The Tracial Class Property for Crossed Products by Finite Group Actions

Abstract

We define the concept of tracial 𝒞-algebra of 𝐶^∗-algebras, which generalize the concept of local 𝒞-algebra of 𝐶^∗-algebras given by H. Osaka and N. C. Phillips. Let 𝒞 be any class of separable unital 𝐶^∗-algebras. Let 𝐴 be an infinite dimensional simple unital tracial 𝒞-algebra with the (SP)-property, and let 𝛼∶𝐺→Aut(𝐴) be an action of a finite group 𝐺 on 𝐴 which has the tracial Rokhlin property. Then 𝐴×_𝛼𝐺 is a simple unital tracial 𝒞-algebra.

1. Introduction

In this paper, our purpose is to prove that certain classes of separable unital 𝐶^∗-algebras are closed under crossed products by finite group actions with the tracial Rokhlin property.

The term “tracial” has been widely used to describe the properties of 𝐶^∗-algebras since Lin introduced the concept of tracial rank of 𝐶^∗-algebras in [1]. The notion of tracial rank was motivated by the Elliott program of classification of nuclear 𝐶^∗-algebras. 𝐶^∗-algebras with tracial rank no more than 𝑘 for some 𝑘∈ℕ are 𝐶^∗-algebras that can be locally approximated by 𝐶^∗-subalgebras in ℐ^(𝑘) after cutting out a “small” approximately central projection 𝑝. The term “tracial” come from the fact that, in good cases, the projection 𝑝 is “small” if 𝜏(𝑝)<𝜀 for every tracial state 𝜏 on 𝐴. The 𝐶^∗-algebras of tracial rank zero can be determined by 𝐾-theory and hence can be classified. For example, Lin proved that if a simple separable amenable unital 𝐶^∗-algebra 𝐴 has tracial rank zero and satisfies the Universal Coefficient Theorem, then 𝐴 is a simple AH-algebra with slow dimension growth and with real rank zero [2, 3]. In [4], Fang discovered the classification of certain nonsimple 𝐶^∗-algebras with tracial rank zero.

These successes suggest that one consider “tracial” versions of other 𝐶^∗-algebra concepts. In [5], Yao and Hu introduced the concept of tracial real rank of 𝐶^∗-algebras. In [6], Fan and Fang introduced the concept of tracial stable rank of 𝐶^∗-algebras. In [7, 8], Elliott and Niu and Fang and Fan studied the general concept of tracial approximation of properties of 𝐶^∗-algebras. The concept of the Rokhlin property in ergodic theory was adapted to the context of von Neumann algebras by Connes [9]. Then Herman and Ocneanu [10] and Rørdam [11] and Kishimoto [12] introduced the Rokhlin property to a much more general context of 𝐶^∗-algebras. In [13], Phillips introduced the concept of tracial Rokhlin property of finite group actions, which is more universal than the Rokhlin property. In [14], Osaka and Phillips introduced the concepts of local class property and approximate class property of unital 𝐶^∗-algebras and proved that these two properties are closed under crossed products by finite actions with the Rokhlin property.

Inspired by these papers, we introduce the concept of tracial class property of 𝐶^∗-algebras and prove that, for appropriate classes of 𝐶^∗-algebras, the tracial class property is closed under crossed products by finite group actions with the tracial Rokhlin property. As consequences, we get analogs of results in [13–18] such as the following ones. Let 𝐴 be a separable simple unital 𝐶^∗-algebra, and let 𝛼 be an action of a finite group 𝐺 on 𝐴 which has the tracial Rokhlin property. If 𝐴 is an AF-algebra, then 𝐴×_𝛼𝐺 has tracial rank zero. If 𝐴 is an A𝕋-algebra with the (SP)-property, then 𝐴×_𝛼𝐺 has tracial rank no more than one. If 𝐴 has stable rank one and real rank zero, then the induced crossed product 𝐴×_𝛼𝐺 has these two properties.

2. Definitions and Preliminaries

We denote by ℐ⁽⁰⁾ the class of finite dimensional 𝐶^∗-algebras and by ℐ^(𝑘) the class of 𝐶^∗-algebras with the form 𝑝(𝐶(𝑋)⊗𝐹)𝑝, where 𝐹∈ℐ⁽⁰⁾, 𝑋 is a finite CW complex with dimension 𝑘, and 𝑝∈𝐶(𝑋)⊗𝐹 is a projection.

Let 𝑝,𝑞 be projections in 𝐴 and 𝑎∈𝐴₊. If 𝑝 is Murray-von Neumann equivalent to 𝑞, then we write [𝑝]=[𝑞]. If 𝑝 is Murray-von Neumann equivalent to a subprojection of 𝑎𝐴𝑎, then we write [𝑝]≤[𝑎].

Let 𝐴 be a 𝐶^∗-algebra, and let ℱ be a subset of 𝐴,𝑎,𝑏,𝑥∈𝐴,𝜀>0. If ‖𝑎−𝑏‖<𝜀; then we write 𝑎≈_𝜀𝑏. If there exists an element 𝑦∈ℱ such that ‖𝑥−𝑦‖<𝜀, then we write 𝑥∈_𝜀ℱ.

Definition 2.1 (see [19, Definition  3.6.2], [5, Definition 1.4,], and [6, Definition 2.1]). Let 𝐴 be a simple unital 𝐶^∗-algebra and 𝑘∈ℕ. 𝐴 is said to have tracial rank no more than 𝑘; write TR(𝐴)≤𝑘; (tracial real rank zero, write TRR(𝐴)=0; tracial stable rank one, write Tsr(𝐴)=1), if for any 𝜀>0, any finite subset ℱ⊆𝐴 and any nonzero positive element 𝑏∈𝐴, there exist a nonzero projection 𝑝∈𝐴 and a 𝐶^∗-subalgebra 𝐵⊆𝐴 with 1_𝐵=𝑝 and 𝐵∈ℐ^(𝑘) (RR(𝐵)=0; tsr(𝐵)=1, resp.) such that(1)‖𝑝𝑎−𝑎𝑝‖<𝜀 for any 𝑎∈ℱ,(2)𝑝𝑎𝑝∈_𝜀𝐵 for all 𝑎∈ℱ,(3)[1−𝑝]≤[𝑏].

If, furthermore, TR(𝐴)𝑘−1, then we say TR(𝐴)=𝑘.

Lemma 2.2 (see [5, Theorem 3.3], [6, Theorem 3.3]). Let 𝐴 be a simple unital 𝐶^∗-algebra. If TRR(𝐴)=0, then RR(𝐴)=0. If Tsr(𝐴)=1 and has the (SP)-property, then tsr(𝐴)=1.

Definition 2.3 (see [13, Definition 1.2]). Let 𝐴 be an infinite dimensional finite simple separable unital 𝐶^∗-algebra, and let 𝛼∶𝐺→Aut(𝐴) be an action of a finite group 𝐺 on 𝐴. We say that 𝛼 has the tracial Rokhlin property if, for every 𝜀>0, every finite set ℱ⊆𝐴, every positive element 𝑏∈𝐴, there are mutually orthogonal projections {𝑒_𝑔∶𝑔∈𝐺} such that(1)‖𝛼_𝑔(𝑒_ℎ)−𝑒_𝑔ℎ‖<𝜀 for all 𝑔,ℎ∈𝐺,(2)‖𝑒_𝑔𝑎−𝑎𝑒_𝑔‖<𝜀 for all 𝑔∈𝐺 and all 𝑎∈ℱ,(3)with 𝑒=∑_𝑔∈𝐺𝑒_𝑔, [1−𝑒]≤[𝑏].

Lemma 2.4 (see [13, Corollary 1.6]). Let 𝐴 be an infinite dimensional finite simple separable unital 𝐶^∗-algebra, and let 𝛼∶𝐺→Aut(𝐴) be an action of a finite group 𝐺 on 𝐴 which has the tracial Rokhlin property. Then 𝐴×_𝛼𝐺 is simple.

Lemma 2.5 (see [20, Theorem 4.2]). Let 𝐴 be a simple unital 𝐶^∗-algebra with the (SP)-property, and let 𝛼∶𝐺→Aut(𝐴) be an action of a discrete group 𝐺 on 𝐴. Suppose that the normal subgroup 𝑁={𝑔∈𝐺∣𝛼_𝑔isinneron𝐴} of 𝐺 is finite; then any nonzero hereditary 𝐶^∗-subalgebra of the crossed product 𝐴×_𝛼𝐺 has a nonzero projection which is Murray-von Neumann equivalent to a projection in 𝐴×_𝛼𝑁.

If the action 𝛼∶𝐺→Aut(𝐴) has the tracial Rokhlin property, then each 𝛼_𝑔 is outer for all 𝑔∈𝐺⧵{1}. So 𝑁={𝑔∈𝐺∣𝛼_𝑔isinneron𝐴}={1}. Since 𝐴×_𝛼𝑁=𝐴×_𝛼{1}≅𝐴, by Lemma 2.5 we have the following lemma.

Lemma 2.6. Let 𝐴 be an infinite dimensional finite simple separable unital 𝐶^∗-algebra with the (SP)-property, and let 𝛼∶𝐺→Aut(𝐴) be an action of a finite group 𝐺 on 𝐴 which has the tracial Rokhlin property; then any nonzero hereditary 𝐶^∗-subalgebra of the crossed product 𝐴×_𝛼𝐺 has a nonzero projection which is Murray-von Neumann equivalent to a projection in 𝐴.

Lemma 2.7 (see [19, Lemma 3.5.6]). Let 𝐴 be a simple 𝐶^∗-algebra with the (SP)-property, and let 𝑝,𝑞∈𝐴 be two nonzero projections. Then there are nonzero projections 𝑝₁≤𝑝,𝑞₁≤𝑞 such that [𝑝₁]=[𝑞₁].

Definition 2.8 (see [14, Definition 1.1]). Let 𝒞 be a class of separable unital 𝐶^∗-algebras. We say that 𝒞 is finitely saturated if the following closure conditions hold.(1)If 𝐴∈𝒞 and 𝐵≅𝐴, then 𝐵∈𝒞.(2)If 𝐴_𝑖∈𝒞 for 𝑖=1,2,…,𝑛, then ⨁^𝑛_𝑘=1𝐴_𝑘∈𝒞.(3)If 𝐴∈𝒞 and 𝑛∈ℕ, then 𝑀_𝑛(𝐴)∈𝒞.(4)If 𝐴∈𝒞 and 𝑝∈𝐴 is a nonzero projection, then 𝑝𝐴𝑝∈𝒞.

Moreover, the finite saturation of a class 𝒞 is the smallest finitely saturated class which contains 𝒞.

Definition 2.9 (see [14, Definition 1.2]). Let 𝒞 be a class of separable unital 𝐶^∗-algebras. We say that 𝒞 is flexible if.(1)for every 𝐴∈𝒞, every 𝑛∈ℕ, and every nonzero projection 𝑝∈𝑀_𝑛(𝐴), the corner 𝑝𝑀_𝑛(𝐴)𝑝 is semiprojective and finitely generated;(2)for every 𝐴∈𝒞 and every ideal 𝐼⊆𝐴, there is an increasing sequence 𝐼₁⊆𝐼₂⊆⋯ of ideals of 𝐴 such that ∪^∞_𝑛=1𝐼_𝑛=𝐼 and such that for every 𝑛 the 𝐶^∗-algebra 𝐴/𝐼_𝑛 is in the finite saturation of 𝒞.

Example 2.10. (1) Let 𝒞={⨁^𝑛_𝑖=1𝑀_𝑘(𝑖)∣𝑛,𝑘(𝑖)∈ℕ}; that is, 𝒞 contains all finite dimensional algebras. 𝒞 is finitely saturated and flexible.
(2) Let 𝒞={⨁^𝑛_𝑖=1𝐶(𝑋_𝑖,𝑀_𝑘(𝑖))∣𝑛,𝑘(𝑖)∈ℕ;each𝑋_𝑖isaclosedsubsetofthecircle}. We can show that 𝒞 is finitely saturated and flexible.
(3) Let 𝒞={𝑓∈⨁^𝑛_𝑖=1𝐶([0,1],𝑀_𝑘(𝑖))∣𝑛,𝑘(𝑖)∈ℕ,𝑓(0)isscalar}. We can also show that 𝒞 is finitely saturated and flexible.
(4) For some 𝑑∈ℕ, let 𝒞_𝑑 contain all the 𝐶^∗-algebras ⨁^𝑛_𝑖=1𝑝_𝑖𝐶(𝑋_𝑖,𝑀_𝑘(𝑖))𝑝_𝑖, where 𝑛,𝑘(𝑖)∈ℕ, each 𝑝_𝑖 is a nonzero projection in 𝐶(𝑋_𝑖,𝑀_𝑘(𝑖)), and each 𝑋_𝑖 is a compact metric space with covering dimension at most 𝑑. The class 𝒞_𝑑 is not flexible for 𝑑≠0 (see [14] Example 2.9).

Definition 2.11 (see [16, Definition 1.4]). Let 𝒞 be a class of separable unital 𝐶^∗-algebras. A unital approximate 𝒞-algebra is a 𝐶^∗-algebra which is isomorphic to an inductive limit lim_𝑛→∞(𝐴_𝑛,𝜙_𝑛), where each 𝐴_𝑛 is in the finite saturation of 𝒞 and each homomorphism 𝜙_𝑛∶𝐴_𝑛→𝐴_𝑛+1 is unital.

Definition 2.12 (see [14, Definition 1.5]). Let 𝒞 be a class of separable unital 𝐶^∗-algebras. Let 𝐴 be a separable unital 𝐶^∗-algebra. We say that 𝐴 is a unital local 𝒞-algebra if, for every 𝜀>0 and every finite subset ℱ⊂𝐴, there is a 𝐶^∗-algebra 𝐵 in the finite saturation of 𝒞 and a ∗-homomorphism 𝜙∶𝐵→𝐴 such that 𝑎∈_𝜀𝜙(𝐵) for all 𝑎∈ℱ.

By [14] Proposition 1.6, if 𝒞 is a finitely saturated flexible class of separable unital 𝐶^∗-algebras, then every unital local 𝒞-algebra is a unital approximate 𝒞-algebra. The converse is clear.

Let 𝒞 be a class as (1) of Example 2.10. Then a unital AF-algebra is a unital approximate 𝒞-algebra and is a unital local 𝒞-algebra.

Let 𝒞 be a class as (2) of Example 2.10. Then a unital A𝕋-algebra is a unital approximate 𝒞-algebra and is a unital local 𝒞-algebra.

Definition 2.13. Let 𝐴 be a simple unital 𝐶^∗-algebra, and let 𝒞 be a class of separable unital 𝐶^∗-algebra. We say that 𝐴 is a tracial 𝒞-algebra if, for any 𝜀>0, any finite subset ℱ⊂𝐴, and any nonzero positive element 𝑏∈𝐴, there exist a nonzero projection 𝑝∈𝐴, a 𝐶^∗-algebra 𝐵 in the finite saturation of 𝒞, and a ∗-homomorphism 𝜙∶𝐵→𝐴 with 1_𝜙(𝐵)=𝑝, such that(1)‖𝑝𝑎−𝑎𝑝‖<𝜀 for any 𝑎∈ℱ,(2)𝑝𝑎𝑝∈_𝜀𝜙(𝐵) for all 𝑎∈ℱ,(3)[1−𝑝]≤[𝑏] in 𝐴.

Using the similar proof of Lemma 3.6.5 of [19] about the tracial rank of unital hereditary 𝐶^∗-subalgebras of a simple unital 𝐶^∗-algebra, we get the following one.

Lemma 2.14. Let 𝒞 be any finitely saturated class of separable unital 𝐶^∗-algebras. Let 𝑝 be a projection in a simple unital 𝐶^∗-algebra 𝐴 with the (SP)-property. If 𝐴 is a tracial 𝒞-algebra, so also is 𝑝𝐴𝑝.

For 𝑛∈ℕ, 𝛿>0, a unital 𝐶^∗-algebra 𝐴, if 𝑤_𝑖,𝑗, for 1≤𝑖,𝑗≤𝑛, are elements of 𝐴, such that ‖𝑤_𝑖,𝑗‖≤1 for 1≤𝑖,𝑗≤𝑛, such that ‖𝑤^∗_𝑖,𝑗−𝑤_𝑗,𝑖‖<𝛿 for 1≤𝑖,𝑗≤𝑛, such that ‖𝑤_𝑖1,𝑗1𝑤_𝑖2,𝑗2−𝛿_𝑗1,𝑖2𝑤_𝑖1,𝑗2‖<𝛿 for 1≤𝑖₁,𝑖₂,𝑗₁,𝑗₂≤𝑛, and such that 𝑤_𝑖,𝑖 are mutually orthogonal projections, we say that 𝑤_𝑖,𝑗(1≤𝑖,𝑗≤𝑛) form a 𝛿-approximate system of 𝑛×𝑛 matrix units in 𝐴.

By perturbation of projections (see Theorem 2.5.9 of [19]), we have Lemma 2.15.

Lemma 2.15. For any 𝑛∈ℕ, any 𝜀>0, there exists 𝛿=𝛿(𝑛,𝜀)>0 such that, whenever (𝑓_𝑖,𝑗)_{1≤𝑖,𝑗≤𝑛} is a system of matrix units for 𝑀_𝑛, whenever 𝐵 is a unital 𝐶^∗-algebra, and whenever 𝑤_𝑖,𝑗, for 1≤𝑖,𝑗≤𝑛, are elements of 𝐵 which form a 𝛿-approximate system of 𝑛×𝑛 matrix units, then there exists a ∗-homomorphism 𝜙∶𝑀_𝑛→𝐵 such that 𝜙(𝑓_𝑖,𝑖)=𝑤_𝑖,𝑖 for 1≤𝑖≤𝑛 and ‖𝜙(𝑓_𝑖,𝑗)−𝑤_𝑖,𝑗‖<𝜀 for 1≤𝑖,𝑗≤𝑛.

3. Main Results

Theorem 3.1. Let 𝒞 be any class of separable unital 𝐶^∗-algebras. Let 𝐴 be an infinite dimensional finite simple unital tracial 𝒞-algebra with the (SP)-property, and let 𝛼∶𝐺→Aut(𝐴) be an action of a finite group 𝐺 on 𝐴 which has the tracial Rokhlin property. Then 𝐴×_𝛼𝐺 is a simple unital tracial 𝒞-algebra.

Proof. By Lemma 2.4, 𝐴×_𝛼𝐺 is a simple unital 𝐶^∗-algebra. By Definition 2.13, it suffices to show the following.
For any 𝜀>0, any finite subset ℱ=ℱ₀∪{𝑢_𝑔∣𝑔∈𝐺}⊂𝐴×_𝛼𝐺, where ℱ₀ is a finite subset of the unit ball of 𝐴 and 𝑢_𝑔∈𝐴×_𝛼𝐺 is the canonical unitary implementing the automorphism 𝛼_𝑔, and any nonzero positive element 𝑏∈𝐴×_𝛼𝐺, there exist a nonzero projection 𝑝∈𝐴×_𝛼𝐺, a 𝐶^∗-algebra 𝐵 in the finite saturation of 𝒞, and a ∗-homomorphism 𝜙∶𝐵→𝐴×_𝛼𝐺 with 1_𝜙(𝐵)=𝑝, such that(1)‖𝑝𝑎−𝑎𝑝‖<𝜀 for any 𝑎∈ℱ,(2)𝑝𝑎𝑝∈_𝜀𝜙(𝐵) for all 𝑎∈ℱ,(3)[1−𝑝]≤[𝑏] in 𝐴×_𝛼𝐺.
By Lemma 2.6, there exists a nonzero projection 𝑞∈𝐴 such that [𝑞]≤[𝑏] in 𝐴×_𝛼𝐺.
Since 𝐴 is an infinite dimensional simple unital 𝐶^∗-algebra with the (SP)-property, by [19, Lemma 3.5.7], there exist orthogonal nonzero projections 𝑞₁,𝑞₂∈𝐴 such that 𝑞₁+𝑞₂≤𝑞.
Set 𝑛=card(𝐺) and set 𝜀₀=𝜀/48𝑛. Choose 𝛿>0 according to Lemma 2.15 for 𝑛 given above and 𝜀₀ in place of 𝜀. Moreover we may require 𝛿<min{𝜀/72𝑛,𝜀/(24𝑛(𝑛−1))}.
Apply Definition 2.3 with ℱ₀ given above, with 𝛿 in place of 𝜀, with 𝑞₁ in place of 𝑏. There exist mutually orthogonal projections 𝑒_𝑔∈𝐴 for 𝑔∈𝐺 such that(1^′)‖𝛼_𝑔(𝑒_ℎ)−𝑒_𝑔ℎ‖<𝛿 for all 𝑔,ℎ∈𝐺,(2^′)‖𝑒_𝑔𝑎−𝑎𝑒_𝑔‖<𝛿 for all 𝑔∈𝐺 and all 𝑎∈ℱ₀,(3^′)[1−𝑒]≤[𝑞₁] in 𝐴, where 𝑒=∑_𝑔∈𝐺𝑒_𝑔.
By Lemma 2.7, there are nonzero projections 𝑣₁,𝑣₂∈𝐴 such that 𝑣₁≤𝑒₁,𝑣₂≤𝑞₂ and [𝑣₁]=[𝑣₂].
Define 𝑤_𝑔,ℎ=𝑢_𝑔ℎ⁻¹𝑒_ℎ for 𝑔,ℎ∈𝐺. By the proof of Theorem 2.2 of [14], we can estimate that 𝑤_𝑔,ℎ(𝑔,ℎ∈𝐺) form a 𝛿-approximate system of 𝑛×𝑛 matrix units in 𝐴×_𝛼𝐺. Moreover, ∑_𝑔∈𝐺𝑤_𝑔,𝑔=∑_𝑔∈𝐺𝑒_𝑔=𝑒.
Let (𝑓_𝑔,ℎ)_{𝑔,ℎ∈𝐺} be a system of matrix units for 𝑀_𝑛. By Lemma 2.15, there exists a ∗-homomorphism 𝜙₀∶𝑀_𝑛→𝐴×_𝛼𝐺 such that 𝜙₀𝑓_𝑔,ℎ−𝑤_𝑔,ℎ<𝜀₀ (3.1) for all 𝑔,ℎ∈𝐺, and 𝜙₀(𝑓_𝑔,𝑔)=𝑒_𝑔 for all 𝑔∈𝐺.
Set 𝐸=𝑀_𝑛⊗𝑒₁𝐴𝑒₁. Define an injective unital ∗-homomorphism 𝜙₁∶𝐸→𝑒(𝐴×_𝛼𝐺)𝑒 by 𝜙₁𝑓_𝑔,ℎ⊗𝑎=𝜙₀𝑓_𝑔,1𝑎𝜙₀𝑓_1,ℎ (3.2) for all 𝑔,ℎ∈𝐺 and 𝑎∈𝑒₁𝐴𝑒₁. Then 𝜙₁1_𝑀𝑛⊗𝑒₁=_𝑔∈𝐺𝑒_𝑔=𝑒,𝜙₁𝑓_1,1⊗𝑎=𝑎 (3.3) for all 𝑎∈𝑒₁𝐴𝑒₁ and 𝜙₁𝑓_𝑔,ℎ⊗𝑒₁=𝜙₀𝑓_𝑔,1𝑒₁𝜙₀𝑓_1,ℎ=𝜙₀𝑓_𝑔,1𝜙₀𝑓_1,1𝜙₀𝑓_1,ℎ=𝜙₀𝑓_𝑔,ℎ=𝑒_𝑔𝜙₀𝑓_𝑔,ℎ𝑒_ℎ. (3.4)
By (2^′), for all 𝑎∈ℱ₀, we have ‖𝑎𝑒−𝑒𝑎‖≤_𝑔∈𝐺𝑎𝑒_𝑔−𝑒_𝑔𝑎<𝑛𝛿. (3.5)
By (1^′), for all 𝑔∈𝐺, we get 𝑢_𝑔𝑒−𝑒𝑢_𝑔≤𝑢_𝑔𝑒𝑢_𝑔−1−𝑒=_ℎ∈𝐺𝛼_𝑔𝑒_ℎ−_ℎ∈𝐺𝑒_𝑔ℎ≤𝑛𝛿. (3.6)
For all 𝑔∈𝐺, we have 𝑒𝑢_𝑔𝑒−_ℎ∈𝐺𝜙₁𝑓_𝑔ℎ,ℎ⊗𝑒₁≤𝑒𝑢_𝑔𝑒−𝑢_𝑔𝑒+𝑢_𝑔𝑒−_ℎ∈𝐺𝜙₁𝑓_𝑔ℎ,ℎ⊗𝑒₁<𝑛𝛿+𝑢_𝑔𝑒−_ℎ∈𝐺𝜙₁𝑓_𝑔ℎ,ℎ⊗𝑒₁=𝑛𝛿+_ℎ∈𝐺𝑢_𝑔𝑒_ℎ−_ℎ∈𝐺𝜙₁𝑓_𝑔ℎ,ℎ⊗𝑒₁=𝑛𝛿+_ℎ∈𝐺𝑤_𝑔ℎ,ℎ−_ℎ∈𝐺𝜙₀𝑓_𝑔ℎ,ℎ<𝑛𝛿+𝑛𝜀₀<5𝜀.144 (3.7) That is, for all 𝑔∈𝐺, we have 𝑒𝑢_𝑔𝑒∈_5𝜀/144𝜙₁(𝐸). (3.8)
Set 𝑏=∑_𝑔∈𝐺𝑓_𝑔,𝑔⊗𝑒₁𝛼_𝑔−1(𝑎)𝑒₁; then 𝑏∈𝐸. Using ‖𝑒_𝑔𝑎𝑒_ℎ−𝑎𝑒_𝑔𝑒_ℎ‖<𝛿, we get 𝑒𝑎𝑒−_𝑔∈𝐺𝑒_𝑔𝑎𝑒_𝑔≤_𝑔≠ℎ𝑒_𝑔𝑎𝑒_ℎ<𝑛(𝑛−1)𝛿. (3.9)
We also have 𝜙₀𝑓_𝑔,1𝑒₁−𝑢_𝑔𝑒₁≤𝜙₀𝑓_𝑔,1−𝑢_𝑔𝑒₁=𝜙₀𝑓_𝑔,1−𝑤_𝑔,1<𝜀₀,𝜙₀𝑓_1,𝑔−𝑒₁𝑢_𝑔−1≤𝜙₀𝑓_1,𝑔−𝑢_𝑔−1𝑒_𝑔+𝑢_𝑔−1𝑒_𝑔−𝑒₁𝑢_𝑔−1<𝜀₀𝑒+𝛿,₁𝛼_𝑔−1(𝑎)𝑒₁−𝛼_𝑔−1𝑒_𝑔𝑎𝑒_𝑔<2𝛿. (3.10)
Then, for all 𝑎∈ℱ₀, we have 𝑒𝑎𝑒−𝜙₁(𝑏)=𝑒𝑎𝑒−𝜙_1𝑔∈𝐺𝑓_𝑔,𝑔⊗𝑒₁𝛼_𝑔−1(𝑎)𝑒₁=𝑒𝑎𝑒−_𝑔∈𝐺𝜙₀𝑓_𝑔,1𝑒₁𝛼_𝑔−1(𝑎)𝑒₁𝜙₀𝑓_1,𝑔<𝑒𝑎𝑒−_𝑔∈𝐺𝑢_𝑔𝑒₁𝛼_𝑔−1(𝑎)𝑒₁𝑢_𝑔−1+2𝑛𝜀₀<+𝑛𝛿𝑒𝑎𝑒−_𝑔∈𝐺𝑢_𝑔𝛼_𝑔−1𝑒_𝑔𝑎𝑒_𝑔𝑢_𝑔−1+3𝑛𝛿+2𝑛𝜀₀=𝑒𝑎𝑒−_𝑔∈𝐺𝑒_𝑔𝑎𝑒_𝑔+3𝑛𝛿+2𝑛𝜀₀<𝑛(𝑛−1)𝛿+3𝑛𝛿+2𝑛𝜀₀<𝜀+𝜀24+𝜀24=𝜀248. (3.11) That is, for all 𝑎∈ℱ₀, 𝑒𝑎𝑒∈_𝜀/8𝜙₁(𝐸). (3.12)
By (3.8) and (3.12), we can write 𝑒𝑎𝑒∈_𝜀/8𝜙₁(𝐸) (3.13) for all 𝑎∈ℱ.
Write ℱ=𝑏∣𝑏∈𝐸,𝜙₁(𝑏)<𝜀−𝑒𝑎𝑒8for𝑎∈ℱ. (3.14)
By Lemma 2.14, 𝐸 is a simple unital tracial 𝒞-algebra. Apply Definition 2.13 with ℱ given above, with 𝜀/8 in place of 𝜀 and 𝑓_1,1⊗𝑣₁ in place of 𝑎. There exist a nonzero projection 𝑝₀∈𝐸, a 𝐶^∗-algebra 𝐵 in the finite saturation of 𝒞, and a ∗-homomorphism 𝜓₀∶𝐵→𝐸 with 1_𝜓0(𝐵)=𝑝₀, such that(1′′)‖𝑝₀𝑏−𝑏𝑝₀‖<𝜀/8 for any 𝑏∈ℱ,(2′′)𝑝₀𝑏𝑝₀∈_𝜀/8𝜓₀(𝐵) for all 𝑏∈ℱ,(3′′)[1_𝐸−𝑝₀]≤[𝑓_1,1⊗𝑣₁] in 𝐸.
Set 𝑝=𝜙₁(𝑝₀) and 𝜙=𝜙₁∘𝜓₀∶𝐵→𝑒(𝐴×_𝛼𝐺)𝑒.
For every 𝑎∈ℱ, there exists 𝑏∈ℱ such that 𝜙₁(𝑏)≈_𝜀/8𝑒𝑎𝑒. Then 𝑝𝑎=𝑝𝑒𝑎≈_𝑛𝛿𝑝𝑒𝑎𝑒≈_𝜀/8𝑝𝜙₁(𝑏)=𝜙₁𝑝₀𝑏≈_𝜀/8𝜙₁𝑏𝑝₀≈_{𝑛𝛿+𝜀/4}𝑎𝑝. (3.15) That is, 𝜀𝑝𝑎−𝑎𝑝<2𝑛𝛿+2<𝜀. (3.16)
Let 𝑐∈𝐵 such that 𝑝₀𝑏𝑝₀≈_𝜀/8𝜓₀(𝑐). Then 𝑝𝑎𝑝=𝑝𝑒𝑎𝑒𝑝≈_𝜀/8𝑝𝜙₁(𝑏)𝑝=𝜙₁𝑝₀𝑏𝑝₀≈_𝜀/8𝜙₁𝜓₀(𝑐)=𝜙(𝑐). (3.17) Hence, 𝑝𝑎𝑝∈_𝜀𝜙(𝐵). (3.18)
By (3^′′), in 𝐴×_𝛼𝐺, [𝑒−𝑝]=[𝜙₁(1_𝐸−𝑝₀)]≤[𝜙₁(𝑓_1,1⊗𝑣₁)]=[𝑣₁]=[𝑣₂]. Therefore, 1−𝑝=[1−𝑒]+𝑒−𝑝≤𝑞₁+𝑣₂≤𝑞₁+𝑞₂≤𝑞≤[𝑏]. (3.19)
From (3.16), (3.18), and (3.19), 𝐴×_𝛼𝐺 is a simple unital tracial 𝒞-algebra.

Corollary 3.2. Let 𝐴 be an infinite dimensional separable simple unital 𝐶^∗-algebra, and let 𝛼∶𝐺→Aut(𝐴) be an action of a finite group 𝐺 on 𝐴 which has the tracial Rokhlin property. If 𝐴 is an AF-algebra, then the induced crossed product 𝐴×_𝛼𝐺 has tracial rank zero. If 𝐴 is an A𝕋-algebra with the (SP)-property, then the induced crossed product 𝐴×_𝛼𝐺 has tracial rank no more than one.

Proof. If 𝐴 is an AF-algebra, then 𝐴 is a unital local 𝒞-algebra, where 𝒞 is a class of 𝐶^∗-algebras satisfying condition (1) of Example 2.10. By Theorem 3.1, we know that 𝐴×_𝛼𝐺 is a simple unital tracial 𝒞-algebra. By the definition of tracial rank zero, TR(𝐴×_𝛼𝐺)=0.
If 𝐴 is an A𝕋-algebra, then 𝐴 is a unital local 𝒞-algebra, where 𝒞 is a class of 𝐶^∗-algebra satisfying condition (2) of Example 2.10. By Theorem 3.1, we know that 𝐴×_𝛼𝐺 is a simple unital tracial 𝒞-algebra. Since the covering dimension of closed subsets of the circle is no more than one, by the definition of tracial rank, TR(𝐴×_𝛼𝐺)≤1.

It should be noted that the AF-part was proved by Phillips in [13] Theorem 2.6.

Corollary 3.3. Let 𝐴 be an infinite dimensional finite separable simple unital 𝐶^∗-algebra with the (SP)-property, and let 𝛼∶𝐺→Aut(𝐴) be an action of a finite group 𝐺 on 𝐴 which has the tracial Rokhlin property. If 𝐴 has stable rank one, then the induced crossed product 𝐴×_𝛼𝐺 has stable rank one. If 𝐴 has real rank zero, then the induced crossed product 𝐴×_𝛼𝐺 has real rank zero.

Proof. Let 𝒞 be the class of all separable unital 𝐶^∗-algebras with stable rank one. By Theorems 3.1.2, 3.18, and 3.19 in [19], we have that 𝒞 is finitely saturated and satisfies condition (2) of Definition 2.9. By Theorem 3.1, the crossed product 𝐴×_𝛼𝐺 is a simple unital tracial 𝒞-algebra, that is, for any 𝜀>0, any finite subset ℱ⊂𝐴×_𝛼𝐺, and any nonzero positive element 𝑏∈𝐴×_𝛼𝐺, there exist a nonzero projection 𝑝∈𝐴×_𝛼𝐺, a 𝐶^∗-algebra 𝐵 in 𝒞, and a ∗-homomorphism 𝜙∶𝐵→𝐴×_𝛼𝐺 with 1_𝜙(𝐵)=𝑝, such that(1)‖𝑝𝑎−𝑎𝑝‖<𝜀 for any 𝑎∈ℱ,(2)𝑝𝑎𝑝∈_𝜀𝜙(𝐵) for all 𝑎∈ℱ,(3)[1−𝑝]≤[𝑏] in 𝐴×_𝛼𝐺.
Hence, Tsr(𝐴×_𝛼𝐺)=1. By Lemma 2.2, tsr(𝐴×_𝛼𝐺)=1.
Let 𝒞 be the class of all separable unital 𝐶^∗-algebras with real rank zero. We can use the same argument to show that the crossed product 𝐴×_𝛼𝐺 is a simple unital tracial 𝒞-algebra. Hence TRR(𝐴×_𝛼𝐺)=0. By Lemma 2.2, RR(𝐴×_𝛼𝐺)=0.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11071188) and Zhejiang Provincial Natural Science Foundation of China (LQ12A01004). The authors would like to express their hearty thanks to the referees for their very helpful comments and suggestions.