e). For an aligned description the starts are

Their length-k intervals also cover [0,e): the middle blocks cover [i,i+(H_0-1)k), the first edge covers the beginning since 0\le i<k, and the last edge covers the end since i+(H_0-1)k\ge e-k. All of these blocks lie within the macro-interval. For each time in it, choose one of its covering blocks. The fixed vertex and the offset within that block select one fixed member of \mathcal U by Equation (5.8).

Figure 1 shows the two edge overlaps and the fallback partition for one macro-interval.

Doing this in all uJ macro-intervals shows that the whole preimage of the chosen target open set is contained in one fixed member of \mathcal U^{[0,uN)}. The assertion is about the whole preimage; no injectivity of \Phi_u is required.

A finite subcomplex of smaller dimension. The target description now retains every member of the orbit-cover information. It remains to bound the dimension of a single subcomplex containing its image; this is where the pointwise boundary count produces the strict decrease in cover rate. Equip the products of copies of S with their product cells, the joins with the joins of those cells and their endpoint cells, and \mathcal K_u with the resulting product cell structure. These are finite polytopal complexes and admit compatible simplicial subdivisions. In a join, the carrier dimension is the dimension of the retained branch at an endpoint and is the sum of the two carrier dimensions plus one when both weights are positive. Product carrier dimensions add.

Call a macro-interval good at x if w(h^{T_{b,q}}x)=1, and bad otherwise. Apply Equation (5.9) to y=h^{bN}x in each of the u windows. The number B_u(x) of bad macro-intervals satisfies

Each macro-interval has (H_0-1)g replacement slots. Consequently at most b_0un of the boundary slots in Equation (5.11) can belong to bad intervals. If S_u^{\mathrm{good}}(x) is the number of boundary slots in good intervals, then

At a good interval the join is at its aligned endpoint. The two unchanged edge factors cost at most 2m, and Equation (5.12) bounds the other factors. Thus its carrier dimension is at most

At a bad interval we may ignore all successful slots. Its carrier dimension is at most (2H_0+1)m+1: this bounds the sum of the aligned and fallback dimensions plus one, and also bounds the fallback endpoint. Using Equations (5.16) and (5.17), the carrier dimension of \Phi_u(x) is therefore at most

It follows that \Phi_u(X) lies in the finite subcomplex consisting of all cells of dimension at most \lfloor uNE_k\rfloor, together with their faces. In particular this is an actual subcomplex dimension bound, not a bound on dimensions of fibers or on a varying family of images.

Restrict the finite target open cover constructed above to this subcomplex, triangulate it compatibly, and apply Corollary 4.1. We obtain

By Equation (4.3), d is the infimum of all normalized cover orders, so even one such u gives a contradiction. This proves the theorem.□

Proof of Corollary 1.1. Put A=C(X)\rtimes_h\Z. Minimality and infinitude imply that h is free: a periodic orbit would be a finite nonempty closed invariant subset of X, and hence all of X. Niu’s upper bound therefore gives

in the extended nonnegative reals [10, Theorem 5.6].

Let \mathcal R be the set of finite comparison bounds in the definition of the radius of comparison. For every r\in\mathcal R, Corollary 5.1 and the supremum over all finite open covers give

If \mathcal R is nonempty, taking its infimum yields \mdim(X,h)/2\le\rc(A). This passage does not require the infimum itself to belong to \mathcal R. If \mathcal R is empty, the same lower inequality holds because \rc(A)=\infty by definition. Together with Equation (5.18), this proves the equality.

Explicitly, when \mdim(X,h)=\infty, Corollary 5.1 excludes every finite comparison bound, so \rc(A)=\infty. When \mdim(X,h)=0, Equation (5.18) and nonnegativity give \rc(A)=0. The argument above covers all intervening finite values with the same trace normalization.□

Proof of Corollary 1.2. Write A=C(X)\rtimes_h\Z as in the corollary. This crossed product is simple and nuclear [3, Section 3]. It is also separable and unital, since C(X) is separable and unital and the acting group is countable. The canonical inclusion C(X)\subset A makes A infinite-dimensional because X is infinite.

Lindenstrauss–Weiss proved that the small boundary property implies zero mean dimension [9, Theorem 5.4]. Lindenstrauss proved the reverse implication for systems with an infinite minimal factor [8, Theorem 6.2]; here the identity map supplies that factor. These directions are also recalled together in [3, Theorem 2.3]. Elliott–Niu’s Theorem 4.7 gives Jiang–Su stability from zero mean dimension.

For the converse, suppose that A is Jiang–Su stable. Take a,b\in M_\infty(A)_+ with d_\tau(a)<d_\tau(b) for every normalized trace \tau on A, and place both elements in a common finite corner M_q(A). This algebra is simple, exact, unital and Jiang–Su stable. Every normalized trace on M_q(A) is \tau\otimes\operatorname{tr}_q for a normalized trace \tau on A, where \operatorname{tr}_q=\operatorname{Tr}_q/q. Thus, for c\in M_q(A)_+, its rank function for that normalized matrix trace is

where the right side uses the unnormalized matrix extension fixed in Equation (1.1). The strict rank inequalities therefore hold for every normalized trace on M_q(A). Rørdam’s comparison theorem [14, Corollary 4.6] gives a\precsim_{M_q(A)}b, hence a\precsim b in the convention of this paper. Thus 0 itself is a comparison bound, so \rc(A)=0, and Corollary 1.1 gives \mdim(X,h)=0.

Finally, the crossed product satisfies all the hypotheses of [1, Theorems A and B]: it is simple, separable, unital, nuclear and infinite-dimensional. Finite nuclear dimension implies Jiang–Su stability by Winter’s theorem [17]; the converse, with nuclear dimension at most one, is supplied by Castillejos–Evington–Tikuisis–White–Winter. Thus Jiang–Su stability, finite nuclear dimension and nuclear dimension at most one are equivalent for A. This proves the five equivalences. If \mdim(X,h)>0, including the value \infty, their contrapositives give the final assertion.□