Certain 𝐾_𝟎-Monoid Properties Preserved by Tracial Approximation

Abstract

We show that the following 𝐾_𝟎-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.

1. Introduction

The Elliott conjecture asserts that all nuclear, separable 𝐶^∗-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 1_𝐵=𝑝 and 𝐵∈Ω, such that(1)‖𝑥𝑝−𝑝𝑥‖<𝜀 for all 𝑥∈𝐹,(2)𝑝𝑥𝑝∈_𝜀𝐵 for all 𝑥∈𝐹, (3)1−𝑝 is Murray-von Neumann equivalent to a projection in 𝑎𝐴𝑎.

The question of the behavior of 𝐶^∗-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 following 𝐾₀-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

Let 𝑎 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^∗=𝑃_𝑎, where 𝑃_𝑎 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,…,𝑏_𝑛∈Her(𝑏) such that [𝑎]≤[𝑏_𝑖], 𝑖=1,2,…,𝑛.

Let 0<𝜎₁<𝜎₂≤1 be two positive numbers. Define 𝑓^𝜎2_𝜎1(𝑡)=1,if𝑡≥𝜎₂,𝑡−𝜎₁𝜎₂−𝜎₁,if𝜎₁≤𝑡≤𝜎₂,0,if0<𝑡≤𝜎₁. (2.1)

Let Ω 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 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 1_𝐵=𝑝 and 𝐵∈Ω, such that(1)‖𝑥𝑝−𝑝𝑥‖<𝜀 for all 𝑥∈𝐹,(2)𝑝𝑥𝑝∈_𝜀𝐵 for all 𝑥∈𝐹,(3)[1−𝑝]≤[𝑎].

Definition 2.2 (see [5]). Let Ω 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 1_𝐵=𝑝, such that(1)‖𝑥𝑝−𝑝𝑥‖<𝜀 for all 𝑥∈𝐹,(2)𝑝𝑥𝑝∈_𝜀𝐵 for all 𝑥∈𝐹, ‖𝑝𝑎𝑝‖≥‖𝑎‖−𝜀,(3)𝑛[𝑓^𝜎2_𝜎1((1−𝑝)𝑎(1−𝑝))]≤[𝑓^𝜎4_𝜎3(𝑝𝑎𝑝)].

Lemma 2.3 (see [2]). If the class Ω 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]). Let Ω 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 projections 𝑝,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 is𝑉(𝐴)∶=𝑝∣𝑝=𝑝^∗=𝑝²∈𝑀_∞(𝐴). (2.2) Addition in 𝑉(𝐴) is defined by𝑝+𝑞∶=diag𝑝,𝑞. (2.3)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: if 𝑥,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.

If 𝑥,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 element 𝑢 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.

Let 𝑀 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 monoid 𝑀 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 monoid 𝑀 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.

Let 𝑀 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 monoid 𝑀 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 monoid 𝑀 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 monoid 𝑀 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. Let Ω 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 exists 𝑑≪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,1_𝐵=𝑟 such that(1)‖𝑥𝑟−𝑟𝑥‖<𝜀 for all 𝑥∈𝐹,(2)𝑟𝑥𝑟∈_𝜀𝐵 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 that𝑝−𝑝₁−𝑝₂<𝜀,𝑞−𝑞₁−𝑞₂<𝜀,𝑒−𝑒₁−𝑒₂<𝜀. (3.1) Therefore, we have 𝑝=𝑝₁+𝑝₂,𝑞=𝑞₁+𝑞₂,[𝑒]=𝑒₁+𝑒₂,𝑝₁+𝑒₁≤𝑞₁+𝑒₁,𝑝₂+𝑒₂≤𝑒₂+𝑞₂. (3.2)
Since 𝐵∈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).
For 𝐺={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,1_𝐶=𝑠 such that
(1′)‖𝑥𝑠−𝑠𝑥‖<𝜀 for all 𝑥∈𝐺,(2′)𝑠𝑥𝑠∈_𝜀𝐶 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 𝑝₂−𝑝₃−𝑝₄<𝜀,𝑞₂−𝑞₃−𝑞₄<𝜀,𝑒₂−𝑒₃−𝑒₄<𝜀. (3.3) Therefore, we have 𝑝₂=𝑝₃+𝑝₄,𝑞₂=𝑞₃+𝑞₄,𝑒₂=𝑒₃+𝑒₄,𝑝₃+𝑒₃≤𝑒₃+𝑞₃,𝑝₄+𝑒₄≤𝑒₄+𝑞₄. (3.4)
Since 𝐶∈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 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. (3.5)
Since [𝑓]+[𝑔]+[𝑒₁]+[𝑒₃]=[𝑒₁]+[𝑒₃], therefore [𝑓]+[𝑔]+[𝑒]=[𝑒], that is, [𝑓]+[𝑔]≪[𝑒].

Theorem 3.2. Let Ω 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 exist 𝑎 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,1_𝐵=𝑟 such that(1)‖𝑝𝑟−𝑟𝑝‖<𝜀,(2)𝑟𝑝𝑟∈_𝜀𝐵.
By (1) and (2), there exist projections 𝑝₁∈𝐵 and 𝑝₂∈(1−𝑟)𝐴(1−𝑟) such that 𝑝−𝑝₁−𝑝₂<𝜀. (3.6) Therefore, we have 𝑝=𝑝₁+𝑝₂. (3.7)
Since 𝐵∈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).
For 𝐺={𝑝₂,𝑒,𝑓}, any 𝜀>0, since 𝐴∈𝑇𝐴Ω, there exist a projection 𝑠∈𝐴 and a 𝐶^∗-subalgebra 𝐶⊆𝐴 with 𝐶∈Ω, 1_𝐶=𝑠 such that
(1′)‖𝑥𝑠−𝑠𝑥‖<𝜀 for all 𝑥∈𝐺,(2′)𝑠𝑥𝑠∈_𝜀𝐶 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 𝑝₂−𝑝₃−𝑝₄<𝜀. (3.8) Therefore, we have 𝑝₂=𝑝₃+𝑝₄. (3.9)
Since 𝐶∈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 𝑝=𝑝₁+𝑝₃+𝑝₄=2[𝑒]+3𝑓+2𝑔+3[ℎ]+𝑝₄=2𝑒−𝑣^∗𝑣+2𝑣^∗𝑣+𝑣𝑣^∗+3𝑓+2𝑔+3[ℎ]=2𝑒−𝑣^∗𝑣+3𝑣^∗𝑣+3𝑓+2𝑔+3[ℎ]=2𝑒−𝑣^∗𝑣+𝑔+3𝑣^∗𝑣+𝑓+[ℎ]. (3.10)

Theorem 3.3. Let Ω 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 that 𝑥=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, 1_𝐵=𝑟 such that(1)‖𝑥𝑟−𝑟𝑥‖<𝜀 for all 𝑥∈𝐹,(2)𝑟𝑥𝑟∈_𝜀𝐵 for all 𝑥∈𝐹,(3)[𝑓^𝜎2_𝜎1((1−𝑟)𝑝(1−𝑟))]≤[𝑓^𝜎4_𝜎3(𝑟𝑝𝑟)].
By (1) and (2), there exist projections 𝑝₁,𝑞₁∈𝐵 and 𝑝₂,𝑞₂∈(1−𝑟)𝐴(1−𝑟) such that 𝑝−𝑝₁−𝑝₂<𝜀,𝑞−𝑞₁−𝑞₂<𝜀. (3.11) Therefore, we have 𝑝=𝑝₁+𝑝₂,𝑞=𝑞₁+𝑞₂,𝑝₁+𝑝₁=𝑝₁+𝑞₁,𝑝₂+𝑝₂=𝑝₂+𝑞₂. (3.12)
Since 𝐵∈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 𝑝=𝑝₁+𝑝₂=𝑝₁−𝑣𝑣^∗+𝑣𝑣^∗+𝑝₂=𝑝₁−𝑣𝑣^∗+𝑝₂+𝑝₂=𝑝₁−𝑣𝑣^∗+𝑝₂+𝑞₂=𝑝₁+𝑞₂=𝑞₁+𝑞₂=𝑞. (3.13)

Theorem 3.4. Let Ω 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 that 𝑎≤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,1_𝐵=𝑟 such that(1)‖𝑥𝑟−𝑟𝑥‖<𝜀 for all 𝑥∈𝐹,(2)𝑟𝑥𝑟∈_𝜀𝐵 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 𝑝−𝑝₁−𝑝₂<𝜀,𝑞−𝑞₁−𝑞₂<𝜀,𝑡−𝑡₁−𝑡₂<𝜀. (3.14) Therefore, we have 𝑝=𝑝₁+𝑝₂,𝑞=𝑞₁+𝑞₂,[𝑡]=𝑡₁+𝑡₂,𝑝₁+𝑡₁≤𝑞₁+𝑡₁,𝑝₂+𝑡₂≤𝑡₂+𝑞₂,𝑡₁⪯𝑞₁,𝑡₂⪯𝑞₂. (3.15) Since 𝐵∈Ω and 𝑉(𝐵) is separative, we have [𝑝₁]≤[𝑞₁] in 𝑉(𝐵).
For 𝐺={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,1_𝐶=𝑤 such that(1′)‖𝑥𝑤−𝑤𝑥‖<𝜀 for all 𝑥∈𝐺,(2′)𝑤𝑥𝑤∈_𝜀𝐶 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 𝑝₂−𝑝₃−𝑝₄<𝜀,𝑞₂−𝑞₃−𝑞₄<𝜀,𝑡₂−𝑡₃−𝑡₄<𝜀. (3.16) Therefore, we have 𝑝₂=𝑝₃+𝑝₄,𝑞₂=𝑞₃+𝑞₄,𝑡₂=𝑡₃+𝑡₄,𝑝₃+𝑡₃≤𝑡₃+𝑞₃,𝑝₄+𝑡₄≤𝑡₄+𝑞₄,𝑡₃⪯𝑞₃,𝑡₄⪯𝑞₄. (3.17)
Since 𝐶∈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 𝑝=𝑝₁+𝑝₃+𝑝₄≤𝑝₁−𝑣^∗𝑣+𝑣𝑣^∗+𝑝₃+𝑝₄≤𝑝₁−𝑣^∗𝑣+𝑞₄+𝑝₃+𝑡₄≤𝑝₁+𝑞₃+𝑞₄≤𝑞₁+𝑞₃+𝑞₄=𝑞. (3.18)
Secondly, with the same methods and technique, we can show that 𝑎=𝑏 for any 𝑎,𝑏,𝑐∈𝑉(𝐴) with 𝑎+𝑐=𝑏+𝑐 and 𝑐⪯𝑎,𝑏.

Theorem 3.5. Let Ω 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 that 𝑎+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,1_𝐵=𝑟 such that(1)‖𝑥𝑟−𝑟𝑥‖<𝜀 for all 𝑥∈𝐹,(2)𝑟𝑥𝑟∈_𝜀𝐵 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 𝑝−𝑝₁−𝑝₂<𝜀,𝑞−𝑞₁−𝑞₂<𝜀,𝑒−𝑒₁−𝑒₂<𝜀,𝑓−𝑓₁−𝑓₂<𝜀. (3.19) Therefore, we have 𝑝=𝑝₁+𝑝₂,𝑞=𝑞₁+𝑞₂,[𝑒]=𝑒₁+𝑒₂,𝑓=𝑓₁+𝑓₂,𝑝₁+𝑒₁≤𝑞₁+𝑒₁,𝑝₂+𝑒₂≤𝑒₂+𝑞₂,𝑒₁≤𝑓₁,𝑒₂≤𝑓₂. (3.20) 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 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. (3.21) 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.

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).