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 ×α G 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 [<xref ref-type="bibr" rid="B10">19</xref>, Definition 3.6.2], [<xref ref-type="bibr" rid="B20">5</xref>, Definition 1.4,], and [<xref ref-type="bibr" rid="B3">6</xref>, 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 1B=p and B∈ℐ(k) (RR(B)=0; tsr(B)=1, resp.) such that

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 [<xref ref-type="bibr" rid="B17">13</xref>, 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 {eg:g∈G} such that

∥αg(eh)-egh∥<ε for all g,h∈G,

∥ega-aeg∥<ε for all g∈G and all a∈ℱ,

with e=∑g∈Geg, [1-e]≤[b].

Lemma 2.4 (see [<xref ref-type="bibr" rid="B17">13</xref>, 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 [<xref ref-type="bibr" rid="B8">20</xref>, 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∣αg isinneron
A} 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 αg is outer for all g∈G∖{1}. So N={g∈G∣αgisinneronA}={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 [<xref ref-type="bibr" rid="B10">19</xref>, Lemma <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M159"><mml:mn>3.5</mml:mn><mml:mo>.</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>]).

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 [<xref ref-type="bibr" rid="B16">14</xref>, 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.

If A∈𝒞 and B≅A, then B∈𝒞.

If Ai∈𝒞 for i=1,2,…,n, then ⨁k=1nAk∈𝒞.

If A∈𝒞 and n∈ℕ, then Mn(A)∈𝒞.

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 [<xref ref-type="bibr" rid="B16">14</xref>, Definition 1.2]).

Let 𝒞 be a class of separable unital C*-algebras. We say that 𝒞 is flexible if.

for every A∈𝒞, every n∈ℕ, and every nonzero projection p∈Mn(A), the corner pMn(A)p is semiprojective and finitely generated;

for every A∈𝒞 and every ideal I⊆A, there is an increasing sequence I1⊆I2⊆⋯ of ideals of A such that ∪n=1∞In¯=I and such that for every n the C*-algebra A/In is in the finite saturation of 𝒞.

Example 2.10.

(1) Let 𝒞={⨁i=1nMk(i)∣n,k(i)∈ℕ}; that is, 𝒞 contains all finite dimensional algebras. 𝒞 is finitely saturated and flexible.

(2) Let 𝒞={⨁i=1nC(Xi,Mk(i))∣n,k(i)∈ℕ;eachXiisaclosedsubsetofthecircle}. We can show that 𝒞 is finitely saturated and flexible.

(3) Let 𝒞={f∈⨁i=1nC([0,1],Mk(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 𝒞d contain all the C*-algebras ⨁i=1npiC(Xi,Mk(i))pi, where n,k(i)∈ℕ, each pi is a nonzero projection in C(Xi,Mk(i)), and each Xi is a compact metric space with covering dimension at most d. The class 𝒞d is not flexible for d≠0 (see [14] Example 2.9).

Definition 2.11 (see [<ext-link ext-link-type="uri" xlink:href="16">16</ext-link>, 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 limn→∞(An,ϕn), where each An is in the finite saturation of 𝒞 and each homomorphism ϕn:An→An+1 is unital.

Definition 2.12 (see [<xref ref-type="bibr" rid="B16">14</xref>, 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 1ϕ(B)=p, such that

∥pa-ap∥<ε for any a∈ℱ,

pap∈εϕ(B) for all a∈ℱ,

[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 wi,j, for 1≤i,j≤n, are elements of A, such that ∥wi,j∥≤1 for 1≤i,j≤n, such that ∥wi,j*-wj,i∥<δ for 1≤i,j≤n, such that ∥wi1,j1wi2,j2-δj1,i2wi1,j2∥<δ for 1≤i1,i2,j1,j2≤n, and such that wi,i are mutually orthogonal projections, we say that wi,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 (fi,j)1≤i,j≤n is a system of matrix units for Mn, whenever B is a unital C*-algebra, and whenever wi,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:Mn→B such that ϕ(fi,i)=wi,i for 1≤i≤n and ∥ϕ(fi,j)-wi,j∥<ε for 1≤i,j≤n.

3. Main ResultsTheorem 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∪{ug∣g∈G}⊂A×αG, where ℱ0 is a finite subset of the unit ball of A and ug∈A×αG is the canonical unitary implementing the automorphism αg, 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 1ϕ(B)=p, such that

∥pa-ap∥<ε for any a∈ℱ,

pap∈εϕ(B) for all a∈ℱ,

[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 eg∈A for g∈G such that

∥αg(eh)-egh∥<δ for all g,h∈G,

∥ega-aeg∥<δ for all g∈G and all a∈ℱ0,

[1-e]≤[q1] in A, where e=∑g∈Geg.

By Lemma 2.7, there are nonzero projections v1,v2∈A such that v1≤e1,v2≤q2 and [v1]=[v2].

Define wg,h=ugh-1eh for g,h∈G. By the proof of Theorem 2.2 of [14], we can estimate that wg,h(g,h∈G) form a δ-approximate system of n×n matrix units in A×αG. Moreover, ∑g∈Gwg,g=∑g∈Geg=e.

Let (fg,h)g,h∈G be a system of matrix units for Mn. By Lemma 2.15, there exists a *-homomorphism ϕ0:Mn→A×αG such that
(3.1)∥ϕ0(fg,h)-wg,h∥<ε0
for all g,h∈G, and ϕ0(fg,g)=eg for all g∈G.

Set E=Mn⊗e1Ae1. Define an injective unital *-homomorphism ϕ1:E→e(A×αG)e by
(3.2)ϕ1(fg,h⊗a)=ϕ0(fg,1)aϕ0(f1,h)
for all g,h∈G and a∈e1Ae1. Then
(3.3)ϕ1(1Mn⊗e1)=∑g∈Geg=e,ϕ1(f1,1⊗a)=a
for all a∈e1Ae1 and
(3.4)ϕ1(fg,h⊗e1)=ϕ0(fg,1)e1ϕ0(f1,h)=ϕ0(fg,1)ϕ0(f1,1)ϕ0(f1,h)=ϕ0(fg,h)=egϕ0(fg,h)eh.

By (2′), for all a∈ℱ0, we have
(3.5)∥ae-ea∥≤∑g∈G∥aeg-ega∥<nδ.

By (1′), for all g∈G, we get
(3.6)∥uge-eug∥≤∥ugeug-1-e∥=∥∑h∈Gαg(eh)-∑h∈Gegh∥≤nδ.

For all g∈G, we have
(3.7)∥euge-∑h∈Gϕ1(fgh,h⊗e1)∥≤∥euge-uge∥+∥uge-∑h∈Gϕ1(fgh,h⊗e1)∥<nδ+∥uge-∑h∈Gϕ1(fgh,h⊗e1)∥=nδ+∥∑h∈Gugeh-∑h∈Gϕ1(fgh,h⊗e1)∥=nδ+∥∑h∈Gwgh,h-∑h∈Gϕ0(fgh,h)∥<nδ+nε0<5ε144.
That is, for all g∈G, we have
(3.8)euge∈5ε/144ϕ1(E).

Set b=∑g∈Gfg,g⊗e1αg-1(a)e1; then b∈E. Using ∥egaeh-aegeh∥<δ, we get
(3.9)∥eae-∑g∈Gegaeg∥≤∑g≠h∥egaeh∥<n(n-1)δ.

We also have
(3.10)∥ϕ0(fg,1)e1-uge1∥≤∥ϕ0(fg,1)-uge1∥=∥ϕ0(fg,1)-wg,1∥<ε0,∥ϕ0(f1,g)-e1ug-1∥≤∥ϕ0(f1,g)-ug-1eg∥+∥ug-1eg-e1ug-1∥<ε0+δ,∥e1αg-1(a)e1-αg-1(egaeg)∥<2δ.

Then, for all a∈ℱ0, we have
(3.11)∥eae-ϕ1(b)∥=∥eae-ϕ1(∑g∈Gfg,g⊗e1αg-1(a)e1)∥=∥eae-∑g∈Gϕ0(fg,1)e1αg-1(a)e1ϕ0(f1,g)∥<∥eae-∑g∈Guge1αg-1(a)e1ug-1∥+2nε0+nδ<∥eae-∑g∈Gugαg-1(egaeg)ug-1∥+3nδ+2nε0=∥eae-∑g∈Gegaeg∥+3nδ+2nε0<n(n-1)δ+3nδ+2nε0<ε24+ε24+ε24=ε8.
That is, for all a∈ℱ0,
(3.12)eae∈ε/8ϕ1(E).

By (3.8) and (3.12), we can write
(3.13)eae∈ε/8ϕ1(E)
for all a∈ℱ.

Write
(3.14)ℱ~={b∣b∈E,∥ϕ1(b)-eae∥<ε8fora∈ℱ}.

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 f1,1⊗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 1ψ0(B)=p0, such that

∥p0b-bp0∥<ε/8 for any b∈ℱ~,

p0bp0∈ε/8ψ0(B) for all b∈ℱ~,

[1E-p0]≤[f1,1⊗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
(3.15)pa=pea≈nδpeae≈ε/8pϕ1(b)=ϕ1(p0b)≈ε/8ϕ1(bp0)≈nδ+ε/4ap.
That is,
(3.16)∥pa-ap∥<2nδ+ε2<ε.

Let c∈B such that p0bp0≈ε/8ψ0(c). Then
(3.17)pap=peaep≈ε/8pϕ1(b)p=ϕ1(p0bp0)≈ε/8ϕ1((ψ0(c)))=ϕ(c).
Hence,
(3.18)pap∈εϕ(B).

By (3′′), in A×αG, [e-p]=[ϕ1(1E-p0)]≤[ϕ1(f1,1⊗v1)]=[v1]=[v2]. Therefore,
(3.19)[1-p]=[1-e]+[e-p]≤[q1]+[v2]≤[q1]+[q2]≤[q]≤[b].

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 1ϕ(B)=p, such that

∥pa-ap∥<ε for any a∈ℱ,

pap∈εϕ(B) for all a∈ℱ,

[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.

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