Abstract and Applied Analysis

Volume 2011 (2011), Article ID 503679, 10 pages

http://dx.doi.org/10.1155/2011/503679

## Certain -Monoid Properties Preserved by Tracial Approximation

Department of Mathematics, Shanghai Maritime University, 1550 Haigang Avenue New Harbor City, Shanghai 201306, China

Received 31 August 2011; Revised 16 October 2011; Accepted 30 October 2011

Academic Editor: Giovanni Galdi

Copyright © 2011 Qingzhai Fan. 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 show that the following -monoid properties of -algebras in the class are inherited by simple unital -algebras in the class : (1) pseudocancellation property, (2) weakly divisible, (3) strongly separative, (4) separative, and (5) preminimal.

#### 1. Introduction

The Elliott conjecture asserts that all nuclear, separable -algebras are classified up to isomorphism by an invariant, called the Elliott invariant. A first version of the Elliott conjecture might be said to have begun with the K-theoretical classification of AF-algebras in [1]. Since then, many classes of -algebras have been found to be classified by the Elliott invariant. Among them, one important class is the class of simple unital AH-algebras. A very important axiomatic version of the classification of AH-algebras without dimension growth was given by H. Lin. Instead of assuming inductive limit structure, he started with a certain abstract approximation property and showed that -algebras with this abstract approximation property and certain additional properties are AH-algebras without dimension growth. More precisely, Lin introduced the class of tracially approximate interval algebras.

Following the notion of Lin on the tracial approximation by interval algebras, Elliott and Niu in [2] considered tracial approximation by more general -algebras. Let be a class of unital -algebras. Then, the class of -algebras which can be tracially approximated by -algebra in , denoted by , is defined as follows. A simple unital -algebra is said to belong to the class , if, for any , any finite subset , and any nonzero element , there exist a nonzero projection and a -subalgebra of with and , such that(1) for all ,(2) for all , (3) is Murray-von Neumann equivalent to a projection in .

The question of the behavior of -algebra properties under passage from a class to the class is interesting and sometimes important. In fact, the property of having tracial states, the property of being of stable rank one, and the property that the strict order on projections is determined by traces were used in the proof of the classification theorem in [2], and [3] by Elliott and Niu.

In this paper, we show that the following -monoid properties of -algebras in the class are inherited by simple unital -algebras in the class :(1)pseudocancellation property,(2)weakly divisible,(3)strongly separative,(4)separative,(5)preminimal.

#### 2. Preliminaries and Definitions

Let and be two positive elements in a -algebra . We write (cf. Definition 3.5.2 in [4]), if there exists a partial isometry such that, for every , , , , where is the range projection of in and . We write if . Let be a positive integer. We write , if there are mutually orthogonal positive elements such that , .

Let be two positive numbers. Define

Let be a class of unital -algebras. Then, the class of -algebras which can be tracially approximated by -algebras in is denoted by .

*Definition 2.1 (see [2]). *A simple unital *-*algebra is said to belong to the class if, for any *, *any finite subset *,* and any nonzero element *, *there exist a nonzero projection and a *-*subalgebra of with and *, *such that(1) for all ,(2) for all ,(3).

*Definition 2.2 (see [5]). *Let be a class of unital -algebras. A unital *-*algebra is said to have property if, for any positive numbers *, *any *,* any finite subset *, *any nonzero positive element *,* and any integer *, *there exist a nonzero projection *, *and a -subalgebra of with and *, *such that(1) for all ,(2) for all , ,(3).

Lemma 2.3 (see [2]). *If the class is closed under tensoring with matrix algebras or closed under taking unital hereditary -subalgebras, then is closed under passing to matrix algebras or unital hereditary -subalgebras.*

Theorem 2.4 (see [5]). *Let be a class of unital -algebras such that is closed under taking unital hereditary -subalgebras and closed taking finite direct sums. Let be a simple unital -algebra. Then, the following are equivalent: *(1)*,*(2)* has property .*

Call projections equivalent, denoted , when there is a partial isometry such that , . The equivalent classes are denoted by , and the set of all these is Addition in is defined by becomes an abelian monoid, and we call the -monoid of .

All abelian monoids have a natural preorder, the algebraic ordering, defined as follow: if , we write if there is a in such that . In the case of , the algebraic ordering is given by Murray-von Neumann subequivalence, that is, if and only if there is a projection such that . We also write, as is customary, to mean that is subequivalent to .

If , we will write if there is a nonzero element in , such that .

Let us recall that an element in a monoid is an order unit provided , and, for any in , there is such that .

Let be an order monoid and . We write if and only if there exists an integer such that . We write if and only if .

We say that a monoid is conical if only when . Note that, for any -algebra , the monoid is conical.

We say that an order monoid has the pseudocancellation property when it satisfies the statement that, for any with , there exists such that .

Let be a monoid. An element in will be termed weakly divisible if there exist and in such that . We say that is weakly divisible if every element is weakly divisible. We say that has weak divisible for order units if every unit is weakly divisible.

We say that an order monoid is said to be strongly separative when it satisfies the statement that, for any such that , we have .

*Definition 2.5 (see [6]). *We say that an order monoid is preminimal when it satisfies both following statements:(1) for any with and ,(2) for any with and .

*Definition 2.6 (see [6]). *We say that an order monoid is separative when it satisfies both following statements:(1) for any with and ,(2) for any with and .

#### 3. The Main Results

Theorem 3.1. *Let be a class of unital -algebras such that for any the -monoid has the pseudocancellation property. Then, the -monoid has the pseudocancellation property for any simple unital -algebra .*

*Proof. *We need to show that there exists such that for any with . By Lemma 2.3, we may assume that , , for some projections . For , any , since , there exist a projection and a -subalgebra with such that(1) for all ,(2) for all .

By and , there exist projections and such that
Therefore, we have

Since and has the pseudocancellation property, we may assume that there exists a projection such that and in .

For , any , since , there exist a projection and a -subalgebra with such that

for all , for all ,.

By and , there exist projections and such that
Therefore, we have

Since and has the pseudocancellation property, we may assume that there exists a projection such that and in .

By , we have , there exists a partial isometry such that , .

Therefore, we have

Since , therefore , that is, .

Theorem 3.2. *Let be a class of unital -algebras such that, for any , the -monoid is weakly divisible. Then, the -monoid is weakly divisible for any simple unital -algebra .*

*Proof. *We need to show that there exist and in such that for any . By Lemma 2.3, we may assume that for some projection . For , any , since , there exist a projection and a -subalgebra with such that(1),(2).

By and , there exist projections and such that
Therefore, we have

Since and is weakly divisible, we may assume that there exist projections such that in .

For , any , since , there exist a projection and a -subalgebra with , such that

for all , for all ,.

By and , there exist projections and such that
Therefore, we have

Since and is weakly divisible, we may assume that there exist projections such that in .

By , we have , and there exist a partial isometry such that , .

Therefore, we have

Theorem 3.3. *Let be a class of unital -algebras such that, for any , the -monoid is strongly separative. Then, the -monoid is strongly separative for any simple unital -algebra .*

*Proof. *We need to show that for any with . By Lemma 2.3, we may assume that , for some projections . For , any , any positive numbers , since , by Theorem 2.4, there exist a projection and a -subalgebra with , such that(1) for all ,(2) for all ,(3).

By and , there exist projections and such that
Therefore, we have

Since and is strongly separative, we have in .

By , we have , there exists a partial isometry such that , . Therefore, we have

Theorem 3.4. *Let be a class of unital -algebras such that, for any , the -monoid is separative. Then, the -monoid is separative for any simple unital -algebra .*

*Proof. *We prove this theorem by two steps.

Firstly, we need to show that for any with and . By Lemma 2.3, we may assume that , , for some projections . For , any , since , there exist a projection and a -subalgebra with such that(1) for all ,(2) for all .

By and , there exist projections and such that
Therefore, we have
Since and is separative, we have in .

For , any , since , there exist a projection and a -subalgebra with such that for all , for all ,.

By and , there exist projections and such that
Therefore, we have

Since and is separative, we have in .

By , we have , there exists a partial isometry such that , .

Therefore, we have

Secondly, with the same methods and technique, we can show that for any with and .

Theorem 3.5. *Let be a class of unital -algebras such that, for any , the -monoid is a preminimal monoid. Then, the -monoid is a preminimal monoid for any simple unital -algebra .*

*Proof. *We prove this theorem by two steps.

Firstly, we need to show that for any with and . By Lemma 2.3, we may assume that , , , for some projections . For , any , since , there exist a projection and a -subalgebra with such that(1) for all ,(2) for all .

By and , there exist projections and such that
Therefore, we have
Since and is preminimal, we have in .

Since , there exists a partial isometry such that . Therefore, we have
Secondly, with the same methods and technique, we can show that for any with and .

#### Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 11101268) and Science or Technology Program of Shanghai Maritime University (no. 20110052).

#### References

- G. A. Elliott, “On the classification of inductive limits of sequences of semisimple finite-dimensional algebras,”
*Journal of Algebra*, vol. 38, no. 1, pp. 29–44, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. A. Elliott and Z. Niu, “On tracial approximation,”
*Journal of Functional Analysis*, vol. 254, no. 2, pp. 396–440, 2008. View at Publisher · View at Google Scholar - Z. Niu,
*A classification of certain tracially approximately subhomogeneous C*-algebras*, Ph.D. thesis, University of Toronto, 2005. - H. Lin,
*An Introduction to the Classification of Amenable C*-Algebras*, World Scientific, River Edge, NJ, USA, 2001. - Q. Fan and X. Fang, “Non-simple tracial approximation,”
*Houston Journal of Mathematics*, vol. 37, pp. 1249–1263, 2011. View at Google Scholar - F. Wehrung, “Restricted injectivity, transfer property and decompositions of separative positively ordered monoids,”
*Communications in Algebra*, vol. 22, no. 5, pp. 1747–1781, 1994. View at Publisher · View at Google Scholar