Abstract and Applied Analysis

Volume 2012 (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

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

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

Received 11 September 2012; Accepted 14 October 2012

Academic Editor: Toka Diagana

Copyright © 2012 Xinbing Yang and Xiaochun 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.

Abstract.

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

1. Introduction

In this paper, our purpose is to prove that certain classes of separable unital -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 -algebras since Lin introduced the concept of tracial rank of -algebras in [1]. The notion of tracial rank was motivated by the Elliott program of classification of nuclear -algebras. -algebras with tracial rank no more than for some are -algebras that can be locally approximated by -subalgebras in after cutting out a “small” approximately central projection . The term “tracial” come from the fact that, in good cases, the projection is “small” if for every tracial state on . The -algebras of tracial rank zero can be determined by -theory and hence can be classified. For example, Lin proved that if a simple separable amenable unital -algebra has tracial rank zero and satisfies the Universal Coefficient Theorem, then 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 -algebras with tracial rank zero.

These successes suggest that one consider “tracial” versions of other -algebra concepts. In [5], Yao and Hu introduced the concept of tracial real rank of -algebras. In [6], Fan and Fang introduced the concept of tracial stable rank of -algebras. In [7, 8], Elliott and Niu and Fang and Fan studied the general concept of tracial approximation of properties of -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 -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 -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 -algebras and prove that, for appropriate classes of -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 be a separable simple unital -algebra, and let be an action of a finite group on which has the tracial Rokhlin property. If is an AF-algebra, then has tracial rank zero. If is an A-algebra with the (SP)-property, then has tracial rank no more than one. If has stable rank one and real rank zero, then the induced crossed product has these two properties.

2. Definitions and Preliminaries

We denote by the class of finite dimensional -algebras and by the class of -algebras with the form , where , is a finite CW complex with dimension , and is a projection.

Let be projections in and . If is Murray-von Neumann equivalent to , then we write . If is Murray-von Neumann equivalent to a subprojection of , then we write .

Let be a -algebra, and let be a subset of . If ; then we write . If there exists an element such that , then we write .

Definition 2.1 (see [19, Definition  3.6.2], [5, Definition 1.4,], and [6, Definition 2.1]). Let be a simple unital -algebra and . is said to have tracial rank no more than ; write ; (tracial real rank zero, write ; tracial stable rank one, write ), if for any , any finite subset and any nonzero positive element , there exist a nonzero projection and a -subalgebra with and (; , resp.) such that

(1) for any ,(2) for all ,(3).

If, furthermore, , then we say .

Lemma 2.2 (see [5, Theorem 3.3], [6, Theorem 3.3]). Let be a simple unital -algebra. If , then . If and has the (SP)-property, then .

Definition 2.3 (see [13, Definition 1.2]). Let be an infinite dimensional finite simple separable unital -algebra, and let be an action of a finite group on . We say that has the tracial Rokhlin property if, for every , every finite set , every positive element , there are mutually orthogonal projections such that

(1) for all ,(2) for all and all ,(3)with , .

Lemma 2.4 (see [13, Corollary 1.6]). Let be an infinite dimensional finite simple separable unital -algebra, and let be an action of a finite group on which has the tracial Rokhlin property. Then is simple.

Lemma 2.5 (see [20, Theorem 4.2]). Let be a simple unital -algebra with the (SP)-property, and let be an action of a discrete group on . Suppose that the normal subgroup of is finite; then any nonzero hereditary -subalgebra of the crossed product has a nonzero projection which is Murray-von Neumann equivalent to a projection in .

If the action has the tracial Rokhlin property, then each is outer for all . So . Since , by Lemma 2.5 we have the following lemma.

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

Lemma 2.7 (see [19, Lemma ]). Let be a simple -algebra with the (SP)-property, and let be two nonzero projections. Then there are nonzero projections such that .

Definition 2.8 (see [14, Definition 1.1]). Let be a class of separable unital -algebras. We say that is finitely saturated if the following closure conditions hold.

(1)If and , then .(2)If for , then .(3)If and , then .(4)If and is a nonzero projection, then .

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

Definition 2.9 (see [14, Definition 1.2]). Let be a class of separable unital -algebras. We say that is flexible if.

(1)for every , every , and every nonzero projection , the corner is semiprojective and finitely generated;(2)for every and every ideal , there is an increasing sequence of ideals of such that and such that for every the -algebra is in the finite saturation of .

Example 2.10. Let ; that is, contains all finite dimensional algebras. is finitely saturated and flexible.

Let . We can show that is finitely saturated and flexible.

Let . We can also show that is finitely saturated and flexible.

For some , let contain all the -algebras , where , each is a nonzero projection in , and each is a compact metric space with covering dimension at most . The class is not flexible for (see [14] Example 2.9).

Definition 2.11 (see [16, Definition 1.4]). Let be a class of separable unital -algebras. A unital approximate -algebra is a -algebra which is isomorphic to an inductive limit , where each is in the finite saturation of and each homomorphism is unital.

Definition 2.12 (see [14, Definition 1.5]). Let be a class of separable unital -algebras. Let be a separable unital -algebra. We say that is a unital local -algebra if, for every and every finite subset , there is a -algebra in the finite saturation of and a -homomorphism such that for all .

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

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

Let be a class as of Example 2.10. Then a unital A-algebra is a unital approximate -algebra and is a unital local -algebra.

Definition 2.13. Let be a simple unital -algebra, and let be a class of separable unital -algebra. We say that is a tracial -algebra if, for any , any finite subset , and any nonzero positive element , there exist a nonzero projection , a -algebra in the finite saturation of , and a -homomorphism with , such that

(1) for any ,(2) for all ,(3) in .

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

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

For , , a unital -algebra , if , for , are elements of , such that for , such that for , such that for , and such that are mutually orthogonal projections, we say that form a -approximate system of matrix units in .

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

Lemma 2.15. For any , any , there exists such that, whenever is a system of matrix units for , whenever is a unital -algebra, and whenever , for , are elements of which form a -approximate system of matrix units, then there exists a -homomorphism such that for and for .

3. Main Results

Theorem 3.1. Let be any class of separable unital -algebras. Let be an infinite dimensional finite simple unital tracial -algebra with the (SP)-property, and let be an action of a finite group on which has the tracial Rokhlin property. Then is a simple unital tracial -algebra.

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

For any , any finite subset , where is a finite subset of the unit ball of and is the canonical unitary implementing the automorphism , and any nonzero positive element , there exist a nonzero projection , a -algebra in the finite saturation of , and a -homomorphism with , such that

(1) for any ,(2) for all ,(3) in .

By Lemma 2.6, there exists a nonzero projection such that in .

Since is an infinite dimensional simple unital -algebra with the (SP)-property, by [19, Lemma ], there exist orthogonal nonzero projections such that .

Set and set . Choose according to Lemma 2.15 for given above and in place of . Moreover we may require .

Apply Definition 2.3 with given above, with in place of , with in place of . There exist mutually orthogonal projections for such that

) for all ,() for all and all ,() in , where .

By Lemma 2.7, there are nonzero projections such that and .

Define for . By the proof of Theorem 2.2 of [14], we can estimate that form a -approximate system of matrix units in . Moreover, .

Let be a system of matrix units for . By Lemma 2.15, there exists a -homomorphism such that

for all , and for all .

Set . Define an injective unital -homomorphism by

for all and . Then

for all and

By (), for all , we have

By (), for all , we get

For all , we have

That is, for all , we have

Set ; then . Using , we get

We also have

Then, for all , we have

That is, for all ,

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

for all .

Write

By Lemma 2.14, is a simple unital tracial -algebra. Apply Definition 2.13 with given above, with in place of and in place of . There exist a nonzero projection , a -algebra in the finite saturation of , and a -homomorphism with , such that

(1′′) for any ,(2′′) for all ,(3′′) in .

Set and .

For every , there exists such that . Then

That is,

Let such that . Then

Hence,

By (), in , . Therefore,

From (3.16), (3.18), and (3.19), is a simple unital tracial -algebra.

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

Proof. If is an AF-algebra, then is a unital local -algebra, where is a class of -algebras satisfying condition of Example 2.10. By Theorem 3.1, we know that is a simple unital tracial -algebra. By the definition of tracial rank zero, .

If is an A-algebra, then is a unital local -algebra, where is a class of -algebra satisfying condition of Example 2.10. By Theorem 3.1, we know that is a simple unital tracial -algebra. Since the covering dimension of closed subsets of the circle is no more than one, by the definition of tracial rank, .

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

Corollary 3.3. Let be an infinite dimensional finite separable simple unital -algebra with the (SP)-property, and let be an action of a finite group on which has the tracial Rokhlin property. If has stable rank one, then the induced crossed product has stable rank one. If has real rank zero, then the induced crossed product has real rank zero.

Proof. Let be the class of all separable unital -algebras with stable rank one. By Theorems , 3.18, and 3.19 in [19], we have that is finitely saturated and satisfies condition of Definition 2.9. By Theorem 3.1, the crossed product is a simple unital tracial -algebra, that is, for any , any finite subset , and any nonzero positive element , there exist a nonzero projection , a -algebra in , and a -homomorphism with , such that

(1) for any ,(2) for all ,(3) in .

Hence, . By Lemma 2.2, .

Let be the class of all separable unital -algebras with real rank zero. We can use the same argument to show that the crossed product is a simple unital tracial -algebra. Hence . By Lemma 2.2, .

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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