# CertainK0-Monoid Properties Preserved by Tracial Approximation

Authors: Qingzhai Fan
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We show that the followingK0-monoid properties of C*-algebras in the class Ω are inherited by simple unital C*-algebras in the class TAΩ: (1) pseudocancellation property, (2) weakly divisible, (3) strongly separative, (4) separative, and (5) preminimal.


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## Body

## 1. Introduction

The Elliott conjecture asserts that all nuclear, separableC*-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 C*-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 C*-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 C*-algebras. Let Ω be a class of unital C*-algebras. Then, the class of C*-algebras which can be tracially approximated by C*-algebra in Ω, denoted by TAΩ, is defined as follows. A simple unital C*-algebra A is said to belong to the class TAΩ, if, for any ε>0, any finite subset F⊆A, and any nonzero element a≥0, there exist a nonzero projection p∈A and a C*-subalgebra B of A with 1B=p and B∈Ω, such that(1)
∥xp-px∥<ε for all x∈F,(2)
pxp∈Bε for all x∈F,(3)
1-p is Murray-von Neumann equivalent to a projection in aAa¯.The question of the behavior ofC*-algebra properties under passage from a class Ω to the class TAΩ 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 followingK0-monoid properties of C*-algebras in the class Ω are inherited by simple unital C*-algebras in the class TAΩ:(1)
pseudocancellation property,(2)
weakly divisible,(3)
strongly separative,(4)
separative,(5)
preminimal.


## 2. Preliminaries and Definitions

Leta and b be two positive elements in a C*-algebra A. We write [a]≤[b] (cf. Definition  3.5.2 in [4]), if there exists a partial isometry v∈A** such that, for every c∈Her(a), v*c, cv∈A, vv*=Pa, where Pa is the range projection of a in A** and v*cv∈Her(b). We write [a]=[b] if v*Her(a)v=Her(b). Let n be a positive integer. We write n[a]≤[b], if there are n mutually orthogonal positive elements b1,b2,…,bn∈Her(b) such that [a]≤[bi], i=1,2,…,n.Let0<σ1<σ2≤1 be two positive numbers. Define
(2.1)fσ1σ2(t)={1,ift≥σ2,t-σ1σ2-σ1,ifσ1≤t≤σ2,0,if0<t≤σ1.LetΩ be a class of unital C*-algebras. Then, the class of C*-algebras which can be tracially approximated by C*-algebras in Ω is denoted by TAΩ.Definition 2.1 (see [2]).
A simple unitalC*-algebra A is said to belong to the class TAΩ if, for any ε>0,any finite subset F⊆A, and any nonzero element a≥0,there exist a nonzero projection p∈A and a C*-subalgebra B of A with 1B=p and B∈Ω,such that(1)
∥xp-px∥<ε for all x∈F,(2)
pxp∈Bε for all x∈F,(3)
[1-p]≤[a].Definition 2.2 (see [5]).
LetΩ be a class of unital C*-algebras. A unital C*-algebra A is said to have property (III) if, for any positive numbers 0<σ3<σ4<σ1<σ2<1,any ε>0, any finite subset F⊆A,any nonzero positive element a, and any integer n>0,there exist a nonzero projection p∈A,and a C*-subalgebra B of A with B∈Ω and 1B=p,such that(1)
∥xp-px∥<ε for all x∈F,(2)
pxp∈Bε for all x∈F, ∥pap∥≥∥a∥-ε,(3)
n[fσ1σ2((1-p)a(1-p))]≤[fσ3σ4(pap)].Lemma 2.3 (see [2]).
If the classΩ is closed under tensoring with matrix algebras or closed under taking unital hereditary  C*-subalgebras, then  TAΩ is closed under passing to matrix algebras or unital hereditary  C*-subalgebras.Theorem 2.4 (see [5]).
LetΩ be a class of unital  C*-algebras such that  Ω is closed under taking unital hereditary  C*-subalgebras and closed taking finite direct sums. Let  A be a simple unital  C*-algebra. Then, the following are equivalent: (1)
A∈TAΩ,(2)
A has property (III).Call projectionsp,q∈M∞(A) equivalent, denoted p~q, when there is a partial isometry v∈M∞(A) such that p=v*v, q=vv*. The equivalent classes are denoted by [·], and the set of all these is(2.2)V(A):={[p]∣p=p*=p2∈M∞(A)}.
Addition in V(A) is defined by(2.3)[p]+[q]≔[diag⁡(p,q)].V(A) becomes an abelian monoid, and we call V(A) the K0-monoid of A.All abelian monoids have a natural preorder, the algebraic ordering, defined as follow: ifx,y∈M, we write x≤y if there is a z in M such that x+z=y. In the case of V(A), the algebraic ordering is given by Murray-von Neumann subequivalence, that is, [p]≤[q] if and only if there is a projection p′≤q such that p~p′. We also write, as is customary, p⪯q to mean that p is subequivalent to q.Ifx,y∈M, we will write x≤*y if there is a nonzero element z in M, such that x+z=y.Let us recall that an elementu in a monoid M is an order unit provided u≠0, and, for any x in M, there is n∈ℕ such that x≤nu.LetM be an order monoid and x,y∈M. We write x⪯y if and only if there exists an integer n>0 such that x≤ny. We write x≪y if and only if x+y=y.We say that a monoidM is conical if x+y=0 only when x=y=0. Note that, for any C*-algebra A, the monoid V(A) is conical.We say that an order monoidM has the pseudocancellation property when it satisfies the statement that, for any a,b,c,d with a+c≤b+c, there exists d≪c such that a≤b+d.LetM be a monoid. An element x in M will be termed weakly divisible if there exist a and b in M such that x=2a+3b. We say that M is weakly divisible if every element is weakly divisible. We say that M has weak divisible for order units if every unit is weakly divisible.We say that an order monoidM is said to be strongly separative when it satisfies the statement that, for any x,y∈M such that 2x=x+y, we have x=y.Definition 2.5 (see [6]).
We say that an order monoidM is preminimal when it satisfies both following statements:(1)
a+d≤b+d for any a,b,c,d with a+c≤b+c and c≤d,(2)
a+d=b+d for any a,b,c,d with a+c=b+c and c≤d.Definition 2.6 (see [6]).
We say that an order monoidM is separative when it satisfies both following statements:(1)
a≤b for any a,b,c with a+c≤b+c and c⪯b,(2)
a=b for any a,b,c with a+c=b+c and c⪯a,b.


## 3. The Main Results

Theorem 3.1.
LetΩ be a class of unital  C*-algebras such that for any  B∈Ω the  K0-monoid  V(B) has the pseudocancellation property. Then, the  K0-monoid  V(A) has the pseudocancellation property for any simple unital  C*-algebra  A∈TAΩ.Proof.
We need to show that there existsd≪c such that a≤b+d for any a,b,c,d∈V(A) with a+c≤b+c. By Lemma 2.3, we may assume that a=[p], b=[q], c=[e] for some projections p,q,e∈proj(A). For F={p,q,e}, any ε>0, since A∈TAΩ, there exist a projection r∈A and a C*-subalgebra B⊆A with B∈Ω,1B=r such that(1)
∥xr-rx∥<ε for all x∈F,(2)
rxr∈Bε for all x∈F.
By(1) and (2), there exist projections p1,q1,e1∈B and p2,q2,e2∈(1-r)A(1-r) such that(3.1)‖p-p1-p2‖<ε,‖q-q1-q2‖<ε,‖e-e1-e2‖<ε.
Therefore, we have
(3.2)[p]=[p1]+[p2],[q]=[q1]+[q2],[e]=[e1]+[e2],[p1]+[e1]≤[q1]+[e1],[p2]+[e2]≤[e2]+[q2].
SinceB∈Ω and V(B) has the pseudocancellation property, we may assume that there exists a projection f∈A such that [f]≪[e1] and [p1]≤[q1]+[f] in V(B).
ForG={p2,q2,e2,f}, any ε>0, since A∈TAΩ, there exist a projection s∈A and a C*-subalgebra C⊆A with C∈Ω,1C=s such that
(1′)
∥xs-sx∥<ε for all x∈G,(2′)
sxs∈Cε for all x∈G,(3′)
[1-s]≤[f].
By(1′) and (2′), there exist projections p3,q3,e3∈C and p4,q4,e4∈(1-s)A(1-s) such that
(3.3)‖p2-p3-p4‖<ε,‖q2-q3-q4‖<ε,‖e2-e3-e4‖<ε.
Therefore, we have
(3.4)[p2]=[p3]+[p4],[q2]=[q3]+[q4],[e2]=[e3]+[e4],[p3]+[e3]≤[e3]+[q3],[p4]+[e4]≤[e4]+[q4].
SinceC∈Ω and V(C) has the pseudocancellation property, we may assume that there exists a projection g∈A such that [g]≪[e3] and [p3]≤[q3]+[g] in V(C).
By(3′), we have [e4]≤[1-s]≤[f], there exists a partial isometry v∈A such that vv*=e4, v*v≤f.
Therefore, we have(3.5)[p]=[p1]+[p3]+[p4]≤[q1]+[f]+[q3]+[g]+[p4]≤[q1]+[f-v*v]+[vv*]+[q3]+[g]+[p4]≤[q1]+[f-v*v]+[q4]+[e4]+[g]+[q3]≤[q1]+[f]+[q3]+[q4]+[g]≤[q]+[f]+[g].
Since[f]+[g]+[e1]+[e3]=[e1]+[e3], therefore [f]+[g]+[e]=[e], that is, [f]+[g]≪[e].Theorem 3.2.
LetΩ be a class of unital  C*-algebras such that, for any  B∈Ω, the  K0-monoid  V(B) is weakly divisible. Then, the  K0-monoid  V(A) is weakly divisible for any simple unital  C*-algebra  A∈TAΩ.Proof.
We need to show that there exista and b in V(A) such that x=2a+3b for any x∈V(A). By Lemma 2.3, we may assume that x=[p] for some projection p∈proj(A). For F={p}, any ε>0, since A∈TAΩ, there exist a projection r∈A and a C*-subalgebra B⊆A with B∈Ω,1B=r such that(1)
∥pr-rp∥<ε,(2)
rpr∈Bε.
By(1) and (2), there exist projections p1∈B and p2∈(1-r)A(1-r) such that
(3.6)‖p-p1-p2‖<ε.
Therefore, we have
(3.7)[p]=[p1]+[p2].
SinceB∈Ω and V(B) is weakly divisible, we may assume that there exist projections e,f∈B such that [p1]=2[e]+3[f] in V(B).
ForG={p2,e,f}, any ε>0, since A∈TAΩ, there exist a projection s∈A and a C*-subalgebra C⊆A with C∈Ω, 1C=s such that
(1′)
∥xs-sx∥<ε for all x∈G,(2′)
sxs∈Cε for all x∈G,(3′)
3[1-s]≤[e].
By(1′) and (2′), there exist projections p3∈C and p4∈(1-s)A(1-s) such that
(3.8)‖p2-p3-p4‖<ε.
Therefore, we have
(3.9)[p2]=[p3]+[p4].
SinceC∈Ω and V(C) is weakly divisible, we may assume that there exist projections g,h∈B such that [p3]=2[g]+3[h] in V(C).
By(3′), we have 3[p4]≤[e], and there exist a partial isometry v∈A such that vv*=p4, v*v≤e.
Therefore, we have(3.10)[p]=[p1]+[p3]+[p4]=2[e]+3[f]+2[g]+3[h]+[p4]=2[e-v*v]+2[v*v]+[vv*]+3[f]+2[g]+3[h]=2[e-v*v]+3[v*v]+3[f]+2[g]+3[h]=2([e-v*v]+[g])+3([v*v]+[f]+[h]).Theorem 3.3.
LetΩ be a class of unital  C*-algebras such that, for any  B∈Ω, the  K0-monoid  V(B) is strongly separative. Then, the  K0-monoid  V(A) is strongly separative for any simple unital  C*-algebra  A∈TAΩ.Proof.
We need to show thatx=y for any x,y∈V(A) with 2x=x+y. By Lemma 2.3, we may assume that x=[p], y=[q] for some projections p,q∈proj(A). For F={p,q}, any ε>0, any positive numbers 0<σ3<σ4<σ1<σ2<1, since A∈TAΩ, by Theorem 2.4, there exist a projection r∈A and a C*-subalgebra B⊆A with B∈Ω, 1B=r such that(1)
∥xr-rx∥<ε for all x∈F,(2)
rxr∈Bε for all x∈F,(3)
[fσ1σ2((1-r)p(1-r))]≤[fσ3σ4(rpr)].
By(1) and (2), there exist projections p1,q1∈B and p2,q2∈(1-r)A(1-r) such that
(3.11)‖p-p1-p2‖<ε,‖q-q1-q2‖<ε.
Therefore, we have
(3.12)[p]=[p1]+[p2],[q]=[q1]+[q2],[p1]+[p1]=[p1]+[q1],[p2]+[p2]=[p2]+[q2].
SinceB∈Ω and V(B) is strongly separative, we have [p1]=[q1] in V(B).
By(3), we have [p2]≤[p1], there exists a partial isometry v∈A such that v*v=p2, vv*≤p1. Therefore, we have
(3.13)[p]=[p1]+[p2]=[p1-vv*]+[vv*]+[p2]=[p1-vv*]+[p2]+[p2]=[p1-vv*]+[p2]+[q2]=[p1]+[q2]=[q1]+[q2]=[q].Theorem 3.4.
LetΩ be a class of unital  C*-algebras such that, for any  B∈Ω, the  K0-monoid  V(B) is separative. Then, the  K0-monoid  V(A) is separative for any simple unital  C*-algebra  A∈TAΩ.Proof.
We prove this theorem by two steps.
Firstly, we need to show thata≤b for any a,b,c∈V(A) with a+c≤b+c and c⪯b. By Lemma 2.3, we may assume that a=[p], b=[q], c=[t] for some projections p,q,t∈proj(A). For F={p,q,t}, any ε>0, since A∈TAΩ, there exist a projection r∈A and a C*-subalgebra B⊆A with B∈Ω,1B=r such that(1)
∥xr-rx∥<ε for all x∈F,(2)
rxr∈Bε for all x∈F.
By(1) and (2), there exist projections p1,q1,t1∈B and p2,q2,t2∈(1-r)A(1-r) such that
(3.14)‖p-p1-p2‖<ε,‖q-q1-q2‖<ε,‖t-t1-t2‖<ε.
Therefore, we have
(3.15)[p]=[p1]+[p2],[q]=[q1]+[q2],[t]=[t1]+[t2],[p1]+[t1]≤[q1]+[t1],[p2]+[t2]≤[t2]+[q2],[t1]⪯[q1],[t2]⪯[q2].
Since B∈Ω and V(B) is separative, we have [p1]≤[q1] in V(B).
ForG={p1,p2,q2,t2}, any ε>0, since A∈TAΩ, there exist a projection w∈A and a C*-subalgebra C⊆A with C∈Ω,1C=w such that(1′)
∥xw-wx∥<ε for all x∈G,(2′)
wxw∈Cε for all x∈G,(3′)
2[1-w]≤[p1].
By(1′) and (2′), there exist projections p3,q3,t3∈C and p4,q4,t4∈(1-w)A(1-w) such that
(3.16)‖p2-p3-p4‖<ε,‖q2-q3-q4‖<ε,‖t2-t3-t4‖<ε.
Therefore, we have
(3.17)[p2]=[p3]+[p4],[q2]=[q3]+[q4],[t2]=[t3]+[t4],[p3]+[t3]≤[t3]+[q3],[p4]+[t4]≤[t4]+[q4],[t3]⪯[q3],[t4]⪯[q4].
SinceC∈Ω and V(C) is separative, we have [p3]≤[q3] in V(C).
By(3′), we have [t4]≤[1-w]<*[p1], there exists a partial isometry v∈A such that vv*=t4, v*v≤p1.
Therefore, we have(3.18)[p]=[p1]+[p3]+[p4]≤[p1-v*v]+[vv*]+[p3]+[p4]≤[p1-v*v]+[q4]+[p3]+[t4]≤[p1]+[q3]+[q4]≤[q1]+[q3]+[q4]=[q].
Secondly, with the same methods and technique, we can show thata=b for any a,b,c∈V(A) with a+c=b+c and c⪯a,b.Theorem 3.5.
LetΩ be a class of unital  C*-algebras such that, for any  B∈Ω, the  K0-monoid  V(B) is a preminimal monoid. Then, the  K0-monoid  V(A) is a preminimal monoid for any simple unital  C*-algebra  A∈TAΩ.Proof.
We prove this theorem by two steps.
Firstly, we need to show thata+d≤b+d for any a,b,c,d∈V(A) with a+c≤b+c and c≤d. By Lemma 2.3, we may assume that a=[p], b=[q], c=[e], d=[f] for some projections p,q,e,f∈proj(A). For F={p,q,e,f}, any ε>0, since A∈TAΩ, there exist a projection r∈A and a C*-subalgebra B⊆A with B∈Ω,1B=r such that(1)
∥xr-rx∥<ε for all x∈F,(2)
rxr∈Bε for all x∈F.
By(1) and (2), there exist projections p1,q1,e1,f1∈B and p2,q2,e2,f2∈(1-r)A(1-r) such that
(3.19)‖p-p1-p2‖<ε,‖q-q1-q2‖<ε,‖e-e1-e2‖<ε,‖f-f1-f2‖<ε.
Therefore, we have
(3.20)[p]=[p1]+[p2],[q]=[q1]+[q2],[e]=[e1]+[e2],[f]=[f1]+[f2],[p1]+[e1]≤[q1]+[e1],[p2]+[e2]≤[e2]+[q2],[e1]≤[f1],[e2]≤[f2].
Since B∈Ω and V(B) is preminimal, we have [p1]+[f1]≤[q1]+[f1] in V(B).
Since[e2]≤[f2], there exists a partial isometry v∈A such that vv*=e2,v*v≤f2. Therefore, we have
(3.21)[p]+[f]=[p1]+[p2]+[f1]+[f2]=[p1]+[p2]+[f1]+[f2-v*v]+[v*v]≤[p1]+[q2]+[f1]+[f2-v*v]+[e2]≤[p1]+[q2]+[f1]+[f2]≤[q1]+[q2]+[f1]+[f2]=[q]+[f].
Secondly, with the same methods and technique, we can show that a+c=b+c for any a,b,c,d∈V(A) with a+d=b+d and c≤d.



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