Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C*-algebras

TL;DR

This paper establishes a relationship between the comparison radius of crossed product C*-algebras and the mean topological dimension of free minimal dynamical systems, under certain Rokhlin and comparison conditions.

Contribution

It proves that the comparison radius is at most half the mean topological dimension when the system has a uniform Rokhlin property and Cuntz comparison on open sets.

Findings

Comparison radius is at most half the mean topological dimension.

Conditions are satisfied for $ ext{Z}$ actions and certain extensions.

Uses Cuntz comparison and subgroupoid techniques.

Abstract

Let $(X, \Gamma)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $\Gamma$ is a discrete amenable group. It is shown that, if $(X, \Gamma)$ has a version of Rokhlin property (uniform Rokhlin property) and if $\mathrm{C}(X)\rtimes\Gamma$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes \Gamma$ is at most half of the mean topological dimension of $(X, \Gamma)$. These two conditions are shown to be satisfied if $\Gamma = \mathbb Z$ or if $(X, \Gamma)$ is an extension of a free Cantor system and $\Gamma$ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids.

Equations

$$A_1\to A_2\to \cdots \to \underrightarrow{\lim }{A}_i,$$

$$\underset{i\to \infty }{\mathrm{liminf}}\frac{\dim (X_i)}{n_i},$$

$$x_{n}b{y}_n\to a,\mathrm{as}n\to \infty .$$

$$E,F\subseteq X\mathrm{with}\mu (E)<\lambda \mu (F)$$

$$[E]\le m[F]$$

$$\mathrm{rc}(\mathrm{C}(X)⋊\Gamma )\le \frac{1}{2}\mathrm{mdim}(X,\Gamma ).$$

$$Y\gamma =Y,\gamma \in \Gamma ,$$

$$U\cup {\sigma }^{-1}(U)\cup \cdots \cup {\sigma }^{-n}(U)=X.$$

$$\mu (E)=\mu (E\gamma ),\gamma \in \Gamma .$$

$$\frac{\left|FK\Delta F\right|}{\left|F\right|}<\epsilon .$$

$${\mathrm{int}}_K(F)=\{\gamma \in F:\gamma K\subseteq F\}.$$

$$\left|F\setminus {\mathrm{int}}_K(F)\right|\le \left|K\right|\left|FK\setminus F\right|\le \left|K\right|\left|FK\Delta F\right|,$$

$$\frac{\left|F\setminus {\mathrm{int}}_K(F)\right|}{\left|F\right|}<\epsilon .$$

$$\mathrm{ocap}(E):=\underset{n\to \infty }{\lim }\frac{1}{\left|{\Gamma }_n\right|}\underset{x\in X}{\sup }\sum _{\gamma \in {\Gamma }_n}{\chi }_E(x\gamma ),$$

$$D(\mathcal{U})=\min \{\mathrm{ord}(\mathcal{V}):\text{V is an open cover of X and V⪯U}\},$$

$$\mathrm{ord}(\mathcal{V})=-1+\underset{x\in X}{\sup }\sum _{V\in \mathcal{V}}{\chi }_V(x),$$

$$\mathrm{mdim}(X,\Gamma ):=\underset{\mathcal{U}}{\sup }\underset{n\to \infty }{\lim }\frac{1}{\left|{\Gamma }_n\right|}D(\underset{\gamma \in {\Gamma }_n}{\bigvee }{\gamma }^{-1}(\mathcal{U})),$$

$$\{U\cap V:U\in \alpha ,V\in \beta \}$$

$$\mathrm{mdim}(X,\sigma )=\underset{\mathcal{U}}{\sup }\underset{n\to \infty }{\lim }\frac{1}{n}D(\mathcal{U}\vee {\sigma }^{-1}(\mathcal{U})\vee \cdots \vee {\sigma }^{-n+1}(\mathcal{U})).$$

$$\text{C*}\{f,u_{\gamma };u_{\gamma }f{u}_{\gamma }^{\ast }=f(\cdot \gamma )=f\circ \gamma ,u_{{\gamma }_1}{u}_{{\gamma }_2}^{\ast }=u_{{\gamma }_1{\gamma }_2^{-1}},u_e=1,f\in \mathrm{C}(X),\gamma ,{\gamma }_1,{\gamma }_2\in \Gamma \}.$$

$$\underset{n\to \infty }{\lim }{x}_{n}b{y}_n=a,$$

$${\mathrm{M}}_{\infty }(A)=\bigcup _{n=1}^{\infty }{\mathrm{M}}_n(A),$$

$${\mathrm{d}}_{\tau }(a):=\underset{n\to \infty }{\lim }(\tau \otimes \mathrm{Tr})(a^{\frac{1}{n}})={\mu }_{\tau \otimes \mathrm{Tr}}(\mathrm{sp}(a)\cap (0,+\infty )),a\in {\mathrm{M}}_{\infty }(A{)}^{+},$$

$${\mathrm{d}}_{\tau }(a)\le {\mathrm{d}}_{\tau }(b).$$

$${\mathrm{d}}_{{\tau }_{\mu }}=\mu (f^{-1}(0,+\infty )),$$

$${\tau }_{\mu }(f)=\int f\mathrm{d}\mu ,f\in \mathrm{C}(X).$$

$$E=f^{-1}(0,+\infty )\mathrm{and}F=g^{-1}(0,+\infty ).$$

$$E={\phi }_E^{-1}(0,+\infty ).$$

$$(a-\epsilon {)}_{+}=f(a)\in A,$$

$${\mathrm{d}}_{\tau }(a)+r<{\mathrm{d}}_{\tau }(b),\tau \in \mathrm{T}(A),$$

Full text

These two conditions are shown to be satisfied if $\Gamma=\mathbb{Z}$ or if $(X,\Gamma)$ is an extension of a free Cantor system and $\Gamma$ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids.

The research is partially supported by a Simons Collaboration Grant (Grant #317222) and partially supported by an NSF grant (DMS-1800882)

1. 1 Introduction
2. 2 Notation and preliminaries
3. 3 Uniform Rokhlin property and structure of $\mathrm{C}(X)\rtimes\Gamma$
4. 4 Comparison of open sets, comparison radius, and mean topological dimension
5. 5 Recursive subhomogeneous C*-algebras with diagonal maps
6. 6 Small subgroupoids and recursive subhomogeneous C*-algebras
7. 7 Comparison of diagonal elements and comparison of open sets
8. 8 Lower semicontinuous set-valued functions and small subgroupoids

1. Introduction

Consider a topological dynamical system $(X,\Gamma)$, where $X$ is a compact Hausdorff space and $\Gamma$ is a discrete amenable group. The mean (topological) dimension of $(X,\Gamma)$, denoted by $\mathrm{mdim}(X,\Gamma)$, was introduced by Gromov ([21]), and then was developed and studied systematically by Lindenstrauss and Weiss ([32]). It is a numerical invariant, taking value in $[0,+\infty]$, to measure the complexity of $(X,\Gamma)$ in terms of dimension growth with respect to partial orbits.

On the other hand, the concept of dimension growth also appears in the classification theory of C*-algebras, and it is used to provide a condition for a certain C*-algebra to be classified by its Elliott invariant. For example, consider a unital inductive system of C*-algebras

[TABLE]

where each $A_{i}=\mathrm{M}_{n_{i}}(\mathrm{C}(X_{i}))$ is the algebra of $\mathrm{M}_{n_{i}}(\mathbb{C})$-valued continuous functions on some compact separable Hausdorff space $X_{i}$; its dimension growth is defined as

[TABLE]

where $\mathrm{dim}(X_{i})$ is the topological covering dimension of $X_{i}$. If the dimension growth is [math], then the isomorphism classes of the limit C*-algebras $\varinjlim A_{i}$ are classified by their Elliott invariant ([19], [9], [42]). (See [20], [11] [14], and [10] for further developments of the classification of C*-algebras in this direction.)

For a general abstract C*-algebra, it might not have an obvious inductive limit decomposition in terms of homogeneous or subhomogeneous C*-algebras as the example above. In such cases, comparison radius, introduced by Toms ([40]) and denoted by $\mathrm{rc}(A)$, plays a role as the dimension growth of such a C*-algebra $A$.

Let us start with some comparison theory of C*-algebras. Let $a,b$ be positive elements of a matrix algebra over $A$. Then $a$ is said to be Cuntz sub-equivalent to $b$ if there are sequences $(x_{n}),(y_{n})$ in a matrix algebra over $A$ such that

[TABLE]

In the case that $a,b$ are projections, the Cuntz sub-equivalence relation recovers the classical Murray-von Neumann sub-equivalence relation; and moreover, if $A=\mathrm{C}(X)$, $a$ is Cuntz sub-equivalent to $b$ if and only if the vector bundle over $X$ induced by $a$ is isomorphic to a sub-bundle of the the vector bundle induced by $b$.

If $a$ is Cuntz sub-equivalent to $b$, then the rank function induced by $a$ is always dominated by the rank function induced by $b$, just like the case above, the rank of the vector bundle induced by $a$ is at most the rank of the vector bundle induced by $b$. The converse is not true in general, as there are many examples of a pair of vector bundles such that the bundle with smaller rank is not isomorphic to a sub-bundle of the other one.

However, if the gap between the ranks of two (complex) vector bundles is sufficiently large (larger than $\frac{1}{2}\mathrm{dim(X)}$), then the converse is true; that is, the vector bundle with small dimension is isomorphic to a sub-bundle of the vector bundle with larger dimension.

The comparison radius is the infimum of all the gaps of the ranks so that the converse does hold (see Definition 2.13), and a typical example is that the comparison radius of $\mathrm{M}_{n}(\mathrm{C}(X))$ is at most $\frac{1}{2}\frac{\mathrm{dim}(X)}{n}$, that is, half of the dimension ratio of $\mathrm{M}_{n}(\mathrm{C}(X))$. Thus, in general, the comparison radius is regarded as a version of dimension growth for an abstract C*-algebra.

For the given topological dynamical system $(X,\Gamma)$, the natural C*-algebra to be considered is the crossed product C*-algebra $\mathrm{C}(X)\rtimes\Gamma$. It would be interesting to compare the mean dimension of $(X,\Gamma)$ (the dynamical dimension growth) with the comparison radius of $\mathrm{C}(X)\rtimes\Gamma$ (the C*-algebraic dimension growth), and this is the main motivation of this paper. Indeed, Phillips and Toms conjectured that the comparison radius of $\mathrm{C}(X)\rtimes\Gamma$ should equal half of the mean dimension of $(X,\Gamma)$.

Many results in this direction have been obtained: For free minimal $\mathbb{Z}$-actions, if $X$ is finite dimensional (hence the dynamical system has mean dimension zero), it was shown in [43] that the algebra $\mathrm{C}(X)\rtimes\mathbb{Z}$ has finite nuclear dimension; therefore the C*-algebra has strict comparison of positive elements, and the radius of comparison is zero. Still with the assumption that $X$ is finite dimensional, this result was generalized to free minimal actions by $\mathbb{Z}^{d}$ in [39] or by a group with comparison property in [28].

Without assuming $X$ to be finite dimensional, for a free minimal $\mathbb{Z}$-action with zero mean dimension, it was shown in [15] that the crossed product C*-algebra absorbs the Jiang-Su algebra (and therefore has finite nuclear dimension by [16]). In particular, this implies that the comparison radius is [math]. For free minimal $\mathbb{Z}$-actions without zero mean dimension, Phillips showed in [36] that the radius of comparison of $\mathrm{C}(X)\rtimes\mathbb{Z}$ is at most $1+36\mathrm{mdim}(X,\sigma,\mathbb{Z})$.

The argument in [43], [15], or [36] relies on the Putnam’s orbit-cutting algebra (or the large sub-algebra) $A_{y}$; and in the case of zero mean dimension, the argument in [15] also heavily depends on the small boundary property (which is equivalent to mean dimension zero in the case of $\mathbb{Z}$-actions).

However, beyond the $\mathbb{Z}$-action case, it is not clear in general how to construct large sub-algebras; moreover, once the dynamical system does not have mean dimension zero, the small boundary property does not hold anymore.

To deal with these difficulties, in this paper we consider the following properties of the dynamical systems and the crossed product C*-algebras:

• •

Uniform Rokhlin Property (URP). The topological dynamical system $(X,\Gamma)$ is said to have Uniform Rokhlin Property (URP) if there exist disjoint towers with shapes arbitrarily invariant and all level sets are open, such that the complement of the towers has arbitrarily small orbit capacity (see Definition 3.1).

• •

Cuntz comparison of open sets (COS). Consider the C*-algebra $\mathrm{C}(X)\rtimes\Gamma$. Note that any open set $E\subseteq X$ represents a Cuntz equivalence class of $\mathrm{C}(X)$ and hence a Cuntz equivalence class of $\mathrm{C}(X)\rtimes\Gamma$. Denote it by $[E]$. Then the dynamical system $(X,\Gamma)$ is said to have Cuntz comparison of open sets (COS) if there are $\lambda\in\mathbb{R}^{+}$ and $m\in\mathbb{N}$ such that for any open sets

[TABLE]

for all ergodic measures $\mu$, one has that

[TABLE]

in the Cuntz semigroup of $\mathrm{C}(X)\rtimes\Gamma$. (See Definition 4.1.)

A consequence of the (URP) is that the crossed product C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ can be (weakly) tracially approximated by homogeneous C*-algebras with dimension ratio at most the mean dimension of $(X,\Gamma)$ (see Theorem 3.9); roughly speaking, any element of $\mathrm{C}(X)\rtimes\Gamma$ can be approximately decomposed as the (not necessary orthogonal) sum of an element in a homogeneous C*-algebra with dimension ratio at most $\mathrm{mdim}(X,\Gamma)$ and an element which is uniformly small under all traces (i.e., under all ergodic measures).

This tracial approximation property, together with Cuntz comparison of open sets, implies that the radius of comparison of the crossed product C*-algebra is at most half of the mean dimension:

Theorem** (Theorem 4.8).**

Let $(X,\Gamma)$ be a free and minimal topological dynamical system satisfying the (URP) and (COS). Then,

[TABLE]

All minimal $\mathbb{Z}$-actions have the (URP) (see Lemma 3.6); and if $(X,\Gamma)$ is an extension of a free minimal system with small boundary property, it has the (URP) (see Corollary 3.8).

To investigate when $\mathrm{C}(X)\rtimes\Gamma$ has the (COS), small subgroupoids of $X\rtimes\Gamma$ are considered. These are the open and relatively compact subgroupoids of $X\rtimes\Gamma$, and they can be regarded as a local version of the orbit-cutting subalgebra $A_{y}$. It is well known that the C*-algebra of a small subgroupoid is subhomogeneous (i.e., the dimensions of its irreducible representations are uniformly bounded).

It turns out that the C*-algebra of a small subgroupoid is rather special: it is a recursive subhomogeneous C*-algebra with diagonal maps (see Theorem 6.15); and with a revised argument of [15], the Cuntz class of diagonal elements of such a C*-algebra are shown to be determined by their ranks (see Theorem 7.8), provided that the dimensions of the irreducible representations are sufficiently large, but regardless of the topological dimension of its spectrum.

This comparison property of diagonal elements leads to a Cuntz comparison of open sets for $(X,\Gamma)$: if $\Gamma$ is amenable and $X\rtimes\Gamma$ has small subgroupoids with each orbit arbitrarily invariant, then the dynamical system $(X,\Gamma)$ has the Cuntz-comparison property on open sets (see Corollary 7.13). Such small subgroupoids always exist if

• •

$\Gamma=\mathbb{Z}$, or

• •

$(X,\Gamma)$ is an extension of a free Cantor system and $\Gamma$ has subexponential growth.

Therefore, in these cases, the dynamical system $(X,\Gamma)$ has the (COS) (see Corollary 7.9 and Corollary 8.11); in particular, the estimate (1.1) holds (see Corollary 7.10 and Corollary 8.12).

Remark 1.1*.*

In [33], using the adding-one-dimension and going-down argument of [23], the (COS) and (URP) (and hence the estimate (1.1)) are obtained for arbitrary minimal and free $\mathbb{Z}^{d}$-actions. Hence (1.1) holds for all minimal free $\mathbb{Z}^{d}$-actions.

In [34], still under the assumption of the (COS) and (URP), it is shown that $\mathrm{mdim}(X,\Gamma)=0$ implies that the C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ is classified by its Elliott invariant.

2. Notation and preliminaries

2.1. Topological Dynamical Systems

Definition 2.1**.**

A topological dynamical system $(X,\Gamma)$ consists of a separable compact Hausdorff space $X$, a discrete group $\Gamma$, and a homomorphism $\Gamma\to\mathrm{Homeo}(X)$, where $\mathrm{Homeo}(X)$ is the group of homeomorphisms of $X$, acting on $X$ from the right. In this paper, we frequently omit the word topological, and just refer it as a dynamical system.

The dynamical system $(X,\Gamma)$ is said to be free if $x\gamma=x$ implies $\gamma=e$, where $x\in X$ and $\gamma\in\Gamma$.

A closed set $Y\subseteq X$ is said to be invariant if

[TABLE]

and the dynamical system $(X,\Gamma)$ is said to be minimal if $\varnothing$ and $X$ are the only invariant closed subsets.

If $\Gamma=\mathbb{Z}$, then $\sigma$ is induced by a single homeomorphism of $X$, which is still denoted by $\sigma$. In this case, the dynamical system is denoted by $(X,\sigma)$.

Remark 2.2*.*

In the case $\Gamma=\mathbb{Z}$, it is well known that $(X,\sigma)$ is minimal if and only if one of the following holds:

1. (1)

if $Y\subseteq X$ is closed and $\sigma(Y)\subseteq Y$, then $Y=X$ or $Y=\varnothing$; 2. (2)

for any $x\in X$, the forward orbit $\{x,\sigma(x),\sigma^{2}(x)...\}$ is dense; 3. (3)

for any $x\in X$, the orbit $\{...,\sigma^{-1}(x),x,\sigma(x),...\}$ is dense; 4. (4)

for any non-empty open set $U\subseteq X$, there is $n\in\mathbb{N}$ such that

[TABLE]

Definition 2.3**.**

A Borel measure $\mu$ on $X$ is invariant if for any Borel set $E\subseteq X$, one has

[TABLE]

Denote by $\mathcal{M}_{1}(X,\Gamma)$ the set of all invariant Borel probability measures on $X$. It is a Choquet simplex under the weak* topology.

Definition 2.4**.**

Let $\Gamma$ be a (countable) discrete group. Let $K\subseteq\Gamma$ be a finite set and let $\delta>0$. Then a finite set $F\subseteq\Gamma$ is said to be $(K,\varepsilon)$-invariant if

[TABLE]

The group $\Gamma$ is amenable if there is a sequence $(\Gamma_{n})$ of finite subsets of $\Gamma$ such that for any $(K,\varepsilon)$, the set $\Gamma_{n}$ is $(K,\varepsilon)$-invariant if $n$ is sufficiently large. The sequence $(\Gamma_{n})$ is called a Følner sequence.

The $K$-interior of a finite set $F\subseteq\Gamma$ is defined as

[TABLE]

Note that

[TABLE]

and hence for any $\varepsilon>0$, if $F$ is $(K,\frac{\varepsilon}{\left|K\right|})$-invariant, then

[TABLE]

Definition 2.5** (see [32]).**

Consider a topological dynamical system $(X,\Gamma)$, where $\Gamma$ is amenable, and let $E\subseteq X$. The orbit capacity of $E$ is defined by

[TABLE]

where $(\Gamma_{n})$ is a Følner sequence, and $\chi_{E}$ is the characteristic function of $E$. The limit always exists and is independent from the choice of the Følner sequence $(\Gamma_{n})$.

Remark 2.6*.*

Orbit capacity has the following properties:

1. (1)

If $E$ is a closed set, then $\mathrm{ocap}(E)\leq\varepsilon$ if and only if $\mu(E)\leq\varepsilon$, $\mu\in\mathcal{M}_{1}(X,\Gamma)$. 2. (2)

The orbit capacity is semicontinuous in the sense that for any closed set $D\subseteq X$ and any $\varepsilon>0$, there is an open neighbourhood $U\supseteq D$ such that $\mathrm{ocap}(U)<\mathrm{ocap}(D)+\varepsilon$.

Definition 2.7** (see [21] and [32]).**

Let $\mathcal{U}$ be an open cover of $X$. Define

[TABLE]

where

[TABLE]

and $\mathcal{V}\preceq\mathcal{U}$ means that, for any $V\in\mathcal{V}$, there is $U\in\mathcal{U}$ with $V\subseteq U$.

Consider a topological dynamical system $(X,\Gamma)$, where $\Gamma$ is a discrete amenable group. The mean topological dimension is defined by

[TABLE]

where $\mathcal{U}$ runs over all finite open covers of $X$, $(\Gamma_{n})$ is a Følner sequence (the limit is independent from the choice of $(\Gamma_{n})$), and $\alpha\vee\beta$ denotes the open cover

[TABLE]

for any open covers $\alpha$ and $\beta$.

Note that in the case $\Gamma=\mathbb{Z}$, one has

[TABLE]

Remark 2.8*.*

It is shown in [5] that, if $\Gamma$ is a countable infinite amenable group, then each number in $[0,+\infty]$ can be realized as the mean dimension of a minimal dynamical system $(X,\Gamma)$.

2.2. Crossed product C*-algebras

Consider a topological dynamical system $(X,\Gamma)$. The (full) crossed product C*-algebra $A=\mathrm{C}(X)\rtimes\Gamma$ is defined to be the universal C*-algebra

[TABLE]

The C*-algebra $A$ is nuclear (Corollary 7.18 of [44]) if $\Gamma$ is amenable. If, moreover, $(X,\Gamma)$ is minimal and topologically free, the C*-algebra $A$ is simple (Theorem 5.16 of [8] and Théorème 5.15 of [45]), i.e., $A$ has no non-trivial two-sided ideals.

2.3. Comparison for positive elements of a C*-algebra

Definition 2.9**.**

Let $A$ be a C*-algebra, and let $a,b\in A^{+}$. We say that $a$ is Cuntz sub-equivalent to $b$, denoted by $a\precsim b$, if there are $x_{n}$, $y_{n}$, $n=1,2,...$, such that

[TABLE]

and we say that $a$ is Cuntz equivalent to $b$ if $a\precsim b$ and $b\precsim a$.

Denote by $\mathrm{M}_{n}(A)$ the C*-algebra of $n\times n$ matrices over $A$. Regard $\mathrm{M}_{n}(A)$ as the upper-left conner of $\mathrm{M}_{n+1}(A)$, denote by

[TABLE]

the algebra of all finite matrices over $A$, and denote by $\mathrm{Tr}$ the standard (unbounded) trace of $\mathrm{M}_{\infty}(A)$.

Let $\tau:A\to\mathbb{C}$ be a tracial state. Then define the rank function

[TABLE]

where $\mu_{\tau\otimes\mathrm{Tr}}$ is the Borel measure induced by ${\tau\otimes\mathrm{Tr}}$ on $\mathrm{sp}(a)$, the spectrum of $a$.

It is well known that for any $a,b\in\mathrm{M}_{n}(A)^{+}$ for some $n\in\mathbb{N}$, if $a\precsim b$, then

[TABLE]

Example 2.10*.*

Consider $f\in\mathrm{C}(X)^{+}$ and let $\mu$ be a Borel probability measure on $X$. Then

[TABLE]

where $\tau_{\mu}$ is the trace of $\mathrm{C}(X)$ defined by

[TABLE]

Let $f,g\in\mathrm{C}(X)$ be positive functions, and define the open sets

[TABLE]

Then $f\precsim g$ if, and only if, $E\subseteq F$. That is, the Cuntz equivalence classes of $f$ and $g$ are determined by their open supports.

For each open set $E\subseteq X$, pick a continuous function $\varphi_{E}:X\to[0,+\infty)$ satisfying

[TABLE]

For instance, one can pick $\varphi_{E}(x)=d(x,X\setminus E)$, where $d$ is a compatible metric on $X$. The notation $\varphi_{E}$ will be used throughout the paper. Note that the Cuntz equivalence class of $\varphi_{E}$ is independent of the choice of the individual function $\varphi_{E}$.

Definition 2.11**.**

Let $a\in A^{+}$, where $A$ is a C*-algebra, and let $\varepsilon>0$. Define

[TABLE]

where $f(t)=\max\{t-\varepsilon,0\}$.

The following is a frequently used fact on Cuntz comparison.

Lemma 2.12** (Section 2 of [38]).**

Let $a,b$ be positive elements of a C-algebra $A$. Then $a\precsim b$ if and only if $(a-\varepsilon)_{+}\precsim b$ for all $\varepsilon>0$.*

Definition 2.13** (Definition 6.1 of [40]).**

The radius of comparison of a unital C*-algebra $A$, denoted by $\mathrm{rc}(A)$, is the infimum of the set of real numbers $r>0$ such that if $a,b\in\mathrm{M}_{\infty}(A)^{+}$ satisfy

[TABLE]

then $a\precsim b$, where $\mathrm{T}(A)$ is the simplex of tracial states. (In [40], the radius of comparison is defined in terms of quasitraces instead of traces; but since all the algebras considered in this paper are nuclear, by [24], any quasitrace actually is a trace.)

Example 2.14*.*

Let $X$ be a compact Hausdorff space. Then

[TABLE]

where $\mathrm{dim(X)}$ is the topological covering dimension of $X$ (a lower bound of $\mathrm{rc}(\mathrm{C}(X))$ in terms of cohomological dimension is given in [12]).

The main purpose of this paper is to investigate whether the dynamical version of (2.2) holds, that is, whether one has

[TABLE]

3. Uniform Rokhlin property and structure of $\mathrm{C}(X)\rtimes\Gamma$

Let us first consider a Rokhlin property for a dynamical system $(X,\Gamma)$. It turns out that this Rokhlin property implies that the crossed-product C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ can be weakly tracially approximated by (not necessary unital) homogeneous C*-algebras.

3.1. Uniform Rokhlin Property

Definition 3.1**.**

A topological dynamical system $(X,\Gamma)$, where $\Gamma$ is a discrete amenable group, is said to have Uniform Rokhlin Property (URP) if for any $\varepsilon>0$ and any finite set $K\subseteq\Gamma$, there exist closed sets $B_{1},B_{2},...,B_{S}\subseteq X$ and $(K,\varepsilon)$-invariant sets $\Gamma_{1},\Gamma_{2},...,\Gamma_{S}\subseteq\Gamma$ such that

[TABLE]

are mutually disjoint and

[TABLE]

In fact, in the definition of the (URP), the base sets $B_{1},...,B_{S}$ can also be assumed to be open:

Lemma 3.2**.**

A topological dynamical system $(X,\Gamma)$ has (URP) if, and only if, it satisfies Definition 3.1 but with $B_{1}$, …, $B_{S}$ being open sets, instead.

Proof.

Let $(K,\varepsilon)$ be given. Assume there exist closed sets $B_{1},B_{2},...,B_{S}\subseteq X$ and $(K,\varepsilon)$-invariant sets $\Gamma_{1},\Gamma_{2},...,\Gamma_{S}\subseteq\Gamma$ such that

[TABLE]

are mutually disjoint and

[TABLE]

Since each $B_{s}$, $s=1,...,S$, is compact, one can choose an open neighbourhood $B_{s}^{\prime}$ of $B_{s}$ such that

[TABLE]

are mutually disjoint. Then

[TABLE]

and thus satisfies Definition 3.1 with open base sets.

For the converse, assume there exist open sets $B_{1},B_{2},...,B_{S}\subseteq X$ and $(K,\varepsilon)$-invariant sets $\Gamma_{1},\Gamma_{2},...,\Gamma_{S}\subseteq\Gamma$ such that

[TABLE]

are mutually disjoint and

[TABLE]

Consider the closed set $X\setminus\bigsqcup_{s=1}^{S}\bigsqcup_{\gamma\in\Gamma_{s}}B_{s}\gamma$. It has an open neighbourhood $U$ such that $\mathrm{ocap}(U)<\varepsilon$ (see Remark 2.6), and there is a closed subset $B_{s}^{\prime}\subseteq B_{s}$, $s=1,...,S$, such that

[TABLE]

Indeed, $B_{s}^{\prime}$ can be chosen as $\bigcup_{\gamma\in\Gamma_{s}}(U^{c}\cap(B_{s}\gamma))\gamma^{-1}$ (note that $U^{c}$ is closed in $X$, $U^{c}\subseteq\bigsqcup_{s=1}^{S}\bigsqcup_{\gamma\in\Gamma_{s}}B_{s}\gamma$, and each $B_{s}\gamma$ is open; so each set $U^{c}\cap(B_{s}\gamma)$ is closed in $X$). Then the closed sets $B_{s}^{\prime}$ satisfy Definition 3.1 (with base sets closed). ∎

Theorem 3.3** (Theorem 5.5 of [28]).**

If $(X,\Gamma)$ is free and has small boundary property, then $(X,\Gamma)$ has the (URP).

Remark 3.4*.*

In the case that $\Gamma=\mathbb{Z}^{d}$, it follows from Theorem 1.10.1 of [22], that if $(X,\mathbb{Z}^{d})$ is an extension of free dynamical system with the small boundary property, then it has the Topological Rokhlin Property in the sense of 1.9 of [22] (actually, the proof of 1.10.1 of [22] shows that the dynamical system $(X,\mathbb{Z}^{d})$ has the (URP)).

Remark 3.5*.*

The level sets of the towers obtained in [28] actually have arbitrarily small diameter, and this is referred by the authors as the property of almost finiteness in measure (Definition 3.5 of [28]). This property is shown to be equivalent to the small boundary property (Theorem 5.5 of [28]). In the case that $X$ is a Cantor set, the result is also obtained in [6].

It follows from Corollary 3.4 of [31] that any free minimal $\mathbb{Z}$-action has the (URP):

Lemma 3.6**.**

Let $\sigma:X\to X$ be a homeomorphism, and assume that $(X,\sigma)$ is an extension of a free minimal system. Then, for any $\varepsilon>0$, any $N\in\mathbb{N}$, there is a closed set $B\subseteq X$ such that

[TABLE]

are disjoint and

[TABLE]

In particular, $(X,\sigma)$ has the (URP) (with $S=1$).

Proof.

Let $\varepsilon>0$ and $N\in\mathbb{N}$ be arbitrary. It follows from Corollary 3.4 of [31] that there is a continuous function $n:X\to[0,+\infty)$ such that

[TABLE]

Consider the level sets

[TABLE]

and the (open) set

[TABLE]

where

[TABLE]

Note that

[TABLE]

[TABLE]

and

[TABLE]

Define

[TABLE]

and define

[TABLE]

For each $l=0,1,...,N-1$, since $E^{\prime}_{iN+l}$, $i=0,1,...,$ is eventually empty, the set $E_{l}$ is closed, and hence the set $B$ is closed as well. Since

[TABLE]

and $E_{0}$, $E_{1}$, …, $E_{N-1}$ are mutually disjoint, one has that

[TABLE]

are mutually disjoint. Thus, $B$, $\sigma(B)$, …, $\sigma^{N-1}(B)$ form a tower.

Let us estimate $\mathrm{ocap}(X\setminus\bigsqcup_{l=0}^{N-1}\sigma^{l}(B))$. Note that

[TABLE]

and hence by (3.1),

[TABLE]

Therefore, for any $l=1,2,...,N-1$, one has

[TABLE]

and thus, together with (3.2),

[TABLE]

as desired. ∎

Lemma 3.7**.**

If a dynamical system $(X,\Gamma)$ is an extension of a dynamical system $(Y,\Gamma)$ which is free and has the (URP), then $(X,\Gamma)$ has the (URP).

Proof.

Denote the quotient map from $(X,\Gamma)$ to $(Y,\Gamma)$ by $\pi$. Note that for any set $E\subseteq Y$, one has $\mathrm{ocap}(\pi^{-1}(E))\leq\mathrm{ocap}(E)$. Since $(Y,\Gamma)$ has the (URP), for any $\varepsilon>0$ and any finite set $K\subseteq\Gamma$, there exist closed sets $B_{1},B_{2},...,B_{S}\subseteq X$ and $(K,\varepsilon)$-invariant sets $\Gamma_{1},\Gamma_{2},...,\Gamma_{S}\subseteq\Gamma$ such that

[TABLE]

are mutually disjoint and

[TABLE]

Then the closed sets

[TABLE]

together with $\Gamma_{s}$, $s=1,2,...,S$, form disjoint Rokhlin towers of $(X,\Gamma)$ with complement of orbit capacity at most $\varepsilon$. ∎

Since any free topological dynamical system with the small boundary property has the (URP) (Theorem 5.5 of [28]; see Theorem 3.3 above), one has the following corollary.

Corollary 3.8**.**

If a topological dynamical system $(X,\Gamma)$ is an extension of a free dynamical system with small boundary property, then $(X,\Gamma)$ has the (URP). In particular, if $(X,\Gamma)$ is an extension of a free minimal Cantor system, $(X,\Gamma)$ has the (URP).

3.2. A tracial approximation structure for $\mathrm{C}(X)\rtimes\Gamma$

Considering a topological dynamical system $(X,\Gamma)$ with the (URP), let us show that the C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ can be (weakly) tracially approximated by (not necessarily unital) homogeneous C*-algebras with dimension ratio almost dominated by the mean dimension of $(X,\Gamma)$.

Theorem 3.9**.**

Let $(X,\Gamma)$ be a dynamical system with the (URP). The C-algebra $A:=\mathrm{C}(X)\rtimes\Gamma$ has the following property: For any finite set $\{f_{1},f_{2},...,f_{n}\}\subseteq\mathrm{M}_{m}(A)$ (where $m\in\mathbb{N}$), $h\in\mathrm{C}(X)^{+}$ with $h(x)\geq\frac{3}{4}$, $x\in F$ for a closed set $F\subseteq X$, and any $\delta>0$, there exist a positive element $p\in\mathrm{C}(X)\subseteq A$ with $\left\|p\right\|\leq 1$, a sub-C*-algebra $C\subseteq A$ with $C\cong\bigoplus_{s=1}^{S}\mathrm{M}_{K_{s}}(\mathrm{C}_{0}(Z_{s}))$ for some $K_{1},...,K_{S}\in\mathbb{N}$ and some locally compact Hausdorff spaces $Z_{1},...,Z_{S}$ together with compact subsets $[Z_{s}]\subseteq Z_{s}$, $s=1,...,S$, $\{f^{\prime}_{1},f_{2}^{\prime},...,f_{n}^{\prime}\}\subseteq\mathrm{M}_{m}(A)$, and $h^{\prime}\in\mathrm{C}(X)^{+}$ such that if $p_{m}:=p\otimes 1_{m}\in\mathrm{M}_{m}(A)\cong A\otimes\mathrm{M}_{m}(\mathbb{C})$, then*

1. (1)

$\left\|h-h^{\prime}\right\|<\delta$, $\left\|f_{i}-f^{\prime}_{i}\right\|<\delta$, $i=1,2,...,n$; 2. (2)

$\left\|p_{m}f^{\prime}_{i}-f_{i}^{\prime}p_{m}\right\|<\delta$, $i=1,2,...,n$; 3. (3)

$p\in C$, $ph^{\prime}p\in C$, and $p_{m}f^{\prime}_{i}p_{m}\in\mathrm{M}_{m}(C)$, $i=1,2,...,n$; 4. (4)

$\frac{\mathrm{dim}([Z_{s}])}{K_{s}}<\mathrm{mdim}(X,\Gamma)+\delta$, $s=1,...,S$; 5. (5)

$\mathrm{\mu}(X\setminus p^{-1}(1))<\delta$, $\mu\in\mathcal{M}_{1}(X,\Gamma)$; 6. (6)

under the isomorphism $C\cong\bigoplus_{s=1}^{S}\mathrm{M}_{K_{s}}(\mathrm{C}_{0}(Z_{s}))$, one has

[TABLE]

where $\mu$ runs through $\mathcal{M}_{1}(X,\Gamma)$; 7. (7)

under the isomorphism $C\cong\bigoplus_{s=1}^{S}\mathrm{M}_{K_{s}}(\mathrm{C}_{0}(Z_{s}))$, the element $p$ has the form

[TABLE]

where $p_{s,i}:Z_{s}\to[0,1]$, and

[TABLE]

In particular, one has

[TABLE]

and

[TABLE] 8. (8)

still under the isomorphism $C\cong\bigoplus_{s=1}^{S}\mathrm{M}_{K_{s}}(\mathrm{C}_{0}(Z_{s}))$, any diagonal element of $C$ actually is in $\mathrm{C}(X)$, and if $f\in C^{+}$ is an diagonal element satisfying $f|_{[Z_{s}]}=1_{K_{s}}$, $s=1,...,S$, then, as an element of $\mathrm{C}(X)$,

[TABLE]

Before proving Theorem 3.9, we have the following two lemmas on partition of unity, which are elementary and might be well known.

Lemma 3.10**.**

Let $V_{1},...,V_{n}$ be open subsets of $X$, where $X$ is a separable locally compact Hausdorff space. Let $D$ be a compact subset of $\bigcup_{i=1}^{n}V_{i}$. Then there are continuous functions $\phi_{i}:X\to[0,1]$, $i=1,...,n$, such that

1. (1)

$\mathrm{supp}(\phi_{i})\subseteq V_{i}$, $i=1,...,n$, 2. (2)

$\sum_{i=1}^{n}\phi_{i}(x)=1$, $x\in D$.

Proof.

Pick open sets $V_{i}^{\prime}\subseteq V_{i}$, $i=1,...,n$, such that $\overline{V^{\prime}_{i}}\subseteq V_{i}$ and $D\subseteq\bigcup_{i=1}^{n}V_{i}^{\prime}$. Pick continuous functions $g_{i}:X\to[0,1]$ such that

[TABLE]

Define

[TABLE]

It is clear that $\mathrm{supp}(\phi_{i})\subseteq V_{i}$, $i=1,...,n$. Note that

[TABLE]

and hence by (3.3),

[TABLE]

as desired. ∎

Lemma 3.11**.**

Let $X$ be a separable compact Hausdorff space, and let $\mathcal{V}$ and $\mathcal{W}$ be two finite collections of open subsets such that $\mathcal{V}\cup\mathcal{W}$ forms a cover of $X$. Assume there are continuous functions $\phi_{V}:X\to[0,1]$, $V\in\mathcal{V}$ such that

1. (1)

$\mathrm{supp}(\phi_{V})\subseteq V$, $V\in\mathcal{V}$, 2. (2)

$\sum_{V\in\mathcal{V}}\phi_{V}(x)\leq 1$, $x\in X$, and 3. (3)

$\mathrm{int}((\sum_{V\in\mathcal{V}}\phi_{V})^{-1}(1))\cup\bigcup_{W\in\mathcal{W}}W=X$.

Then, there are continuous functions $\phi_{W}$, $W\in\mathcal{W}$, such that

[TABLE]

form a partition of unity subordinate to $\mathcal{V}\cup\mathcal{W}$.

Proof.

Consider the function

[TABLE]

and list $\mathcal{W}=\{W_{1},W_{2},...,W_{n}\}$. Since $\mathrm{int}(g^{-1}(1))\cup\bigcup_{W\in\mathcal{W}}W=X$, there are open sets $W^{\prime}_{1}$, $W^{\prime}_{2}$, …, $W^{\prime}_{n}$ such that $\overline{W^{\prime}_{1}}\subseteq W_{1}$, $\overline{W^{\prime}_{2}}\subseteq W_{2}$, …, $\overline{W^{\prime}_{2}}\subseteq W_{2}$ and

[TABLE]

Pick continuous functions $g_{1}$, $g_{2}$, …, $g_{n}$ such that $g_{i}|_{W^{\prime}_{i}}=1$, $g_{i}|_{W_{i}^{c}}=0$, $i=1,2,...,n$.

Define

[TABLE]

It is clear that $\mathrm{supp}(h_{i})\subseteq W_{i}$. Moreover,

[TABLE]

Then

[TABLE]

and hence

[TABLE]

Note that

[TABLE]

and then by (3.4),

[TABLE]

Then

[TABLE]

form a partition of unity subordinate to $\mathcal{V}\cup\mathcal{W}$, as desired. ∎

The following lemma particularly asserts that each Rokhlin tower corresponds to a matrix algebra over the C*-algebra of the base set.

Lemma 3.12**.**

Let $A$ be a C-algebra. Let $u_{1},u_{2},...,u_{n}\in A$ be unitaries with $u_{1}=1$, and let $\mathcal{G}\subseteq A$ be a finite set. Assume that*

[TABLE]

Then there is an isomorphism

[TABLE]

under which

[TABLE]

Proof.

Denote by $C=\mathrm{C^{*}}\{u^{*}_{i}a:i=1,2,...,n,\ a\in\mathcal{G}\}\subseteq A$, and consider

[TABLE]

Note that $D$ is isomorphic to $\mathrm{C^{*}}(\mathcal{G})$ by (3.5).

Embed $A$ into the enveloping von Neumann algebra $A^{\prime\prime}$. Let $a$ be a strictly positive element of $\textrm{C*}(\mathcal{G})$ with norm $1$, and let $p$ denote the w*-limit of $a^{\frac{1}{n}}$, $n=1,2,...$, in $A^{\prime\prime}$. Then $p$ is an (open) projection with $pa=a$, $a\in A$.

Consider

[TABLE]

Then, it follows from (3.5) that $v_{i}v^{*}_{j}=\delta_{i,j}p$; that is, the elements $\{v_{1},v_{2},...,v_{n}\}$ form a system matrix unit, and thus the C*-algebra generated by $\{v_{1},v_{2},...,v_{n}\}$ is isomorphic to $\mathrm{M}_{n}(\mathbb{C})$.

For any $a\in\textrm{C*}(\mathcal{G})$, by (3.5), one has

[TABLE]

[TABLE]

and

[TABLE]

That is, $D$ and $\{v_{1},v_{2},...,v_{n}\}$ commute, and $u^{*}_{1}pu_{1}+\cdots+u^{*}_{n}pu_{n}$, which equals $v^{*}_{1}v_{1}+\cdots+v^{*}_{n}v_{n}$, acts on $D$ as the unit. Therefore the C*-algebra generated by $D$ and $\{v_{1},v_{2},...,v_{n}\}$ is unital and is isomorphic to the tensor product $\tilde{D}\otimes\mathrm{M}_{n}(\mathbb{C})\cong\mathrm{M}_{n}(\tilde{D})$, where $\tilde{D}$ is the unitization of $D$.

Since

[TABLE]

the C*-algebra $C$ is a sub-C*-algebra of $\tilde{D}\otimes\mathrm{M}_{n}(\mathbb{C})$ generated by products of $1_{\tilde{D}}\otimes\mathrm{M}_{n}(\mathbb{C})$ and $D\otimes 1_{\mathrm{M}_{n}(\mathbb{C})}$, which is exactly the sub-C*-algebra $D\otimes\mathrm{M}_{n}(\mathbb{C})\cong\mathrm{M}_{n}(D)$. ∎

Proof of Theorem 3.9.

It is enough to show the theorem for $m=1$, i.e., $f_{1},...,f_{n}\in A$. For the case $m\neq 1$, write $f_{i}=(f^{(i)}_{j,k})$ where $f^{(i)}_{j,k}\in A$, and then apply the theorem with $\{f^{(i)}_{j,k}:1\leq i\leq n,1\leq j,k\leq m\}$ and $\frac{\delta}{m^{2}}$ in place of $\{f_{1},f_{2},...,f_{n}\}$ and $\delta$ respectively, together with the given function $h$ and the given closed set $F$.

Without loss of generality, one may assume

[TABLE]

for some finite set $\mathcal{N}\subseteq\Gamma$ with $e\in\mathcal{N}=\mathcal{N}^{-1}$, and some $f_{i,\gamma}\in\mathrm{C}(X)$. Denote by

[TABLE]

Since $X$ is compact, there is an open cover $\mathcal{O}$ such that

[TABLE]

and

[TABLE]

Pick a natural number

[TABLE]

and pick a finite set $K\subseteq\Gamma$ and $\varepsilon>0$ so that if a finite set $\Gamma_{0}\subseteq\Gamma$ is $(K,\varepsilon)$-invariant, then

[TABLE]

For any $x\in X$, denote by $\delta_{x}$ the Dirac measure concentrated at $x$; and for any finite set $\Gamma_{0}\subseteq\Gamma$, denote by $\delta_{x,\Gamma_{0}}=\frac{1}{\left|\Gamma_{0}\right|}\sum_{\gamma\in\Gamma_{0}}\delta_{x\gamma}$.

One may then again assume $K$ is sufficiently large and $\varepsilon$ is sufficiently small so that if a finite set $\Gamma_{0}\subseteq\Gamma$ is $(K,\varepsilon)$-invariant, then

[TABLE]

and

[TABLE]

Since $(X,\Gamma)$ has the (URP), there exist closed sets $B_{1},B_{2},...,B_{S}\subset X$ and $(K,\varepsilon)$-invariant sets $\Gamma_{1},\Gamma_{2},...,\Gamma_{S}\subseteq\Gamma$ such that

[TABLE]

are mutually disjoint and

[TABLE]

For each $\Gamma_{s}$, $s=1,2,...,S$, pick $\mathcal{U}_{s}\preceq\bigvee_{\gamma\in\Gamma_{s}}\mathcal{O}\gamma$ such that

[TABLE]

For each $B_{s}$, $s=1,2,...,S$, choose open sets $U^{s}_{3}\supseteq U^{s}_{2}\supseteq U^{s}_{1}\supseteq B_{s}$ such that $U^{s}_{3}\supseteq\overline{U^{s}_{2}}$, $U^{s}_{2}\supseteq\overline{U^{s}_{1}}$, and

[TABLE]

are disjoint.

Consider

[TABLE]

It is an open cover of $B_{s}$ with order at most $\mathrm{ord}(\mathcal{U}_{s})$. Also consider

[TABLE]

Then their union

[TABLE]

forms an open cover of $U^{s}_{3}$, and denote this open cover by $\mathcal{V}_{s}$.

Pick an open cover $\mathcal{W}$ of the closed set $X\setminus\bigsqcup_{s=1}^{S}\bigsqcup_{\gamma\in\Gamma_{s}}U^{s}_{3}\gamma$ such that

[TABLE]

[TABLE]

and

[TABLE]

Then

[TABLE]

forms an open cover of $X$.

Pick an open set ${U^{s}_{3}}^{\prime}\subseteq U^{s}_{3}$ such that $\overline{{U^{s}_{3}}^{\prime}}\subseteq U_{3}$ and

[TABLE]

By Lemma 3.10, there are functions $\phi_{V}:X\to[0,1]$, $V\in\mathcal{V}_{s}$, satisfying

1. (1)

$\mathrm{supp}(\phi_{V})\subseteq V$, $V\in\mathcal{V}_{s}$, 2. (2)

$\sum_{V\in\mathcal{V}_{s}}\phi_{V}(x)=1$, $x\in{U^{s}_{3}}^{\prime}$.

Translate these functions to $\mathcal{V}_{s}\gamma$, $\gamma\in\Gamma_{s}$, and by Lemma 3.11, there is a partition of unity subordinate to $\mathcal{X}=\mathcal{W}\cup\{V\gamma:V\in\mathcal{V}_{s},\ \gamma\in\Gamma_{s},\ s=1,...,S,\}$ in the form of

[TABLE]

Consider the sub-C*-algebra

[TABLE]

A straightforward calculation shows that if $\gamma_{1}\neq\gamma_{2}$, then, for any $V_{1},V_{2}\in\mathcal{V}_{s}$,

[TABLE]

Hence, by Lemma 3.12, $C$ is isomorphic to

[TABLE]

and the set of diagonal elements of $C$ under this isomorphism consists of

[TABLE]

For each $s=1,2,...,S$, consider the set

[TABLE]

and

[TABLE]

Note that

[TABLE]

and by Lemma 4.3 of [15],

[TABLE]

In particular,

[TABLE]

and this verifies Property (4).

Now, let $f\in C^{+}$ be a diagonal element satisfying $f|_{[Z_{s}]}=1_{K_{s}}$, $s=1,...,S$, with respect to the isomorphism above. Then, by (3.13) and (3.14), one has that, as an element of $\mathrm{C}(X)$,

[TABLE]

and by (3.10),

[TABLE]

This, together with (3.13), verifies Property (8).

For each $s=1,2,...,S$, define

[TABLE]

It is clear that $0\leq\chi_{s}(x)\leq 1$, $x\in X$,

[TABLE]

For each $\Gamma_{s}$, $s=1,2,...,S$, define the subsets (see Definition 2.4 for the notation $\mathrm{int}_{K}(F)$)

[TABLE]

Then, for any $\gamma\in\mathcal{N}$, one has

[TABLE]

Indeed, pick an arbitrary $\gamma^{\prime}\in\Gamma_{s,l}$. By the construction, one has

[TABLE]

Therefore

[TABLE]

and hence $\gamma^{\prime}\gamma\in\mathrm{int}_{\mathcal{N}^{l-1}}\Gamma_{s}$ (since $e\in\mathcal{N}^{l-1}$).

Thus, to show (3.16), one only has to show that $\gamma^{\prime}\gamma\notin\mathrm{int}_{\mathcal{N}^{l+2}}\Gamma_{s}$. Suppose $\gamma^{\prime}\gamma\mathcal{N}^{l+2}\subseteq\Gamma_{s}$. Since $\mathcal{N}$ is symmetric, one has $\gamma^{-1}\in\mathcal{N}$; hence $\mathcal{N}^{l+1}\subseteq\gamma\mathcal{N}^{l+2}$ and

[TABLE]

which contradicts (3.17).

Also note that for each $\gamma\in\mathcal{N}$,

[TABLE]

For each $\gamma\in\Gamma_{s}$, define

[TABLE]

By (3.16) and (3.18), the function $\ell$ satisfies

[TABLE]

Define

[TABLE]

Then, by (3.15) (and (3.9)),

[TABLE]

and this proves Property (5).

It follows from Lemma 3.12 that under the isomorphism $C\cong\bigoplus_{s=1}^{S}\mathrm{M}_{\left|\Gamma_{s}\right|}(\textrm{C*}\{\phi_{V}:V\in\mathcal{V}_{s}\}),$ one has

[TABLE]

Denote the diagonal functions by $p_{s,i}$, $s=1,...,S$, $i=1,...,K_{s}$.

Note that $p|_{B_{s}\gamma}=1$, $\gamma\in\Gamma_{s,L+1}$. Therefore, for any $z=\prod_{V\in\mathcal{V}_{s}}\phi_{V}(x)\in[Z_{s}]$, $s=1,...,S$, where $x\in B_{s}$, one has

[TABLE]

This verifies Property (7).

Note that, by the construction of $C$ (see (3.12)),

[TABLE]

Also note that, for each $\gamma^{\prime}\in\mathcal{N}$ and each $\gamma\in\Gamma_{s,l}$, $l=1,2,...,L+1$, one has that $\gamma\gamma^{\prime}\in\Gamma_{s}$. Hence

[TABLE]

and therefore,

[TABLE]

Also note that, for each $\gamma^{\prime}\in\mathcal{N}$, by (3.19) and (3.7),

[TABLE]

and hence

[TABLE]

For each $U\in\mathcal{X}$, pick a point $x_{U}\in U$. For each $f_{i}$, define

[TABLE]

and

[TABLE]

Then, by (3.6) and (3.11),

[TABLE]

and the same argument shows that

[TABLE]

This verified Properties (1).

Moreover, note that for any $U\in\mathcal{X}$, either (in the case that $U\in\mathcal{W}$)

[TABLE]

or $U$ is contained inside some $U_{3}^{s}\gamma$. Since

[TABLE]

for any $W\in\mathcal{W}$, and since

[TABLE]

one has

[TABLE]

since $\phi_{U}\circ\gamma^{-1}=u^{*}_{\gamma}\phi_{U}u_{\gamma}\in C$, $\gamma\in\Gamma_{s}$, $U\in\mathcal{V}_{s}$ and $pu_{\gamma^{\prime}}p\in C$, $\gamma^{\prime}\in\mathcal{N}$ (see (3.20)). The same argument shows that $ph^{\prime}p\in C$. This verifies Properties (3). By (3.21) and (3.2), Property (2) follows.

For Property (6), pick an arbitrary $x\in B_{s}$. By (3.8),

[TABLE]

Since $\left\|h-h^{\prime}\right\|<\delta<\frac{1}{2}$, and $p|_{B_{s}\gamma}=1$, $\gamma\in\Gamma_{s,L+1}$, one has that

[TABLE]

and hence

[TABLE]

Thus, regarding $(ph^{\prime}p-\frac{1}{4})_{+}$ as an element of $C\cong\bigoplus_{s=1}^{S}\mathrm{M}_{\left|\Gamma_{s}\right|}(\mathrm{C}_{0}(Z_{s}))$, one has (by (3.23)) that, for any $z\in[Z_{s}]$ (with $z=\prod_{V\in\mathcal{V}_{s}}\phi_{V}(x)$ for some $x\in B_{s}$),

[TABLE]

as desired. ∎

4. Comparison of open sets, comparison radius, and mean topological dimension

Definition 4.1**.**

Consider a topological dynamical system $(X,\Gamma)$, where $X$ is a compact Hausdorff space and $\Gamma$ is a discrete amenable group. It is said to have $(\lambda,m)$-Cuntz-comparison on open sets, where $\lambda\in(0,1]$ and $m\in\mathbb{N}$, if for any open sets $E,F\subseteq X$ with

[TABLE]

then

[TABLE]

The dynamical system $(X,\Gamma)$ is said to have Cuntz comparison on Open Sets (COS) if it has $(\lambda,m)$-Cuntz-comparison on open sets for some $\lambda$ and $m$.

Let $(X,\Gamma)$ be a minimal free dynamical system. Assume that $(X,\Gamma)$ has the (URP) and (COS). The main result of this section is that the comparison radius of $\mathrm{C}(X)\rtimes\Gamma$ is at most $\frac{1}{2}\mathrm{mdim}(X,\Gamma)$ (see Theorem 4.8).

First, one needs some preparations. Let $a\in A^{+}$, where $A$ is a C*-algebra, and let $\varepsilon>0$. Recall that (Definition 2.11)

[TABLE]

where $f(t)=\max\{t-\varepsilon,0\}$. Note

[TABLE]

and also note that, by Proposition 2.2 of [38], if $\|a-b\|<\varepsilon$, then $(a-\varepsilon)_{+}\precsim b.$ These facts are used throughout this section.

Let $(X,\Gamma)$ be a free dynamical system, and assume that $\mathbb{E}:\mathrm{C}(X)\rtimes\Gamma\to\mathrm{C}(X)$ is a conditional expectation, where $\mathrm{C}(X)\rtimes\Gamma$ is a crossed-product C*-algebra. Then

[TABLE]

Indeed, let $\gamma\in\Gamma\setminus\{e\}$ and consider $\mathbb{E}(u_{\gamma})\in\mathrm{C}(X)$. Note that for all $g\in\mathrm{C}(X)$, since $u_{\gamma}^{*}gu_{\gamma}\in\mathrm{C}(X)$, one has

[TABLE]

Assume $\mathbb{E}(u_{\gamma})\neq 0$. Then there is $x_{0}\in X$ such that $\mathbb{E}(u_{\gamma})(x_{0})\neq 0$. Since $(X,\Gamma)$ is free, one has $x_{0}\neq x_{0}\gamma^{-1}$. Pick $g\in\mathrm{C}(X)$ such that $g(x_{0})=1$ and $g(x_{0}\gamma^{-1})=0$. Then

[TABLE]

which is a contradiction.

Then, one has the following lemma, which is similar to Lemma 3.1 of [13].

Lemma 4.2**.**

Let $(X,\Gamma)$ be a free topological dynamical system, where $\Gamma$ is a discrete group. Let $\mathrm{C}(X)\rtimes\Gamma$ be a crossed product C-algebra with a faithful conditional expectation $\mathbb{E}:\mathrm{C}(X)\rtimes\Gamma\to\mathrm{C}(X)$ (the faithful conditional expectation exists if $\Gamma$ is amenable, see, for instance, Proposition 4.1.9 of [1]). Let $a\in\mathrm{C}(X)\rtimes\Gamma$ be a nonzero positive element. Then, there is a positive nonzero element $h\in\mathrm{C}(X)$ such that $h\precsim a$.*

Proof.

Since $\mathbb{E}$ is faithful, one may assume that $\left\|\mathbb{E}(a)\right\|=1$. One asserts that for arbitrary $\varepsilon>0$, there is $p\in\mathrm{C}(X)$ such that $\left\|p\right\|=1$ and

[TABLE]

Then $h:=(p^{2}-2\varepsilon)_{+}\precsim pap\precsim a$ satisfies the statement of the lemma.

Let us prove the assertion. Without loss of generality, one may assume

[TABLE]

Since $(X,\Gamma)$ is free, by (4.1), one has that $f_{e}=\mathbb{E}(a)$. Since $\left\|f_{e}\right\|=1$, there is a point $x\in X$ such that $f_{e}(x)=1$; since the action is free, there is $p\in\mathrm{C}(X)^{+}$ such that $\left\|p\right\|=1$, $pf_{e}\approx_{\varepsilon}p$, and

[TABLE]

Then $p$ is the desired function. ∎

Lemma 4.3**.**

Let $(X,\Gamma)$ be a dynamical system with the (URP), and assume that $\left|\Gamma\right|=\infty$.

Then, for any $r\in(0,+\infty)$ and any $\varepsilon>0$, there is a positive element $h\in\mathrm{M}_{\infty}(\mathrm{C}(X))$ such that

[TABLE]

Moreover, $h$ can be chosen to have the form $h=\mathrm{diag}\{h_{0},\underbrace{1,...,1}_{\lfloor r\rfloor}\}$ for some $h_{0}\in\mathrm{C}(X)$ together with a closed set $F\subseteq X$ such that $h_{0}(x)=1$, $x\in F$, and

[TABLE]

where $\mu$ runs through $\mathcal{M}_{1}(X,\Gamma)$, $\lfloor r\rfloor$ is the integer part of $r$, and $\{r\}$ is the decimal part of $r$.

Proof.

First note that since $\left|\Gamma\right|=\infty$, it has the property that $\left|\Gamma_{n}\right|\to\infty$ for any Følner sequence $\Gamma_{n}$, $n=1,2,...$ . Otherwise, if there is a Følner sequence $(\Gamma_{n})$ and $M>0$ such that $\left|\Gamma_{n}\right|<M$, $n=1,2,...$, then, for any finite $K\subseteq\Gamma$ with $\left|K\right|>M$ and $e\in K$, since $\Gamma_{n}$ is $(K,\frac{1}{2M})$-invariant for sufficiently large $n$ and $\left|\Gamma_{n}\right|<M$, one has that $\Gamma_{n}K=\Gamma_{n}$ for sufficiently large $n$. But $\left|K\right|\leq\left|\Gamma_{n}K\right|=\left|\Gamma_{n}\right|<M$, which contradicts to the choice of $K$.

Also note that one only has to prove the lemma for a rational number $r=\frac{m}{n}$ with $m,n\in\mathbb{N}$ and $\frac{1}{n}<\varepsilon$ (one does not require that $(m,n)=1$).

Assume $r\leq 1$. Since $(X,\Gamma)$ has the (URP), there exist closed sets $B_{1},B_{2},...,B_{S}\subseteq X$ and $(K,\varepsilon)$-invariant sets $\Gamma_{1},\Gamma_{2},...,\Gamma_{S}\subseteq\Gamma$ such that

1. (1)

$B_{s}\gamma$, $\gamma\in\Gamma_{s}$, $s=1,2,...,S$ are disjoint, and 2. (2)

$\mathrm{ocap}(X\setminus\bigsqcup_{s=1}^{S}\bigsqcup_{\gamma\in\Gamma_{s}}B_{s}\gamma)<\varepsilon.$

Note that $\Gamma_{1}$, …, $\Gamma_{S}$ can be chosen so that

[TABLE]

Pick open sets $U_{s}\supseteq B_{s}$, $s=1,2,...,S$, so that

[TABLE]

are disjoint. By (4.2), there are $\Gamma^{\prime}_{s}\subseteq\Gamma_{s}$, $s=1,2,...,S$, such that

[TABLE]

Put

[TABLE]

Then, by (4.3), for any $\mu\in\mathcal{M}_{1}(X,\Gamma)$,

[TABLE]

and

[TABLE]

Then the function $h:=\varphi_{U}$ satisfies the lemma (with $d=1$ and $F_{1}=\bigsqcup_{s=1}^{S}\bigsqcup_{\gamma\in\Gamma_{s}^{\prime}}B_{s}\gamma$).

For general $r$, pick $h_{0}$ to satisfy the lemma for $\{r\}$; and the element

[TABLE]

satisfies the lemma. ∎

Definition 4.4**.**

A positive element $a$ of a C*-algebra $A$ is said to be compact if $\mathrm{sp}(a)\cap(0,\delta)=\varnothing$ for some $\delta>0$.

Note that if $a$ is a compact element, then $a$ is Cuntz equivalent to the spectral projection $\chi_{(\frac{\delta}{2},+\infty)}(a)$ where $\delta$ satisfies $\mathrm{sp}(a)\cap(0,\delta)=\varnothing$.

Lemma 4.5**.**

Let $A$ be a finite C-algebra, and let $a,b$ be nonzero positive elements of $A$ with $a\precsim b$. Assume either*

1. (1)

$b$* is not compact, or* 2. (2)

$a$* is not compact, or,* 3. (3)

$a$* is not Cuntz equivalent to $b$.*

Then, for any $\varepsilon>0$, there are nonzero positive elements $b_{1},b_{2}$ such that

1. (1)

$b_{1}\perp b_{2}$, 2. (2)

$b_{1},b_{2}\in\overline{bAb}$, and 3. (3)

$(a-\varepsilon)_{+}\precsim b_{2}$.

Proof.

Assume that $\mathrm{sp}(b)\cap(0,\delta)\neq\varnothing$ for all $\delta>0$ (that is, $b$ is not a compact element). Then, for the arbitrarily given $\varepsilon$, there is $\delta$ such that $(a-\varepsilon)_{+}\precsim(b-\delta)_{+}$. Then, $b_{1}:=f_{\delta}(b)$ and $b_{2}:=(b-\delta)_{+}$, where $f_{\delta}(t)=\max\{0,t(\delta-t)\}$, are desired elements.

Now, assume that $a$ is not compact, and one may also assume that $b$ is a compact element, and without loss of generality, one may assume that $b$ is a nonzero projection. For the arbitrarily given $\varepsilon>0$, there is $a^{\prime}\in bAb$ such that $(a-\frac{\varepsilon}{2})_{+}$ is Cuntz equivalent to $a^{\prime}$ (in particular, $(a-\frac{\varepsilon}{2})_{+}\precsim a^{\prime}$).

If $(a-\frac{\varepsilon}{2})_{+}$ is not a compact element, then $a^{\prime}$ is not a compact element neither. By the argument for the case that $b$ is not a compact element, there are $b_{1},b_{2}$ such that

1. (1)

$b_{1}\perp b_{2}$, 2. (2)

$b_{1},b_{2}\in\overline{a^{\prime}Aa^{\prime}}\subseteq\overline{bAb}$, and 3. (3)

$((a-\frac{\varepsilon}{2})_{+}-\frac{\varepsilon}{2})_{+}\precsim b_{2}$.

Since $(a-\varepsilon)_{+}\precsim((a-\frac{\varepsilon}{2})_{+}-\frac{\varepsilon}{2})_{+}$, the elements $b_{1},b_{2}$ satisfy the lemma.

If $(a-\frac{\varepsilon}{2})_{+}$ is a compact element (so $(a-\frac{\varepsilon}{2})_{+}$ and $a^{\prime}$ can be assumed to be projections), since $a$ is assumed not to be a compact element, one has that $(a-\frac{\varepsilon}{2})_{+}\precsim b$ but not equivalent to $b$, and hence $a^{\prime}\neq b$ (since $A$ is finite). Hence the elements $b_{1}:=b-a^{\prime}$ ($b$ is assumed to be a projection) and $b_{2}:=a^{\prime}$ satisfy the lemma.

For the remaining case that $a$ and $b$ are non-equivalent compact elements, one may assume that $a,b$ are projections such that $a<b$ but $a\neq b$. Then $b_{1}:=b-a$ and $b_{2}:=a$ satisfy the lemma. ∎

Lemma 4.6**.**

For any $\varepsilon\in(0,1)$ and any nonzero $x=(x_{i,j})\in\mathrm{M}_{L}(A)$ (for some $L\in{\mathbb{N}}$), where $A$ is a C-algebra, there is $\delta(\varepsilon,\left\|x\right\|,L)>0$, such that for any $a,b,p\in A^{+}$ with $\left\|a\right\|,\left\|b\right\|,\left\|p\right\|\leq 1$ satisfying*

1. (1)

$\left\|a\otimes 1_{M}-x^{*}(b\otimes 1_{N})x\right\|<\varepsilon$, for some $M,N\in\mathbb{N}$, 2. (2)

$\left\|px_{i,j}-x_{i,j}p\right\|<\delta$, and 3. (3)

$p$, $pap$, $pbp$, $px_{i,j}p$ are in a sub-C-algebra $C\subseteq A$,*

there is $c\in C$ such that

1. (1)

$(pap-102\varepsilon)_{+}\otimes 1_{M}\precsim c\otimes 1_{N}$* in $C$, and* 2. (2)

$c\precsim p(b-\frac{\varepsilon^{3}}{8\max\{1,\left\|x\right\|^{6}\}})_{+}p$* in $A$.*

Proof.

Set

[TABLE]

For each $t_{0}\in(0,+\infty)$, define the positive function

[TABLE]

Choose $\delta\in(0,\frac{\varepsilon}{2L^{2}})$ sufficiently small such that

[TABLE]

for any $y\in A^{+}$ with $\left\|x_{i,j}y-yx_{i,j}\right\|<\delta$ and $\left\|y\right\|\leq 1$. Note that the choice of $\delta$ only depends on $\varepsilon$, $\left\|x\right\|$, and $L$. Indeed, pick a polynomial $P(t)=c_{1}t+\cdots+c_{n}t^{n}$ such that

[TABLE]

Set

[TABLE]

Then

[TABLE]

if $\left\|x_{i,j}y-yx_{i,j}\right\|<\delta$, and hence

[TABLE]

One asserts that this $\delta$ satisfies the conclusion of the lemma.

Let $a,b,p\in A^{+}$ satisfy the conditions of the lemma. Then, by (4.4), with

[TABLE]

($\bar{h}_{\varepsilon_{1}}$ is a well-defined continuous function on $\mathbb{R}$), one has

[TABLE]

Hence, together with the assumption of the lemma,

[TABLE]

Since $p\in C$ and $px_{i,j}p\in C$, $i,j=1,...,L$, one has that

[TABLE]

(the functions $\bar{h}_{\varepsilon_{1}}$ and ${h}^{\frac{1}{2}}_{\varepsilon_{1}}$ take value [math] at [math]), and hence $z\in\mathrm{M}_{L}(C)$, and

[TABLE]

Since

[TABLE]

one has

[TABLE]

and hence

[TABLE]

By (4.5), one has

[TABLE]

and hence

[TABLE]

Therefore,

[TABLE]

in the algebra $\mathrm{M}_{\infty}(C)$. Since

[TABLE]

one has

[TABLE]

in the algebra $A$. Then $c:=(pbp-\varepsilon_{0})_{+}\in C$ has the desired property. ∎

The following lemma is well know. A proof is included for reader’s convenience.

Lemma 4.7**.**

Let $A$ be a simple non-elementary C-algebra, and let $n\in\mathbb{N}$ be arbitrary. Then there are nonzero positive elements $e_{1},...,e_{n}\in A$ which are mutually orthogonal and mutually Cuntz equivalent.*

Proof.

Since $A$ is non-elementary, it contains a positive element with spectrum containing infinitely many points, and hence there are nonzero positive elements $e_{1},e_{2},...,e_{n}\in A$ which are mutually orthogonal.

Consider $e_{1}$ and $e_{2}$. Since $A$ is simple, for any $\varepsilon>0$, there are $x_{1},...,x_{n}$, $y_{1},...,y_{n}$ for some $n\in\mathbb{N}$ such that $x_{1}e_{1}y_{1}+\cdots+x_{n}e_{1}y_{n}\approx_{\varepsilon}e_{2}.$ In particular, with $\varepsilon$ sufficiently small, there is $x\in A$ such that $z_{1}:=e_{1}xe_{2}\neq 0$. Note that

[TABLE]

With the polar decomposition of $z_{1}$ and a functional calculus, one may assume that there is a positive element $e_{1}^{\prime}$ with norm one such that

[TABLE]

Replacing $e_{1},e_{2}$ by $z_{1}z_{1}^{*}$ and $z_{1}^{*}z_{1}$ respectively, one obtains nonzero positive elements

[TABLE]

which are mutually orthogonal and $e_{1}=z_{1}z_{1}^{*},$ $e_{2}=z_{1}^{*}z_{1}$. Moreover, there is a positive element $e^{\prime}_{1}$ with norm one such that

[TABLE]

Using the simplicity again, the same argument as above shows that there is $z_{2}\in A\setminus\{0\}$ such that

[TABLE]

and, moreover, there is a positive element $e_{1}^{\prime\prime}$ with norm one such that

[TABLE]

Note that

[TABLE]

and

[TABLE]

Therefore, replacing $e_{1},e_{1}^{\prime},e_{2},e_{3}$, and $z_{1}$ by $z_{2}z_{2}^{*},e_{1}^{\prime\prime},((z_{2}z_{2}^{*})^{\frac{1}{2}}z_{1})^{*}((z_{2}z_{2}^{*})^{\frac{1}{2}}z_{1}),z^{*}_{2}z_{2}$, and $(z_{2}z_{2}^{*})^{\frac{1}{2}}z_{1}$ respectively, one obtains positive elements

[TABLE]

which are mutually orthogonal and $e_{1}=z_{1}z_{1}^{*},$ $e_{2}=z_{1}^{*}z_{1}$, and $e_{1}=z_{2}z_{2}^{*},$ $e_{3}=z_{2}^{*}z_{2}$. Moreover, there is a positive element $e^{\prime}_{1}$ with norm one such that $e_{1}e_{1}^{\prime}=e_{1}^{\prime}$.

Repeating this argument finitely many times, one obtains nonzero positive elements

[TABLE]

which are mutually orthogonal and elements $z_{1},...,z_{n-1}$ such that

[TABLE]

Then the elements $e_{1},e_{2},...,e_{n}$ clearly satisfy the lemma. ∎

Theorem 4.8**.**

Let $(X,\Gamma)$ be a free and minimal topological dynamical system satisfying the (URP) and (COS).

Let $a,b$ be non-zero positive elements of $\mathrm{M}_{\infty}(\mathrm{C}(X)\rtimes\Gamma)$ such that

[TABLE]

for some $r>\frac{1}{2}\mathrm{mdim}(X,\Gamma)$. Then $a\precsim b$. In other words,

[TABLE]

Proof.

One only has to prove the statement for $\left|\Gamma\right|=\infty$. In the case that $\Gamma$ is a finite group, since the action is minimal, one has that the system $(X,\Gamma)$ is conjugate to the action of $\Gamma$ on itself by translation. Hence $(X,\Gamma)$ has mean topological dimension zero and $\mathrm{C}(X)\rtimes\Gamma\cong\mathrm{M}_{\left|\Gamma\right|}(\mathbb{C})$, which has zero radius of comparison. In particular, the statement of the theorem holds.

Let us assume that $\left|\Gamma\right|=\infty$. Moreover, without loss of generality, one may assume that $\left\|a\right\|=\left\|b\right\|=1$.

Pick $r^{\prime}\in(\frac{1}{2}\mathrm{mdim}(X,\Gamma),r)$ and some $\delta^{\prime}>0$ such that

[TABLE]

Since $\left|\Gamma\right|=\infty$, by Lemma 4.3 (with $r=\frac{1}{2}\mathrm{mdim}(X,\Gamma)+2\delta^{\prime}$ and $\varepsilon=\delta^{\prime}$), there is a positive element $h\in\mathrm{M}^{+}_{\infty}(\mathrm{C}(X))$ such that

[TABLE]

and

[TABLE]

Note that

[TABLE]

for any $\tau\in\mathrm{T}(\mathrm{C}(X)\rtimes\Gamma)$. Moreover, by Lemma 4.3, $h$ can be chosen such that

[TABLE]

for a positive function $h_{0}\in\mathrm{C}(X)$, some integer $d$, and $h_{0}(x)=1$ on a closed set $F$ with

[TABLE]

where $\mu$ runs through $\mathcal{M}_{1}(X,\Gamma)$.

Pick $m\in\mathbb{N}$ such that $a,b,h\in\mathrm{M}_{m}(\mathrm{C}(X)\rtimes\Gamma)$.

Since $\mathrm{C}(X)\rtimes\Gamma$ is nuclear, all quasitraces are traces. By Theorem 4.3 of [38], any lower semicontinuous dimension function of $\mathrm{C}(X)\rtimes\Gamma$ has the form $\mathrm{d}_{\tau}$ for some $\tau\in\mathrm{T}(\mathrm{C}(X)\rtimes\Gamma)$.

Since $(X,\Gamma)$ is minimal, the C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ is simple. Hence the Cuntz class of $b$ is a strong order unit of the Cuntz semigroup of $\mathrm{C}(X)\rtimes\Gamma$ (see Proposition 4.2 of [3]). By (4.7) and the proof of Proposition 3.2 of [38], there is $N\in\mathbb{N}$ such that

[TABLE]

Moreover, one may assume that $(a\oplus h)\otimes 1_{N+1}$ is not Cuntz equivalent to $b\otimes 1_{N}$.

Let $\varepsilon\in(0,\frac{1}{408})$ be arbitrary, by Lemma 4.5, there are nonzero positive elements $b_{1}$ and $b_{2}$ such that

1. (1)

$b_{1}\perp b_{2}$, 2. (2)

$b_{1},b_{2}\in\overline{bAb}$, and 3. (3)

$((a-\varepsilon)_{+}\oplus(h-\varepsilon)_{+})\otimes 1_{N+1}\precsim b_{2}\otimes 1_{N}.$

By (3), there is $\delta>0$ such that

[TABLE]

Since $(X,\Gamma)$ has the (COS), the C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ has $(\lambda,m^{\prime})$-Cuntz-comparison of open sets for some $\lambda\in\mathbb{R}^{+}$ and $m^{\prime}\in\mathbb{N}$. Fix $\lambda$ and $m^{\prime}$.

Consider the nonzero positive element $b_{1}$. Since $\mathrm{C}(X)\rtimes\Gamma$ is simple and non-elementary, the hereditary C*-algebra generated by $b_{1}$ is also simple and non-elementary. By Lemma 4.7, there are mutually orthogonal nonzero positive elements $b_{1}^{(1)},...,b_{1}^{(m^{\prime})}$ in the hereditary C*-algebra generated by $b_{1}$ such that $b_{1}^{(1)},...,b_{1}^{(m^{\prime})}$ are mutually Cuntz equivalent. By Lemma 4.2, there exists a nonzero positive element $\tilde{b}_{1}\in\mathrm{C}(X)$ such that $\tilde{b}_{1}\precsim b_{1}^{(1)}$, and hence there are nonzero positive elements $b_{1,1},b_{1,2}\in\mathrm{C}(X)$ such that

[TABLE]

and

[TABLE]

By (4.9), there is $x=(x_{i,j})\in\mathrm{M}_{L}(\mathrm{M}_{m}(A))$ (for some $L\in\mathbb{N}$) such that

[TABLE]

Let $\delta^{\prime\prime}=\delta(\varepsilon,\left\|x\right\|,L)$ be the constant of Lemma 4.6 with respect to $x$ and $\varepsilon$. Moreover, $\delta^{\prime\prime}$ can be chosen so that

[TABLE]

and if $\left\|pc-cp\right\|<\delta^{\prime\prime}$ for some positive elements $p,c$ in some C*-algebra with $\left\|c\right\|,\left\|p\right\|\leq 1$, then

[TABLE]

Applying Theorem 3.9 with $\{(a-2\varepsilon)_{+},\ (b_{2}-\delta)_{+},\ x_{i,j}:1\leq i,j\leq L\}$ in place of $\{f_{1},f_{2},...,f_{n}\}$, $(h_{0}-2\varepsilon)_{+}$ in place of $h$ (note that the value of $(h_{0}-2\varepsilon)_{+}$ at any point of $F$ is at least $\frac{3}{4}$), and $\min\{\frac{\delta^{\prime}}{4(d+1)},\delta^{\prime\prime},\frac{1}{m}\lambda\{\inf_{\nu}\{\nu(b_{1,s}^{-1}(0,+\infty)):\nu\in\mathcal{M}_{1}(X,\Gamma)\}:s=1,2\}$ in place of $\delta$, there are

[TABLE]

for some $p^{\prime},h_{0}^{\prime}\in\mathrm{C}(X)^{+}$ with $\left\|p^{\prime}\right\|\leq 1$, and a sub-C*-algebra $C\subseteq A$ and $A\cong\bigoplus_{s=1}^{S}\mathrm{M}_{K_{s}}(\mathrm{C}_{0}(Z_{s}))$ for locally compact metrizable spaces $Z_{s}$ together with a compact set $[Z_{s}]\subseteq Z_{s}$ such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

and, under the isomorphism $C\cong\bigoplus_{s=1}^{S}\mathrm{M}_{K_{s}}(\mathrm{C}_{0}(Z_{s}))$,

[TABLE]

and

[TABLE]

Moreover, still under the isomorphism $C\cong\bigoplus_{s=1}^{S}\mathrm{M}_{K_{s}}(\mathrm{C}_{0}(Z_{s}))$, if $f\in C^{+}$ is an diagonal element satisfying $f|_{Z_{s}}=1_{K_{s}}$, $s=1,...,S$, then $f\in\mathrm{C}(X)$, and

[TABLE]

Note that, by (4.20), (4.21) and (4.8), for any $z\in[Z_{s}]$,

[TABLE]

Then, since $(ph^{\prime}p-102\varepsilon)_{+}\succsim(ph^{\prime}p-\frac{1}{4})_{+}$ ($\varepsilon$ is assumed to be at most $\frac{1}{408}$), by (4.19), one has that for any $z\in[Z_{s}]$,

[TABLE]

Since $\mathrm{C}(X)\rtimes\Gamma$ has $(\lambda,m^{\prime})$-comparison on open sets, it follows from (4.18) that

[TABLE]

Note that, without loss of generality, one may assume that $\left\|x\right\|=\left\|x^{\prime}\right\|$. By (4.11), (4.13), and (4.14),

[TABLE]

together with (4.15) and (4.17), it follows from Lemma 4.6 that there is $c\in\mathrm{M}_{m}(C)$

[TABLE]

in $\mathrm{M}_{m}(C)$ and

[TABLE]

in $\mathrm{M}_{m}(A)$. By (4.25),

[TABLE]

and hence

[TABLE]

Therefore, by (4),

[TABLE]

and by Theorem 4.6 of [41],

[TABLE]

There is then a positive central element $g\in\mathrm{M}_{m}(C)$ with $g=g^{\prime}\otimes 1_{m}$ for some $g^{\prime}\in\mathrm{C}(X)^{+}$ (and $g^{\prime}\in C$) such that $\left\|g^{\prime}\right\|=1$, $g^{\prime}|_{[Z_{s}]}=1_{K_{s}}$, and

[TABLE]

in $\mathrm{M}_{m}(C)$. Since $g^{\prime}|_{[Z_{s}]}=1_{K_{s}}$, $s=1,...,S$, by (4.22) and $(\lambda,m^{\prime})$-comparison, one has

[TABLE]

Note that

[TABLE]

Therefore, together with (4.15) and (4.12),

[TABLE]

and hence

[TABLE]

Since $\varepsilon$ is arbitrarily small, one has that $a\precsim b$, as desired. ∎

5. Recursive subhomogeneous C*-algebras with diagonal maps

In the rest of paper, let us investigate Cuntz-comparison of Open Sets (COS) for a given dynamical system $(X,\Gamma)$. Let us start with a special class of concrete C*-algebras which will appear naturally as the C*-algebras of small subgroupoids of a transformation groupoid (see Section 6). Any C*-algebra in this class enjoys a comparison property of diagonal elements (see Theorem 7.8), and this eventually leads to the (COS) for $(X,\Gamma)$ if small subgroupoids with arbitrary large orbits exist (see Corollary 7.13).

Recall that

Definition 5.1** ([35]).**

The class of recursive subhomogeneous C*-algebras (RSH algebras) is the smallest class $\mathcal{R}$ of C*-algebras which is closed under isomorphism and such that:

1. (1)

if $X$ is a compact Hausdorff space and $n\geq 1$, then $\mathrm{M}_{n}(\mathrm{C}(X))\in\mathcal{R}$, 2. (2)

$\mathcal{R}$ is closed under the following pull back construction: if $A\in\mathcal{R}$, $X$ is a compact Hausdorff space, $X^{(0)}\subseteq X$ is closed, $\varphi:A\to\mathrm{M}_{n}(\mathrm{C}(X^{(0)}))$ is any unital homomorphism, and $\rho:\mathrm{M}_{n}(\mathrm{C}(X))\to\mathrm{M}_{n}(\mathrm{C}(X^{(0)}))$ is the restriction homomorphism, then the pullback

[TABLE]

is in $\mathcal{R}$.

From the definition, it is clear that any recursive subhomogeneous C*-algebra can be written in the form

[TABLE]

with $C_{k}=\mathrm{M}_{n(k)}(\mathrm{C}(X_{k}))$ for compact Hausdorff spaces $X_{k}$ and positive integers $n(k)$, with $C_{k}^{(0)}=\mathrm{M}_{n(k)}(\mathrm{C}(X_{k}^{(0)}))$ for compact subsets $X_{k}^{(0)}\subseteq X_{k}$ (possible empty), and where the maps $C_{k}\to C_{k}^{(0)}$ are always the restriction maps. An expression of this type will be referred to as a decomposition of $A$, and the notation used here will be referred to as the standard notation for a decomposition.

The RSH algebras considered in this paper will have the gluing maps $\varphi$ being of diagonal type:

Definition 5.2**.**

Let

[TABLE]

be an RSH algebra. A homomorphism $\phi:A\to\mathrm{M}_{n}(\mathrm{C}(Y))$, where $Y$ is a compact metrizable space, is said to be of diagonal type if there is a partition

[TABLE]

such that for each $Y_{i}$, there are continuous maps $\lambda_{1}^{(i)}:Y_{i}\to X_{t_{1}}$, $\lambda_{2}^{(i)}:Y_{i}\to X_{t_{2}}$, …, $\lambda_{s_{i}}^{(i)}:Y_{i}\to X_{t_{s_{i}}}$ for some $s_{i}\in\mathbb{N}$ such that

[TABLE]

where $(f_{0},f_{1},...,f_{K})\in A.$

Consider the crossed product C*-algebra $\mathrm{C}(X)\rtimes\mathbb{Z}$, and consider a closed set $Y\subseteq X$ with nonempty interior. The Putnam sub-C*-algebra $A_{Y}\subseteq\mathrm{C}(X)\rtimes\mathbb{Z}$ is an RSH with diagonal maps in a natural way. The construction was introduced in [25] for $X$ being the Cantor set and then was generalized in [29] for a general $X$. (More examples will be constructed in the next section using groupoids.)

Definition 5.3**.**

Let $Y$ be a closed subset of $X$ with non-empty interior. The C*-algebra $A_{Y}$ is defined by

[TABLE]

Let us collect some basic properties of this sub-C*-algebra:

Consider the first return times

[TABLE]

Since $\sigma$ is minimal, $X$ is compact, and $Y$ has a non-empty interior, this set of numbers is finite; let us write it as

[TABLE]

for some $K\in\mathbb{N}$. Since $X$ is an infinite set and $\sigma$ is minimal, the first return time $J_{1}$ is arbitrarily large if $Y$ is sufficiently small.

For each $1\leq k\leq K$, consider the (locally compact—see below) subset of $X$

[TABLE]

Then the sets

[TABLE]

—which are naturally listed as shown—form a partition of $X$. This is often called a Rokhlin partition.

Lemma 5.4** ([29]; see also [30] or Lemma 2.15 of [36]).**

*In terms of the notation introduced above, one has that, for each $1\leq k\leq K$,

$\mathrm{(1)}$ the set $Z_{1}\cup\cdots\cup Z_{k}$ is closed (and so $Z_{k}$ is locally compact),

$\mathrm{(2)}$ the set $\overline{Z_{k}}\cap(Z_{1}\cup\cdots\cup Z_{k-1})$ is the disjoint union of the subsets*

[TABLE]

where $J_{t_{1}}+J_{t_{2}}+\cdots+J_{t_{s}}=J_{k}$ for some $1\leq t_{1},t_{2},...,t_{s}\leq k$.

A quite explicit description of the subalgebra $A_{Y}$ of the crossed product, a C*-algebra of type I, was obtained by Q. Lin ([29]). It is a subhomogeneous algebra of order at most $J_{K}$. In fact, it is an RSH algebra with all gluing maps being diagonal.

Theorem 5.5** ([29]; see also [30] or Theorem 2.22 of [36]).**

In terms of the notation introduced above, one has that the C-algebra $A_{Y}$ is isomorphic to the sub-C*-algebra of $\bigoplus_{k=1}^{K}\mathrm{M}_{J_{k}}(\mathrm{C}(\overline{Z_{k}}))$ consisting of the elements $(F_{1},...,F_{K})$ with*

[TABLE]

whenever

[TABLE]

where $J_{t_{1}}+J_{t_{2}}+\cdots+J_{t_{s}}=J_{k}$.

Moreover, for any $f,g\in\mathrm{C}(X)$ with $g|_{Y}=0$, the images of $f,ug\in A_{Y}$ in this identification are

[TABLE]

and

[TABLE]

respectively.

6. Small subgroupoids and recursive subhomogeneous C*-algebras

In this section, let us consider a class of small subgroupoid of the transformation groupoid $X\rtimes\Gamma$. It turns out that the C*-algebra of a such subgroupoid is an RSH algebra with diagonal maps.

Consider a topological dynamical system $(X,\Gamma)$. Recall that (see Example 1.2.a of [37]) the transformation groupoid $X\rtimes\Gamma$ is defined by taking $X\times\Gamma$ with

1. (1)

$\mathrm{r}(x,\gamma)=(x,e)$, $\mathrm{s}(x,\gamma)=(x\gamma,e)$, $(x,\gamma)\in X\times\Gamma$, 2. (2)

$(x,\gamma_{1})(y,\gamma_{2})=(x,\gamma_{1}\gamma_{2})$ if $x\gamma_{1}=y$, and 3. (3)

$(x,\gamma)^{-1}=(x\gamma,\gamma^{-1})$, $(x,\gamma)\in X\times\Gamma$.

The topology on $X\rtimes\Gamma$ is the product topology on $X\times\Gamma$. Note that $X$, regarded as $\{(x,e):x\in X\}$, is the unit space of $X\rtimes\Gamma$, and $X\rtimes\Gamma$ is principle if $(X,\Gamma)$ is free.

Definition 6.1**.**

A small subgroupoid $\mathcal{G}\subseteq X\rtimes\Gamma$ is a subgroupoid which is open, relatively compact, and $\{(x,e):x\in X\}\subseteq\mathcal{G}.$

It is well known that the C*-algebra $\textrm{C*}(\mathcal{G})\subseteq\textrm{C*}(X\rtimes\Gamma)$ is subhomogeneous if $\mathcal{G}$ is small subgroupoid. Using the orbit structure of $\mathcal{G}$, let us show that $\textrm{C*}(\mathcal{G})$ actually is an RSH-algebra with diagonal maps.

Consider

[TABLE]

Since $\mathcal{G}$ is relatively compact, one has that

[TABLE]

Since the unit space of the subgroupoid $\mathcal{G}$ is $X$, it induces an equivalence relation on $X$ by

[TABLE]

For each $x\in X$, define

[TABLE]

It is clear that

[TABLE]

Definition 6.2**.**

Let $\mathcal{G}\subseteq X\rtimes\Gamma$ be any subgroupoid, and let $x\in X$. Define the shape of $x$ to be

[TABLE]

Remark 6.3*.*

Note that since $(x,e)\in\mathcal{G}$, $x\in X$, one has

[TABLE]

The function $\mathrm{S}$ has the following properties.

Lemma 6.4**.**

Let $(X,\Gamma)$ be a free dynamical system, and let $\mathcal{G}$ be a relatively compact subgroupoid of $X\rtimes\Gamma$. Then

1. (1)

$|\mathrm{Orbit}_{\mathcal{G}}(x)|=|\mathrm{S}(x)|$, $x\in X$; 2. (2)

$\mathrm{Orbit}_{\mathcal{G}}(x)=x\mathrm{S}(x)$, $x\in X$; 3. (3)

if $(x,\gamma)\in\mathcal{G}$, then

[TABLE]

i.e., if $y\sim_{\mathcal{G}}x$, then $\mathrm{S}(y)=\gamma^{-1}\mathrm{S}(x),$ where $y=x\gamma$.

Proof.

Note that $y\in\mathrm{Orbit}_{\mathcal{G}}(x)$, if and only if, there is $\gamma\in\Gamma$ such that $(x,\gamma)\in\mathcal{G}$ (hence $\gamma\in\mathrm{S}(x)$) and $y=x\gamma$; and the latter condition is equivalent to $y\in x\mathrm{S}(x)$. This proves (2), and (1) follows from (2) by the freeness of the action.

Let $\gamma^{\prime}\in\mathrm{S}(x\gamma)$, then $(x\gamma,\gamma^{\prime})\in\mathcal{G}$. Since $(x,\gamma)\in\mathcal{G}$, one has that $(x,\gamma\gamma^{\prime})=(x,\gamma)(x\gamma,\gamma^{\prime})\in\mathcal{G}$, and hence $\gamma\gamma^{\prime}\in\mathrm{S}(x)$ and $\gamma^{\prime}\in\gamma^{-1}\mathrm{S}(x)$. Therefore $\mathrm{S}(x\gamma)\subseteq\gamma^{-1}\mathrm{S}(x);$

Now, if $\gamma^{\prime}\in\gamma^{-1}\mathrm{S}(x)$, then $\gamma\gamma^{\prime}\in\mathrm{S}(x)$, and hence $(x,\gamma\gamma^{\prime})\in\mathcal{G}$. Since $(x,\gamma)\in\mathcal{G}$, one has

[TABLE]

and hence $\gamma^{\prime}\in\mathrm{S}(x\gamma)$, as desired. ∎

Definition 6.5**.**

Let $\mathcal{F}\subseteq\Gamma$ be a finite set containing $e$. Define

[TABLE]

Lemma 6.6**.**

Let $\mathcal{G}\subseteq X\rtimes\Gamma$ be open and relatively compact subgroupoid. Then, the function $x\mapsto\mathrm{S}(x)$ is lower semicontinuous in the following sense: for any $x$, there is an open set $U\ni x$ such that

[TABLE]

Hence, for any finite subset $\mathcal{F}\subseteq\Gamma$, if $x_{n}\in Z_{\mathcal{F}}$ and $x_{n}\to x_{\infty}\in X$, then

[TABLE]

Proof.

Write $\mathrm{S}(x)=\{r_{1},r_{2},...,r_{n}\}$. Since $\mathcal{G}$ is open, there is $U\ni x$ such that

[TABLE]

In particular this implies $\gamma_{i}\in\mathrm{S}(x^{\prime})$, $i=1,2,...,n$, as desired. ∎

Definition 6.7**.**

Let $x\in X$. Define

[TABLE]

Remark 6.8*.*

Note that if $x\in Z_{\mathcal{F}}$ where $\mathcal{F}\subseteq\Gamma$ is a finite set, then the map $(x,\gamma)\mapsto\gamma$ induces a one-to-one correspondence between $\mathcal{G}^{x}$ and $\mathrm{S}(x)=\mathcal{F}$.

For a small subgroupoid $\mathcal{G}$, there is a one-to-one correspondence between the orbits and the equivalence classes of the irreducible representations of $\textrm{C*}(\mathcal{G})$.

Lemma 6.9** (Proposition 3.8 of [2]).**

Let $(X,\Gamma)$ be a free dynamical system, and let $\mathcal{G}\subseteq X\rtimes\Gamma$ be a relatively compact subgroupoid. Let $x\in X$. The map

[TABLE]

where $\{E_{g,h}\}_{\mathcal{G}^{x}}$ are the standard matrix units of $\mathcal{G}^{x}$, induces an irreducible representation of $\textrm{C*}(\mathcal{G}_{0})$ on $\mathbb{C}^{|\mathcal{G}^{x}|}$ (if $\mathcal{G}^{x}\neq\varnothing$), still denoted by $\pi_{x}$. Moreover, any irreducible representation of $\textrm{C*}(\mathcal{G}_{0})$ arises in this way, and $x\sim_{\mathcal{G}}y$ if and only if $\pi_{x}$ is unitarily equivalent to $\pi_{y}$.

Lemma 6.10**.**

Let $\mathcal{G}\subseteq X\rtimes\Gamma$ be a relatively compact subgroupoid. For each finite set $\mathcal{F}\subseteq\Gamma$, there is a homeomorphism

[TABLE]

Proof.

Define the map

[TABLE]

and it is the desired homeomorphism. ∎

Lemma 6.11**.**

Let $(X,\Gamma)$ be a free dynamical system, and let $\mathcal{G}\subseteq X\rtimes\Gamma$ be a relatively compact subgroupoid. Let $\mathcal{F}\subseteq\Gamma$ be a finite set. Then there is a homomorphism

[TABLE]

by

[TABLE]

where $\{E_{\gamma_{1},\gamma_{2}}:\gamma_{1},\gamma_{2}\in\mathcal{F}\}$ are the standard matrix units over $\mathcal{F}$ and $f\in\mathrm{C_{0}}(\mathcal{G})$ being regarded as a function on $X\rtimes\Gamma$.

Proof.

By Lemma 6.10, one has that $\mathcal{G}^{Z_{\mathcal{F}}}$ is homeomorphic to $\mathcal{Z}_{\mathcal{F}}\times\mathcal{F}$. Then, for each $g\in\mathcal{G}^{Z_{\mathcal{F}}}$, there is a unique $\gamma_{g}\in\mathcal{F}$ such that

[TABLE]

By Lemma 6.9, for any $x\in X$, the map

[TABLE]

induces an irreducible representation of $\textrm{C*}(\mathcal{G})$. Then, if $x\in Z_{\mathcal{F}}$, one has

[TABLE]

Regarding $x\mapsto E_{(x,\gamma_{1}),(x,\gamma_{2})}$ as the constant function $E_{\gamma_{1},\gamma_{2}}$, one has the homomorphism

[TABLE]

Note the function $\pi_{Z_{\mathcal{F}}}(f)$ can be continuously extended to $\partial Z_{\mathcal{F}}$, and the evaluation at each $x\in\partial Z_{\mathcal{F}}$ is still a (but non-irreducible) representation of $\textrm{C*}(\mathcal{G})$. ∎

Lemma 6.12**.**

Let $(X,\Gamma)$ be a free dynamical system, and let $\mathcal{G}\subseteq X\rtimes\Gamma$ be a relatively compact subgroupoid with unit space $X$. Since the unit space of $\mathcal{G}$ is $X$, one has the decomposition

[TABLE]

where $\mathcal{F}_{1},...,\mathcal{F}_{L}$ are finite subsets of $\Gamma$. Then the map

[TABLE]

is an injection.

Proof.

Note that any irreducible representation of $\mathrm{C^{*}}(\mathcal{G})$ factors though $\Phi$, and hence $\Phi$ must be injective. ∎

Lemma 6.13**.**

Let $(X,\Gamma)$ be a free dynamical system, and let $\mathcal{G}\subseteq X\rtimes\Gamma$ be a small subgroupoid. Let $\mathcal{F}\subseteq\Gamma$ be a finite set with $Z_{\mathcal{F}}\neq\varnothing$, and let $x\in\overline{Z_{\mathcal{F}}}$. Then there are finite sets $\mathcal{F}_{1},\mathcal{F}_{2},...,\mathcal{F}_{s}$ and $\gamma_{1},\gamma_{2},...,\gamma_{s}\in\mathcal{F}$ such that $e\in\mathcal{F}_{i}$, $i=1,2,...,s$,

[TABLE]

and

[TABLE]

The decomposition of $\mathcal{F}$ is unique (for the given $x$).

Proof.

Since the unit space of $\mathcal{G}$ is $X$, there is a partition

[TABLE]

such that $y\sim_{\mathcal{G}}z$ if and only if $y,z\in S_{i}$ for some $S_{i}$.

One claims that $S_{i}=[y]$, $i=1,2,...,s$, whenever $y\in S_{i}$. Then the lemma follows.

For each $S_{i}$ and $y\in S_{i}$, let $z\in X$ be $z\sim_{\mathcal{G}}y$. Since $z\sim_{\mathcal{G}}y$, there is $\gamma\in\Gamma$ such that $z=(y,\gamma)\in\mathcal{G}$. Since $\mathcal{G}$ is open, one has that $(y^{\prime},\gamma)\in\mathcal{G}$ if $y^{\prime}$ is sufficiently close to $y$. Since $y\in x\mathcal{F}$, there is $\gamma^{\prime}\in\mathcal{F}$ such that $y=x\gamma^{\prime}$. Note that $x\in\overline{Z_{\mathcal{F}}}$, there is a sequence $(x_{n})\subseteq Z_{\mathcal{F}}$ with $x_{n}\to x$ as $n\to\infty$, and hence $x_{n}\gamma^{\prime}\to x\gamma^{\prime}=y$, as $n\to\infty$. In particular, $(x_{n}\gamma^{\prime},\gamma)\in\mathcal{G}$ for sufficiently large $n$. Noting that $(x_{n},\gamma^{\prime})\in\mathcal{G}$ (since $x_{n}\in Z_{\mathcal{F}}$ and $\gamma^{\prime}\in\mathcal{F}$), one has

[TABLE]

Hence $\gamma^{\prime}\gamma\in\mathcal{F}$, and $z=y\gamma=x\gamma^{\prime}\gamma\in x\mathcal{F},$ as desired. ∎

Corollary 6.14**.**

Let $(X,\Gamma)$ be a free dynamical system, and let $\mathcal{G}\subseteq X\rtimes\Gamma$ be a small groupoid. With the notation as above, one has that for any $f\in\textrm{C*}(\mathcal{G})$,

[TABLE]

Proof.

Note that if $\gamma_{g}\in\gamma_{i}\mathcal{F}_{i}$ and $\gamma_{h}\in\gamma_{j}\mathcal{F}_{j}$ with $i\neq j$, then $(x\gamma_{h},\gamma_{h}^{-1}\gamma_{g})\notin\mathcal{G}$. Otherwise, $x\gamma_{h}\sim_{\mathcal{G}}x\gamma_{g}$, which contradicts the fact that $\mathrm{S}(x\gamma_{i})=\mathcal{F}_{i}$ and $\mathrm{S}(x\gamma_{j})=\mathcal{F}_{j}$. In particular, for any $f\in\mathrm{C}_{0}(\mathcal{G})$, one has

[TABLE]

It then follows from Lemma 6.11 that for any $f\in\mathrm{C}_{0}(\mathcal{G})$,

[TABLE]

as desired. ∎

Theorem 6.15**.**

Let $(X,\Gamma)$ be a free dynamical system, let $\mathcal{G}\subseteq X\rtimes\Gamma$ be a small subgroupoid, and let $E\subseteq X$ be an open set. Then there is an isomorphism

[TABLE]

where $C_{i}=\mathrm{M}_{n_{i}}(\mathrm{C}(Z_{i}))$ for a set $Z_{i}$ which is a disjoint union of finitely many closed subsets of $X$, and $C_{i}^{(0)}=\mathrm{M}_{n_{i}}(\mathrm{C}(Z^{(0)}_{i}))$ with $Z^{(0)}_{i}$ a closed subset of $Z_{i}$, such that

1. (1)

$n_{0}<n_{1}<\cdots<n_{K}$, 2. (2)

all the maps $\eta_{i}:A_{i}\to C_{i+1}^{(0)}$ are of diagonal type, where

[TABLE] 3. (3)

under this isomorphism and the RSH-decomposition, the element $\varphi_{E}\in\mathrm{C^{*}}(\mathcal{G})$ is a diagonal matrix on each $Z_{i}$ and

[TABLE]

where $\varphi_{E}$ is defined in (2.1).

Proof.

Decompose $X$ into

[TABLE]

where $\mathcal{F}_{1},...,\mathcal{F}_{L}$ are finite subsets of $\Gamma$. By Lemma 6.12, there is an embedding

[TABLE]

Reindex $\mathcal{F}_{i}$ into

[TABLE]

so that

[TABLE]

and

[TABLE]

Put

[TABLE]

Note that the sets $\overline{Z_{\mathcal{F}_{i,j}}}$, $j=1,...,l_{i}$, considered as subsets of $X$, might have non-empty intersections; but one takes the abstract disjoint union in the construction of $Z_{i}$. Also note that

[TABLE]

and $Z_{i}^{(0)}$ is a closed subset of $Z_{i}$ (follows from Lemma 6.6).

By Lemma 6.13, each $\overline{Z_{\mathcal{F}_{i,j}}}\setminus Z_{\mathcal{F}_{i,j}}$ has a partition

[TABLE]

such that for each $(Z_{i,j}^{(0)})_{d}$, there is a decomposition

[TABLE]

with $\gamma_{1},...,\gamma_{s_{d}}\in\mathcal{F}_{i,j}$ such that

[TABLE]

and the decomposition of $\mathcal{F}_{i,j}$ is different for each $d$.

For each $(Z_{i,j}^{(0)})_{d}$, $d=1,...,d_{i,j}$, consider the map

[TABLE]

and with all of these maps, one has the RSH algebra

[TABLE]

with $C_{i}=\mathrm{M}_{n_{i}}(\mathrm{C}(Z_{i}))$, $C_{i}^{(0)}=\mathrm{M}_{n_{i}}(\mathrm{C}(Z_{i}^{(0)}))$, and all the maps $A_{i}\to C_{i+1}^{(0)}$ are of diagonal type.

By Corollary 6.14, one has $\Phi(\textrm{C*}(\mathcal{G}))\subseteq A$. One the other hand, the embedding map $\Phi$ induces a one-to-one correspondence between the irreducible representations of $\textrm{C*}(\mathcal{G})$ and the irreducible representations of $A$. Thus $\textrm{C*}(\mathcal{G})$ is a rich subalgebra of $A$, and hence $\Phi(\textrm{C*}(\mathcal{G}))=A$ (see Theorem 11.1.6 of [4]).

Consider the open set $E$ and the function $\varphi_{E}\in\mathrm{C}(X)$. Then, for each $x\in{Z_{\mathcal{F}_{i,j}}}$,

[TABLE]

as desired. ∎

7. Comparison of diagonal elements and comparison of open sets

In this section, let us show that the Cuntz comparison of the diagonal elements in an RSH-algebra with diagonal maps is roughly determined by their ranks (see Theorem 7.8), regardless of the dimensions of the base spaces. As a consequence, the Cuntz comparison of open subsets of $X$ is then determined by the partial orbit or by the measures if there exist small subgroupoids with arbitrarily large orbit (see Corollary 7.12 and Corollary 7.13). The proof of the main result of this section (Theorem 7.8) follows closely the argument of Theorem 4.5 of [15].

First, recall the following comparison theorem for general RSH algebras, .

Theorem 7.1** (Theorem 4.6 of [41]).**

Let $A$ be a recursive subhomogeneous C-algebras with a decomposition*

[TABLE]

where

[TABLE]

with $X^{(0)}_{i}$ a closed subset of $X_{i}$. Let $a,b\in A^{+}$ be positive elements satisfying

[TABLE]

for any $i=0,1,...,l$. Then $a\precsim b$.

Using this theorem, one has the following comparison result.

Proposition 7.2**.**

Let $A$ be a separable RSH algebra with a fixed decomposition

[TABLE]

and let $a,b$ be positive elements of $A$. Write

[TABLE]

with $X^{(0)}_{i}$ a closed subset of $X_{i}$.

For each $i=0,1,...,l$, list

[TABLE]

and set

[TABLE]

and

[TABLE]

$j=0,1,...,s_{i}.$**

Assume that for each $i=1,2,...,l$ and each $j=0,1,...,s_{i}$,

[TABLE]

Then $a\precsim b$ in $A$.

Proof.

Let us construct a new RSH decomposition of $A$:

[TABLE]

with $D_{i}=\mathrm{M}_{m_{i}}(\mathrm{C}(Y_{i}))$ and $D_{i}^{(0)}=\mathrm{M}_{m_{i}}(\mathrm{C}(Y^{(0)}_{i}))$ with $Y^{(0)}_{i}$ a closed subset of $Y_{i}$ such that

[TABLE]

in this new RSH decomposition. Then, it follows from Theorem 7.1 that $a\precsim b$.

To building the new RSH decomposition, let us start with

[TABLE]

Since the rank function is lower semicontinuous, one has that

[TABLE]

are closed. (Note that $W_{0,j}=Z_{0,0}\cup\cdots\cup Z_{0,j}$.)

Set

[TABLE]

with the map $D_{0,0}\to D^{(0)}_{0,1}$ being the identity map. Then it is clear that

[TABLE]

Putting

[TABLE]

and repeating this construction, one obtains the new RSH decomposition of $C_{0}=\mathrm{M}_{n_{0}}(\mathrm{C}(X_{0}))$ by

[TABLE]

where

[TABLE]

It clearly satisfies (7.1).

Now, assume that one has a desired decomposition of

[TABLE]

let us construct the decomposition for $A_{i}\oplus_{C_{i+1}^{(0)}}C_{i+1}$ which satisfies (7.1).

Write

[TABLE]

and denote by $\rho:A_{i}\to C_{i+1}^{(0)}$ the homomorphism associated with the (original) RSH decomposition of $A$.

Set

[TABLE]

and

[TABLE]

For the next stage, consider

[TABLE]

and

[TABLE]

where $(\rho(f)|_{X_{i+1}^{(0)}\cap W_{i+1,1}},g)$ is regarded as the (well defined) function on

[TABLE]

which is $\rho(f)$ on $X_{i+1}^{(0)}\cap W_{i+1,1}$ and is $g$ on $W_{i+1,0}$.

At $j$-th stage, set

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

is regarded as the function on

[TABLE]

which is $\rho(f)$ on $X_{i+1}^{(0)}\cap W_{i+1,j}$ and is $g_{j-1}$ on $W_{i+1,j-1}$. It is well defined.

In this new RSH-decomposition, note that the base spaces are $W_{i+1,j}$ together with the closed subset $(X_{i}^{(0)}\cap W_{i+1,j})\cup W_{i+1,j-1}$. Note that

[TABLE]

and hence

[TABLE]

and it satisfies the condition (7.1). ∎

Definition 7.3**.**

Let $A$ be a separable RSH algebra with a fixed decomposition

[TABLE]

Write

[TABLE]

with $X^{(0)}_{k}$ a closed subset of $X_{k}$. Let $\mathcal{F}\subseteq A$ be a finite set such that if

[TABLE]

then

[TABLE]

Then, for each $k=0,1,...,K$, define the map $\Phi_{\mathcal{F},k}:X_{k}\to\mathbb{R}^{n_{k}^{2}\left|\mathcal{F}\right|}$ by

[TABLE]

The following lemma shows that in an RSH algebra $A$ with diagonal maps, there always exist finitely many elements of $A$ such that these elements span the whole $\mathrm{M}_{n}(\mathbb{C})$ under any irreducible representation of $A$ but the topological dimension of the values of their entries is controlled by the dimension of the irreducible representations of $A$. These elements play the same role as that of the element $uH$ in the proof of Theorem 4.5 of [15].

Lemma 7.4**.**

Let $A$ be a separable RSH algebra with a fixed decomposition

[TABLE]

Write

[TABLE]

with $X^{(0)}_{i}$ a closed subset of $X_{i}$. Assume

1. (1)

$n_{0}<n_{1}<\cdots<n_{K}$, and 2. (2)

all the maps

[TABLE]

are of diagonal type, where

[TABLE]

Then there are finite sets

[TABLE]

*such that *

1. (1)

[TABLE] 2. (2)

there are continuous functions

[TABLE]

such that $\chi_{i}^{-1}(0)=X_{i}^{(0)}$ and

[TABLE]

where $\{e_{s,t}:1\leq s,t\leq n_{i}\}$ are the standard matrix units of $\mathrm{M}_{n_{i}}(\mathbb{C})\subseteq\mathrm{M}_{n_{i}}(\mathrm{C}(X_{i}))$; 3. (3)

and there are polyhedrons

[TABLE]

such that

[TABLE]

and

[TABLE]

for any $0\leq k\leq K$.

Proof.

Let us construct $\mathcal{E}_{k}$, $k=0,1,...,K$, recursively. Consider

[TABLE]

and put

[TABLE]

where $e_{i,j}^{(0)}$ are matrix units of $\mathrm{M}_{n_{0}}(\mathbb{C})\subseteq\mathrm{M}_{n_{0}}(\mathrm{C}(X_{0}))$. Since $\mathcal{E}_{0}$ consists of constant functions, $\Phi_{\mathcal{E}_{0},0}$ is constant. Then it is clear that

[TABLE]

where $\mathrm{pt}$ is the constant value of $\Phi_{\mathcal{E}_{0},0}$, satisfy (7.2), (7.3), and (7.4) (with $\chi_{0}$ the constant function $1$).

Assume

[TABLE]

are constructed to satisfy (7.2), (7.3), and (7.4). Consider

[TABLE]

If $X_{k}^{(0)}=\varnothing$, then $A_{k}:=A_{k-1}\oplus C_{k}$. Define

[TABLE]

where $e_{i,j}^{(k)}$ are matrix units of $\mathrm{M}_{n_{k}}(\mathbb{C})\subseteq\mathrm{M}_{n_{k}}(\mathrm{C}(X_{k}))=C_{k}$. Then

[TABLE]

where $\mathcal{E}_{k-1}$ and $\mathcal{E}_{k}^{\prime}$ are regarded as subsets of $A_{k-1}\oplus C_{k}$ naturally, and

[TABLE]

where $\mathrm{pt}$ is the constant value of $\Phi_{\mathcal{E}_{k},k}$ on $X_{k}$, satisfy (7.2), (7.3), and (7.4) (with $\chi_{k}$ the constant function $1$).

Assume that $X_{k}^{(0)}\neq\varnothing$. Since the map $\eta$ is of diagonal type, there exist $l\in\mathbb{N}$ and a partition

[TABLE]

such that for each $Y_{i}$, there are continuous maps

[TABLE]

for some $s_{i}\in\mathbb{N}$ such that

[TABLE]

where $(f_{0},f_{1},...,f_{k-1})\in A_{k-1}.$ Then it is clear that

[TABLE]

Let us estimate the dimension of $V_{k,i}^{\prime}$. Let $(m_{0},m_{1},...,m_{k-1})$ be the multiplicities of the map $\eta|_{Y_{i}}$. Note that

[TABLE]

and

[TABLE]

Since $n_{k}>n_{k-1}$, one has that

[TABLE]

Hence, for each $i=1,2,...,k$, one has

[TABLE]

Put

[TABLE]

By (7.5), one has

[TABLE]

Also note that

[TABLE]

For each element $e\in\mathcal{E}_{k-1}\subseteq A_{k-1}$, consider $\eta(e)\in C_{k}^{(0)}$. Since $V_{k}^{\prime}$ is a polyhedron, it is a neighborhood retraction; hence there is an open set $U\supseteq X_{k}^{0}$ such that there is an extension of $\eta(e)$ to $U$, denoted by $e^{\prime}$, such that

[TABLE]

Pick a continuous function $\chi_{k}:X_{k}\to[0,1]$ such that

[TABLE]

Define

[TABLE]

Set

[TABLE]

and

[TABLE]

where $e_{i,j}^{(k)}$ are matrix units of $\mathrm{M}_{n_{k}}(\mathbb{C})\subseteq\mathrm{M}_{n_{k}}(\mathrm{C}(X_{k}))$. Then

[TABLE]

satisfies (7.2), (7.3), and (7.4) (for $A_{k}$).

Indeed, since $\mathcal{E}^{\prime}_{k}\subseteq\mathcal{E}_{k}$, (7.2) is satisfied. Define

[TABLE]

Then, by (7.7),

[TABLE]

and by (7.6),

[TABLE]

So, (7.3) and (7.4) are satisfied (for $A_{k}$). By the induction, the desired finite sets $\mathcal{E}\subseteq A$ exist. ∎

Remark 7.5*.*

The estimation which is needed later in Theorem 7.8 on the dimension of $V_{i}$ is actually

[TABLE]

But the stronger version (7.3) is needed for the induction argument.

Lemma 7.6**.**

Consider $n\times n$ matrices

[TABLE]

and

[TABLE]

where $\alpha,\beta\in\mathbb{R}\setminus\{0\}$ and $e_{s,t}$ is the matrix with $(s,t)$-entry $1$ and all other entries [math].

If there is a unitary $U\in\mathrm{M}_{n}(\mathbb{C})$ satisfying

[TABLE]

then

[TABLE]

Proof.

Note that

[TABLE]

and hence $\alpha=\beta$. Consider a pair of matrices $(a_{i,j}^{(k)}),(b_{i,j}^{(k)})$. Then, for any $1\leq s,t\leq n$,

[TABLE]

Since $\alpha=\beta\neq 0$, one has that $a_{s,t}^{(k)}=b_{s,t}^{(k)}$, as desired. ∎

Lemma 7.7** (Lemma 4.3 of [15]).**

Let $X$ be a second countable locally compact Hausdorff space, and let $S$ be a sub-C-algebra of $\mathrm{M}_{n}(\mathrm{C_{0}}(X))$.*

Suppose that there exist a topological space $\Delta$ and a surjective continuous map $\xi:X\to\Delta$ such that

1. (1)

for any $x_{1},x_{2}\in X$,

[TABLE]

if and only if

[TABLE]

where $\pi_{x}$ is the standard irreducible representation of $\mathrm{M}_{n}(\mathrm{C_{0}}(X))$ at $x\in X$, 2. (2)

for any sequence $x_{i},i=1,2,...,$ in $X$, any $x$ in $X$, and any $g\in S$, if

[TABLE]

then

[TABLE]

and 3. (3)

$\pi_{x}(S)=\mathrm{M}_{n}(\mathbb{C})$, for any $x\in X$.

Then there is an isomorphism $\phi:\mathrm{M}_{n}(\mathrm{C}_{0}(\Delta))\to S$. Moreover, under this isomorphism, if $X^{(0)}$ is a closed subset of $X$ and $\pi_{X^{(0)}}$ is the restriction map, there is a commutative diagram

[TABLE]

where $\Delta^{(0)}=\xi(X^{(0)})$ and $\pi_{\Delta^{(0)}}$ is the restriction map.

Proof.

For each $f\in S$, define a function $\tilde{f}:\Delta\to\mathrm{M}_{n}(\mathbb{C})$ by

[TABLE]

By Condition (2), $\tilde{f}$ is well defined, and $\tilde{f}$ is continuous. Moreover, $\tilde{f}$ vanishes at infinity. To see this, note that, if $z_{i}\in\Delta$ with $z_{i}\to\infty$, then, since $\xi$ is surjective, there are $x_{i}\in X$ with $\xi(x_{i})=z_{i}$. Then $x_{i}\to\infty$. Otherwise, there is a subsequence, say $(x_{i_{k}})$, converging to a point $x\in X$. Since $\xi$ is continuous, one has that

[TABLE]

which contradicts the assumption $z_{i}\to\infty$. Hence

[TABLE]

and $\tilde{f}\in\mathrm{M}_{n}(\mathrm{C_{0}}(\Delta))$.

Moreover, it is clear that the map $f\mapsto\tilde{f}$ is an injective homomorphism, and thus one can regard $S$ as a sub-C*-algebra of $\mathrm{M}_{n}(\mathrm{C_{0}}(\Delta))$. It follows from Conditions (1) and (3) that $S$ is a rich sub-C*-algebra of $\mathrm{M}_{n}(\mathrm{C_{0}}(\Delta))$ in the sense of Dixmier (11.1.1 of [4]), and therefore $S=\mathrm{M}_{n}(\mathrm{C}_{0}(\Delta))$ by Proposition 11.1.6 of [4] (or by Theorem 7.2 of [26]).

Note that the above construction also induces an isomorphism

[TABLE]

by $f\mapsto\tilde{f}$ with $\tilde{f}(\xi(x))=f(x)$. Also note that for any $f\in\mathrm{M}_{n}(\mathrm{C}(\Delta))$, and any $x\in X^{(0)}$, one has

[TABLE]

Thus, the diagram commutes. ∎

The following is the main result of this section. Its proof is in the same line as that of Theorem 4.5 of [15]: One considers the sub-C*-algebra generated by the given elements $a$, $b$, and the finite set $\mathcal{E}_{K}$ obtained from Lemma 7.4; then the resulting sub-C*-algebra actually has the dimension gap property of Proposition 7.2, and the comparison between $a$ and $b$ follows.

Theorem 7.8**.**

Let $A$ be a separable RSH algebra with a fixed decomposition

[TABLE]

and let $a,b$ be positive elements of $A$. Write

[TABLE]

with $X^{(0)}_{i}$ a closed subset of $X_{i}$. Assume

1. (1)

$n_{0}<n_{1}<\cdots<n_{K}$, 2. (2)

$a$* and $b$ are diagonal matrices on each $X_{i}$,* 3. (3)

all the maps $\eta_{i}:A_{i}\to C_{i+1}^{(0)}$, $i=0,...,K-1$, are of diagonal type, where

[TABLE] 4. (4)

for each $i=0,1,...,K$, and each $x\in X_{i}\setminus X_{i}^{(0)}$,

[TABLE]

Then $a\precsim b$.

Proof.

Write

[TABLE]

with $a_{i},b_{i}\in C_{i}$, $i=0,1,...,K$.

Let us construct sub-C*-algebras

[TABLE]

together with closed subsets $\Delta_{i}^{(0)}\subseteq\Delta_{i}$ ($\Delta_{0}^{(0)}=\varnothing$) satisfying

[TABLE]

where $D_{i}^{(0)}=\mathrm{M}_{n_{i}}(\mathrm{C}(\Delta_{i}^{(0)}))$, such that

[TABLE]

and if

[TABLE]

and define

[TABLE]

where $j=0,1,...,s_{i},$ then

[TABLE]

Then, by (7.8) and (7.9),

[TABLE]

is a sub-C*-algebra of $A$ with $a,b\in S$; by (7.10) and Proposition 7.2, one has $a\precsim b$, and this proves the theorem.

Let us construct the desired sub-C*-algebras $D_{i}$ recursively. By Lemma 7.4, there exist finite sets $\mathcal{E}_{k}\subseteq A_{k}$, where

[TABLE]

such that

[TABLE]

[TABLE]

and there is a polyhedron $V_{k}\subseteq\mathbb{R}^{n_{k}^{2}\left|\mathcal{E}_{k}\right|}$ such that

[TABLE]

and

[TABLE]

Define

[TABLE]

It clearly satisfies (7.9) (with $i=0$). (7.8) is also satisfied with $\Delta_{0}^{(0)}=\varnothing$. Set

[TABLE]

and consider the map

[TABLE]

Set

[TABLE]

Then, for any $x\in X_{0}$, the restriction of $\pi_{x}$ to $D_{0}$ is still irreducible, and by Lemma 7.6, one has that for any $x_{1},x_{2}\in X_{0}$,

[TABLE]

By Lemma 7.7,

[TABLE]

Write

[TABLE]

where $j=0,1,...,s_{0}$, and

[TABLE]

where $j=0,1,...,s_{0}.$ Then

[TABLE]

Since $a_{0}$ and $b_{0}$ are diagonal and

[TABLE]

by the construction of $\Phi_{{\{a,b\}\cup\mathcal{E}_{0}},0}$ ((7.15)), one has

[TABLE]

Hence

[TABLE]

and for any $x\in Z_{0,j}$,

[TABLE]

Thus, the C*-algebra $D_{0}$ satisfies (7.10) (with $i=0$).

Let us assume that $D_{0},D_{1},...,D_{k-1}$ are constructed to satisfy (7.8) (with $i=k-1$), (7.9) (with $i=k-1$), and (7.10) (with $i=k-1$), and also assume that

[TABLE]

Let us construct $D_{k}$. Define

[TABLE]

where $e_{s,t},1\leq s,t\leq n_{k}$ are (constant) matrix units of $\mathrm{M}_{n_{k}}(\mathbb{C})\subseteq\mathrm{M}_{n_{k}}(\mathrm{C}(X_{k}))$. Set

[TABLE]

and consider

[TABLE]

and set

[TABLE]

Then, for any $x\in X_{k}$, the restriction of $\pi_{x}$ to $D_{k}$ is still irreducible, and by Lemma 7.6, for any $x_{1},x_{2}\in X_{k}$,

[TABLE]

By Lemma 7.7, one has that $D_{k}\cong\mathrm{M}_{n_{k}}(\mathrm{C}(\Delta_{k})$ and

[TABLE]

By (7.11),

[TABLE]

Then, by (7) and $\mathcal{E}_{k}\in D_{k}$, one has

[TABLE]

and hence (7.8) is satisfied (with $i=k$).

Write

[TABLE]

where $j=0,1,...,s_{k}$ and

[TABLE]

where $j=0,1,...,s_{k}.$ Then

[TABLE]

Since $a_{k}$ and $b_{k}$ are diagonal and

[TABLE]

by the construction of $\Phi_{{\{a,b\}\cup\mathcal{E}_{k}\cup\{e_{s,t}\}},k}$ ((7)), one has

[TABLE]

Hence

[TABLE]

and for any $x\in Z_{k,j}$,

[TABLE]

Thus, the C*-algebra $D_{k}$ satisfies (7.10) (with $i=k$).

Put

[TABLE]

and consider

[TABLE]

For the induction, we also need to show

[TABLE]

(Note that

[TABLE]

since $\mathcal{E}_{k}|_{(X_{0},...,X_{k-1})}=\mathcal{E}_{k-1}$).

Indeed, set

[TABLE]

and let us show that actually

[TABLE]

Pick any point

[TABLE]

Since $\mathcal{E}_{k}\subseteq B$ and (7.12), the restriction of $\pi_{x}$ to $B$ is still irreducible (with the same dimension), and hence any irreducible representation of $B$ or

[TABLE]

actually is a restriction of some $\pi_{x}$. In particular, the restriction of any irreducible representation of

[TABLE]

to $B$ is still irreducible.

Let $\pi_{x_{1}}$ and $\pi_{x_{2}}$ be irreducible representations of $B$ which are not equivalent. If $x_{1}$ and $x_{2}$ are in different components, say, in $X_{i}\setminus X_{i}^{(0)}$ and $X_{j}\setminus X_{j}^{(0)}$, $i\neq j$, respectively, then the the restrictions of $\pi_{x_{1}}$ and $\pi_{x_{2}}$ to $B$ are not equivalent since they have different dimensions.

Assume that $x_{1},x_{2}\in X_{i}\setminus X_{i}^{(0)}$ with $i<k$. In particular, $\pi_{x_{1}}$ and $\pi_{x_{2}}$ are irreducible representations of

[TABLE]

Since $\pi_{x_{1}}$ and $\pi_{x_{2}}$ are not equivalent, one has that

[TABLE]

Since the restriction of $\mathcal{E}_{k}$ to $X_{i}$ is the same as the restriction of $\mathcal{E}_{k-1}$ to $X_{i}$, one has

[TABLE]

Hence, by Lemma 7.6, the restrictions of $\pi_{x_{1}}$ and $\pi_{x_{2}}$ to $B$ are not equivalent.

Assume $x_{1},x_{2}\in X_{k}\setminus X_{k}^{(0)}$. If the restriction of $\pi_{x_{1}}$ and $\pi_{x_{2}}$ to $B$ are equivalent, then, by Lemma 7.6,

[TABLE]

Since matrix units $\{e_{s,t}\}$ are constant functions, one then has

[TABLE]

So, $\pi_{x_{1}}$ and $\pi_{x_{2}}$ are equivalent as representations of $D_{k}$ and hence are equivalent, which contradicts the assumption. Therefore, the restrictions of $\pi_{x_{1}}$ and $\pi_{x_{2}}$ to $B$ are not equivalent. Hence $B$ is a rich sub-C*-algebra, and therefore,

[TABLE]

by Proposition 11.1.6 of [4].

Therefore, by induction, there are C*-algebras $D_{0},D_{1},...,D_{K}$ satisfying (7.8), (7.9) and (7.10), and hence $a\precsim b$, as desired. ∎

The following are several corollaries of Theorem 7.8

Corollary 7.9**.**

Let $X$ be compact metrizable, and let $\sigma:X\to X$ be a minimal free homeomorphism. Suppose $E,F\subseteq X$ are open sets satisfying

[TABLE]

Then

[TABLE]

in $\mathrm{C}(X)\rtimes_{\sigma}\mathbb{Z}$.

Proof.

Since $\sigma$ is minimal, there is $\delta>0$ such that

[TABLE]

Let $\varepsilon>0$ be arbitrary. One asserts that there is $N_{0}>\frac{4}{\delta}$ such that for any $x\in X$ and any $N>N_{0}$, one has

[TABLE]

and

[TABLE]

Indeed, if the assertion were not true, there are $(k_{n})\subseteq\mathbb{N}$ and $(x_{k_{n}})\subseteq X$ such that $k_{n}\to\infty$ as $n\to\infty$, and for all $n$, one has

[TABLE]

or

[TABLE]

Consider the discrete probability measures

[TABLE]

where $\delta_{x}$ is the Dirac measure concentrated at $x$. Pick an accumulation point $\mu_{\infty}$ in the weak*-topology. Note that $\mu_{\infty}\in\mathcal{M}_{1}(X,\sigma)$. Passing to a subsequence, one assumes that $\mu_{n}\to\mu_{\infty}$.

Assume that (7) holds for infinitely many $n$, pick a closed set $E^{\prime}$ such that

[TABLE]

and then

[TABLE]

which contradicts to (7.20).

Assume that (7.22) holds for infinitely many $n$. Since $F$ is open, then

[TABLE]

which contradicts the choice of $\delta$. This proves the assertion.

Consider the C*-algebra $A_{Y}\subseteq\mathrm{C}(X)\rtimes_{\sigma}\mathbb{Z}$, where $Y$ is a closed subset with nonempty interior. By Theorem 5.5, $A_{Y}$ is an RSH algebra with diagonal maps, and the canonical RSH decomposition of $A_{Y}$ satisfies Conditions (1), (2), and (3) of Theorem 7.8. With $Y$ sufficiently small, one can assume that the heights of the Rokhlin towers in the decomposition of $A_{Y}$ are at least $N_{0}$ (so that $\frac{1}{N_{0}}<\frac{\delta}{4}$), and therefore, by the assertion, Conditions (4) of Theorem 7.8 is also satisfied. Thus, it follows from Theorem 7.8 that $(\varphi_{E}-\varepsilon)_{+}\precsim\varphi_{F}$. Since $\varepsilon$ is arbitrary, one has $\varphi_{E}\precsim\varphi_{F}$. ∎

Corollary 7.10**.**

Let $X$ be a separable compact Hausdorff space, and let $\sigma:X\to X$ be a minimal free homeomorphism. Then

[TABLE]

Proof.

By Lemma 3.6, $(X,\sigma)$ has the (URP). By Corollary 7.9, $\mathrm{C}(X)\rtimes\mathbb{Z}$ has $(\frac{1}{4},1)$-Cuntz-comparison on open sets. The statement then follows from Theorem 4.8. ∎

Remark 7.11*.*

Corollary 7.10 is generalized in [33] to $\mathbb{Z}^{d}$-actions.

Corollary 7.12**.**

Let $(X,\Gamma)$ be a free dynamical system, and let $\mathcal{G}\subseteq X\rtimes\Gamma$ be a small subgroupoid. Let $E,F\subseteq X$ be open sets such that

[TABLE]

and

[TABLE]

Then, $\varphi_{E}\precsim\varphi_{F}$ in $\textrm{C*}(\mathcal{G})$.

Proof.

This follows directly from Theorem 6.15 and Theorem 7.8. ∎

Corollary 7.13**.**

Let $(X,\Gamma)$ is a minimal free dynamical system, where $\Gamma$ is amenable. Assume that $(X,\Gamma)$ has the property that for any finite set $K\subseteq\Gamma$ and any $\varepsilon>0$, there is a small subgroupoid $\mathcal{G}\subseteq X\rtimes\Gamma$ such that $\mathrm{Orbit}_{\mathcal{G}}(x)$ is $(K,\varepsilon)$-invariant for any $x\in X$. Then, for any open sets $E,F\subseteq X$ with

[TABLE]

one has that

[TABLE]

In other words, $\mathrm{C}(X)\rtimes\Gamma$ has $(\frac{1}{4},1)$-Cuntz-comparison of open sets.

Proof.

Without loss of generality, one may assume that $\left|\Gamma\right|=\infty$. Otherwise, the C*-algebra $\mathrm{C}(X)\rtimes\Gamma$ is isomorphic to $\mathrm{M}_{\left|\Gamma\right|}(\mathbb{C})$, and the statement holds.

Since $\sigma$ is minimal, there is $\delta>0$ such that

[TABLE]

Let $\varepsilon^{\prime}>0$ be arbitrary. With the same argument as that of Corollary 7.9, it follows from (7.23) that there exists $(K,\varepsilon)$ such that if $\Gamma_{0}\subseteq\Gamma$ is $(K,\varepsilon)$-invariant, then for any $x\in X$, one has

[TABLE]

and

[TABLE]

If $\mathcal{G}\subseteq X\rtimes\Gamma$ is a small subgroupoid with all orbits $(K,\varepsilon)$-invariant, it then follows from Corollary 7.12 that $(\varphi_{E}-\varepsilon^{\prime})_{+}\precsim\varphi_{F}$. Since $\varepsilon^{\prime}$ is arbitrary, one has $\varphi_{E}\precsim\varphi_{F}$. ∎

8. Lower semicontinuous set-valued functions and small subgroupoids

In this section, let us show that if there is an equivariant lower semicontinuous set-valued function on $X$ (in particular, if lower semicontinuous dynamical tiling exists), then there always exists a small subgroupoid associated to this function (Theorem 8.3).

Definition 8.1**.**

Consider a topological dynamical system $(X,\Gamma)$. A shape function with domain $\Omega$, where $\Omega$ is an open subset of $X$, is a set-valued function

[TABLE]

such that

1. (1)

$e\in S(x)$, $x\in\Omega$, 2. (2)

$S(x)$ is uniformly bounded in the sense that there is a finite set $M\subseteq\Gamma$ such that

[TABLE] 3. (3)

the function $S$ is lower semicontinuous in the sense that for any $x\in\Omega$, there is an open neighbourhood $U\ni x$ such that

[TABLE]

and 4. (4)

the function $S$ is equivariant in the sense that

[TABLE]

Example 8.2*.*

Consider the dynamical system $(X,\sigma)$, where $\sigma:X\to X$ is a minimal homeomorphism. Let $Y\subseteq X$ be a closed set with non-empty interior. For each $x\in X$, define the positive first return time and negative first return time of $x$ to be

[TABLE]

and

[TABLE]

Then

[TABLE]

is a shape function (with domain $X$).

Actually, let

[TABLE]

be the Rokhlin partition associated to $Y$ (see Section 5). Then

[TABLE]

Theorem 8.3**.**

Let $(X,\Gamma)$ be a topological dynamical system, and let $S$ be a shape function with domain $\Omega$. Then there is an open and relatively compact subgroupoid $\mathcal{G}\subseteq X\rtimes\Gamma$ such that the unit space is $\{(x,e):x\in\Omega\}$ and

[TABLE]

Proof.

For each finite subset $\mathcal{F}\subseteq\Gamma$, define

[TABLE]

Since $x\mapsto\mathcal{S}(x)$ is equivariant, there are finite subsets $F_{1},F_{2},...,F_{n}\subseteq\Gamma$ and a partition of $\Omega$

[TABLE]

For each $\gamma\in\Gamma$, define

[TABLE]

Note that

[TABLE]

Let us verify that $\Omega\setminus Y_{\gamma}$ is open.

Assume that $x\in\Omega\setminus Y_{\gamma}$, then

[TABLE]

and let us assume that

[TABLE]

for some $\gamma_{i}\in F^{-1}_{i}\cap F^{-1}_{i}\gamma$. That is

[TABLE]

Since $x\mapsto\mathcal{S}(x)$ is lower semicontinuous, there is an open set $U\ni x$ such that

[TABLE]

Let us show that $U\subseteq\Omega\setminus Y_{\gamma}$, and hence $\Omega\setminus Y_{\gamma}$ is open.

For each $y\in U$, there is $F_{j}$ and $c\in\Gamma$ such that

[TABLE]

Since $e\in F_{j}c$, one has $c\in F_{j}^{-1}$. Also note that

[TABLE]

and $e\in F_{i}\gamma_{i}\gamma^{-1}$ (since $\gamma_{i}\in F_{i}^{-1}\gamma$); one has $e\in F_{j}c\gamma^{-1}$ and hence $c\in F_{j}^{-1}\gamma$. That is

[TABLE]

In particular,

[TABLE]

This shows that $\Omega\setminus Y_{\gamma}$ is open.

Define

[TABLE]

Since $\Omega\setminus Y_{\gamma}$ is open, $\mathcal{G}$ is an open (and relatively compact) subset containing $(x,e)$, $x\in\Omega$. Let us show that $\mathcal{G}$ is actually a subgroupoid.

Let $(x,\gamma)\in\mathcal{G}$. Then there are $F_{i}$ and $\gamma_{i}\in F_{i}^{-1}\cap F_{i}^{-1}\gamma$ such that

[TABLE]

and hence $x\in Z_{F_{i}}\gamma_{i}\gamma^{-1}$. Since $\gamma_{i}\in F_{i}^{-1}\cap F_{i}^{-1}\gamma$, one has

[TABLE]

and therefore,

[TABLE]

(note that $x\gamma\in\Omega$).

Let $x\in X$, $\gamma_{1},\gamma_{2}\in\Gamma$ with

[TABLE]

Then, there are

[TABLE]

such that

[TABLE]

Since $\gamma_{i_{2}}\in F_{i_{2}}^{-1}\gamma_{2}$, one has

[TABLE]

and hence

[TABLE]

Noting that

[TABLE]

and

[TABLE]

one has that $F_{i_{1}}=F_{i_{2}}$. So, let us denote both $F_{i_{1}}$ and $F_{i_{2}}$ by $F_{i}$.

Since

[TABLE]

one has

[TABLE]

Since

[TABLE]

one has

[TABLE]

and hence

[TABLE]

Therefore,

[TABLE]

Thus $\mathcal{G}$ is a subgroupoid of $X\rtimes\Gamma$.

It is clear that the unit space of $\mathcal{G}$ is $\{(x,e):x\in\Omega\}$. Let $x\in\Omega$, and let us calculate the $\mathcal{G}$-orbit of $x$. Assume that $x\in Z_{F_{i}}$ for some $F_{i}$. Then

[TABLE]

If $(x,\gamma)\in\mathcal{G}$ for some $\gamma\in\Gamma$, then

[TABLE]

Assume that $x\gamma\in Z_{F_{j}}\gamma_{j}$ for some $\gamma_{j}\in F_{j}^{-1}\cap F_{j}^{-1}\gamma$. Then, $x\in Z_{F_{j}}\gamma_{j}\gamma^{-1}$. Note

[TABLE]

one has that $x\in Z_{F_{j}}F_{j}^{-1}$, and therefore

[TABLE]

On the other hand, for any $\gamma\in F_{i}$, it is clear that $(x,\gamma)\in\mathcal{G}$. Therefore

[TABLE]

and hence

[TABLE]

∎

Remark 8.4*.*

By considering

[TABLE]

one extends $\mathcal{G}$ to a (open and relatively compact) subgroupoid with unit space $\{(x,e):x\in X\}$ without changing orbit of each $x\in\Omega$.

Definition 8.5**.**

A free dynamical system $(X,\Gamma)$ is said to have lower semicontinuous dynamical tiling property (LscT) if for any finite set $K\subseteq\Gamma$ and any $\varepsilon>0$, there is a partition valued function $\mathcal{T}:X\to\mathcal{P}(\Gamma)$ such that

1. (1)

there are finite sets $F_{1},F_{2},...,F_{n}\subseteq\Gamma$ such that for each $x\in X$, the partition $\mathcal{T}(x)$ is a tiling of $\Gamma$ with tiles $F_{1},F_{2},...,F_{n}$, and $F_{1},F_{2},...,F_{n}\subseteq\Gamma$ are $(K,\varepsilon)$ invariant, 2. (2)

the map $\mathcal{T}$ is equivariant, i.e.,

[TABLE] 3. (3)

the function $\mathcal{T}$ is lower semicontinuous, i.e., for any finite set $F\subseteq\Gamma$ and any $x\in X$, there is a neighbourhood $U\ni x$ such that

[TABLE]

The system $(X,\Gamma)$ is said to have continuous dynamical tiling property (CT) if Condition (3) is strengthen to

1. (3’)

the function $\mathcal{T}$ is continuous, i.e., for any finite set $F\subseteq\Gamma$ and any $x\in X$, there is a neighbourhood $U\ni x$ such that

[TABLE]

where $\mathcal{P}(x)|_{F}$ and $\mathcal{P}(y)|_{F}$ denote partitions of $F$ induced by $\mathcal{P}(x)$ and $\mathcal{P}(y)$, respectively.

The following lemma is straightforward.

Lemma 8.6**.**

If a free dynamical system $(Y,\Gamma)$ has (LscT) (or (CT)), and if $(X,\Gamma)$ is an extension of $(Y,\Gamma)$, then $(X,\Gamma)$ has (LscT) (or (CT)).

Once the dynamical system $(X,\Gamma)$ has the (LscT), then arbitrarily invariant shape functions (hence small subgroupoids) exist:

Corollary 8.7**.**

Let $(X,\Gamma)$ be a free dynamical system with (LscT). Then, for any finite set $K\subseteq\Gamma$ and any $\varepsilon>0$, there is a small subgroupoid $\mathcal{G}\subseteq X\rtimes\Gamma$ such that for any $x\in X$, the $\mathcal{G}$-orbit $\mathrm{Orbit}_{\mathcal{G}}(x)$ is $(K,\varepsilon)$-invariant.

Proof.

For each $x\in X$, define

[TABLE]

where $(\mathcal{T}(x))_{e}$ is the tile of $\mathcal{T}(x)$ containing $e$. Then $x\mapsto S(x)$ is a shape function with domain $X$ in the sense of Definition 8.1, and the statement follows directly from Theorem 8.3. ∎

Together with Corollary 7.13, one has

Corollary 8.8**.**

Let $(X,\Gamma)$ be a free dynamical system with (LscT). Then for any open sets $E,F\subseteq X$ with

[TABLE]

one has

[TABLE]

The topological-dynamical version of subequivalence and comparison have a long history, see, for example, [17] and [18]. The following definitions can be found in [27] and [7].

Definition 8.9**.**

Let $G$ be a countable amenable group.

1. (1)

Let $G$ act on a zero-dimensional compact metric space $X$. For two clopen sets $A,B\subseteq X$, we say that $A$ is subequivalent to $B$ (and write $A\precsim B$), if there exists a finite partition $A=\bigsqcup_{i=1}^{k}A_{i}$ of $A$ into clopen sets and there are elements $g_{1},g_{2},...,g_{k}$ of $G$ such that $g_{1}(A_{1})$, $g_{2}(A_{2})$, …, $g_{k}(A_{k})$ are disjoint subsets of $B$. We say that the action admits comparison if for any pair of clopen subsets $A,B$ of $X$, the condition that for each invariant measure $\mu$ on $X$ we have $\mu(A)<\mu(B)$, implies $A\precsim B$. 2. (2)

If every action of $G$ on any zero-dimensional compact metric space admits comparison then we will say that $G$ has the comparison property.

Also recall

Theorem 8.10** (Theorem 5.11 and Theorem 6.2 of [7]).**

Any amenable group with subexponential growth has the comparison property. Any free Cantor system $(\Omega,\Gamma)$ with $\Gamma$ an amenable group with the comparison property has (CT).

Then, for extensions of a Cantor system, one has the following Cuntz comparison of open sets.

Corollary 8.11**.**

If $\Gamma$ has comparison property in sense of Definition 8.9 (in particular, if $\Gamma$ has subexponential growth) and $(X,\Gamma)$ has a free Cantor factor, then, for any open sets $E,F\subseteq X$ with

[TABLE]

one has that

[TABLE]

In other words, $\mathrm{C}(X)\rtimes\Gamma$ has $(\frac{1}{4},1)$-Cuntz-comparison of open sets.

Proof.

Assume that $(X,\Gamma)$ is an extension of a free Cantor system $(\Omega,\Gamma)$. By Theorem 8.10, the factor $(\Omega,\Gamma)$ has the (CT), and then, by Lemma 8.6, $(X,\Gamma)$ has the (CT) (in particular, the (LscT)). The statement follows from Corollary 8.8. ∎

Corollary 8.12**.**

Let $(X,\Gamma)$ be a minimal free topological dynamical system which is an extension of a free Cantor system. If $\Gamma$ has the comparison property in sense of Definition 8.9 (in particular, if $\Gamma$ has subexponential growth), then

[TABLE]

Proof.

By Corollary 3.8, $(X,\Gamma)$ has the (URP). Since $\Gamma$ is assumed to have the comparison property (Definition 8.9), it follows from Corollary 8.11 that $\mathrm{C}(X)\rtimes\Gamma$ has $(\frac{1}{4},1)$-Cuntz-comparison on open sets. Then the statement follows from Theorem 4.8. ∎