Scholarly article on topic 'The Tracial Class Property for Crossed Products by Finite Group Actions'

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Academic research paper on topic "The Tracial Class Property for Crossed Products by Finite Group Actions"

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 745369,10 pages doi:10.1155/2012/745369

Research Article

The Tracial Class Property for Crossed Products by Finite Group Actions

Xinbing Yang1 and Xiaochun Fang2

1 Department of Mathematics, Zhejiang Normal University, Zhejiang, Jinhua 321004, China

2 Department of Mathematics, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Xiaochun Fang, xfang@mail.tongji.edu.cn Received 11 September 2012; Accepted 14 October 2012 Academic Editor: Toka Diagana

Copyright © 2012 X. Yang and X. Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 a : G ^ Aut(A) be an action of a finite group G on A which has the tracial Rokhlin property. Then A xa 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 1(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 t (p) < e for every tracial state t 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

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 Rordam [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 a be an action of a finite group G on A which has the tracial Rokhlin property. If A is an AF-algebra, then A xa G has tracial rank zero. If A is an AT-algebra with the (SP)-property, then A xa G has tracial rank no more than one. If A has stable rank one and real rank zero, then the induced crossed product A xa G has these two properties.

2. Definitions and Preliminaries

We denote by 1(0) the class of finite dimensional C*-algebras and by 1(k) the class of C*-algebras with the form p(C(X)®F)p, where F € 1(0), X is a finite CW complex with dimension k, and p € C(X) 0 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, e > 0. If ||a - b\\ < e; then we write a ae b. If there exists an element y € F such that ||x - y\\ < e, then we write

X €e F.

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 e > 0, any finite subset F C A and any nonzero positive element b € A, there exist a nonzero projection p € A and a C*-subalgebra B c A with 1B = p and B € 1(k) (RR(B) = 0; tsr(B) = 1, resp.) such that

(1) \\pa - ap\\ < e for any a €F,

(2) pap €£ B 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 a : G ^ Aut(A) be an action of a finite group G on A. We say that a has the tracial Rokhlin property if, for every e > 0, every finite set F ^ A, every positive element b £ A, there are mutually orthogonal projections {eg : g £ G} such that

(1) ||ag(eh) - egh|| < e for all g,h £ G,

(2) ||ega - aeg|| < e for all g £ G and all a £F,

(3) with e = Xg£G eg, [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 a : G ^ Aut (A) be an action of a finite group G on A which has the tracial Rokhlin property. Then A xa G is simple.

Lemma 2.5 (see [20, Theorem 4.2]). Let A be a simple unital C*-algebra with the (SP)-property, and let a : G ^ Aut(A) be an action of a discrete group G on A. Suppose that the normal subgroup N = {g £ G | ag is inner on A} of G is finite; then any nonzero hereditary C*-subalgebra of the crossed product A xa G has a nonzero projection which is Murray-von Neumann equivalent to a projection in A N.

If the action a : G ^ Aut(A) has the tracial Rokhlin property, then each ag is outer for all g £ G \ {1}. So N = {g £ G | ag is inner on A} = {1}. Since A x„N = A Xa {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 a : 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 xa G 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 p\ < p, q < 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 Ai £ C for i = 1,2,...,n, then ®nk=lAk £ C.

(3) If A £C and n £ N, then Mn(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 € Mn(A), the corner pMn(A)p is semiprojective and finitely generated;

(2) for every A €C and every ideal I c A, there is an increasing sequence I1 c I2 c ••• of ideals of A such that U^=1In = 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 = {0n=1Mk(i) | n,k(i) € N}; that is, C contains all finite dimensional algebras. C is finitely saturated and flexible.

(2) Let C = {0"iC(Xi,Mk(i)) | n,k(i) € N; each Xi is a closed subset of the circle}. We can show that C is finitely saturated and flexible.

(3) Let C = {/ € 0n=1C([0,1],Mk(i)) | n,k(i) € N,/(0) is scalar}. We can also show that C is finitely saturated and flexible.

(4) For some d € N, let Cd contain all the C*-algebras 0n=1piC(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 [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 limn^^(An,$n), where each An is in the finite saturation of C and each homomorphism $n : An ^ An+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 e > 0 and every finite subset F c A, there is a C*-algebra B in the finite saturation of C and a *-homomorphism $ : B ^ A such that a €e $(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 e > 0, any finite subset F c 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 $ : B ^ A with 1$(B) = p, such that

(1) \pa - ap\ < e for any a €F,

(2) pap €£ $(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, 6 > 0, a unital C*-algebra A, if wij, for 1 < i, j < n, are elements of A, such that \\wij|| < 1 for 1 < i, j < n, such that \\w*j - w^W <6 for 1 < i, j < n, such that \\wiuh wi2j2 - 6j1ii2 wi1rj2 W <6 for 1 < i1,iï,j1,jï < n, and such that wiii are mutually orthogonal projections, we say that wiij (1 < i,j < n) form a 6-approximate system of n x 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 e> 0, there exists 6 = 6(n,e) > 0 such that, whenever (fiij)1<irj<n is a system of matrix units for Mn, whenever B is a unital C*-algebra, and whenever i, j < n, are elements of B which form a 6-approximate system of n x n matrix units, then there exists a *-homomorphism $ : Mn ^ B such that $(firi) = wiri for 1 < i < n and Wtyfbj) - wi,j\\ < efor 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 a : G ^ Aut (A) be an action of a finite group G on A which has the tracial Rokhlin property. Then A xaG is a simple unital tracial C-algebra.

Proof. By Lemma 2.4, A xa G is a simple unital C*-algebra. By Definition 2.13, it suffices to show the following.

For any e > 0, any finite subset F = Fo U {ug | g £ G} c A xa G, where Fo is a finite subset of the unit ball of A and Ug £ A xa G is the canonical unitary implementing the automorphism ag, and any nonzero positive element b £ A xa G, there exist a nonzero projection p £ A xa G, a C*-algebra B in the finite saturation of C, and a *-homomorphism $ : B ^ A xa G with 1$(B) = p, such that

(1) ||pa - ap|| < e for any a £F,

(2) pap £e $(B) for all a £F,

(3) [1 - p] < [b] in A xa G.

By Lemma 2.6, there exists a nonzero projection q £ A such that [q] < [b] in A xa G. 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 qi, q2 £ A such that qi + q2 < q.

Set n = card(G) and set e0 = e/48n. Choose 6 > 0 according to Lemma 2.15 for n given above and e0 in place of e. Moreover we may require 6 < min{e/72n,e/(24n(n - 1))}.

Apply Definition 2.3 with Fo given above, with 6 in place of e, with q1 in place of b. There exist mutually orthogonal projections eg £ A for g £ G such that

(1') ||ag(eh) - egh|| <6 for all g,h £ G,

(2') ||ega - aeg|| < 6 for all g £ G and all a £ Fo,

(3') [1 - e] < [q1] in A, where e = XgeG eg.

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

Define wgh = ugh-1 eh for g,h £ G. By the proof of Theorem 2.2 of [14], we can estimate that wg,h (g, h £ G) form a 6-approximate system of n x n matrix units in A xa G. Moreover,

XgeG wg,g = geG eg = e.

Let (/g,h)grheG be a system of matrix units for Mn. By Lemma 2.15, there exists a *-homomorphism $0 : Mn ^ A xa G such that

\\tyo(fg,h) - wg,h\\ <eo

for all g,h e G, and ty0(fg,g) = eg for all g e G.

Set E = Mn 0 e1Ae1. Define an injective unital *-homomorphism ty1 : E ^ e(A xa G)e

$i(fg,h 0 a) = tyofg,1)atyo fh for all g,h e G and a e e1 Ae1. Then

tyi(lMn 0e1) = Yueg = e, faifu 0 a) = a

for all a e e1Ae1 and

ty1fg,h 0e0 = fa(fg,0e1fa(f1 h) = fa(fg,0tyo(fu)tyo(f1,h) = fa(fg,h) = egfa(fg,h)eh.

By (2'), for all a e F0, we have

|ae - ea|| < ^ \\aeg - ega\\ < nö.

By (1'), for all g e G, we get

Uge - eug\\ < \\MgeMg-1 - e\\ =

IX (eh) - X egh

heG heG

< nö.

For all g e G, we have

euge - ^ $1(fgh,h 0 e1)

< \\euge - uge\\ +

Uge - ^h(fgh,h 0e1)

< nö +

Uge - ^ ty1(fgh,h 0 e1)

^^ugeh - ^ty1(fgh,h 0 e1) heG heG

^wgh,h - Xty0(fgh,h)

heG heG

< nö + ne0 <

5e 144.

Abstract and Applied Analysis That is, for all g € G, we have

euge e5e/144ty1 (E).

Set b = XgeG fg,g 0 e1ag-1 (a)e1; then b e E. Using ||egaeh - aegeh|| < ö, we get

eae - ^egae

< y, \\egaeh\\ < n(n - 1)ö.

We also have

\\ty0(fg,1)e1 - uge1\\ < W^C/gO - %e1\\ = \\ty0(fg,1) - wg,1\ < so, \\tyo(f1,g) - e1ug-1 \\ < \\tyo(f1,g) - ug-1 eg\\ + \\ug-1 eg - e1ug-1 \\ < so + ö, (3.10) \\e1«g-1 (a)e1 - ag-1 (egaeg) \\ < 2ö.

Then, for all a € Fo, we have

|eae - ty1(b) \\ =

- tyAZ fg,

g 0 e1ag-1 (a)e1

eae - ^tyo (fg,1)e!«g-1 (a)e1tyo (f1,g)

eae - ^uge1ag-1 (a)e1ug-1

- X ugag-1(egaeg)ug-1

+ 2ns0 + nö

+ 3nö + 2ns0

(3.11)

That is, for all a e Fo,

eae - ^egae

■ 3nö + 2nso

eeee < n(n - 1)ö + 3nö + 2nso < — + tt + tt = - .

o 24 24 24 8

eae e£/8ty1(E).

(3.12)

By (3.8) and (3.12), we can write

eae e£/8ty1(E)

(3.13)

for all a € F.

F =\ b | b € E, || $1 (b) - eae\\ <8 for a e^. (3.14)

By Lemma 2.14, E is a simple unital tracial C-algebra. Apply Definition 2.13 with F given above, with e/8 in place of e and f1r1 ® 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 f0: B ^ E with 1V0(B) = p0, such that

(1'') ||p0b - bp0|| < e/8 for any b £ F,

(2'') p0bp0 ££/8 f0(B) for all b £ F,

(3'') [1e - pa] < [fn ® V1] in E.

Set p = $1 (p0) and $ = $1 o f0: B ^ e(A xa G)e.

For every a £F, there exists b £ F such that $1 (b) «e/8 eae. Then

pa = pea «n6 peae «e/8 p$1(b) = $1(p0b) «e/8 $1 (bp0) «n6+e/4 ap. (3.15)

That is,

||pa - ap|| < 2n6 + 2 < e.

(3.16)

Let c £ B such that p0bp0 «e/8 ^0(c). Then

pap = peaep «e/8 p$1(b)p = $^p0bp^ «e/8 $1((^0(c))) = $(c). (3.17)

Hence,

pap £e $(B). (3.18)

By (3''),in A xa G, [e - p] = [$1 (1e - p0)] < [$1 (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 xa G is a simple unital tracial C-algebra. □

Corollary 3.2. Let A be an infinite dimensional separable simple unital C*-algebra, and let a : 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 xaG has tracial rank zero. If A is an AT-algebra with the (SP)-property, then the induced crossed product A xa G 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 xa G is a simple unital tracial C-algebra. By the definition of tracial rank zero, TR(A xa G) = 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 xa G 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 xa G) < 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 a : 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 xa G has stable rank one. If A has real rank zero, then the induced crossed product A xa G 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 xa G is a simple unital tracial C-algebra, that is, for any e > 0, any finite subset F c A xa G, and any nonzero positive element b € A xa G, there exist a nonzero projection p € A xa G, a C*-algebra B in C, and a *-homomorphism $ : B ^ A xa G with 1$(B) = p, such that

(1) ||pa - ap|| < e for any a €F,

(2) pap €e $(B) for all a €F,

(3) [1 - p] < [b] in A xaG.

Hence, Tsr(A xa G) = 1. By Lemma 2.2, tsr(A xa G) = 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 xa G is a simple unital tracial C-algebra. Hence TRR(A xa G) = 0. By Lemma 2.2, RR(A xa G) = 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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