The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

TL;DR

This paper investigates how the radius of comparison and the Cuntz semigroup behave under finite group actions with the Rokhlin property on simple unital C*-algebras, establishing bounds and isomorphisms.

Contribution

It proves bounds on the radius of comparison for fixed point algebras and crossed products, and shows isomorphisms of the Cuntz semigroup parts under such actions.

Findings

rc(A^α) ≤ rc(A)

rc(C*(G, A, α)) ≤ (1/|G|) rc(A)

Constructed example with specific radius of comparison values

Abstract

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let \alpha \colon G \to Aut (A) be an action of G on A which has the weak tracial Rokhlin property. Let A^{\alpha} be the fixed point algebra. Then the radius of comparison satisfies rc (A^{\alpha}) \leq rc (A) and rc ( C* (G, A, \alpha) ) \leq ( 1 / card (G) ) rc (A). The inclusion of A^{\alpha} in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^{\alpha}) to the fixed points of the purely positive part of Cu (A), and the purely positive part of Cu ( C* (G, A, \alpha) ) is isomorphic to this semigroup. We construct an example in which G is the two element group, A is a simple unital AH algebra, \alpha has the Rokhlin property, rc (A) > 0, rc (A^{\alpha}) = rc (A), and rc (C* (G, A, \alpha)) = (1/2) rc (A).

Equations

$${\operatorname{rc}}(A^{\alpha})\leq{\operatorname{rc}}(A)\qquad{\mbox{and}}\qquad{\operatorname{rc}}\bigl{(}C^{*}(G,A,\alpha)\bigr{)}\leq\frac{1}{{\operatorname{card}}(G)}\cdot{\operatorname{rc}}(A).$$

$$\underset{n\to \infty }{\lim }{v}_{n}b{v}_n^{\ast }=a.$$

$$\big{(}(a-\varepsilon_{1})_{+}-\varepsilon_{2}\big{)}_{+}=\big{(}a-(\varepsilon_{1}+\varepsilon_{2})\big{)}_{+}.$$

$$\big{(}a-(\varepsilon_{1}+\varepsilon_{2})\big{)}_{+}\precsim_{A}\big{(}(1-g)a(1-g)-\varepsilon_{1}\big{)}_{+}\oplus\Big{(}g-\frac{\varepsilon_{2}}{2}\Big{)}_{+}.$$

$$(h-\varepsilon_{2})_{+}\sim_{A}\big{(}g-[1-(1-\varepsilon_{2})^{1/2}]\big{)}_{+}.$$

$$\big{(}g-1+(1-\varepsilon_{2})^{1/2}\big{)}_{+}\precsim_{A}\Big{(}g-\frac{\varepsilon_{2}}{2}\Big{)}_{+}.$$

$$\displaystyle\big{(}a-(\varepsilon_{1}+\varepsilon_{2})\big{)}_{+}$$

$$\displaystyle\sim_{A}\bigl{(}(1-g)a(1-g)-\varepsilon_{1}\bigr{)}_{+}\oplus\bigl{(}h^{1/2}ah^{1/2}-\varepsilon_{2}\bigr{)}_{+}$$

$${\precsim }_{A}b\oplus (h-{\epsilon }_2{)}_{+}$$

$$\displaystyle\sim b\oplus\bigl{(}g-1+(1-\varepsilon_{2})^{1/2}\bigr{)}_{+}\precsim_{A}b\oplus\Big{(}g-\frac{\varepsilon_{2}}{2}\Big{)}_{+}.$$

$$\Vert a-v_n(b-\delta {)}_{+}{v}_n^{\ast }\Vert <\frac{1}{n}.$$

$$\Big{(}a-\frac{1}{n}\Big{)}_{+}=d_{n}v_{n}(b-\delta)_{+}v_{n}^{*}d_{n}^{*}.$$

$$\Big{(}a-\frac{1}{n}\Big{)}_{+}=w_{n}bw_{n}^{*}\qquad\mbox{and}\qquad\|w_{n}\|\leq\Big{\|}\Big{(}a-\frac{1}{n}\Big{)}_{+}\Big{\|}^{1/2}\delta^{-1/2}.$$

$$\tau (x)={\tau }_2(x\otimes e_{1,1}).$$

$$N(\tau)=\sup\bigl{(}\bigl{\{}\tau(a)\colon{\mbox{$a\in A_{+}$ and $\|a\|\leq 1$}}\bigr{\}}\bigr{)}.$$

$$J_{\tau }=\{x\in A:\tau (x^{\ast }x)=0\}$$

$$(x+y{)}^{\ast }(x+y)\le (x+y{)}^{\ast }(x+y)+(x-y{)}^{\ast }(x-y)=2x^{\ast }x+2y^{\ast }y,$$

$$0\leq\tau\bigl{(}(x+y)^{*}(x+y)\bigr{)}\leq\tau\bigl{(}2x^{*}x+2y^{*}y\bigr{)}\leq 4\bigl{(}\tau(x^{*}x)+\tau(y^{*}y)\bigr{)}=0.$$

$$0\le \tau ((ax{)}^{\ast }(ax))\le \Vert a^{\ast }a\Vert \tau (x^{\ast }x)=0,$$

$$d_{\tau }(a)=\underset{n\to \infty }{\lim }\tau (a^{1/n})$$

$${\operatorname{rc}}(A)=\inf\big{(}\big{\{}r\in[0,\infty)\colon{\mbox{$A$ has $r$-comparison}}\big{\}}\big{)}$$

$${\operatorname{rc}}\bigl{(}pM_{n}(A)p\bigr{)}\leq\frac{1}{\lambda}\cdot{\operatorname{rc}}(A).$$

$$d_{\tau }(a)+\frac{{\tau }_n(p)}{\lambda }\cdot \mathrm{rc}(A)<d_{\tau }(b)$$

$$\lambda=\inf\bigl{(}\bigl{\{}\tau_{n}(p)\colon\tau\in{\operatorname{QT}}(A)\bigr{\}}\bigr{)}\qquad{\mbox{and}}\qquad\eta=\sup\bigl{(}\bigl{\{}\tau_{n}(p)\colon\tau\in{\operatorname{QT}}(A)\bigr{\}}\bigr{)}.$$

$$\frac{1}{\eta}\cdot{\operatorname{rc}}(A)\leq{\operatorname{rc}}\big{(}pM_{n}(A)p\big{)}\leq\frac{1}{\lambda}\cdot{\operatorname{rc}}(A).$$

$${\operatorname{rc}}(A)\leq\left(\frac{1}{\inf\bigl{(}\bigl{\{}\sigma_{m}(q)\colon\sigma\in{\operatorname{QT}}(pM_{n}(A)p)\bigr{\}}\bigr{)}}\right){\operatorname{rc}}(pM_{n}(A)p).$$

$$\mathrm{QT}(p{M}_n(A)p)=\left\{{\tau }_n(p{)}^{-1}{\tau }_n{|}_{p{M}_n(A)p}:\tau \in \mathrm{QT}(A)\right\}.$$

$$\inf\bigl{(}\bigl{\{}\sigma_{m}(q)\colon\sigma\in{\operatorname{QT}}(pM_{n}(A)p)\bigr{\}}\bigr{)}=\frac{1}{\sup\bigl{(}\bigl{\{}\tau_{n}(p)\colon\tau\in{\operatorname{QT}}(A)\bigr{\}}\bigr{)}}=\frac{1}{\eta}.$$

$$a\in A_{+},\Vert a\Vert \le M,\Vert x\Vert \le M,\text{mbox}and\Vert ax-xa\Vert <\delta ,$$

$${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m,{\beta }_1,{\beta }_2,\dots ,{\beta }_n\in \mathbb{R}$$

$$|f_0(\lambda )-f(\lambda )|<\frac{\epsilon }{3(C+1)}\text{mbox}and|g_0(\lambda )-g(\lambda )|<\frac{\epsilon }{3(C+1)}$$

$$R=\max\big{(}|\alpha_{1}|,|\alpha_{2}|,\ldots,|\alpha_{m}|,|\beta_{1}|,|\beta_{2}|,\ldots,|\beta_{n}|\big{)}+1\quad{\mbox{and}}\quad\delta=\frac{\varepsilon}{3mnR^{2}M^{m+n}}.$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Cuntz subequivalence

4. 2.2 Quasitraces on C*-algebras

5. 2.3 Radius of comparison

6. 2.4 The radius of comparison of a corner

7. 2.5 Approximation lemmas

8. 3 Injectivity of ${\operatorname{Cu}}_{+}(A^{\alpha})\to{\operatorname{Cu}}_{+}(A)^{\alpha}$

9. 4 Radius of comparison of the fixed point algebra and crossed product

10. 5 Surjectivity of ${\operatorname{Cu}}_{+}(A^{\alpha})\to{\operatorname{Cu}}_{+}(A)^{\alpha}$

11. 6 An example

12. 7 Open problems

13. 8 Acknowledgments

1. Introduction

We prove that if $G$ is a finite group, $A$ is an infinite-dimensional stably finite simple unital C*-algebra, and $\alpha\colon G\to{\operatorname{Aut}}(A)$ is an action of $G$ on $A$ which has the weak tracial Rokhlin property, then the radii of comparison (see below for further discussion) of $A$, the crossed product, and the fixed point algebra are related by

[TABLE]

See Theorem 4.1 and Theorem 4.5. These inequalities fail for general pointwise outer actions; see Example 6.22.

In fact, we prove a much stronger result, relating the Cuntz semigroups (see below and Section 2 for further discussion): the inclusion of $A^{\alpha}$ in $A$ induces an isomorphism from the subsemigroup ${\operatorname{Cu}}_{+}(A^{\alpha})\subseteq{\operatorname{Cu}}(A^{\alpha})$ consisting of zero and the purely positive elements (recalled in Definition 3.8) to the fixed points of ${\operatorname{Cu}}_{+}(A)$ under the action induced by $\alpha$. By Example 4.7, the restriction to the purely positive part is necessary. If $A$ has stable rank one, then one can use ${\operatorname{W}}(A)$ in place of ${\operatorname{Cu}}(A)$. We consider this to be a striking result, since the Cuntz semigroup is often considered to be too complicated to compute for C*-algebras without strict comparison.

We further give an example of an infinite-dimensional stably finite simple unital C*-algebra $A$ and an action $\alpha\colon{\mathbb{Z}}/2{\mathbb{Z}}\to{\operatorname{Aut}}(A)$ which even has the Rokhlin property, and for which the radii of comparison of $A$, the crossed product, and the fixed point algebra are all strictly positive. It is initially not obvious that such an example should exist. The algebra $A$ even has stable rank one.

Along the way, we estimate (Theorem 2.18) the radius of comparison of a corner of a simple unital C*-algebra. This result is surely known, but we have not found it in the literature. We also prove (Lemma 4.4) that if $A$ is simple and unital, $G$ has order $n$, and $\alpha\colon G\to{\operatorname{Aut}}(A)$ has the weak tracial Rokhlin property, then every quasitrace on $C^{*}(G,A,\alpha)$ takes the value $\frac{1}{n}$ on the average of the unitaries in the crossed product which correspond to the elements of $G$.

The importance of the Cuntz semigroup has become apparent in work related to the Elliott classification program. See [3] for a survey of many aspects of the Cuntz semigroup. It is generally large and complicated; roughly speaking, among simple nuclear C*-algebras, the classifiable ones are those whose Cuntz semigroups are easily accessible. With the near completion of the Elliott program, attention is turning to nonclassifiable C*-algebras, and the Cuntz semigroup is the main additional available invariant. Given its complexity, it is somewhat surprising that there is such a strong connection between the Cuntz semigroup of a simple C*-algebra and the Cuntz semigroup of its crossed product by a weak tracial Rokhlin action. It seems, also by comparison with [38], that the purely positive part of the Cuntz semigroup does not see differences which are “small in trace”.

The radius of comparison is a numerical invariant, based on the Cuntz semigroup, which was introduced in Section 6 of [44] to distinguish examples of nonisomorphic simple separable unital AH algebras with the same Elliott invariant. Its importance goes well beyond this application. For example, it is now conjectured that if $h$ is a minimal homeomorphism of a compact metric space $X$, then {\operatorname{rc}}\bigl{(}C^{*}({\mathbb{Z}},X,h)\bigr{)} is equal to half the mean dimension of $h$; mean dimension is an invariant introduced in dynamics which at the time had no apparent connection with C*-algebras. The radius of comparison also plays a key role in a recent example of a simple separable unital AH algebra whose Elliott invariant has an automorphism not implemented by any automorphism of the algebra [22].

The weak tracial Rokhlin property (Definition 2.2 of [15]; see Definition 3.2 below) is a generalization of the tracial Rokhlin property (Definition 1.2 of [35]) which uses positive elements instead of projections. It is a slight modification of the generalized tracial Rokhlin property of Definition 5.2 of [20]. It is much more common than the Rokhlin property, any of the higher dimensional Rokhlin properties with commuting towers, or even the tracial Rokhlin property. See Example 3.12 of [36] for a collection of examples of actions of finite groups which have the tracial Rokhlin property (and hence the weak tracial Rokhlin property) but not the Rokhlin property. It is shown in [2] that if $A$ is a simple C*-algebra which is tracially ${\mathcal{Z}}$-absorbing, and if the minimal tensor product $A^{\otimes n}$ of $n$ copies of $A$ is finite, then the permutation action of $S_{n}$ on $A^{\otimes n}$ has the weak tracial Rokhlin property. Using ${\mathcal{Z}}^{\otimes n}\cong{\mathcal{Z}}$, one gets in particular an action of $S_{n}$ on ${\mathcal{Z}}$ which has the weak tracial Rokhlin property. (This was proved for the generalized tracial Rokhlin property in Example 5.10 of [20].) However, by Corollary 4.8(1) of [21], there is no action on ${\mathcal{Z}}$ which has any higher dimensional Rokhlin property with commuting towers.

The Rokhlin property case of our Cuntz semigroup is already known, even for nonunital C*-algebras (Theorem 4.1 of [16]). That paper does not consider the radius of comparison, does not consider ${\operatorname{W}}(A)$, and gives no example like that in our Section 6, in which the algebra is simple and has nonzero radius of comparison. Moreover, as pointed out above, the weak tracial Rokhlin property is much more common than the Rokhlin property.

Something close to the case ${\operatorname{rc}}(A)=0$ of our radius of comparison result is also already known. Let $A$ be a simple separable nuclear unital C*-algebra. If ${\operatorname{rc}}(A)=0$, and if one assumes that the set of extreme points of ${\operatorname{T}}(A)$ is compact and finite-dimensional then $A$ is ${\mathcal{Z}}$-stable (Corollary 7.9 of [29]; Corollary 1.2 of [43]; Corollary 4.7 of [45]) and, in particular, tracially ${\mathcal{Z}}$-absorbing in the sense of Definition 2.1 of [20]. If $G$ is finite and $\alpha\colon G\to{\operatorname{Aut}}(A)$ has the generalized tracial Rokhlin property (Definition 5.2 of [20]), then $C^{*}(G,A,\alpha)$ is tracially ${\mathcal{Z}}$-absorbing by Theorem 5.6 of [20]. Therefore $A$ has strict comparison by Theorem 3.3 of [20]. (The group need not be finite; see Definition 6.1 of [20] and Theorem 6.7 of [20] for results for ${\mathbb{Z}}$, and [30] for some results for actions of countable amenable groups.)

Despite the relative abundance of actions with the weak tracial Rokhlin property, it is not obvious that there are actions with this property on stably finite simple unital C*-algebras with strictly positive radius of comparison. Getting the Rokhlin property seems even harder. For example, according to Theorems 3.4 and 3.5 of [24], if in addition $A$ is nuclear, satisfies the Universal Coefficient Theorem, and has tracial rank zero (or is a unital Kirchberg algebra satisfying the Universal Coefficient Theorem), and $\alpha\colon G\to{\operatorname{Aut}}(A)$ is an action of $G$ on $A$ which has the Rokhlin property and is trivial on K-theory, then $A$ is stable under tensoring with the ${\operatorname{card}}(G)^{\infty}$ UHF algebra. The same conclusion, under somewhat different hypotheses, is obtained in Theorem 5.10 of [16]. One might naively expect something like this to be true more generally. In fact, though, we exhibit an action $\alpha\colon G\to{\operatorname{Aut}}(A)$ with the Rokhlin property (not just the weak tracial Rokhlin property), in which $A$ is a simple unital AH algebra, $G={\mathbb{Z}}/2{\mathbb{Z}}$ (although a similar construction will work for any finite group), and $A$, $A^{\alpha}$, and $C^{*}(G,A,\alpha)$ all have finite but nonzero radius of comparison.

When $G$ has order $n$ and $\alpha\colon G\to{\operatorname{Aut}}(A)$ is an action of $G$ on $A$ which has the Rokhlin property, the usual method of proving properties of $C^{*}(G,A,\alpha)$ is local approximation by algebras of the form $M_{n}(eAe)$ for suitable projections $e\in A$. See Theorem 3.2 of [32]; there are a number of applications of this method in that paper. Weaker versions of this are true for versions of the weak tracial Rokhlin property, and are implicit in Sections 5 and 6 of [20] and in [30]. This method does not seem to work even for the Rokhlin property case of our radius of comparison result; the best we could get this way is {\operatorname{rc}}\bigl{(}C^{*}(G,A,\alpha)\bigr{)}\leq{\operatorname{rc}}(A). The difficulty is with ${\operatorname{rc}}(eAe)$. Instead, we first prove that ${\operatorname{rc}}(A^{\alpha})\leq{\operatorname{rc}}(A)$. Using the notation of Definition 2.14 and Definition 2.15 below, suppose we have $a,b\in(A^{\alpha})_{+}$ with $d_{\tau}(a)+{\operatorname{rc}}(A)<d_{\tau}(b)$ for every normalized quasitrace $\tau$ on $A^{\alpha}$. This applies in particular for normalized quasitraces $\tau$ on $A$, so $a\precsim_{A}b$. Thus, there is $v\in A$ such that $\|vbv^{*}-a\|$ is small. Now use the Rokhlin property to “average” $v$ over $G$, as described in Remark 10.3.9 and Exercise 10.3.10 of [17], getting $w\in A^{\alpha}$ such that $\|wbw^{*}-a\|$ is small. The generalization to the weak tracial Rokhlin property uses the same idea, but is considerably more technical, and requires generalizations of some of the Cuntz comparison results in [38].

The construction of an action with the Rokhlin property is a modification of the idea used in [22] to find an automorphism of the Elliott invariant of a simple AH algebra which does not lift to an automorphism of the algebra. The construction there “merged” two direct systems whose direct limits had different radii of comparison but the same Elliott invariant. This “merging” was done by adding a very small number of maps which go from one of the original systems to the other. Here, we “merge” two copies of the same direct system, with the system having been chosen so that its direct limit has large radius of comparison. The action exchanges the two copies of this system. The Rokhlin projections are, roughly speaking, the identities of the algebras in the two original systems.

This paper is organized as follows. Section 2 contains information on the Cuntz semigroup, Cuntz comparison, quasitraces, and the radius of comparison. It also contains several approximation results which are used repeatedly. Some of this material is new or at least not in the literature, and some definitions and results are stated here for the convenience of the reader and for easy reference. In Section 3 we prove injectivity on the purely positive part for the map ${\operatorname{Cu}}(A^{\alpha})\to{\operatorname{Cu}}(A)^{\alpha}$. This is enough to prove the bound on the radius of comparison of a crossed product by an action with the weak tracial Rokhlin property, and our surjectivity result does not seem to help with the reverse inequality, so we prove the bound in Section 4. Section 5 contains our surjectivity result on the purely positive part for the map ${\operatorname{Cu}}(A^{\alpha})\to{\operatorname{Cu}}(A)^{\alpha}$, as well as results on ${\operatorname{W}}(A^{\alpha})\to{\operatorname{W}}(A)^{\alpha}$ when $A$ has stable rank one. Since ${\operatorname{W}}(A)$ is not considered in [16], we also prove the corresponding result for Rokhlin actions on unital but not necessarily simple C*-algebras. In Section 6, we construct the example referred to above. In Section 7, we state a few open problems.

2. Preliminaries

In this section, we collect for easy reference some information on the Cuntz semigroup, quasitraces, and the radius of comparison. A fair amount is already in the literature, but there are several facts we did not find, among them, the estimate in Theorem 2.18 for the radius of comparison of a corner. Lemma 2.6 is definitely new.

2.1. Cuntz subequivalence

Notation 2.1**.**

We use the following standard notation. If $A$ is a C*-algebra, or if $A=M_{\infty}(B)$ for a C*-algebra $B$, we write $A_{+}$ for the set of positive elements of $A$.

Parts (1) and (2) of the following definition are originally from [8]. The usual notation for Cuntz subequivalence is $a\precsim b$. We include $A$ in the notation because we need to use Cuntz subequivalence with respect to subalgebras.

Definition 2.2**.**

Let $A$ be a C*-algebra.

1. (1)

For $a,b\in M_{\infty}(A)_{+}$, we say that $a$ is Cuntz subequivalent to $b$ in $A$, written $a\precsim_{A}b$, if there is a sequence $(v_{n})_{n=1}^{\infty}$ in $M_{\infty}(A)$ such that

[TABLE] 2. (2)

We say that $a$ and $b$ are Cuntz equivalent in $A$, written $a\sim_{A}b$, if $a\precsim_{A}b$ and $b\precsim_{A}a$. This relation is an equivalence relation, and we write $\langle a\rangle_{A}$ for the equivalence class of $a$. We define ${\operatorname{W}}(A)=M_{\infty}(A)_{+}/\sim_{A}$, together with the commutative semigroup operation $\langle a\rangle_{A}+\langle b\rangle_{A}=\langle a\oplus b\rangle_{A}$ and the partial order $\langle a\rangle_{A}\leq\langle b\rangle_{A}$ if $a\precsim_{A}b$. We write [math] for $\langle 0\rangle_{A}$. 3. (3)

We take ${\operatorname{Cu}}(A)={\operatorname{W}}(K\otimes A)$. We write the classes as $\langle a\rangle_{A}$ for $a\in(K\otimes A)_{+}$. 4. (4)

Let $A$ and $B$ be C*-algebras, and let $\varphi\colon A\to B$ be a homomorphism. We use the same letter for the induced maps $M_{n}(A)\to M_{n}(B)$ for $n\in{\mathbb{Z}}_{>0}$ and $M_{\infty}(A)\to M_{\infty}(B)$. We define ${\operatorname{W}}(\varphi)\colon{\operatorname{W}}(A)\to{\operatorname{W}}(B)$ and ${\operatorname{Cu}}(\varphi)\colon{\operatorname{Cu}}(A)\to{\operatorname{Cu}}(B)$ by $\langle a\rangle_{A}\mapsto\langle\varphi(a)\rangle_{B}$ for $a\in M_{\infty}(A)_{+}$ or $a\in(K\otimes A)_{+}$ as appropriate.

Definition 2.3**.**

Let $A$ be a C*-algebra, let $a\in A_{+}$, and let $\varepsilon\geq 0$. Let $f\colon[0,\infty)\to[0,\infty)$ be the function $f(t)=\max(0,\,t-\varepsilon)=(t-\varepsilon)_{+}$. Then, by functional calculus, define $(a-\varepsilon)_{+}=f(a)$.

Part (1) of the following is taken from Proposition 2.4 of [40]. Parts (2) and (3a) are Lemma 2.5(i) and Lemma 2.5(ii) of  [27]. Part (3b) is Lemma 2.2 of [28]. Part (3c) is Corollary 1.6 of [38]. Part (4) is taken from the discussion after Definition 2.3 of [27] and Proposition 2.3(ii) of [12].

Lemma 2.4**.**

Let $A$ be a C*-algebra.

1. (1)

Let $a,b\in A_{+}$. Then the following are equivalent:

1. (a)

$a\precsim_{A}b$. 2. (b)

$(a-\varepsilon)_{+}\precsim_{A}b$ for all $\varepsilon>0$. 3. (c)

For every $\varepsilon>0$ there is $\delta>0$ such that $(a-\varepsilon)_{+}\precsim_{A}(b-\delta)_{+}$. 2. (2)

Let $a\in A_{+}$ and let $\varepsilon_{1},\varepsilon_{2}>0$. Then

[TABLE] 3. (3)

Let $a,b\in A_{+}$ and let $\varepsilon>0$. If $\|a-b\|<\varepsilon$, then:

1. (a)

$(a-\varepsilon)_{+}\precsim_{A}b$. 2. (b)

There is a contraction $d$ in $A$ such that $dbd^{*}=(a-\varepsilon)_{+}$. 3. (c)

For any $\lambda>0$, we have $(a-\lambda-\varepsilon)_{+}\precsim_{A}(b-\lambda)_{+}$. 4. (4)

Let $c\in A$ and let $\lambda\geq 0$. Then $(c^{*}c-\lambda)_{+}\sim_{A}(cc^{*}-\lambda)_{+}$.

Lemma 2.5**.**

Suppose $t\in[0,1]$ and $s\in[0,1)$. Then:

1. (1)

$2t-t^{2}-s>0$ if and only if $t-1+\sqrt{1-s}>0$. 2. (2)

$1-\sqrt{1-s}\geq\frac{s}{2}$.

Proof.

These statements are easy to check. ∎

The following lemma is a generalization of Lemma 1.8 of [38] or Lemma 12.1.5 of [17].

Lemma 2.6**.**

Let $A$ be a unital C*-algebra, let $a,g\in A$ satisfy $0\leq a,g\leq 1$, and let $\varepsilon_{1},\varepsilon_{2}\geq 0$. Then

[TABLE]

Proof.

We may clearly assume $\varepsilon_{2}<1$. Set $h=2g-g^{2}$. Functional calculus and Lemma 2.5(1) imply that

[TABLE]

Since $\varepsilon_{2}\in[0,\,1)$, it follows from Lemma 2.5(2) that $1-\sqrt{1-\varepsilon_{2}}\geq\frac{\varepsilon_{2}}{2}$. So

[TABLE]

Set b=\big{(}(1-g)a(1-g)-\varepsilon_{1}\big{)}_{+}. Using Lemma 1.5 of [38] at the first step, Lemma 2.4(4) at the second step, $\|a\|\leq 1$ and Lemma 1.7 of [38] on the second summand at the third step, (2.1) at the fourth step, and (2.2) at the last step, we get

[TABLE]

This completes the proof. ∎

Let $a,b\in A_{+}$. If $a\precsim_{A}b$ then by definition there is a sequence $(v_{n})_{n=1}^{\infty}$ in $A$ such that $\lim_{{n}\to\infty}v_{n}bv_{n}^{*}=a$. But there need not be a bounded sequence with this property. As a substitute, we have the following result, originally from [2]. We give a proof for the sake of completeness. (There is a similar result in Lemma 2.4(ii) of [28], but there is a gap in the proof.)

Lemma 2.7**.**

Let $A$ be a C*-algebra, let $a,b\in A_{+}$, and let $\delta>0$. If $a\precsim_{A}(b-\delta)_{+}$, then there exists a sequence $(w_{n})_{n\in{\mathbb{Z}}_{>0}}$ in $A$ such that $\|a-w_{n}bw_{n}^{*}\|\to 0$ and $\|w_{n}\|\leq\|a\|^{1/2}\delta^{-1/2}$ for every $n\in{\mathbb{Z}}_{>0}$.

Proof.

Let $n\in{\mathbb{Z}}_{>0}$. Since $a\precsim_{A}(b-\delta)_{+}$, there exists $v_{n}\in A$ such that

[TABLE]

Using Lemma 2.4(3b), we find a contraction $d_{n}\in A$ such that

[TABLE]

Now, applying Lemma 2.4(i) of [28], we get $w_{n}\in A$ such that

[TABLE]

Therefore $w_{n}bw_{n}^{*}\to a$ and $\|w_{n}\|\leq\|a\|^{1/2}\delta^{-1/2}$. ∎

2.2. Quasitraces on C*-algebras

The following definition is from [19]. Parts (1), (2), and (3) correspond to the definition of a quasitrace in [6]. What we and [19] call a quasitrace is called a “2-quasitrace” in [6].

Definition 2.8**.**

Let $A$ be a C*-algebra. A function $\tau\colon A\to\mathbb{C}$ is a quasitrace if the following hold:

1. (1)

$\tau(x^{*}x)=\tau(xx^{*})\geq 0$ for all $x\in A$. 2. (2)

$\tau(a+ib)=\tau(a)+i\tau(b)$ for $a,b\in A_{\mathrm{sa}}$. 3. (3)

$\tau|_{B}$ is linear for every commutative C*-subalgebra $B\subseteq A$. 4. (4)

There is a function $\tau_{2}\colon M_{2}(A)\to\mathbb{C}$ satisfying (1), (2), and (3) with $M_{2}(A)$ in place of $A$, and such that, with $(e_{j,k})_{j,k=1}^{2}$ denoting the standard system of matrix units in $M_{2}(\mathbb{C})$, for all $x\in A$ we have

[TABLE]

A quasitrace $\tau$ on a unital C*-algebra is normalized if $\tau(1)=1$. The set of normalized quasitraces on $A$ is denoted by ${\operatorname{QT}}(A)$.

All quasitraces on a unital exact C*-algebra are traces, by Theorem 5.11 of [19].

Proposition 2.9** ([6]).**

Let $A$ be a stably finite unital C*-algebra. Then ${\operatorname{QT}}(A)\neq\varnothing$.

Proof.

This is in the discussion after Proposition II.4.6 of [6]. ∎

Part (1) of the following proposition is Corollary II.2.3 of [6] and Parts (2) through (6) are taken from Corollary II.2.5 of [6]. That paper uses $\|\tau\|$ instead of $N(\tau)$. We want to avoid conflict with the definition of the norm of a linear functional.

Proposition 2.10** ([6]).**

Let $\tau\colon A\to\mathbb{C}$ be a quasitrace on a C*-algebra $A$, and define

[TABLE]

Then:

1. (1)

$N(\tau)<\infty$. 2. (2)

If $A$ is unital and $\tau\in{\operatorname{QT}}(A)$, then $N(\tau)=1$. 3. (3)

$\tau$ is order-preserving. 4. (4)

If $a,b\in A_{\mathrm{sa}}$, then $|\tau(a)-\tau(b)|\leq N(\tau)\|a-b\|$. 5. (5)

$\tau$ is norm-continuous. 6. (6)

If $a,b\in A_{+}$, then \tau(a+b)\leq 2\big{(}\tau(a)+\tau(b)\big{)}.

Proposition 2.11** ([6]).**

Let $A$ be a C*-algebra and let $\tau$ be a quasitrace on $A$. Then $\tau$ extends uniquely to a quasitrace $\tau_{\infty}$ on $M_{\infty}(A)$ such that, with $(e_{j,k})_{j,k=1}^{\infty}$ denoting the standard system of matrix units in $M_{\infty}(\mathbb{C})$, we have $\tau_{\infty}(a\otimes e_{j,j})=\tau(a)$ for all $a\in A$ and $j\in{\mathbb{Z}}_{>0}$.

We denote the restriction of $\tau_{\infty}$ to $M_{n}(A)$ by $\tau_{n}$. When no confusion is likely, we abbreviate $\tau_{\infty}$ and $\tau_{n}$ to $\tau$.

Proof of Proposition 2.11.

For $M_{n}(A)$ in place of $M_{\infty}(A)$, this is Proposition II.4.1 of [6]. By uniqueness there, for all $n\in{\mathbb{Z}}_{>0}$, the restriction to $M_{n}(A)$ of the extension to $M_{n+1}(A)$ is the extension to $M_{n}(A)$. This implies existence of the extension to $M_{\infty}(A)$, and uniqueness is now immediate. ∎

The following lemma is part of Proposition 3.2 of [19]. (There is a misprint there: it cites Theorem I.1.1 of [6], but apparently Theorem I.1.17 is intended.) Given Proposition 2.10, we can give a simple direct proof, which is the same as for traces except for an extra factor of $2$ in the proof of closure under addition.

Lemma 2.12**.**

Let $\tau$ be a quasitrace on a C*-algebra $A$. Then the set

[TABLE]

is a closed two-sided ideal in $A$.

Proof.

It is obvious that $J_{\tau}$ is closed under scalar multiplication and $x\mapsto x^{*}$.

Let $x,y\in J_{\tau}$. Then

[TABLE]

so, by Proposition 2.10(3) and Proposition 2.10(6),

[TABLE]

Hence $x+y\in J_{\tau}$.

Let $x\in J_{\tau}$ and let $a\in A$. Then, using Proposition 2.10(3),

[TABLE]

so $ax\in J_{\tau}$. Now $xa=(a^{*}x^{*})^{*}\in J_{\tau}$. ∎

We will need Murray-von Neumann equivalence. We use notation which distinguishes it from Cuntz equivalence.

Definition 2.13**.**

Let $A$ be a C*-algebra, and let $p,q\in K\otimes A$ be projections. We say $p$ is Murray-von Neumann subequivalent to $q$, denoted $p\lessapprox q$, if there exists $v\in K\otimes A$ such that $p=vv^{*}$ and $v^{*}v\leq q$. We say that $p$ and $q$ are Murray-von Neumann equivalent, denoted $p\approx q$, if there exists $v\in K\otimes A$ such that $p=vv^{*}$ and $v^{*}v=q$.

It is well known that $p\lessapprox q$ if and only if $p\precsim_{A}q$. However, it is in general not true that $p\sim_{A}q$ implies $p\approx q$. For example, this fails in a purely infinite simple C*-algebra with nonzero $K_{0}$-group. However, if $A$ is stably finite then $p\sim_{A}q$ and $p\approx q$ are equivalent.

2.3. Radius of comparison

The following definition is Definition 12.1.7 of [17].

Definition 2.14**.**

Let $A$ be a unital C*-algebra, and let $\tau\in{\operatorname{QT}}(A)$. Recalling the notation of and after Proposition 2.11, define $d_{\tau}\colon M_{\infty}(A)_{+}\to[0,\infty)$ by

[TABLE]

for $a\in M_{\infty}(A)_{+}$. We also use the same notation for the corresponding functions on ${\operatorname{Cu}}(A)$ and ${\operatorname{W}}(A)$.

The following is Definition 6.1 of [44], except that we allow $r=0$ in (1). This change makes no difference.

Definition 2.15**.**

Let $A$ be a stably finite unital C*-algebra.

1. (1)

Let $r\in[0,\infty)$. We say that $A$ has $r$-comparison if whenever $a,b\in M_{\infty}(A)_{+}$ satisfy $d_{\tau}(a)+r<d_{\tau}(b)$ for all $\tau\in{\operatorname{QT}}(A)$, then $a\precsim_{A}b$. 2. (2)

The radius of comparison of $A$, denoted ${\operatorname{rc}}(A)$, is

[TABLE]

if it exists, and $\infty$ otherwise.

If $A$ is simple, then the infimum in Definition 2.15(2) is attained, that is, $A$ has ${\operatorname{rc}}(A)$-comparison; see Proposition 6.3 of [44]. For exact C*-algebras, one only needs to consider extreme tracial states; see Lemma 2.3 of [11].

By Proposition 6.12 of [38], the radius of comparison of a simple unital C*-algebra is the same whether computed using ${\operatorname{W}}(A)$ or ${\operatorname{Cu}}(A)$.

2.4. The radius of comparison of a corner

We give bounds on the radius of comparison of a full corner in a matrix algebra over a C*-algebra. The result is surely known, but we have not seen a proof in the literature. It will be needed in Section 4 below, to relate ${\operatorname{rc}}(A^{\alpha})$ to ${\operatorname{rc}}(C^{*}(G,A,\alpha))$.

Lemma 2.16**.**

Let $A$ be a stably finite unital C*-algebra, let $n\in{\mathbb{Z}}_{>0}$, and let $p$ be a full projection in $M_{n}(A)$. Then, recalling the notation in and after Proposition 2.11:

1. (1)

$\inf_{\tau\in{\operatorname{QT}}(A)}\tau_{n}(p)>0$. 2. (2)

The map $\theta\colon{\operatorname{QT}}(A)\to{\operatorname{QT}}(pM_{n}(A)p)$, given by $\tau\mapsto\frac{1}{\tau_{n}(p)}\tau_{n}|_{pM_{n}(A)p}$, is bijective.

Proof.

Since $\tau\mapsto\frac{1}{n}\tau_{n}$ is a bijection from ${\operatorname{QT}}(A)$ to {\operatorname{QT}}\bigl{(}M_{n}(A)\bigr{)}, it is easily seen that it suffices to prove the result when $n=1$.

Since $A$ is unital and $p$ is full in $A$, it follows that $ApA=A$. Therefore, using Lemma 2.12, $\tau(p)>0$ for all $\tau\in{\operatorname{QT}}(A)$. Since ${\operatorname{QT}}(A)$ is nonempty (by Proposition 2.9) and compact, and since $\tau\mapsto\tau(p)$ is continuous, (1) follows.

To prove (2), clearly $\frac{1}{\tau(p)}\tau|_{pAp}\in{\operatorname{QT}}(pAp)$. Bijectivity of $\theta$ now follows from Proposition II.4.2 of [6] and Proposition 2.11. ∎

Lemma 2.17**.**

Let $A$ be a stably finite unital C*-algebra, let $n\in{\mathbb{Z}}_{>0}$, and let $p$ be a full projection in $M_{n}(A)$. Recalling the notation in and after Proposition 2.11, if \lambda=\inf\bigl{(}\bigl{\{}\tau_{n}(p)\colon\tau\in{\operatorname{QT}}(A)\bigr{\}}\bigr{)}, then $0<\lambda\leq n$ and

[TABLE]

Proof.

By Lemma 2.16(1) we have $\lambda>0$. Since $p\leq 1_{M_{n}(A)}$ and $\tau_{n}(1_{M_{n}(A)})=n$ for all $\tau\in{\operatorname{QT}}(A)$, and since ${\operatorname{QT}}(A)\neq\varnothing$ by Proposition 2.9, it follows from Proposition 2.10(3) that $\lambda\leq n$.

Now let $m\in{\mathbb{Z}}_{>0}$, let $a,b\in M_{m}(pM_{n}(A)p)_{+}\subseteq M_{mn}(A)_{+}$, and suppose that $d_{\rho}(a)+\frac{1}{\lambda}\cdot{\operatorname{rc}}(A)<d_{\rho}(b)$ for all $\rho\in{\operatorname{QT}}(pM_{n}(A)p)$. By Lemma 2.16(2), this is the same as

[TABLE]

for all $\tau\in{\operatorname{QT}}(A)$. Since $\lambda\leq\tau_{n}(p)$, it follows that $a\precsim_{A}b$, so $a\precsim_{pM_{n}(A)p}b$. ∎

Theorem 2.18**.**

Let $A$ be a stably finite unital C*-algebra, let $n\in{\mathbb{Z}}_{>0}$, and let $p$ be a full projection in $M_{n}(A)$. Recalling the notation in and after Proposition 2.11, define

[TABLE]

Then $0<\lambda\leq\eta\leq n$ and

[TABLE]

Proof.

The parts involving $\lambda$ are Lemma 2.17. Since ${\operatorname{QT}}(A)\neq\varnothing$ (by Proposition 2.9), the relations $\lambda\leq\eta\leq n$ are clear.

Since $p$ is full, there are $m\in{\mathbb{Z}}_{>0}$ and a projection $q\in M_{m}(pM_{n}(A)p)$ such that $1_{A}\approx q$. Then $A\cong qM_{m}(pM_{n}(A)p)q$. Apply Lemma 2.17 with $pM_{n}(A)p$ in place of $A$, with $m$ in place of $n$, and with $q$ in place of $p$. We get

[TABLE]

By Lemma 2.16(2),

[TABLE]

If $\sigma\in{\operatorname{QT}}(pM_{n}(A)p)$ and $\tau\in{\operatorname{QT}}(A)$ is the corresponding quasitrace from (2.4), then, using $q\approx 1_{A}$, we get $\sigma_{m}(q)=\frac{\tau_{mn}(q)}{\tau_{n}(p)}=\frac{1}{\tau_{n}(p)}$. Therefore

[TABLE]

So (2.3) implies that $\frac{1}{\eta}\cdot{\operatorname{rc}}(A)\leq{\operatorname{rc}}(pM_{n}(A)p)$. ∎

2.5. Approximation lemmas

This subsection contains several approximation lemmas which will be needed frequently.

Lemma 2.19**.**

Let $M\in(0,\infty)$, let $f\colon[0,M]\to{\mathbb{C}}$ be continuous, and let $\varepsilon>0$. Then there is $\delta>0$ such that whenever $A$ is a C*-algebra and $a,x\in A$ satisfy

[TABLE]

then $\|f(a)x-xf(a)\|<\varepsilon$.

Proof.

The case $M=1$ is Lemma 2.5 of [4]. The proof of this version is the same. ∎

The statement can also be gotten from Lemma 2.5 of [4] by scaling.

Lemma 2.20**.**

Let $f,g\colon[0,\infty)\to[0,\infty)$ be continuous functions such that $f(0)=g(0)=0$, let $\varepsilon>0$, and let $M\in(0,\infty)$. Then there is $\delta>0$ such that whenever $A$ is a C*-algebra, and $a,b\in A_{+}$ satisfy $\|ab\|<\delta$ and $\|a\|,\,\|b\|\leq M$, then $\|f(a)g(b)\|<\varepsilon$.

This lemma can be proved by approximating $a$ and $b$ by positive elements whose product is zero, but a direct proof seems just as easy.

Proof of Lemma 2.20.

Without loss of generality $M\geq 1$ and $\varepsilon<1$. Set C=\max\big{(}\|f|_{[0,M]}\|_{\infty},\,\|g|_{[0,M]}\|_{\infty}\big{)}. Choose $m,n\in{\mathbb{Z}}_{\geq 0}$ and

[TABLE]

such that the polynomial functions with no constant term, given by $f_{0}(\lambda)=\sum_{k=1}^{m}\alpha_{k}\lambda^{k}$ and $g_{0}(\lambda)=\sum_{l=1}^{n}\beta_{l}\lambda^{l}$ for $\lambda\in[0,M]$, satisfy

[TABLE]

for all $\lambda\in[0,M]$. Without loss of generality $m,n\geq 1$. Define

[TABLE]

Now let $A$, $a$, and $b$ be as in the hypotheses. Using $M\geq 1$ at the second step, we have

[TABLE]

Therefore, since $\varepsilon<1$ implies $\|f_{0}(a)\|\leq\|f(a)\|+1$,

[TABLE]

This completes the proof. ∎

3. Injectivity of

${\operatorname{Cu}}_{+}(A^{\alpha})\to{\operatorname{Cu}}_{+}(A)^{\alpha}$

In this section, we prove that if $G$ is finite, $A$ is unital, stably finite, and simple, and $\alpha\colon G\to{\operatorname{Aut}}(A)$ has the weak tracial Rokhlin property, then the inclusion $A^{\alpha}\to A$ induces an isomorphism from the ordered semigroup of purely positive elements ${\operatorname{Cu}}_{+}(A^{\alpha})\cup\{0\}$ (see Definition 3.8 below) to a subsemigroup of ${\operatorname{Cu}}(A)$. Example 4.7 shows that this result fails if we do not discard the classes of projections.

Notation 3.1**.**

Let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on a unital C*-algebra $A$. For $g\in G$, we let $u_{g}$ be the element of $C_{\mathrm{c}}(G,A,\alpha)$ which takes the value $1$ at $g$ and [math] at the other elements of $G$. We use the same notation for its image in $C^{*}(G,A,\alpha)$. We denote by $A^{\alpha}$ the fixed point algebra, given by

[TABLE]

We extend this notation to the elements of various objects associated with $A$ under the actions induced by $\alpha$, getting, for example, $(K\otimes A)^{\alpha}$, $K_{0}(A)^{\alpha}$, ${\operatorname{W}}(A)^{\alpha}$, ${\operatorname{Cu}}(A)^{\alpha}$, etc.

The following definition, without Condition (4) but also requiring $\|f_{g}\|=1$, appears in Definition 5.2 of [20] under the name generalized tracial Rokhlin property. Definition 2.2 of [15] includes Condition (4) but only has approximate orthogonality of the contractions. By Proposition 3.10 of [14], Definition 2.2 of [15] is equivalent to our definition. Condition (4) is needed to ensure that the trivial action on $\mathbb{C}$ or a purely infinite simple unital C*-algebra does not have the weak tracial Rokhlin property.

Definition 3.2**.**

Let $G$ be a finite group, let $A$ be a simple unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$. We say that $\alpha$ has the weak tracial Rokhlin property if for every $\varepsilon>0$, every finite set $F\subseteq A$, and every positive element $x\in A$ with $\|x\|=1$, there exist orthogonal positive contractions $f_{g}\in A$ for $g\in G$ such that, with $f=\sum_{g\in G}f_{g}$, the following hold:

1. (1)

$\|af_{g}-f_{g}a\|<\varepsilon$ for all $g\in G$ and all $a\in F$. 2. (2)

$\|\alpha_{g}(f_{h})-f_{gh}\|<\varepsilon$ for all $g,h\in G$. 3. (3)

$1-f\precsim_{A}x$. 4. (4)

$\|fxf\|>1-\varepsilon$.

In Definition 3.2, if $G\neq\{1\}$ the algebra $A$ can’t be type I, since $\alpha$ must be pointwise outer. (See Proposition 3.2 of [14].) Therefore $A$ is infinite-dimensional. (For clarity, we often explicitly include infinite-dimensionality in hypotheses anyway.)

Lemma 3.3**.**

Let $G$ be a finite group, let $A$ be an infinite-dimensional simple unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. Then for every $\varepsilon>0$, every finite set $F\subseteq A$, and every positive element $x\in A$ with $\|x\|=1$, there exist positive contractions $e_{g},f_{g}\in A$ for $g\in G$ such that, with $e=\sum_{g\in G}e_{g}$ and $f=\sum_{g\in G}f_{g}$, the following hold:

1. (1)

$\|e_{g}e_{h}\|<\varepsilon$ and $\|f_{g}f_{h}\|<\varepsilon$ for all $g,h\in G$. 2. (2)

$\|ae_{g}-e_{g}a\|<\varepsilon$ and $\|af_{g}-f_{g}a\|<\varepsilon$ for all $g\in G$ and all $a\in F$. 3. (3)

$\alpha_{g}(e_{h})=e_{gh}$ and $\alpha_{g}(f_{h})=f_{gh}$ for all $g,h\in G$. 4. (4)

$(1-f-\varepsilon)_{+}\precsim_{A}x$. 5. (5)

$\|fxf\|>1-\varepsilon$. 6. (6)

$e\in A^{\alpha}$, $f\in A^{\alpha}$, and $\|f\|=1$. 7. (7)

$e_{g}f_{g}=f_{g}$ for all $g\in G$.

For most applications, we do not need the elements $e_{g}$. Also, presumably one can arrange to have $\|e\|\leq 1$, but we don’t need this.

Proof of Lemma 3.3.

Set $n={\operatorname{card}}(G)$.

We may assume $\varepsilon<\frac{1}{2}$. Let $F\subseteq A$ be a finite set, and let $x\in A_{+}$ satisfy $\|x\|=1$. Define

[TABLE]

Define continuous functions $s,t\colon[0,1]\to[0,1]$ by

[TABLE]

Thus, if $c\in A_{+}$ satisfies $\|c\|\leq 1$, then

[TABLE]

Use Lemma 2.20 to choose $\varepsilon_{0}>0$ such that whenever $B$ is a C*-algebra and $a,b\in B_{+}$ satisfy $\|a\|\leq 1$, $\|b\|\leq 1$, and $\|ab\|<\varepsilon_{0}$, then

[TABLE]

Use Lemma 2.19 to choose $\varepsilon_{1}>0$ such that whenever $B$ is a C*-algebra and $a\in B_{+}$ and $z\in B$ satisfy $\|a\|\leq M$, $\|z\|\leq M$, and $\|az-za\|<\varepsilon_{1}$, then

[TABLE]

Define

[TABLE]

Applying Definition 3.2 with with $F$ and $x$ as given and with $\varepsilon^{\prime}$ in place of $\varepsilon$, we get orthogonal positive contractions $d_{g}\in A$ for $g\in G$ such that, with $d=\sum_{g\in G}d_{g}$, the following hold:

1. (8)

$\|ad_{g}-d_{g}a\|<\varepsilon^{\prime}$ for all $g\in G$ and all $a\in F$. 2. (9)

$\|\alpha_{g}(d_{h})-d_{gh}\|<\varepsilon^{\prime}$ for all $g,h\in G$. 3. (10)

$1-d\precsim_{A}x$. 4. (11)

$\|dxd\|>1-\varepsilon^{\prime}$.

Define \nu=\big{\|}\sum_{g\in G}\alpha_{g}(t(d_{1}))\big{\|}. We claim that

[TABLE]

To prove the claim, first use $\|x\|=1$ and $\|d\|\leq 1$ to get

[TABLE]

Second, using the second part of (3.2) at the second step,

[TABLE]

This relation implies that \bigl{|}\nu-\|d\|\bigr{|}<2n\rho, so

[TABLE]

and, using (3.5) and $\varepsilon<\frac{1}{2}$,

[TABLE]

Using (3.6) and (3.7), we get $|\nu-1|<3n\rho$. The claim is proved.

Define $e_{g}=\alpha_{g}(s(d_{1}))$ and $f_{g}=\frac{1}{\nu}\alpha_{g}(t(d_{1}))$ for $g\in G$, and define $e=\sum_{g\in G}e_{g}$ and $f=\sum_{g\in G}f_{g}$. Clearly $\|e_{g}\|\leq 1$ and $\|f_{g}\|\leq 1$ for all $g\in G$, and $\|f\|=1$. Part (3) of the conclusion is immediate. Clearly $e,f\in A^{\alpha}$, so we have (6), and (7) follows from (3.2).

Now we claim that:

1. (12)

$\|f_{g}-d_{g}\|<2(1+3n)\rho$ for all $g\in G$.

To prove the claim, use the second part of (3.2) and (9) at the second step and (3.4) at the third step to get

[TABLE]

as desired.

We prove Part (1) of the conclusion. For $g\neq h$, using $d_{g}d_{h}=0$ at the first step, (9) at the second step, and (3.3) at the third step, we have

[TABLE]

The choice of $\varepsilon_{0}$ and the relations $e_{g}=s(\alpha_{g}(d_{1}))$ and $f_{g}=\nu^{-1}t(\alpha_{g}(d_{1}))$ now imply

[TABLE]

which is (1).

We prove (2). So let $g\in G$ and let $a\in F$. Using (8) and (9) at the second step, and using (3.3) at the third step, we get

[TABLE]

The choice of $\varepsilon_{1}$ implies

[TABLE]

and

[TABLE]

as desired.

To prove (4), we estimate, using (12) at the third step and (3.1) at the last step

[TABLE]

Therefore, using Lemma 2.4(3a) at the first step and (10) at the second step,

[TABLE]

For (5), we estimate, using part of (3.8) at the second step,

[TABLE]

Therefore, using (11) at the second step,

[TABLE]

This completes the proof. ∎

Lemma 3.4**.**

Let $G$ be a finite group, let $A$ be a C*-algebra, let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$, and let $a,b\in\left(A^{\alpha}\right)_{+}$. If $a\in\overline{bAb}$, then $a\in\overline{bA^{\alpha}b}$ and $a\precsim_{A^{\alpha}}b$.

Proof.

Let $\varepsilon>0$. Choose $c\in A$ such that $\|a-bcb\|<\varepsilon$. Then $\|a-b\alpha_{g}(c)b\|<\varepsilon$ for all $g\in G$. So $d=\frac{1}{{\operatorname{card}}(G)}\sum_{g\in G}\alpha_{g}(c)$ satisfies

[TABLE]

Since $\varepsilon>0$ is arbitrary, $a\in\overline{bA^{\alpha}b}$, whence also $a\precsim_{A^{\alpha}}b$. ∎

Lemma 3.5**.**

Let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on a C*-algebra $A$, let $\delta\in\left(0,\,[2\,{\operatorname{card}}(G)]^{-2}\right)$, and let $a,b\in(A^{\alpha})_{+}$ with $\|b\|=1$ and $\|a\|\leq 1$. If $x$ is a positive element in $\overline{bAb}$ with $a\precsim_{A}(x^{2}-\tfrac{1}{2})_{+}$ and $\|x\alpha_{g}(x)\|<\delta$ for all $g\in G\setminus\{1\}$, then there exists $t\in A^{\alpha}$ such that:

1. (1)

$(a-\delta^{1/2})_{+}\precsim_{A^{\alpha}}tt^{*}$. 2. (2)

$t^{*}t\in\overline{bAb}$. 3. (3)

$(a-\delta^{1/2})_{+}\precsim_{A^{\alpha}}b$.

Proof.

To prove (1), set $\eta=\sqrt{\delta}-2\,{\operatorname{card}}(G)\,\delta$. Since $\delta\in\left(0,\,[2\,{\operatorname{card}}(G)]^{-2}\right)$, it follows that $\eta>0$. Since $a\precsim_{A}(x^{2}-\tfrac{1}{2})_{+}$, by Lemma 2.7 there exists $w\in A$ such that $\|a-wx^{2}w^{*}\|<\eta$ and $\|w\|\leq\sqrt{2}$. Using Lemma 2.4(3b), we find $d\in A$ with $\|d\|\leq 1$ such that $(a-\eta)_{+}=dwx^{2}w^{*}d^{*}$. Set $v=dw$. Then

[TABLE]

Define

[TABLE]

Clearly $t\in A^{\alpha}$.

Now we claim that $\|s\|<2\,{\operatorname{card}}(G)\,\delta$. To prove the claim, for every $g,h\in G$ with $g\neq h$ we have $\|x\,\alpha_{g^{-1}h}(x)\|<\delta$. So

[TABLE]

Thus, using (3.9) and (3.10) at the second step,

[TABLE]

The claim follows.

Using (3.9) at the second step and $a\in A^{\alpha}$ (so that $(a-\eta)_{+}\in A^{\alpha}$) at the last step, we get

[TABLE]

It follows that $tt^{*}-(a-\eta)_{+}=s$. Using the claim, we get

[TABLE]

Therefore, using Lemma 2.4(2) at the first step and Lemma 2.4(3a) at the second step,

[TABLE]

This is (1).

To prove (2), we claim that $\alpha_{g}(xv^{*})\,\alpha_{h}(vx)\in\overline{bAb}$ for all $g,h\in G$. Since $x\in\overline{bAb}$, there exists a sequence $(r_{n})_{n\in{\mathbb{Z}}_{>0}}$ in $A$ such that

[TABLE]

So for all $g,h\in G$ we get, using $b\in A^{\alpha}$ at the first step,

[TABLE]

The claim is proved.

Use the definition of $t$ to compute

[TABLE]

By (3.12) and (3.13), we have $t^{*}t\in\overline{bAb}$, which is (2).

Finally, we prove (3). Since $b\in A^{\alpha}$ and $t^{*}t\in\overline{bAb}$, it follows from Lemma 3.4 that $t^{*}t\precsim_{A^{\alpha}}b$. Therefore, using (1) at the first step and Lemma 2.4(4) at the second step,

[TABLE]

This completes the proof of the lemma. ∎

Lemma 3.6**.**

Let $A$ be an infinite-dimensional simple unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ which has the weak tracial Rokhlin property. Then for every $\varepsilon>0$ and $b\in(A^{\alpha})_{+}$ with $\|b\|=1$, there is a positive element $x\in\overline{bAb}$ with $\|x\|=1$ such that $\|x\alpha_{g}(x)\|<\varepsilon$ for all $g\in G\setminus\{1\}$.

Proof.

We may assume $\varepsilon<\frac{1}{2}$. Set

[TABLE]

Applying Definition 3.2 using $F$, $\varepsilon^{\prime}$, and $b^{2}$, we get positive contractions $f_{g}\in A$ for $g\in G$ such that, with $f=\sum_{g\in G}f_{g}$, the following hold:

1. (1)

$\|zf_{g}-f_{g}z\|<\varepsilon^{\prime}$ for all $g\in G$ and $z\in F$. 2. (2)

$\|\alpha_{g}(f_{h})-f_{gh}\|<\varepsilon^{\prime}$ for all $g,h\in G$. 3. (3)

$\|fb^{2}f\|>1-\varepsilon^{\prime}$.

So we have, using at the first step $g\neq h$ (so that $f_{g}f_{h}=0$), and using (1) at the last step,

[TABLE]

Using this and orthogonality of the elements $f_{g}$ for $g\in G$ at the last step, we estimate

[TABLE]

Therefore, by (3),

[TABLE]

It follows that there exists $s\in G$ such that $\|f_{s}b^{2}f_{s}\|>\frac{1}{2}$. Set $y=bf_{s}^{2}b$. Then

[TABLE]

Now define $x=\|y\|^{-1}\cdot y$. We claim that $\|x\alpha_{g}(x)\|<\varepsilon$ for all $g\in G\setminus\{1\}$. To prove the claim, using $b\in A^{\alpha}$ at the first step, $g\neq 1$ (so that $f_{s}f_{gs}=0$) at the second step, $\|b\|=1$ at the third step, and (1) and (2) at the last step, we estimate

[TABLE]

Since $\|y\|^{-1}<2$, we have

[TABLE]

This completes the proof. ∎

Lemma 3.7**.**

Let $A$ be an infinite-dimensional simple unital C*-algebra and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property. Let $a,b\in(A^{\alpha})_{+}$, and suppose that [math] is a limit point of ${\operatorname{sp}}(b)$. Then $a\precsim_{A}b$ if and only if $a\precsim_{A^{\alpha}}b$.

This result holds when $\alpha$ has the Rokhlin property, without the requirement that [math] be a limit point of ${\operatorname{sp}}(b)$; in this case, $A$ can be any unital C*-algebra. See Theorem 4.1(ii) of [16].

Proof of Lemma 3.7.

We need only prove the forwards implication. So assume that $a\precsim_{A}b$.

We may assume $\|a\|\leq 1$ and $\|b\|=1$. Let $\varepsilon>0$. We may assume $\varepsilon<[2{\operatorname{card}}(G)]^{-2}$. Since $a\precsim_{A}b$, there is $\delta>0$ such that $(a-\varepsilon)_{+}\precsim_{A}(b-\delta)_{+}$. We may require $\delta<1$. Set $a^{\prime}=(a-\varepsilon)_{+}$ and $b^{\prime}=(b-\delta)_{+}$. Choose $w\in A$ such that \big{\|}wb^{\prime}w^{*}-a^{\prime}\big{\|}<[40\,{\operatorname{card}}(G)]^{-1}\varepsilon. Since $b^{\prime},a^{\prime}\in A^{\alpha}$, it follows that

[TABLE]

for all $g\in G$. Choose $\lambda\in{\operatorname{sp}}(b)\cap(0,\delta)$. Let $h\colon[0,\infty)\to[0,1]$ be a continuous function such that $h(\lambda)=1$ and ${\operatorname{supp}}(h)\subseteq(0,\delta)$. Then

[TABLE]

Applying Lemma 3.6 with $h(b)$ in place of $b$, we find a positive element $x\in\overline{h(b)Ah(b)}$ with $\|x\|=1$ such that

[TABLE]

for all $g\in G\setminus\{1\}$. Now set

[TABLE]

Define

[TABLE]

Set M=\max\bigl{(}1,\,\max_{z\in F}\|z\|\bigr{)}, and use Lemma 2.20 and Lemma 2.19 to choose $\delta>0$ such that the following hold:

1. (1)

$\delta\leq\varepsilon^{\prime}$. 2. (2)

If $c,d\in A_{+}$ satisfy $\|c\|,\,\|d\|\leq 1$ and $\|cd\|<\delta$, then \big{\|}c^{1/2}d^{1/2}\big{\|}<\varepsilon^{\prime}. 3. (3)

If $c\in A_{+}$ satisfies $\|c\|\leq M$ and $z\in A$ satisfies $\|z\|\leq M$ and $\|cz-zc\|<\delta$, then \big{\|}c^{1/2}z-zc^{1/2}\big{\|}<\varepsilon^{\prime}.

Applying Lemma 3.3 with $F$ as given, with $\delta$ in place of $\varepsilon$, and with $s$ in place of $x$, we get positive contractions $f_{g}\in A$ for $g\in G$ such that, with $f=\sum_{g\in G}f_{g}$, the following hold:

1. (4)

$\|f_{g}f_{h}\|<\delta$ for all $g,h\in G$ with $g\neq h$. 2. (5)

$\|zf_{g}-f_{g}z\|<\delta$ for all $g\in G$ and $z\in F$. 3. (6)

$\alpha_{g}(f_{h})=f_{gh}$ for all $g,h\in G$. 4. (7)

$f\in A^{\alpha}$ and $\|f\|=1$. 5. (8)

$(1-f-\delta)_{+}\precsim_{A}s$.

Then also:

1. (9)

\big{\|}f_{g}^{1/2}f_{h}^{1/2}\big{\|}<\varepsilon^{\prime} for all $g,h\in G$ with $g\neq h$. 2. (10)

\big{\|}zf_{g}^{1/2}-f_{g}^{1/2}z\big{\|}<\varepsilon^{\prime} for all $g\in G$ and $z\in F$. 3. (11)

\alpha_{g}\big{(}f_{h}^{1/2}\big{)}=f_{gh}^{1/2} for all $g,h\in G$.

Since $\delta\leq\varepsilon^{\prime}<\frac{\varepsilon}{8}$, (8) implies that \big{(}1-f-\frac{\varepsilon}{8}\big{)}_{+}\precsim_{A}s\sim\big{(}x^{2}-\tfrac{1}{2}\big{)}_{+}. Applying Lemma 3.5(3) with $h(b)$ in place of $b$, with \big{(}1-f-\frac{\varepsilon}{8}\big{)}_{+} in place of $a$, with $x$ as given, and with $\varepsilon^{2}/64$ in place of $\delta$, we get

[TABLE]

Now define $v=\sum_{g\in G}\alpha_{g}(f_{1}w)$. Clearly $v\in A^{\alpha}$. We claim that

[TABLE]

To prove the claim, define

[TABLE]

It is immediate that

[TABLE]

Also set

[TABLE]

giving \bigl{\|}{\widetilde{f}}\bigr{\|}\leq{\operatorname{card}}(G).

For $g\in G$, by (1) and (5) we have $\|\alpha_{g}(w)f_{g}-f_{g}\alpha_{g}(w)\|<\varepsilon^{\prime}$. Therefore, for all $g\in G$, using (3.15) on the last term at the second step and $\|b^{\prime}\|\leq 1$ at the last step,

[TABLE]

Set $S=\{(g,g,g)\colon g\in G\}\subseteq G^{3}$. Using (3.19), (9), and $\|b^{\prime}\|\leq 1$ at the second step, we get

[TABLE]

Next, we estimate, using $f=\sum_{g\in G}f_{g}$ at the first step, and using (1), (4), and (5) at the third step,

[TABLE]

A similar calculation, this time using (9), (10), and (1), gives

[TABLE]

The next step is to estimate \bigl{\|}v-v_{0}{\widetilde{f}}\bigr{\|}. For all $g\in G$, using (5), (6), and (1), we get

[TABLE]

Now, by (10),

[TABLE]

It follows, using this, (3.22), and (3.18) at the second step, that

[TABLE]

Combining this with (3.20) and (3.21), we now have

[TABLE]

The claim is proved.

The claim implies that

[TABLE]

Applying Lemma 2.6 with $\frac{\varepsilon}{4}$ in place of $\varepsilon_{1}$, with $\frac{\varepsilon}{2}$ in place of $\varepsilon_{2}$, with $a^{\prime}$ in place of $a$, and with $1-f$ in place of $g$, we get

[TABLE]

Using (3.25) at the second step, (3.24) and (3.17) at the third step, and (3.16) at the last step, we have

[TABLE]

Therefore $(a-\varepsilon)_{+}\precsim_{A^{\alpha}}b$. ∎

Definition 3.8**.**

Let $A$ be a C*-algebra. Following the discussion before Corollary 2.24 of [3] and Definition 3.1 of [38], with slight changes in notation, we define

[TABLE]

[TABLE]

The elements of $A_{++}$ are called purely positive.

We recall some properties of ${\operatorname{W}}_{+}(A)$ and ${\operatorname{Cu}}_{+}(A)$.

Lemma 3.9**.**

Let $A$ be a stably finite simple unital C*-algebra. Then:

1. (1)

${\operatorname{Cu}}(A)\setminus{\operatorname{W}}(A)\subseteq{\operatorname{Cu}}_{+}(A)$. 2. (2)

(K\otimes A)_{++}=\big{\{}a\in(K\otimes A)_{+}\colon\mbox{0$is a limit point of${\operatorname{sp}}(a)}\big{\}}. 3. (3)

${\operatorname{W}}_{+}(A)\cup\{0\}$ and ${\operatorname{Cu}}_{+}(A)\cup\{0\}$ are unital subsemigroups of ${\operatorname{W}}(A)$ and ${\operatorname{Cu}}(A)$. 4. (4)

Let $\eta_{1},\eta_{2},\ldots\in{\operatorname{Cu}}_{+}(A)\cup\{0\}$ satisfy $\eta_{1}\leq\eta_{2}\leq\cdots$. Then \sup\big{(}\{\eta_{n}\colon n\in{\mathbb{Z}}_{\geq 0}\}\big{)}, evaluated in ${\operatorname{Cu}}(A)$, is in ${\operatorname{Cu}}_{+}(A)\cup\{0\}$.

Proof.

Parts (1) and (2) are Lemma 3.2 of [38]. Part (3) for ${\operatorname{W}}(A)$ is Corollary 2.9(i) of [34]. For ${\operatorname{Cu}}(A)$ it is Corollary 3.3 of [38]. Part (4) is Lemma 3.5 of [38] (originally Parts (i) and (iv) of Proposition 6.4 of [12]). ∎

There are further interesting properties: still assuming $A$ is stably finite and simple, ${\operatorname{Cu}}_{+}(A)\cup\{0\}$ is absorbing (this follows from Corollary 3.3 of [38]) and, if $A$ is not of type I, has the same functionals as ${\operatorname{Cu}}(A)$ (Lemma 3.8 of [38]).

We will use the following result several times. The main work for the last sentence of the proof is in [1].

Lemma 3.10**.**

Let $A$ be an infinite-dimensional simple unital C*-algebra, let $G$ be a finite group, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. Then, for every $x\in(A^{\alpha})_{+}\setminus\{0\}$, there exists $c\in(A^{\alpha})_{+}$ such that $c\precsim_{A^{\alpha}}x$ and ${\operatorname{sp}}(c)=[0,1]$.

Proof.

The algebra $A$ is not type I, so Theorem 4.1 of [39] implies that $A^{\alpha}$ is not type I. Since $C^{*}(G,A,\alpha)$ is simple, Lemma 4.3(4) below (or [42]) implies that $A^{\alpha}$ is simple. Apply Lemma 2.1 of [38] to ${\overline{xA^{\alpha}x}}$. ∎

Lemma 3.11**.**

Let $A$ be a stably finite simple unital C*-algebra which is not of type I and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property. Let $\iota\colon A^{\alpha}\to A$ be the inclusion map. Then:

1. (1)

The map ${\operatorname{W}}(\iota)\colon{\operatorname{W}}(A^{\alpha})\to{\operatorname{W}}(A)$ induces an isomorphism of ordered semigroups from ${\operatorname{W}}_{+}(A^{\alpha})\cup\{0\}$ to its image in ${\operatorname{W}}(A)$. 2. (2)

The map ${\operatorname{Cu}}(\iota)\colon{\operatorname{Cu}}(A^{\alpha})\to{\operatorname{Cu}}(A)$ induces an isomorphism of ordered semigroups from ${\operatorname{Cu}}_{+}(A^{\alpha})\cup\{0\}$ to its image in ${\operatorname{Cu}}(A)$.

Proof.

In both parts, we need only prove injectivity and order isomorphism.

By Corollary 4.6 of [14], for every $n\in{\mathbb{Z}}_{>0}$ the action $g\mapsto{\operatorname{id}}_{M_{n}}\otimes\alpha_{g}$ of $G$ on $M_{n}(A)$ has the weak tracial Rokhlin property. With ${\operatorname{W}}_{+}(A^{\alpha})$ in place of ${\operatorname{W}}_{+}(A^{\alpha})\cup\{0\}$, Part (1) now follows from Lemma 3.9(2) and Lemma 3.7. Part (1) as stated is then immediate.

We prove (2). It suffices to prove that if $a,b\in(K\otimes A^{\alpha})_{++}$ satisfy $a\precsim_{A}b$, then $a\precsim_{A^{\alpha}}b$. Let $\varepsilon>0$; we prove that $(a-\varepsilon)_{+}\precsim_{A^{\alpha}}b$.

Choose $\delta>0$ such that

[TABLE]

Choose \lambda\in{\operatorname{sp}}(b)\cap\big{(}0,\frac{\delta}{3}\big{)}. Let $h\colon[0,\infty)\to[0,1]$ be a continuous function such that $h(\lambda)=1$ and {\operatorname{supp}}(h)\subseteq\big{(}0,\frac{\delta}{3}\big{)}. Then

[TABLE]

Choose $n\in{\mathbb{Z}}_{>0}$ and $a_{0},b_{0},c_{0}\in M_{n}(A^{\alpha})_{+}$ such that

[TABLE]

It follows from Lemma 2.4(3c) that

[TABLE]

and

[TABLE]

Set c_{1}=\big{(}c_{0}-\frac{1}{3}\big{)}_{+}. Then $\|c_{1}\|>\frac{1}{3}$, so $c_{1}\neq 0$. Since the action induced by $\alpha$ on $M_{n}(A)$ has the weak tracial Rokhlin property, Lemma 3.10 provides $c\in M_{n}(A^{\alpha})_{+}$ such that $c\precsim_{A^{\alpha}}c_{1}$ and ${\operatorname{sp}}(c)=[0,1]$. In particular,

[TABLE]

At the first step combining the second part of (3.28), (3.26), and the first part of (3.29), we get

[TABLE]

Since

[TABLE]

because [math] is a limit point of the spectrum of \big{(}b_{0}-\frac{\delta}{3}\big{)}_{+}\oplus c, and using Lemma 3.9(2), Part (1) and (3.31) imply

[TABLE]

Using, in order, the first part of (3.28), (3.32), (3.30) and the second part of (3.29), and (3.27), we get

[TABLE]

This completes the proof. ∎

Lemma 3.11 fails if we don’t restrict to the purely positive elements. See Example 4.7. We postpone this example, since it uses Lemma 4.3.

4. Radius of comparison of the fixed point algebra

and crossed product

In the next section, we identify the range of the map ${\operatorname{Cu}}_{+}(A^{\alpha})\to{\operatorname{Cu}}(A)$ when $\alpha$ has the weak tracial Rokhlin property: it is ${\operatorname{Cu}}_{+}(A)^{\alpha}$. This information is not needed for our estimate on the radius of comparison, and does not seem to help with the (still open) opposite inequality to the one we prove. So we prove the radius of comparison results now. Then we discuss what happens under weaker hypotheses on the action, and give the example promised at the end of Section 3.

Theorem 4.1**.**

Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. Then ${\operatorname{rc}}(A^{\alpha})\leq{\operatorname{rc}}(A)$.

Proof.

We use Theorem 12.4.4(ii) of [17]. Thus, let $m,n\in{\mathbb{Z}}_{>0}$ satisfy $\frac{m}{n}>{\operatorname{rc}}(A)$. Let $l\in{\mathbb{Z}}_{>0}$, and let $a,b\in(A^{\alpha}\otimes M_{l})_{+}$ with $\|a\|=\|b\|=1$ satisfy

[TABLE]

in ${\operatorname{W}}({A^{\alpha}})$. Corollary 4.6 of [14], the action $\alpha\otimes{\operatorname{id}}_{M_{l}}\colon G\to{\operatorname{Aut}}(A\otimes M_{l})$, defined by

[TABLE]

also has the weak tracial property. We may therefore assume $l=1$.

We must prove that $a\precsim_{A^{\alpha}}b$. Moreover, by Lemma 2.4(1b), it is enough to show that for every $\varepsilon>0$ we have $(a-\varepsilon)_{+}\precsim_{A^{\alpha}}b$.

So let $\varepsilon>0$. Without loss of generality $\varepsilon<\frac{1}{2}$. Choose $k\in{\mathbb{Z}}_{>0}$ such that

[TABLE]

Then in ${\operatorname{W}}({A^{\alpha}})$ we have

[TABLE]

Let $u\in M_{\infty}({A^{\alpha}})_{+}$ be the direct sum of $kn+1$ copies of $a$, let $z\in M_{\infty}({A^{\alpha}})_{+}$ be the direct sum of $kn$ copies of $b$, and let $q\in M_{\infty}({A^{\alpha}})_{+}$ be the direct sum of $km$ copies of $1_{A}$. Then, by definition, $u\oplus q\precsim_{A^{\alpha}}z$. Therefore Lemma 2.4(1c) provides $\delta>0$ such that \big{(}u\oplus q-\varepsilon\big{)}_{+}\precsim_{A^{\alpha}}(z-\delta)_{+}. Since $\varepsilon<\frac{1}{2}$, we have

[TABLE]

so

[TABLE]

Set $a^{\prime}=(a-\varepsilon)_{+}$ and $b^{\prime}=(b-\delta)_{+}$. Then

[TABLE]

Lemma 2.7 of [38] provides positive elements $c\in{A^{\alpha}}$ and $y\in{A^{\alpha}}\setminus\{0\}$ such that

[TABLE]

in ${\operatorname{W}}(A^{\alpha})$. By Lemma 3.10, there is $y_{0}\in(A^{\alpha})_{+}$ such that $y_{0}\precsim_{A^{\alpha}}y$ and ${\operatorname{sp}}(y_{0})=[0,1]$. Replacing $y$ with this element, we may assume that $y$ is purely positive. By (4.2) and (4.3),

[TABLE]

This relation also holds in ${\operatorname{W}}(A)$. For $\tau\in{\operatorname{QT}}(A)$, apply $d_{\tau}$ and divide by $kn+1$ to get

[TABLE]

So $a^{\prime}\precsim_{A}c$ by (4.1). Therefore, using Lemma 3.7 with $c\oplus y$ in place of $b$ at the second step, and using (4.3) at the third step,

[TABLE]

This completes the proof. ∎

Using [16], we get the same conclusion for Rokhlin actions on stably finite unital C*-algebras.

Theorem 4.2**.**

Let $G$ be a finite group, let $A$ be a stably finite unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$ which has the Rokhlin property. Then ${\operatorname{rc}}(A^{\alpha})\leq{\operatorname{rc}}(A)$.

Proof.

We may clearly assume ${\operatorname{rc}}(A)<\infty$. Let $r\in[0,\infty)$ and suppose that $A$ has $r$-comparison. Let $a,b\in M_{\infty}(A^{\alpha})_{+}$ satisfy $d_{\tau}(a)+r<d_{\tau}(b)$ for all $\tau\in{\operatorname{QT}}(A^{\alpha})$. Since every quasitrace on $A$ restricts to a quasitrace on $A^{\alpha}$, we have $d_{\tau}(a)+r<d_{\tau}(b)$ for all $\tau\in{\operatorname{QT}}(A)$. Since $A$ has $r$-comparison, we get $a\precsim_{A}b$. Now $a\precsim_{A^{\alpha}}b$ by Theorem 4.1(ii) of [16]. So ${\operatorname{rc}}(A^{\alpha})\leq r$. Taking the infimum over $r\in[0,\infty)$ such that $A$ has $r$-comparison, we get ${\operatorname{rc}}(A^{\alpha})\leq{\operatorname{rc}}(A)$. ∎

We now turn to the radius of comparison of the crossed product.

Parts (1)–(4) of the following lemma are originally taken from [42]. Since some properties of the projection $p$ are needed in our computations, we give a more detailed statement.

Lemma 4.3**.**

Let $G$ be a finite group, let $A$ be a unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$. Recalling from Notation 3.1 that $(u_{g})_{g\in G}$ is the family of standard unitaries in $C^{*}(G,A,\alpha)$, define $p=\frac{1}{{\operatorname{card}}(G)}\sum_{g\in G}u_{g}$. Then:

1. (1)

$p$ is a projection in $C^{*}(G,A,\alpha)$. 2. (2)

pap=\Big{(}\frac{1}{{\operatorname{card}}(G)}\sum_{g\in G}\alpha_{g}(a)\Big{)}p for all $a\in A$. 3. (3)

If $a\in A^{\alpha}$, then $pap=ap$. 4. (4)

The map $a\mapsto ap$ is an isomorphism from $A^{\alpha}$ to the corner $pC^{*}(G,A,\alpha)p$. 5. (5)

If $C^{*}(G,A,\alpha)$ has stable rank one, then $A^{\alpha}$ has stable rank one. 6. (6)

If $\alpha$ has the Rokhlin property, then $p$ is full in $C^{*}(G,A,\alpha)$.

Proof.

Parts (1)–(4) are computations. (Also see [42].)

Next, if $C^{*}(G,A,\alpha)$ has stable rank one, then Theorem 3.1.8 of [25] implies that $pC^{*}(G,A,\alpha)p$ has stable rank one, so (5) follows from (4).

For (6), let $J\subseteq C^{*}(G,A,\alpha)$ be the closed ideal generated by $p$, and set $I=J\cap A$. Let $E\colon C^{*}(G,A,\alpha)\to A$ be the standard conditional expectation. The proof of Proposition 10.3.13 of [17] shows that $E(J)\subseteq I$. Since $E({\operatorname{card}}(G)\cdot p)=1$, we have $1\in I$, so $1\in J$. ∎

The proof of the following lemma is easier, and well known, for tracial states. For example, the inequality (4.10) is trivial for tracial states, but it seems to require some effort for quasitraces.

Lemma 4.4**.**

Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property, and let \tau\in{\operatorname{QT}}\big{(}C^{*}(G,A,\alpha)\big{)}. Let $p=\frac{1}{{\operatorname{card}}(G)}\sum_{g\in G}u_{g}$, as in Lemma 4.3. Then $\tau(p)=\frac{1}{{\operatorname{card}}(G)}$.

Proof.

Let $\varepsilon>0$. We show that \big{|}\frac{1}{{\operatorname{card}}(G)}-\tau(p)\big{|}<\varepsilon. By Corollary 2.5 of [38], there is $a\in A_{+}\setminus\{0\}$ such that for all $\rho\in{\operatorname{QT}}(A)$,

[TABLE]

Applying Definition 3.2 with $F=\varnothing$, with $[32\,{\operatorname{card}}(G)]^{-1}\varepsilon$ in place of $\varepsilon$, and with $\|a\|^{-1}\cdot a$ in place of $x$, we get orthogonal positive contractions $f_{g}\in A$ for $g\in G$ such that, with $f=\sum_{g\in G}f_{g}$, we have

[TABLE]

and

[TABLE]

for all $g,h\in G$. This inequality, together with $\|f_{g}\|,\|f_{gh}\|\leq 1$, implies

[TABLE]

Now we claim that the following hold:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We prove (4.7). Since ${\operatorname{sp}}(f)\subseteq[0,1]$, we have $1-f^{2}\sim_{A}1-f$, so $1-f^{2}\precsim_{A}a$ by (4.5). Clearly $\tau|_{A}\in{\operatorname{QT}}(A)$. Therefore, using (4.4) at the last step,

[TABLE]

The relation (4.7) follows because $1$ and $f^{2}$ commute.

To prove (4.8), we start with $(1-f^{2})^{1/2}p(1-f^{2})^{1/2}\leq(1-f^{2})$. Then, by Proposition 2.10(3),

[TABLE]

Therefore, using $[p,\,pf^{2}p]=0$ at the second step, the trace property (Definition 2.8(1)) at the third step, (4.11) at the fourth step, and (4.7) at the last step,

[TABLE]

For (4.9), first we estimate

[TABLE]

Therefore, using (4.12) at the last step,

[TABLE]

Now use Proposition 2.10(4) and $N(\tau)=1$ to get

[TABLE]

We have $\tau\left(\sum_{h\in G}f_{h}pf_{h}\right)=\sum_{h\in G}\tau(f_{h}pf_{h})$ since the elements $f_{h}pf_{h}$, for $h\in G$, commute with each other. The trace property (Definition 2.8(1)) gives $\tau(f_{h}pf_{h})=\tau(pf_{h}^{2}p)$ for $h\in G$. This completes the proof of (4.9).

To prove (4.10), set $b=\sum_{g\in G}\alpha_{g}(f^{2}_{1})$. Then, for $h\in G$, using (4.6) at the second step,

[TABLE]

Next, using Lemma 4.3(2) at the first step and (4.13) at the last step,

[TABLE]

Then, using (4.14) at the last step,

[TABLE]

Finally, we get, using $pbp=bp$ (by Lemma 4.3(3), since $b\in A^{\alpha}$), $N(\tau)=1$, and Proposition 2.10(4) at the second step, and using (4.14) and (4.15) at the third step,

[TABLE]

This completes the proof of the claim.

Now we estimate, using (4.7), (4.9), (4.10), and (4.8) at the second step,

[TABLE]

This completes the proof. ∎

Theorem 4.5**.**

Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. Then

[TABLE]

Proof.

By Lemma 4.4 and Lemma 4.3(4), the projection $p\in C^{*}(G,A,\alpha)$ of Lemma 4.3 satisfies $\tau(p)={\operatorname{card}}(G)^{-1}$ for all $\tau\in{\operatorname{QT}}(C^{*}(G,A,\alpha))$ and $A^{\alpha}\cong pC^{*}(G,A,\alpha)p$. The algebra $C^{*}(G,A,\alpha)$ is simple by Corollary 3.3 of [14]. So $p$ is full. Now $C^{*}(G,A,\alpha)$ is stably finite (being stably isomorphic to $A^{\alpha}\subseteq A$), so Theorem 2.18 implies that {\operatorname{rc}}\big{(}C^{*}(G,A,\alpha)\big{)}={\operatorname{card}}(G)^{-1}{\operatorname{rc}}(A^{\alpha}). This is the first part of the conclusion. The second part now follows from Theorem 4.1. ∎

We get the same outcome with the Rokhlin property and for any stably finite unital C*-algebra, not necessarily simple.

Theorem 4.6**.**

Let $G$ be a finite group, let $A$ be a stably finite unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$ which has the Rokhlin property. Then

[TABLE]

Proof.

The proof is the same as that of Theorem 4.5, except that we now use Lemma 4.3(6) rather than simplicity of $C^{*}(G,A,\alpha)$ to deduce that $p$ is full, and we use Theorem 4.2 instead of Theorem 4.1 at the end. ∎

If $G={\mathbb{Z}}/2{\mathbb{Z}}$ and $\alpha$ is the trivial action, then the conclusions of Theorem 4.1 and Theorem 4.2 hold (because $A^{\alpha}=A$) but the conclusions of Theorem 4.5 and Theorem 4.6 generally fail (because $C^{*}(G,A,\alpha)\cong A\oplus A$ and ${\operatorname{rc}}(A\oplus A)={\operatorname{rc}}(A)$). For pointwise outer actions $\alpha$, in fact the conclusions of all these theorems can fail. See Example 6.22.

Example 4.7**.**

We give an example of a stably finite simple separable unital C*-algebra $D$ which is not of type I and an action $\alpha\colon{\mathbb{Z}}/2{\mathbb{Z}}\to{\operatorname{Aut}}(D)$ such that $\alpha$ has the weak tracial Rokhlin property but such that the map ${\operatorname{W}}(\iota)\colon{\operatorname{W}}(D^{\alpha})\to{\operatorname{W}}(D)$ of Lemma 3.11 is not injective. This example also shows that Lemma 3.7 fails when [math] is not a limit point of ${\operatorname{sp}}(b)$. Our algebra $D$ is in fact a UHF algebra, and $\alpha$ actually has the tracial Rokhlin property. This example is therefore a counterexample to Proposition 6.2 and Corollary 6.3 of [33]. (The mistake in [33] is in the use of $g_{\delta}(b)$ in the proof of Proposition 6.2 of [33]. Since $g_{\delta}(0)\neq 0$, $g_{\delta}(b)\not\in{\overline{bPb}}$.)

Let $D$ and $\alpha$ be as in Example 2.8 of [37]. Let \beta\in{\operatorname{Aut}}\bigl{(}C^{*}({\mathbb{Z}}/2{\mathbb{Z}},D,\alpha)\bigr{)} be the automorphism which generates the dual action. As shown there, $\alpha$ has the tracial Rokhlin property but not the Rokhlin property. The algebra $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},D,\alpha)$ has a unique tracial state, which we call $\tau$. It is clearly $\beta$-invariant. The algebra $D$ also has a unique tracial state $\sigma$; necessarily $\sigma=\tau|_{D}$. Moreover, there is \eta_{0}\in K_{0}\bigl{(}C^{*}({\mathbb{Z}}/2{\mathbb{Z}},D,\alpha)\bigr{)} such that $\beta_{*}(\eta_{0})\neq\eta_{0}$.

Set $\eta=\eta_{0}-\beta_{*}(\eta_{0})$. Then $\eta\neq 0$, but, since $\tau\circ\beta=\tau$, we have $\tau_{*}(\eta)=0$. It follows from Lemma 4.3(4) that $D^{\alpha}$ is isomorphic to a full corner of $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},D,\alpha)$. Thus, except for the $K_{0}$-class of the identity element, the Elliott invariants of $D^{\alpha}$ and $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},D,\alpha)$ are isomorphic. In particular, $D^{\alpha}$ has a unique tracial state $\rho$ (necessarily equal to $\sigma|_{D^{\alpha}}$), and there is $\mu\in K_{0}(D^{\alpha})\setminus\{0\}$ such that $\rho_{*}(\mu)=0$.

Choose projections $p,q\in K\otimes D^{\alpha}$ such that $\mu=[p]-[q]$ in $K_{0}(D^{\alpha})$. Since $[p]\neq[q]$ and $D^{\alpha}$ is stably finite, it follows that $\langle p\rangle\neq\langle q\rangle$ in ${\operatorname{Cu}}(D^{\alpha})$. In fact, they are in $W(D^{\alpha})$. Let $\iota\colon D^{\alpha}\to D$ be the inclusion map. Then $\sigma(\iota(p))=\sigma(\iota(q))$. Since $D$ is a UHF algebra, this implies that $\iota_{*}([p])=\iota_{*}([q])$ in $K_{0}(D)$. Therefore ${\operatorname{W}}(\iota)(\langle p\rangle)={\operatorname{W}}(\iota)(\langle q\rangle)$. Thus ${\operatorname{W}}(\iota)$ is not injective. Also, $p\not\precsim_{D^{\alpha}}q$ but $p\precsim_{D}q$.

5. Surjectivity of

${\operatorname{Cu}}_{+}(A^{\alpha})\to{\operatorname{Cu}}_{+}(A)^{\alpha}$

In this section, we prove that if $G$ is finite, $A$ is unital, stably finite, and simple, and $\alpha\colon G\to{\operatorname{Aut}}(A)$ has the weak tracial Rokhlin property, then the inclusion $A^{\alpha}\to A$ induces an isomorphism of the ordered semigroups of purely positive elements ${\operatorname{Cu}}_{+}(A^{\alpha})\cup\{0\}\to{\operatorname{Cu}}_{+}(A)^{\alpha}\cup\{0\}$. If we assume stable rank one, then the conclusion is valid for ${\operatorname{W}}_{+}(A^{\alpha})$ and ${\operatorname{W}}_{+}(A)^{\alpha}$ as well. We also give the corresponding result for ${\operatorname{W}}(A^{\alpha})$ when $A$ is merely unital but $\alpha$ is assumed to have the Rokhlin property. In this case, we need not discard the classes of the projections, just like in Theorem 4.1(ii) of [16] for ${\operatorname{Cu}}(A^{\alpha})$.

Injectivity was proved in Section 3; the content of this section is the proof of surjectivity.

The next lemma produces the following chain of subequivalences, for any $g\in G$:

[TABLE]

Lemma 5.1**.**

Let $A$ be a C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$. Let $a\in(K\otimes A)_{+}$ satisfy $a\sim_{A}\alpha_{g}(a)$ for all $g\in G$ and $\|a\|\leq 1$. Then for every $\varepsilon>0$ there are $m\in{\mathbb{Z}}_{>0}$, $\delta,\delta_{1},\delta_{2},\ldots,\delta_{6}>0$, and $a^{\prime}\in M_{m}(A)_{+}$ with $\|a^{\prime}\|\leq\|a\|$, such that:

1. (1)

$0<\delta<\delta_{1}<\delta_{2}<\delta_{3}<\delta_{4}<\delta_{5}<\delta_{6}<\varepsilon$. 2. (2)

$a^{\prime}\precsim_{A}(a-\delta)_{+}$. 3. (3)

$(\alpha_{g}(a^{\prime})-\delta_{2})_{+}\precsim_{A}(a^{\prime}-\delta_{1})_{+}$ for all $g\in G$. 4. (4)

$(a^{\prime}-\delta_{5})_{+}\precsim_{A}(\alpha_{g}(a^{\prime})-\delta_{4})_{+}$ for all $g\in G$. 5. (5)

$(a-\varepsilon)_{+}\precsim_{A}(a^{\prime}-\delta_{6})_{+}$.

If, in addition, [math] is a limit point of ${\operatorname{sp}}(a)$, then we may also require:

1. (6)

${\operatorname{sp}}(a^{\prime})\cap(0,\delta)\neq\varnothing$.

Proof.

We may clearly assume that $a\neq 0$.

Let $\varepsilon>0$. First use $\alpha_{g}(a)\sim_{A}a$ for $g\in G$ to choose $\beta_{1}>0$ such that $\beta_{1}<\frac{\varepsilon}{4}$ and for all $g\in G$ we have \bigl{(}a-\frac{\varepsilon}{4}\bigr{)}_{+}\precsim_{A}(\alpha_{g}(a)-\beta_{1})_{+}. Similarly, choose $\beta_{2}>0$ such that $\beta_{2}<\frac{\beta_{1}}{4}$ and for all $g\in G$ we have \bigl{(}\alpha_{g}(a)-\frac{\beta_{1}}{4}\bigr{)}_{+}\precsim_{A}(a-\beta_{2})_{+}. Set

[TABLE]

Then, for $g\in G$, by Lemma 2.7 there are $s_{g},t_{g}\in K\otimes A$ such that

[TABLE]

and

[TABLE]

Use Lemma 12.4.5 of [17] to choose $\mu>0$ so small that whenever $B$ is a C*-algebra and $b,c\in B$ satisfy $0\leq b,c\leq 1$ and $\|b-c\|<\mu$, then

[TABLE]

[TABLE]

Define

[TABLE]

If we do not need Part (6) of the conclusion, simply take $\delta=\nu$. Otherwise, since [math] is a limit point of ${\operatorname{sp}}(a)$, we can choose $\delta>0$ such that $\delta\leq\nu$ and $\frac{5\delta}{2}\in{\operatorname{sp}}(a)$. Choose $m\in{\mathbb{Z}}_{>0}$ such that there is $b\in M_{m}(A)_{+}$ with $\|b\|=\|a\|$ and $\|b-a\|<\frac{\delta}{2}$. Define $a^{\prime}=(b-2\delta)_{+}$. Since $\|b-a\|<\delta$, it follows from Lemma 2.4(3c) that $a^{\prime}\precsim_{A}(a-\delta)_{+}$, which is (2). If [math] is a limit point of ${\operatorname{sp}}(a)$, then we arranged to have $\frac{5\delta}{2}\in{\operatorname{sp}}(a)$, so $\|b-a\|<\frac{\delta}{2}$ implies ${\operatorname{sp}}(b)\cap(2\delta,3\delta)\neq\varnothing$. Thus ${\operatorname{sp}}(a^{\prime})\cap(0,\delta)\neq\varnothing$. This is (6).

We have $\|a^{\prime}-a\|<3\delta\leq\mu$. For $g\in G$, it therefore follows from (5.1), using (5.4) and $\|s_{g}\|\leq M$, that

[TABLE]

and similarly, using (5.2), (5.3), and $\|t_{g}\|\leq M$,

[TABLE]

Define

[TABLE]

Then (1) is clear. Moreover, we have

[TABLE]

So (5.5) and (5.6) imply, for $g\in G$,

[TABLE]

which is (4), and

[TABLE]

which is (3). Finally, $\|a^{\prime}-a\|<3\delta\leq\frac{\varepsilon}{4}$, so by Lemma 2.4(3c) we have (a-\varepsilon)_{+}\precsim_{A}\bigl{(}a^{\prime}-\frac{3\varepsilon}{4}\bigr{)}_{+}=(a^{\prime}-\delta_{6})_{+}, which is (5). This completes the proof. ∎

Lemma 5.2**.**

Let $A$ be a simple C*-algebra which is not of type I. Let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ and let $x\in A_{+}\setminus\{0\}$. Then there exists $z\in(A^{\alpha})_{+}\setminus\{0\}$ such that $z\precsim_{A}x$.

Proof.

Set $n={\operatorname{card}}(G)$. By Lemma 2.4 of [38], there are $b_{1},b_{2},\ldots,b_{n}\in A_{+}\setminus\{0\}$ such that, for $j\neq k$,

[TABLE]

Let $y$ be the direct sum of $n$ copies of $b_{1}$. Using Lemma 2.6 of [38], choose $c\in A_{+}\setminus\{0\}$ such that $c\precsim_{A}\alpha_{g^{-1}}(b_{1})$ for all $g\in G$. Then $\alpha_{g}(c)\precsim_{A}b_{1}$ for all $g\in G$. Set $z=\sum_{g\in G}\alpha_{g}(c)$. Clearly $z\in(A^{\alpha})_{+}\setminus\{0\}$. Then

[TABLE]

as desired. ∎

Lemma 5.3**.**

Let $A$ be an infinite-dimensional simple unital C*-algebra and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ which has the weak tracial Rokhlin property. Let $a\in(K\otimes A)_{+}$ satisfy $a\sim_{A}\alpha_{g}(a)$ for all $g\in G$ and assume that [math] is a limit point of ${\operatorname{sp}}(a)$. Then for every $\varepsilon>0$ there exist $\delta>0$, $m\in{\mathbb{Z}}_{>0}$, and $b\in M_{m}(A^{\alpha})_{+}$ such that

[TABLE]

Proof.

We may assume $\|a\|=1$. Set $n={\operatorname{card}}(G)$.

Let $\varepsilon>0$. Apply the version of Lemma 5.1 which assumes [math] is a limit point of ${\operatorname{sp}}(a)$, and let the notation be as in its conclusion.

By Lemma 5.1(6), there is $\lambda\in{\operatorname{sp}}(a^{\prime})\cap(0,\delta)$. Choose a continuous function $h\colon[0,\infty)\to[0,1]$ such that ${\operatorname{supp}}(h)\subseteq(0,\delta)$ and $h(\lambda)=1$. Use Lemma 5.2 to choose $d\in M_{m}(A^{\alpha})_{+}\setminus\{0\}$ such that $d\precsim_{A}h(a^{\prime})$. Since the action induced by $\alpha$ on $M_{m}(A)$ has the weak tracial Rokhlin property (Corollary 4.6 of [14]), Lemma 3.10 provides $s\in M_{m}(A^{\alpha})_{+}$ such that $s\precsim_{A^{\alpha}}d$ and ${\operatorname{sp}}(s)=[0,1]$.

Define

[TABLE]

[TABLE]

For $g\in G$ use Lemma 2.7, Lemma 5.1(3), Lemma 5.1(4), and (5.7) to choose $v_{g},w_{g}\in M_{m}(A)$ such that

[TABLE]

and

[TABLE]

Set

[TABLE]

Since the induced action on $M_{m}(A)$ has the weak tracial Rokhlin property, we can apply Lemma 3.3 with $M_{m}(A)$ in place of $A$, with $F$ as above, with $\varepsilon^{\prime}$ in place of $\varepsilon$, and with $s$ in place of $x$. We get positive contractions $e_{g},f_{g}\in M_{m}(A)$ for $g\in G$ such that, with $e=\sum_{g\in G}e_{g}$ and $f=\sum_{g\in G}f_{g}$, the following hold:

1. (1)

$\|e_{g}e_{h}\|<\varepsilon^{\prime}$ and $\|f_{g}f_{h}\|<\varepsilon^{\prime}$ for all $g,h\in G$. 2. (2)

$\|ze_{g}-e_{g}z\|<\varepsilon^{\prime}$ and $\|zf_{g}-f_{g}z\|<\varepsilon^{\prime}$ for all $g\in G$ and all $z\in F$. 3. (3)

$(1-f-\varepsilon^{\prime})_{+}\precsim_{A}s$. 4. (4)

$\alpha_{g}(e_{h})=e_{gh}$ and $\alpha_{g}(f_{h})=f_{gh}$ for all $g,h\in G$. 5. (5)

$e,f\in A^{\alpha}$ and $\|f\|=1$. 6. (6)

$e_{g}f_{g}=f_{g}$ for all $g\in G$.

Define

[TABLE]

Then

[TABLE]

Further define

[TABLE]

Then $\|c_{0}\|\leq n$ and $\|c\|\leq n$. Also, $c_{0},c\in(A^{\alpha})_{+}$ by (4).

We claim that

[TABLE]

and

[TABLE]

We prove (5.12). First, if $g\neq h$ then, using (2), the second part of (5.9), and (1) at the second step, and (5.7) at the last step,

[TABLE]

Therefore, using the first part of (5.9) and this estimate at the second step,

[TABLE]

This is (5.12).

Now we prove (5.13). First, by (5.11) and (5.8),

[TABLE]

Next, for $g,h,k\in G$ we have, using (2) and the second part of (5.10) at the second step,

[TABLE]

Therefore, by (5.8),

[TABLE]

Set $S=\{(g,g,g)\colon g\in G\}\subseteq G^{3}$. If $g,k\in G$ are distinct, then $\|f_{g}e_{k}\|=\|f_{g}f_{k}e_{k}\|<\varepsilon^{\prime}$ by (6) and (1). Similarly, if $h,k\in G$ are distinct, then $\|e_{k}f_{h}\|<\varepsilon^{\prime}$. In both cases, by (5.10),

[TABLE]

So, by (5.8),

[TABLE]

Meanwhile, by (6) and the first part of (5.10),

[TABLE]

Finally, using (2) and (1), and using (5.8) at the last step,

[TABLE]

Combining (5.14), (5.15), (5.16), (5.17), and (5.18), we get

[TABLE]

which is (5.13). The claim is proved.

Define $b=c\oplus s$, which is in $M_{2m}(A^{\alpha})_{+}$. From (5.12) we get

[TABLE]

Using $s\precsim_{A}d\precsim_{A}h(a^{\prime})$ and $h(a^{\prime})\perp(a^{\prime}-\delta)_{+}$, as well as Lemma 5.1(2), we have

[TABLE]

Using Lemma 5.1(5) at the first step, (5.7) at the second step, Lemma 2.6 at the third step, (5.13) and $\varepsilon^{\prime}<\frac{\rho}{3}$ (by (5.8)) at the fourth step, and (3) at the fifth step, we get

[TABLE]

The last two relations complete the proof. ∎

Lemma 5.4**.**

Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ which has the weak tracial Rokhlin property. Recalling the notation of Definition 3.8, let $a\in(K\otimes A)_{++}$ satisfy $a\sim_{A}\alpha_{g}(a)$ for all $g\in G$. Then there exists $b\in(K\otimes A^{\alpha})_{++}$ such that:

1. (1)

$\langle b\rangle_{A}=\langle a\rangle_{A}$. 2. (2)

There are $\eta_{0},\eta_{1},\ldots\in{\operatorname{W}}_{+}(A^{\alpha})$ such that $\eta_{0}\leq\eta_{1}\leq\cdots$ and $\langle b\rangle_{A^{\alpha}}=\sup_{n\in{\mathbb{Z}}_{\geq 0}}\eta_{n}$.

Proof.

By induction on $n$, we construct sequences $(\varepsilon_{n})_{n\in{\mathbb{Z}}_{\geq 0}}$ in $(0,\infty)$, $(b_{n})_{n\in{\mathbb{Z}}_{\geq 0}}$ in $(K\otimes A^{\alpha})_{+}$, and $(m(n))_{n\in{\mathbb{Z}}_{\geq 0}}$ in ${\mathbb{Z}}_{>0}$, such that $\lim_{n\to\infty}\varepsilon_{n}=0$, $b_{0}\precsim_{A}(a-\varepsilon_{0})_{+}$, and for all $n\in{\mathbb{Z}}_{\geq 0}$ we have

[TABLE]

and

[TABLE]

To begin, set $\varepsilon_{0}=1$. Given $\varepsilon_{n}$ with $n\in{\mathbb{Z}}_{\geq 0}$, apply Lemma 5.3 with $\varepsilon_{n}$ in place of $\varepsilon$, getting $\delta>0$, $m(n+1)\in{\mathbb{Z}}_{>0}$, and $b_{n+1}\in M_{m(n+1)}(A^{\alpha})_{+}$ such that

[TABLE]

Then set \varepsilon_{n+1}=\min\bigl{(}\delta,\frac{\varepsilon_{n}}{2}\bigr{)}. The induction is complete.

We now have $b_{0}\precsim_{A}b_{1}\precsim_{A}b_{2}\precsim_{A}\cdots$. Since ${\operatorname{id}}_{M_{l}}\otimes\alpha$ has the weak tracial Rokhlin property for all $l\in{\mathbb{Z}}_{>0}$ (by Corollary 4.6 of [14]), it follows from Lemma 3.7 that

[TABLE]

By Theorem 4.19 of [3], there exists $b\in(K\otimes A^{\alpha})_{+}$ such that $\langle b\rangle_{A^{\alpha}}=\sup_{n}\,\langle b_{n}\rangle_{A^{\alpha}}$. Therefore, using Theorem 1.16 of [38] at the third step,

[TABLE]

Moreover, for all $n\in{\mathbb{Z}}_{\geq 0}$, we have $(a-\varepsilon_{n})_{+}\precsim_{A}b_{n+1}\precsim_{A}a$. Since $\lim_{n\to\infty}\varepsilon_{n}=0$, it follows from Lemma 1.25(1) of [38] that $\sup_{n}\,\langle b_{n}\rangle_{A}=\langle a\rangle_{A}$. So $\langle b\rangle_{A}=\langle a\rangle_{A}$, which is Part (1) of the conclusion. Part (2) follows by taking $\eta_{n}=\langle b_{n}\rangle_{A^{\alpha}}$ for $n\in{\mathbb{Z}}_{\geq 0}$.

Finally, we prove that $b\in(K\otimes A^{\alpha})_{++}$. If not, then (see Definition 3.8) there is a projection $p\in K\otimes A^{\alpha}$ such that $\langle b\rangle_{A^{\alpha}}=\langle p\rangle_{A^{\alpha}}$. But then $\langle a\rangle_{A}=\langle p\rangle_{A}$, contradicting $a\in(K\otimes A)_{++}$. ∎

Recall the definition of ${\operatorname{Cu}}_{+}(A)$ (Definition 3.8).

Theorem 5.5**.**

Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ which has the weak tracial Rokhlin property. Then the inclusion map $\iota\colon A^{\alpha}\to A$ induces an isomorphism of ordered semigroups ${\operatorname{Cu}}_{+}(\iota)\colon{\operatorname{Cu}}_{+}(A^{\alpha})\cup\{0\}\to{\operatorname{Cu}}_{+}(A)^{\alpha}\cup\{0\}$.

By Theorem 4.1(ii) of [16], if $\alpha$ has the Rokhlin property, this holds for arbitrary $A$ and without restricting to the classes of purely positive elements.

Proof of Theorem 5.5.

It follows from Lemma 3.11(2), Lemma 3.9(2), and simplicity of $A^{\alpha}$ that the map ${\operatorname{Cu}}_{+}(\iota)\colon{\operatorname{Cu}}_{+}(A^{\alpha})\to{\operatorname{Cu}}(A)$ is injective and is an order isomorphism onto its range. It is trivial that the range is contained in ${\operatorname{Cu}}(A)^{\alpha}$, it follows from Lemma 3.9(2) that the range is contained in ${\operatorname{Cu}}_{+}(A)$, and it follows from Lemma 5.4 that the range contains ${\operatorname{Cu}}_{+}(A)^{\alpha}$. So the range is ${\operatorname{Cu}}_{+}(A)^{\alpha}$. The extension to ${\operatorname{Cu}}_{+}(\iota)\colon{\operatorname{Cu}}_{+}(A^{\alpha})\cup\{0\}\to{\operatorname{Cu}}_{+}(A)^{\alpha}\cup\{0\}$ is immediate. ∎

Corollary 5.6**.**

Let $A$ be an infinite-dimensional simple unital C*-algebra. Let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ which has the weak tracial Rokhlin property. Assume that $A^{\alpha}$ has stable rank one. Then the inclusion map $\iota\colon A^{\alpha}\to A$ induces an isomorphism of ordered semigroups ${\operatorname{W}}_{+}(\iota)\colon{\operatorname{W}}_{+}(A^{\alpha})\cup\{0\}\to{\operatorname{W}}_{+}(A)^{\alpha}\cup\{0\}$.

It is presumably true that if $A$ is an infinite-dimensional stably finite simple unital C*-algebra with stable rank one, $G$ is a finite group, and $\alpha\colon G\to{\operatorname{Aut}}(A)$ has the weak tracial Rokhlin property, then $C^{*}(G,A,\alpha)$ and $A^{\alpha}$ have stable rank one. However, this has not been proved, and a proof presumably requires methods like those in [4]. It is known that if $\alpha$ has the tracial Rokhlin property, then $C^{*}(G,A,\alpha)$ has stable rank one. This is claimed in Theorem 3.1 of [13]. We could not follow the proof there, but a proof will appear in [18]. In this case, $A^{\alpha}$ has stable rank one by Lemma 4.3(5).

We need the following fact. It is part of Theorem 5.15 of [3], except that we omit the separability hypothesis there. That hypothesis isn’t actually needed for the proof given there. (The statement in [3] omits “nondecreasing”, but, as one sees from the proof, this hypothesis is intended.)

Proposition 5.7**.**

Let $A$ be a unital C*-algebra with stable rank one. Let $(\eta_{n})_{n\in{\mathbb{Z}}_{\geq 0}}$ be a bounded nondecreasing sequence in ${\operatorname{W}}(A)$. Let $\eta=\sup_{n\in{\mathbb{Z}}_{\geq 0}}\eta_{n}$, evaluated in ${\operatorname{Cu}}(A)$. Then $\eta\in{\operatorname{W}}(A)$.

Proof.

If $A$ is separable, this is contained in Theorem 5.15 of [3]. The only use of separability in the proof of that theorem is in the use of Lemma 5.13 of [3]. One needs to know that the algebra $A_{\infty}$ in that proof has a strictly positive element. It is enough to show that $A_{\infty}$ has a countable approximate identity, which follows from the fact that, using the notation there, $A_{\infty}$ is the countable increasing union of subalgebras $A_{a_{n}}={\overline{a_{n}Aa_{n}}}$, each of which clearly has a countable approximate identity. ∎

Proof of Corollary 5.6.

Since $C^{*}(G,A,\alpha)$ is simple, Theorem 2.8 of [7] and Lemma 4.3(4) imply that $A^{\alpha}$ is stably isomorphic to $C^{*}(G,A,\alpha)$. The algebra $A^{\alpha}$ is stably finite since it has stable rank one, so $C^{*}(G,A,\alpha)$ is stably finite, and therefore its subalgebra $A$ is stably finite. It now follows from Theorem 5.5 that ${\operatorname{W}}_{+}(\iota)\colon{\operatorname{W}}_{+}(A^{\alpha})\cup\{0\}\to{\operatorname{W}}_{+}(A)\cup\{0\}$ is an order isomorphism from ${\operatorname{W}}_{+}(A^{\alpha})\cup\{0\}$ to some subsemigroup of ${\operatorname{Cu}}_{+}(A)\cup\{0\}$ which is contained in \bigl{(}{\operatorname{W}}_{+}(A)\cap{\operatorname{Cu}}_{+}(A)^{\alpha}\bigr{)}\cup\{0\}={\operatorname{W}}_{+}(A)^{\alpha}\cup\{0\}.

Now let \eta\in\bigl{(}{\operatorname{W}}_{+}(A)\cap{\operatorname{Cu}}_{+}(A)^{\alpha}\bigr{)}\cup\{0\}; we show that $\eta$ is in the range of ${\operatorname{W}}_{+}(\iota)$. This is trivial if $\eta=0$. Otherwise, choose $m\in{\mathbb{Z}}_{>0}$ and $a\in M_{m}(A)_{+}$ such that $\langle a\rangle_{A}=\eta$. Apply Lemma 5.4 to $a$, getting $b\in(K\otimes A^{\alpha})_{++}$ and a nondecreasing sequence $(\eta_{n})_{n\in{\mathbb{Z}}_{\geq 0}}$ in ${\operatorname{W}}_{+}(A^{\alpha})$ such that $\langle b\rangle_{A^{\alpha}}=\sup_{n\in{\mathbb{Z}}_{\geq 0}}\eta_{n}$. This sequence is bounded by $\langle 1_{M_{m}(A^{\alpha})}\rangle_{A^{\alpha}}$. So $\langle b\rangle_{A^{\alpha}}\in{\operatorname{W}}(A^{\alpha})$ by Proposition 5.7, and $\langle b\rangle_{A^{\alpha}}\in{\operatorname{Cu}}_{+}(A^{\alpha})$ by Lemma 3.9(4). The conclusion follows. ∎

Corollary 5.8**.**

Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ which has the weak tracial Rokhlin property. Then

[TABLE]

as ordered semigroups. If $A^{\alpha}$ has stable rank one, then

[TABLE]

as ordered semigroups.

Proof.

It suffices to prove that {\operatorname{Cu}}_{+}\bigl{(}C^{*}(G,A,\alpha)\bigr{)}\cong{\operatorname{Cu}}_{+}(A)^{\alpha} and, in the stable rank one case, that {\operatorname{W}}_{+}\bigl{(}C^{*}(G,A,\alpha)\bigr{)}\cong{\operatorname{W}}_{+}(A)^{\alpha}.

Lemma 4.3(4) and simplicity of $C^{*}(G,A,\alpha)$ (Corollary 3.3 of [14]) imply that $A^{\alpha}$ is isomorphic to a full corner of $C^{*}(G,A,\alpha)$. Since $A^{\alpha}$ and $C^{*}(G,A,\alpha)$ are both unital, it is easy to check that there is $n\in{\mathbb{Z}}_{>0}$ such that $C^{*}(G,A,\alpha)$ is isomorphic to a full corner of $M_{n}(A^{\alpha})$. Therefore $M_{\infty}(C^{*}(G,A,\alpha))\cong M_{\infty}(A^{\alpha})$. In particular, $K\otimes A^{\alpha}\cong K\otimes C^{*}(G,A,\alpha)$. Using Theorem 5.5 at the second step, we get

[TABLE]

When $A^{\alpha}$ has stable rank one, the isomorphism {\operatorname{W}}_{+}\bigl{(}C^{*}(G,A,\alpha)\bigr{)}\cong{\operatorname{W}}_{+}(A)^{\alpha} follows similarly, using Corollary 5.6 and $M_{\infty}(C^{*}(G,A,\alpha))\cong M_{\infty}(A^{\alpha})$. ∎

There is an analog of Corollary 5.6 for Rokhlin actions on unital C*-algebras, whose proof uses Theorem 4.1(ii) of [16] instead of our Theorem 5.5.

Proposition 5.9**.**

Let $A$ be a unital C*-algebra with stable rank one. Let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ which has the Rokhlin property. Then the inclusion map $\iota\colon A^{\alpha}\to A$ induces an isomorphism of ordered semigroups ${\operatorname{W}}(\iota)\colon{\operatorname{W}}(A^{\alpha})\to{\operatorname{W}}(A)^{\alpha}$.

We need Proposition 4.1(1) of [32], but without the separability hypothesis there. We give an easy proof directly from Theorem 3.2 of [32].

Proposition 5.10**.**

Let $A$ be a unital C*-algebra with stable rank one. Let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of a finite group $G$ on $A$ which has the Rokhlin property. Then $C^{*}(G,A,\alpha)$ has stable rank one.

Proof.

Let $a\in C^{*}(G,A,\alpha)$ and let $\varepsilon>0$. Use Theorem 3.2 of [32] to choose a projection $f\in A$, an integer $n\in{\mathbb{Z}}_{>0}$, a unital homomorphism $\varphi\colon M_{n}(fAf)\to C^{*}(G,A,\alpha)$, and an element $b\in M_{n}(fAf)$ such that $\|\varphi(b)-a\|<\frac{\varepsilon}{2}$. Combining Theorem 3.1.8 and Theorem 3.1.9(1) of [25], we see that $M_{n}(fAf)$ has stable rank one. Choose $c\in M_{n}(fAf)$ such that $c$ is invertible and $\|c-b\|<\frac{\varepsilon}{2}$. Then $\varphi(c)$ is an invertible element of $C^{*}(G,A,\alpha)$ such that $\|\varphi(c)-a\|<\varepsilon$. ∎

Proof of Proposition 5.9.

The algebra $C^{*}(G,A,\alpha)$ has stable rank one by Proposition 5.10. It now follows from Lemma 4.3(5) that $A^{\alpha}$ has stable rank one.

Theorem 4.1(ii) of [16] implies that ${\operatorname{W}}(\iota)\colon{\operatorname{W}}(A^{\alpha})\to{\operatorname{W}}(A)$ is an order isomorphism from ${\operatorname{W}}(A^{\alpha})$ to some subsemigroup of ${\operatorname{Cu}}(A)$, which is necessarily contained in ${\operatorname{W}}(A)\cap{\operatorname{Cu}}(A)^{\alpha}={\operatorname{W}}(A)^{\alpha}$.

Let $\eta\in{\operatorname{W}}(A)\cap{\operatorname{Cu}}(A)^{\alpha}$; we need to show that $\eta$ is in the range of ${\operatorname{W}}(\iota)$. Choose $m\in{\mathbb{Z}}_{>0}$ and $a\in M_{m}(A)_{+}$ such that $\langle a\rangle_{A}=\eta$. Since $\eta\in{\operatorname{Cu}}(A)^{\alpha}$, by Theorem 4.1(ii) of [16] there is $b\in(K\otimes A^{\alpha})_{+}$ such that $\langle b\rangle_{A}=\eta$. The case $\eta=0$ is trivial, so without loss of generality $\|b\|=1$. We now construct, by induction on $n$, sequences $(\varepsilon_{n})_{n\in{\mathbb{Z}}_{\geq 0}}$ in $(0,\infty)$, $(b_{n})_{n\in{\mathbb{Z}}_{\geq 0}}$ in $(K\otimes A^{\alpha})_{+}$, and $(m(n))_{n\in{\mathbb{Z}}_{\geq 0}}$ in ${\mathbb{Z}}_{>0}$, such that $\lim_{n\to\infty}\varepsilon_{n}=0$, $b_{0}\precsim_{A}(a-\varepsilon_{0})_{+}$, and for all $n\in{\mathbb{Z}}_{>0}$ we have

[TABLE]

To begin, set $\varepsilon_{0}=1$ and $b_{0}=0$. Given $\varepsilon_{n}$ with $n\in{\mathbb{Z}}_{\geq 0}$, set $\varepsilon_{n+1}=\frac{\varepsilon_{n}}{3}$. Choose $m(n+1)\in{\mathbb{Z}}_{>0}$, and $c_{n+1}\in M_{m(n+1)}(A^{\alpha})_{+}$ such that $\|c_{n+1}-b\|<\varepsilon_{n+1}$. Two applications of Lemma 2.4(3c) give

[TABLE]

Set $b_{n+1}=(c_{n+1}-2\varepsilon_{n+1})_{+}$. The induction is complete.

For $n\in{\mathbb{Z}}_{>0}$, set $\eta_{n}=\langle b_{n}\rangle_{A^{\alpha}}$, which is in ${\operatorname{W}}(A^{\alpha})$. Then $(\eta_{n})_{n\in{\mathbb{Z}}_{\geq 0}}$ is a nondecreasing sequence in ${\operatorname{W}}(A^{\alpha})$ and, by Lemma 1.25(1) of [38], we have $\sup_{n\in{\mathbb{Z}}_{\geq 0}}\eta_{n}=\langle b\rangle_{A^{\alpha}}=\eta$. This sequence is bounded by $\langle 1_{M_{m}(A^{\alpha})}\rangle_{A^{\alpha}}$, so Proposition 5.7 now implies $\langle b\rangle_{A^{\alpha}}\in{\operatorname{W}}(A^{\alpha})$. ∎

6. An example

We give an example of a simple AH algebra $A$ with ${\operatorname{rc}}(A)>0$ and an action $\alpha\colon{\mathbb{Z}}/2{\mathbb{Z}}\to{\operatorname{Aut}}(A)$ which has the Rokhlin property. As discussed in the introduction, it is not a priori obvious that such examples should exist, even with the weak tracial Rokhlin property in place of the Rokhlin property. In our example, we get equality in Theorem 4.1 and Theorem 4.5. See Theorem 6.15 and Theorem 6.21. The algebras $A$ and $A^{\alpha}$ have stable rank one (Lemma 6.5 and Corollary 6.7), and the maps ${\operatorname{W}}(A^{\alpha})\to{\operatorname{W}}(A)^{\alpha}$ and ${\operatorname{Cu}}(A^{\alpha})\to{\operatorname{Cu}}(A)^{\alpha}$ are isomorphisms (Corollary 6.6).

The construction is motivated by [22], in which two AH direct systems with simple direct limits are “merged” into a single larger system whose direct limit is still simple but which is “not very far” from the direct sum of the two original direct limits. The “merger” is accomplished by writing the systems side by side, and inserting a very small number of point evaluation maps which go from one of the original systems to the other. In [22], the essential point was that the two systems were very different but that the base spaces were all contractible. Here, we use two copies of the same system. Writing the direct system sideways, our combined system looks like the following diagram, in which the solid arrows represent many partial maps and the dotted arrows represent a small number of point evaluations:

[TABLE]

The order two automorphism exchanges the two rows.

Since we don’t care about contractibility, we can use products of copies of $S^{2}$ instead of cones over such spaces as in [22]. We compute the radius of comparison exactly, instead of just giving bounds as is done in [22].

To keep the notation simple, we carry out only the case of ${\mathbb{Z}}/2{\mathbb{Z}}$ and radius of comparison less than $1$. Modifications of the construction will presumably work for any finite group and give any value of the radius of comparison in $[0,\infty]$.

Construction 6.1**.**

We define the following objects:

1. (1)

For $n\in{\mathbb{Z}}_{\geq 0}$, define

• •

$d(n)=2^{n+1}-1$.

• •

$l(n)=2^{n+1}$.

• •

$r(0)=1$ and $r(n)=\prod_{k=1}^{n}2^{k+1}$.

• •

$s(0)=1$ and $s(n)=\prod_{k=1}^{n}(2^{k+1}-1)$.

• •

u(n)=\frac{s(n)}{r(n)}=\prod_{k=1}^{n}\big{(}1-\frac{1}{2^{k+1}}\big{)}.

• •

$t(0)=0$ and $t(n+1)=d(n+1)t(n)+[r(n)-t(n)]$. 2. (2)

Define $\kappa=\lim_{n\to\infty}u(n)$. 3. (3)

For $n\in{\mathbb{Z}}_{\geq 0}$, define a compact space by $X_{n}=(S^{2})^{s(n)}$. Then the covering dimension of $X_{n}$ is $\dim(X_{n})=2s(n)$. 4. (4)

For $n\in{\mathbb{Z}}_{\geq 0}$ and $\nu=1,2,\ldots,d(n+1)$, let $P^{(n)}_{\nu}\colon X_{n+1}\to X_{n}$ be the $\nu$ coordinate projection. 5. (5)

Choose points $x_{m}\in X_{m}$ for $m\in{\mathbb{Z}}_{\geq 0}$ such that for all $n\in{\mathbb{Z}}_{\geq 0}$, the set

[TABLE]

is dense in $X_{n}$. (The contribution to this set when $m=n$ is $x_{n}$.) 6. (6)

For $n\in{\mathbb{Z}}_{\geq 0}$, define

[TABLE]

When convenient, we identify $A_{n}$ in the obvious ways with

[TABLE] 7. (7)

For $n\in{\mathbb{Z}}_{\geq 0}$, define a unital homomorphism

[TABLE]

by

[TABLE] 8. (8)

For $n\in{\mathbb{Z}}_{\geq 0}$, define $\Lambda_{n+1,\,n}\colon A_{n}\to A_{n+1}$ by $\Lambda_{n+1,\,n}=\lambda_{n}\otimes{\operatorname{id}}_{M_{r(n)}}$. Thus,

[TABLE]

is given by

[TABLE]

for $f,g\in C(X_{n})$ and $c\in M_{r(n)}$. Using standard matrix unit notation, we can also write this definition as

[TABLE]

For $m,n\in{\mathbb{Z}}_{\geq 0}$ with $m\leq n$, now define

[TABLE] 9. (9)

Define

[TABLE]

For $n\in{\mathbb{Z}}_{\geq 0}$, it is clear that $\Lambda_{n+1,\,n}$ is an injective unital homomorphism. Let $\Lambda_{\infty,n}\colon A_{n}\to A$ be the standard map associated with the direct limit. 10. (10)

Write ${\mathbb{Z}}/2{\mathbb{Z}}=\{0,1\}$. For $n\in{\mathbb{Z}}_{\geq 0}$, define $\alpha^{(n)}\colon{\mathbb{Z}}/2{\mathbb{Z}}\to{\operatorname{Aut}}(A_{n})$ by

[TABLE]

for $f,g\in C(X_{n})$ and $c\in M_{r(n)}$. We also write $\alpha^{(n)}$ for the generating automorphism $\alpha^{(n)}_{1}$. We then have the following diagram:

[TABLE]

Lemma 6.2**.**

Assume the notation and choices in Construction 6.1. Then $0\leq t(n)<r(n)$ for all $n\in{\mathbb{Z}}_{\geq 0}$.

Proof.

The statement is true for $n=0$ by definition. Let $n\in{\mathbb{Z}}_{\geq 0}$ and assume $0\leq t(n)<r(n)$. Then

[TABLE]

which implies (using $d(n+1)-1=2^{n+2}-2\geq 0$)

[TABLE]

So $0\leq t(n)<r(n)$ for all $n\in{\mathbb{Z}}_{\geq 0}$ by induction. ∎

Lemma 6.3**.**

Assume the notation and choices in Construction 6.1. Then $(u(n))_{n\in{\mathbb{Z}}_{\geq 0}}$ is strictly decreasing and $0<\kappa<1$.

Proof.

The first statement is clear, as is $\kappa<1$.

To prove that $\kappa>0$, we first observe that if $\beta_{1},\beta_{2}\in[0,1]$ then $(1-\beta_{1})(1-\beta_{2})\geq 1-\beta_{1}-\beta_{2}$. Induction gives an analogous statement for $n$ factors, so that, in particular, $u(n)\geq 1-\sum_{k=1}^{n}\frac{1}{2^{k+1}}$. Letting $n\to\infty$, we get $\kappa\geq\frac{1}{2}$. ∎

Lemma 6.4**.**

In Construction 6.1(10), the diagram (6.3) commutes. Moreover, there is a unique action $\alpha\colon{\mathbb{Z}}/2{\mathbb{Z}}\to{\operatorname{Aut}}(A)$ such that $\alpha=\varinjlim\alpha^{(n)}$, and this action has the Rokhlin property..

Proof.

For the first statement, let $n\in{\mathbb{Z}}_{\geq 0}$. Using 6.1(7) in the second step and 6.1(10) in the third step, for all $f,g\in C(X_{n})$ and for all $c\in M_{r(n)}$ we have

[TABLE]

Existence of $\alpha$ follows immediately. It is immediate that $\alpha^{(n)}$ has the Rokhlin property for all $n\in{\mathbb{Z}}_{\geq 0}$, and it follows easily that $\alpha$ has the Rokhlin property. ∎

Lemma 6.5**.**

Assume the notation and choices in Construction 6.1. Then the C*-algebra $A$ is stably finite and simple, and has stable rank one.

Proof.

Stable finiteness is immediate. For simplicity, it is easy to check that the hypotheses of Proposition 2.1(ii) of [9] hold. For stable rank one, we observe that the direct system in Construction 6.1(9) has diagonal maps in the sense of Definition 2.1 of [10]. Therefore $A$ has stable rank one by Theorem 4.1 of [10]. ∎

Corollary 6.6**.**

Assume the notation and choices in Construction 6.1. Then the maps ${\operatorname{Cu}}(A^{\alpha})\to{\operatorname{Cu}}(A)^{\alpha}$ and ${\operatorname{W}}(A^{\alpha})\to{\operatorname{W}}(A)^{\alpha}$ are isomorphisms.

Proof.

This follows from Theorem 4.1(ii) of [16] and Proposition 5.9, by Lemma 6.5 and Lemma 6.4. ∎

Corollary 6.7**.**

Assume the notation and choices in Construction 6.1. Then $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},A,\alpha)$ and $A^{\alpha}$ have stable rank one.

Proof.

The result for $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},A,\alpha)$ follows from Lemma 6.5, Lemma 6.4, and Proposition 4.1(1) of [32]. The result for $A^{\alpha}$ now follows from Lemma 4.3(5). ∎

Notation 6.8**.**

Let $p\in C(S^{2},M_{2})$ denote the Bott projection, and let $L$ be the tautological line bundle over $S^{2}\cong\mathbb{C}\mathbb{P}^{1}$. (Thus, the range of $p$ is the section space of $L$.) Recall that $X_{0}=S^{2}$. Assuming the notation and choices in Construction 6.1, for $n\in{\mathbb{Z}}_{\geq 0}$ set

[TABLE]

In particular, $p_{0}=(p,0)$ and $p_{0}^{\prime}=(p,p)$.

Lemma 6.9** ([22]).**

The Cartesian product $L^{\times k}$ does not embed in a trivial bundle over $(S^{2})^{k}$ of rank less than $2k$.

Proof.

This is Lemma 1.9 of [22]. ∎

Lemma 6.10**.**

Assume the notation and choices in Construction 6.1, and adopt Notation 6.8. Let $n\in{\mathbb{Z}}_{\geq 0}$. For $j=1,2,\ldots,s(n)$ let $R_{j}^{(n)}\colon(S^{2})^{s(n)}\to S^{2}$ be the $j$ coordinate projection. Then:

1. (1)

There are orthogonal projections $c^{(0)}_{n}$, c^{(1)}_{n},g_{n}\in M_{2r(n)}\big{(}C(X_{n})\big{)} such that

[TABLE]

$c^{(0)}_{n}$ is the direct sum of the projections $p\circ R^{(n)}_{j}$ for $j=1,2,\ldots,s(n)$, $c^{(1)}_{n}$ is a constant projection of rank $r(n)-s(n)-t(n)$, and $g_{n}$ is a constant projection of rank $t(n)$. 2. (2)

For every $n\in{\mathbb{Z}}_{\geq 0}$ and $\tau\in{\operatorname{T}}(A_{n})$ we have $d_{\tau}(p_{n})\leq 1$.

Proof.

We prove the formula in (1) for $p_{n}$ by induction on $n$. The formula for \big{(}{\operatorname{id}}_{M_{2}}\otimes\alpha^{(n)}\big{)}(p_{n}) then follows from the definition of $\alpha^{(n)}$.

The formula holds for $n=0$, since $r(0)=s(0)=1$, $t(0)=0$, and $r(0)-s(0)-t(0)=0$.

Now assume that it is known for $n$. Recall that $\Lambda_{n+1,\,n}=\lambda_{n}\otimes{\operatorname{id}}_{M_{r(n)}}$. (See Construction 6.1(8).) We suppress ${\operatorname{id}}_{M_{2}}$ in the notation. With this convention, first take $(f,g)$ in (6.1) to be \big{(}c_{n}^{(0)},0\big{)}. The first coordinate \Lambda_{n+1,n}\big{(}c_{n}^{(0)},0\big{)}_{1} is of the form required for $c_{n+1}^{(0)}$, while \Lambda_{n+1,n}\big{(}c_{n}^{(0)},0\big{)}_{2} is a constant function of rank $s(n)$. In the same manner, we see that:

• •

\Lambda_{n+1,n}\big{(}c_{n}^{(1)},0\big{)}_{1} is a constant projection of rank $d(n+1)[r(n)-s(n)-t(n)]$.

• •

\Lambda_{n+1,n}\big{(}c_{n}^{(1)},0\big{)}_{2} is a constant projection of rank $r(n)-s(n)-t(n)$.

• •

$\Lambda_{n+1,n}(0,g_{n})_{1}$ is a constant projection of rank $t(n)$.

• •

$\Lambda_{n+1,n}(0,g_{n})_{2}$ is a constant projection of rank $d(n+1)t(n)$.

Putting these together, we get in the first coordinate of $\Lambda_{n+1,n}(p_{n})$ the direct sum of $c_{n+1}^{(0)}$ as described and a constant function of rank

[TABLE]

A computation shows that this expression is equal to $r(n+1)-s(n+1)-t(n+1)$. In the second coordinate we get a constant projection of rank

[TABLE]

This completes the induction.

For Part (2), we may assume that $\tau$ is extreme in ${\operatorname{T}}(A_{n})$. Then there is $x\in X_{n}\amalg X_{n}$ such that $\tau={\operatorname{tr}}_{r(n)}\otimes{\operatorname{ev}}_{x}$. Therefore

[TABLE]

In each case, Lemma 6.2 implies $d_{\tau}(p_{n})\leq 1$. This completes the proof. ∎

Lemma 6.11**.**

Assume the notation and choices in Construction 6.1, and adopt the notation of Notation 6.8. Let $n\in{\mathbb{Z}}_{\geq 0}$. For $j=1,2,\ldots,s(n)$ let $R_{j}^{(n)}\colon(S^{2})^{s(n)}\to S^{2}$ be the $j$ coordinate projection. Then:

1. (1)

There are orthogonal projections $f_{n},h_{n}$ in M_{2r(n)}\big{(}C(X_{n})\big{)} such that $p^{\prime}_{n}=(f_{n}+h_{n},\,f_{n}+h_{n})$, $f_{n}$ is the direct sum of the projections $p\circ R^{(n)}_{j}$ for $j=1,2,\ldots,s(n)$, and $h_{n}$ is a constant projection of rank $r(n)-s(n)$. 2. (2)

For every $n\in{\mathbb{Z}}_{\geq 0}$ and $\tau\in{\operatorname{T}}(A_{n})$, we have $d_{\tau}(p^{\prime}_{n})=1$.

Proof.

We prove Part (1). Using Lemma 6.10(1), Lemma 6.4, and the definition of $\alpha^{(n)}$ in the third step, we get

[TABLE]

Now it is enough to set $f_{n}=c^{(0)}_{n}$ and $h_{n}=c^{(1)}_{n}+g_{n}$.

For Part (2), we may assume that $\tau$ is extreme in ${\operatorname{T}}(A_{n})$. Then there is $x\in X_{n}\amalg X_{n}$ such that $\tau={\operatorname{tr}}_{r(n)}\otimes{\operatorname{ev}}_{x}$. Adding up the ranks given in Part (1), we see that ${\operatorname{rank}}(p^{\prime}_{n}(x))=r(n)$ for all $x\in X_{n}\amalg X_{n}$. The conclusion follows. ∎

Definition 6.12**.**

Let $A$ be a unital C*-algebra and let $p$ be a projection in $M_{\infty}(A)$. We call $p$ trivial if there is $n\in{\mathbb{Z}}_{\geq 0}$ such that $p$ is Murray-von Neumann equivalent to $1_{M_{n}(A)}$. When $n=0$, this means $p=0$.

Corollary 6.13**.**

Adopt the assumptions and notation of Notation 6.8. Let $n\in{\mathbb{Z}}_{\geq 0}$ and let $e=(e_{1},e_{2})$ be a projection in $M_{\infty}(A_{n})\cong M_{\infty}(C(X_{n})\oplus C(X_{n}))$ such that $e_{1}$ is trivial. If there exists $x\in M_{\infty}(A_{n})$ such that $\|xex^{*}-p^{\prime}_{n}\|<\frac{1}{2}$, then ${\operatorname{rank}}(e_{1})\geq r(n)+s(n)$.

Proof.

Recall the line bundle $L$ and the projection $p$ from Notation 6.8. Also recall from Definition 2.13 that we use $\approx$ for Murray-von Neumann equivalence and $\lessapprox$ for Murray-von Neumann subequivalence.

Let $f_{n},h_{n}\in M_{2r(n)}(C(X_{n}))$ be as in Lemma 6.11, and define $q=f_{n}+h_{n}$. The range of $f_{n}$ is isomorphic to the section space of the $s(n)$-dimensional vector bundle $L^{\times s(n)}$ and $q(p^{\prime}_{n}|_{X_{n}\amalg\varnothing})q=q$. Now $\|xex^{*}-p^{\prime}_{n}\|<\frac{1}{2}$ implies

[TABLE]

Since $e$ and $q$ are projections, $q\lessapprox e|_{X_{n}\amalg\varnothing}=e_{1}$. So there is projection $w\in M_{\infty}(C(X_{n}))$ such that $q+w\approx e_{1}$. Also, $\|xex^{*}-p^{\prime}_{n}\|<\frac{1}{2}$ implies that $p^{\prime}_{n}$ is Murray-von Neumann equivalent to a subprojection of $e$. Therefore $h_{n}\lessapprox e_{1}$, so ${\operatorname{rank}}(h_{n})\leq{\operatorname{rank}}(e_{1})$. Take $e_{0}\in M_{\infty}(C(X_{n}))$ to be a trivial projection of rank ${\operatorname{rank}}(e_{1})-{\operatorname{rank}}(h_{n})$ such that $e_{0}\perp h_{n}$. Since $h_{n}$ and $e_{0}$ are trivial, $e_{0}+h_{n}\approx e_{1}$. So

[TABLE]

Define $k={\operatorname{rank}}(f_{n}+w)$. Then $k\geq s(n)$. Now:

• •

Let $E_{1}$ be a vector bundle whose section space is isomorphic to the range of $f_{n}+w$.

• •

Let $E_{2}$ be a trivial vector bundle whose section space is isomorphic to the range of $e_{0}$.

• •

Set $l={\operatorname{rank}}(h_{n})$.

• •

Let $H^{l}$ be a trivial vector bundle whose section space is isomorphic to the range of $h_{n}$.

Putting these together and using Theorem 9.1.5 of [23], we get $f_{n}+w\approx e_{0}$. Therefore $f_{n}\lessapprox e_{0}$. So ${\operatorname{rank}}(e_{0})\geq 2s(n)$ by Lemma 6.9. Since $e_{0}+h_{n}\approx e_{1}$, we have ${\operatorname{rank}}(e_{1})\geq r(n)+s(n)$. ∎

Remark 6.14**.**

We will use results of Niu from [26] to obtain an upper bound on the radius of comparison of our algebra. Niu introduced a notion of mean dimension for a diagonal AH-system, [26, Definition 3.6]. Suppose we are given a direct system of homogeneous algebras of the form

[TABLE]

in which each of the spaces involved is a connected finite CW complex, and the connecting maps are unital diagonal maps. Let $\gamma$ denote the mean dimension of this system, in the sense of Niu. It follows trivially from [26, Definition 3.6] that

[TABLE]

Theorem 6.2 of [26] then states that if $A$ is the direct limit of a system as above, then ${\operatorname{rc}}(A)\leq\frac{\gamma}{2}$. Since the system we are considering here is of this type, Niu’s theorem applies.

Theorem 6.15**.**

Assume the notation and choices in Construction 6.1 and Notation 6.8. Then ${\operatorname{rc}}(A)=\kappa$.

Proof.

Since $\lim_{{n}\to\infty}\frac{\dim(X_{n})}{r(n)}=2\kappa$ and the C*-algebra $A$ was constructed with diagonal maps, we deduce from Remark 6.14 that ${\operatorname{rc}}(A)\leq\kappa$. Now it suffices to prove that ${\operatorname{rc}}(A)\geq\kappa$. Suppose $\rho<\kappa$. We show that $A$ does not have $\rho$-comparison. Choose $n\in{\mathbb{Z}}_{>0}$ such that $1/{r(n)}<\kappa-\rho$. Choose $M\in{\mathbb{Z}}_{\geq 0}$ such that $\rho+1<\frac{M}{r(n)}<\kappa+1$. Let $e\in M_{\infty}(A_{n})$ be a trivial projection of rank $M$. By slight abuse of notation, we use $\Lambda_{m,n}$ to denote the amplified map from $M_{\infty}(A_{n})$ to $M_{\infty}(A_{m})$ as well. For $m>n$, the rank of $\Lambda_{m,n}(e)$ is $M\cdot\frac{r(m)}{r(n)}$.

We claim that the rank of $\Lambda_{m,n}(e)$ is strictly less than $r(m)+s(m)$ for $m>n$. Suppose {\operatorname{rank}}\big{(}\Lambda_{m,n}(e)\big{)}\geq r(m)+s(m). Then, by the choice of $M$,

[TABLE]

Thus $\frac{s(m)}{r(m)}<\kappa$. This contradicts Lemma 6.3 and Construction 6.1(2). So the claim follows.

Now, for any tracial state $\tau$ on $A_{m}$ (and thus for any tracial state on $A$), we have, using Lemma 6.11(2) in the last step,

[TABLE]

On the other hand, if \Lambda_{\infty,0}\big{(}(p,\,p)\big{)}\lessapprox\Lambda_{\infty,n}(e) then, in particular, there exists some $m>n$ and $x\in M_{\infty}(A_{m})$ such that $\|x\Lambda_{m,n}(e)x^{*}-p^{\prime}_{m}\|<\frac{1}{2}$. Using Corollary 6.13, we have

[TABLE]

This is a contradiction, and we have proved that $A$ does not have $\rho$-comparison. ∎

We now determine the radius of comparison of the crossed product in our example. The methods are very similar.

Construction 6.16**.**

Assume the notation and choices in Parts (1), (3), (4), and (5) in Construction 6.1.

1. (1)

For $n\in{\mathbb{Z}}_{\geq 0}$, we define $B_{n}=C(X_{n})\otimes M_{2r(n)}$, identified with $C(X_{n},M_{2})\otimes M_{r(n)}$. 2. (2)

Let $s\in M_{2}$ be the unitary matrix $s=\left(\begin{smallmatrix}0&1\\ 1&0\end{smallmatrix}\right)$. Define $\Lambda^{\prime}_{n+1,n}\colon B_{n}\to B_{n+1}$ by

[TABLE]

for $f\in C(X_{n},M_{2})$ and $c\in M_{r(n)}$. With abuse of notation (the expression $sf(x_{n})s^{*}\cdot 1_{C(X_{n+1})}$ is the constant function $X_{n+1}\to M_{2}$ with value $sf(x_{n})s^{*}$), the analog of (6.2) is

[TABLE]

It is clear that $\Lambda^{\prime}_{n+1,\,n}$ is injective for all $n\in{\mathbb{Z}}_{\geq 0}$. 3. (3)

Define $B=\varinjlim(B_{n},\,\Lambda^{\prime}_{n+1,\,n})$.

Lemma 6.17**.**

Assume the notation and choices in Construction 6.1 and Construction 6.16. Then $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},A,\alpha)\cong B$.

Proof.

For $t\in{\mathbb{Z}}/2{\mathbb{Z}}$, as in Notation 3.1 let $u_{t}$ be the standard unitary in a crossed product by ${\mathbb{Z}}/2{\mathbb{Z}}$. (In this proof, no confusion will be caused by using the same letter in all crossed products.) For $n\in{\mathbb{Z}}_{\geq 0}$, there is a homomorphism

[TABLE]

such that

[TABLE]

for $f,g\in C(X_{n})$, $t\in{\mathbb{Z}}/2{\mathbb{Z}}$, and $c\in M_{r(n)}$. In view of Lemma 6.4, we can apply Theorem 9.4.34 of [17] to get an isomorphism

[TABLE]

On the other hand, we have an isomorphism $\varphi_{n}\colon C^{*}({\mathbb{Z}}/2{\mathbb{Z}},\,A_{n},\,\alpha^{(n)})\to B_{n}$ which is defined for $f_{m},g_{m}\in C(X_{n})$ and $c_{m}\in M_{r(n)}$ for $m=0,1$ by

[TABLE]

Using matrix unit notation, the right hand side is

[TABLE]

Using (6.2) and (6.4), one checks that the diagram

[TABLE]

commutes for every $n\in{\mathbb{Z}}_{\geq 0}$. The result follows. ∎

Notation 6.18**.**

Let $p\in C(X_{0},M_{2})$ be the Bott projection, as in Notation 6.8. Assuming the notation and choices in 6.16, for $n\in{\mathbb{Z}}_{\geq 0}$ set $q_{n}=\Lambda^{\prime}_{n,0}(p)\in B_{n}$. In particular, $q_{0}=p$.

Lemma 6.19**.**

Adopt the assumptions and notation of Notation 6.18. Let $n\in{\mathbb{Z}}_{\geq 0}$ and for $j=1,2,\ldots,s(n)$ let $R_{j}^{(n)}\colon(S^{2})^{s(n)}\to S^{2}$ be the $j$ coordinate projection. Then:

1. (1)

There are orthogonal projections $y_{n},z_{n}$ in M_{2r(n)}\big{(}C(X_{n})\big{)} such that $q_{n}=y_{n}+z_{n}$, $y_{n}$ is the direct sum of the projections $p\circ R^{(n)}_{j}$ for $j=1,2,\ldots,s(n)$, and $z_{n}$ is a constant projection of rank $r(n)-s(n)$. 2. (2)

For every $n\in{\mathbb{Z}}_{\geq 0}$ and $\tau\in{\operatorname{T}}(B_{n})$, we have $d_{\tau}(q_{n})=\frac{1}{2}$.

Proof.

The proof of (1) is very similar to that of Lemma 6.11(1), but simpler because there is only one summand. The basic facts for the induction step are that $\Lambda^{\prime}_{n+1,n}(y_{n})$ is the direct sum of the projections $p\circ R^{(n)}_{j}\circ P^{(n)}_{k}$ for $j=1,2,\ldots,s(n)$ and $k=1,2,\ldots,d(n+1)$, and a constant projection of rank $s(n)$, and that $\Lambda^{\prime}_{n+1,n}(z_{n})$ is a constant projection of rank $l(n+1)[r(n)-s(n)]$. We omit the details.

The proof of (2) is essentially the same as that of Lemma 6.11(2), and is omitted. ∎

Corollary 6.20**.**

Adopt the assumptions and notation of Notation 6.18. Let $n\in{\mathbb{Z}}_{\geq 0}$ and let $e$ be a trivial projection in $M_{\infty}(B_{n})\cong M_{\infty}(C(X_{n}))$. If there exists $x\in M_{\infty}(B_{n})$ such that $\|xex^{*}-q_{n}\|<\frac{1}{2}$ then ${\operatorname{rank}}(e)\geq r(n)+s(n)$.

Proof.

The proof is essentially the same as that of Corollary 6.13. We use Lemma 6.19 and the projections $y_{n}$ and $z_{n}$ instead of Lemma 6.11 and the projections $f_{n}$ and $h_{n}$. ∎

The next result is the analog of Theorem 6.15. It shows that in our example, we get equality in Theorem 4.1 and Theorem 4.5.

Theorem 6.21**.**

Assume the notation and choices in Construction 6.1 and Notation 6.8. Then

[TABLE]

The proof is similar to that of Theorem 6.15. We give details to show where the factor $\frac{1}{2}$ comes from, and for convenient reference in a paper in preparation.

Proof of Theorem 6.21.

We prove the first part of the conclusion. The second part then follows from Theorem 4.5.

Because $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},\,A,\,\alpha)$ is isomorphic to the C*-algebra $B$ by Lemma 6.17, it suffices to show that ${\operatorname{rc}}(B)=\frac{\kappa}{2}$. Since $\lim_{{n}\to\infty}\frac{\dim(X_{n})}{2r(n)}=\kappa$ and the C*-algebra $B$ was constructed with diagonal maps, we deduce from Remark 6.14 that ${\operatorname{rc}}(B)\leq\frac{\kappa}{2}$. Now it suffices to prove that ${\operatorname{rc}}(B)\geq\frac{\kappa}{2}$. Suppose $\rho<\frac{\kappa}{2}$. We show that $B$ does not have $\rho$-comparison. Choose $n\in{\mathbb{Z}}_{>0}$ such that $1/{r(n)}<\frac{\kappa}{2}-\rho$. Choose $M\in{\mathbb{Z}}_{\geq 0}$ such that $\rho+\frac{1}{2}<\frac{M}{2r(n)}<\frac{\kappa}{2}+\frac{1}{2}$. Let $e\in M_{\infty}(B_{n})$ be a trivial projection of rank $M$. By slight abuse of notation, we use $\Lambda^{\prime}_{m,n}$ to denote the amplified map from $M_{\infty}(B_{n})$ to $M_{\infty}(B_{m})$ as well. For $m>n$, the rank of $\Lambda^{\prime}_{m,n}(e)$ is $M\cdot\frac{r(m)}{r(n)}$. We claim that the rank of $\Lambda^{\prime}_{m,n}(e)$ is strictly less than $r(m)+s(m)$ for $m>n$. Suppose {\operatorname{rank}}\big{(}\Lambda^{\prime}_{m,n}(e)\big{)}\geq r(m)+s(m). Then, considering the choice of $M$,

[TABLE]

Thus $\frac{s(m)}{r(m)}<\kappa$. This contradicts Construction 6.1(2). So the claim follows.

Now, for any extreme tracial state $\tau$ on $B_{m}$ (and thus for any trace on $B$), we have, using Lemma 6.19(2) in the last step,

[TABLE]

On the other hand, if $\Lambda^{\prime}_{\infty,0}(p)\lessapprox\Lambda^{\prime}_{\infty,n}(e)$ then, in particular, there exists some $m>n$ and $x\in M_{\infty}(B_{m})$ such that $\|x\Lambda^{\prime}_{m,n}(e)x^{*}-q_{m}\|<\frac{1}{2}$. Using Corollary 6.20, we get

[TABLE]

This is a contradiction, and we have proved that $B$ does not have $\rho$-comparison. ∎

Example 6.22**.**

We show that, in Theorem 4.1 and Theorem 4.5, the weak tracial Rokhlin property can’t be replaced by pointwise outerness.

Let $A$ and $\alpha\colon{\mathbb{Z}}/2{\mathbb{Z}}\to{\operatorname{Aut}}(A)$ be as in Lemma 6.4, set $B=C^{*}({\mathbb{Z}}/2{\mathbb{Z}},\,A,\,\alpha)$, and let $\beta={\widehat{\alpha}}\colon{\mathbb{Z}}/2{\mathbb{Z}}\to{\operatorname{Aut}}(B)$ be the dual action. It follows from Theorem 6.15, Theorem 6.21, and Lemma 6.3 that the inequalities in Theorem 4.1 and Theorem 4.5 fail for the action $\beta$.

We already know that $B$ is simple, and $B$ is stably finite because it is an AH algebra. It remains to show that $\beta$ is pointwise outer. Suppose not. Then in fact $\beta$ is an inner action, that is, given by conjugation by a unitary of order $2$. (See Exercise 8.2.7 of [17].) So $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},\,B,\,\beta)\cong B\oplus B$. But by Takai duality $C^{*}({\mathbb{Z}}/2{\mathbb{Z}},\,B,\,\beta)\cong M_{2}(A)$, which is simple. Pointwise outerness is proved.

7. Open problems

The most obvious problem is whether equality always holds in Theorem 4.1 and Theorem 4.5.

Question 7.1**.**

Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. Does it follow that

[TABLE]

One might even hope that the reverse inequalities

[TABLE]

hold without restrictions on the action. Quite different methods seem to be needed for this question. Suppose, for example, that we were able to prove (7.1) for pointwise outer actions. Suppose $G$ is finite abelian, $\alpha\colon G\to{\operatorname{Aut}}(A)$ is pointwise outer, and, with $B=C^{*}(G,A,\alpha)$, the dual action $\beta={\widehat{\alpha}}\colon{\widehat{G}}\to{\operatorname{Aut}}(B)$ is pointwise outer and $B$ has strict comparison. We would be able to deduce that C^{*}\bigl{(}{\widehat{G}},B,\beta\bigr{)} has strict comparison. This outcome is at least heuristically related to the long standing open question of whether the crossed product of a simple C*-algebra with stable rank one by a finite group again has stable rank one. Indeed, if $B$ is classifiable in the sense of the Elliott program, and the tracial state space has compact finite-dimensional extreme boundary, it would follow (see Corollary 7.9 of [29], Corollary 1.2 of [43], or Corollary 4.7 of [45]) that C^{*}\bigl{(}{\widehat{G}},B,\beta\bigr{)} is ${\mathcal{Z}}$-stable, and therefore from Theorem 6.7 of [41] that C^{*}\bigl{(}{\widehat{G}},B,\beta\bigr{)} has stable rank one. This case of the problem has been solved [31], but the proof depends on major results in the classification program.

In the example in Section 6, the group action on ${\operatorname{T}}(A)$ is highly nontrivial.

Question 7.2**.**

Does there exist an action of a nontrivial finite group with the weak tracial Rokhlin property on a simple separable unital C*-algebra $A$ with ${\operatorname{rc}}(A)>0$ and such that every tracial state is invariant?

One can ask for even more.

Question 7.3**.**

Does there exist an action of a nontrivial finite group with the Rokhlin property on a simple separable unital C*-algebra $A$ with ${\operatorname{rc}}(A)>0$ and unique tracial state?

By combining methods of Villadsen [46] with those of Section 6, one should be able to at least produce an example of a simple separable unital nuclear C*-algebra $A$ and an action $\alpha\colon{\mathbb{Z}}/2{\mathbb{Z}}\to{\operatorname{Aut}}(A)$ such that $A$ does not have stable rank one, $\alpha$ has the Rokhlin property, and $A$ has exactly two extreme tracial states, which are interchanged by the action $\alpha$.

Question 7.4**.**

Let $A$ be an infinite-dimensional simple unital C*-algebra with stable rank one, let $G$ be a finite group, and let $\alpha\colon G\to{\operatorname{Aut}}(A)$ be an action with the weak tracial Rokhlin property. Does it follow that $C^{*}(G,A,\alpha)$ and $A^{\alpha}$ have stable rank one?

This is wanted for improvement of Corollary 5.6.

Question 7.5**.**

Are the stable rank one hypotheses in Proposition 5.9 and Corollary 5.6 really necessary?

That is, assuming the action has the Rokhlin property or weak tracial Rokhlin property as appropriate, does one get isomorphisms

[TABLE]

rather than just

[TABLE]

One possible generalization of the results of this paper is to the nonunital case. This will be treated in [5] (by a different set of authors). Complications include the additional complexity of the definition of the weak tracial Rokhlin property (see Definition 3.1 of [14]), and what to substitute for the conventional definition of the radius of comparison.

8. Acknowledgments

This research was done while the first author was a visiting scholar at the University of Oregon during the period March 2018 to September 2019. He is thankful to that institution for its hospitality. He was partially supported by the University of Tehran. This paper will be part of first author’s PhD.  dissertation.

The research of the third author was partially supported by the Simons Foundation Collaboration Grant for Mathematicians #587103.

The first author thanks Q. Wang for pointing out Corollary II.4.3 in [6]. All three authors would like to thank M. Amini, I. Hirshberg, and S. Jamali for sharing some of their unpublished work with us.

The first author is grateful to M. B.  Asadi for motivating him to study operator algebras in the first year of his Ph.D.  program.