Comparison properties of asymptotically tracially approximation C*-algebras
aa r X i v : . [ m a t h . OA ] J a n COMPARISON PROPERTIES OF ASYMPTOTICALLY TRACIALLYAPPROXIMATION C ∗ -ALGEBRAS QINGZHAI FAN AND XIAOCHUN FANGA
BSTRACT . We show that the following properties of the C ∗ -algebras in a class P areinherited by simple unital C ∗ -algebras in the class of asymptotically tracially in P : (1) β -comparison (in the sense of Kirchberg and Rørdam), (2) n -comparison (in the sense ofWinter).
1. I
NTRODUCTION
Tracial topological rank no more than k was introduced by Lin in [18]. Instead of as-suming inductive limit structure, he started with a certain abstract tracial approximationproperty, and C ∗ -algebras with tracial topological rank no more than one and certain ad-ditional properties are AH algebras without dimension growth which was classified byElliott-Gong in [3] (in the real rank zero case) and classified by Elliott-Gong-Li in [4].This abstract tracial approximation structure has proved to be very important in theclassification of simple amenable C ∗ -algebras. For example, it led to the classification ofunital simple separable amenable C ∗ -algebras with finite nuclear dimension in the UCTclass (see [12], [5], [28]).Examples of Rørdam [26] and Toms [29], relying on techniques pioneered by Villadsen[31], demonstrated that some sort of regularity condition, stronger than nuclearity, is nec-essary in order to have a classification by K -theory and trace. Three regularity conditionshave emerged: finite nuclear dimension, tensorial absorption of the Jiang-Su algebra Z , and algebraic regularity in the Cuntz semigroup. Toms and Winter have conjectured (seee.g. [7]) that these three fundamental properties are equivalent for all separable, simple,unital, amenable C ∗ -algebras.Kirchberg and Rørdam introduced a weaker comparison property and also a property ofa C ∗ -algebra called β -comparison in [16]. The property of n -comparison was introducedby Winter in [32].It is an open problem if Kirchberg’s and Rørdam’s weak comparison and β -comparison,Winter’s m -comparison, and strict comparison all agree for simple unital C ∗ -algebras.(Somewhat confusingly, it is known that m -comparison for a particular m does agree with β -comparison for β = m + 1 .)In order to search a tracial version of Toms-Winter conjecture, also inspire by tracial Z -absorbing C ∗ -algebras which was introduced by Hirshberg and Orovitz in [15], Fu andLin introduced a class of asymptotically tracially approximation C ∗ -algebras in [10].The following definition is not exactly same as definition 3.1 in [10], but by proposition3.8 in [10], the following definition is equivalent to the definition 3.1 in [10].Let P be a class of unital C ∗ -algebras, the class of simple unital C ∗ -algebras which canbe asymptotically tracially in P , denoted by ATA P . Key words C ∗ -algebras, asymptotically tracially approximation, Cuntz semigroup.2000 Mathematics Subject Classification.
Definition 1.1. ( [10] .) A simple unital C ∗ -algebra A is said to belong to the class ATA P if, for any ε > , any finite subset F ⊆ A, and any non-zero element a ≥ , there exista C ∗ -algebra B in P and completely positive contractive linear maps α : A → B and β n : B → A , and γ n : A → A ∩ β n ( B ) ⊥ such that (1) the map α is unital completely positive linear map, β n (1 B ) and γ n (1 A ) are projec-tions and β n (1 B ) + γ n (1 A ) = 1 A for all n ∈ N , (2) k x − γ n ( x ) − β n ( α ( x )) k < ε for all x ∈ F and for all n ∈ N , (3) α is an F - ε approximate embedding, (4) lim n →∞ k β n ( xy ) − β n ( x ) β n ( y ) k = 0 and lim n →∞ k β n ( xy ) k = k x k for all x, y ∈ B , and (5) γ n (1) . a for all n ∈ N . In [10], Fu and Lin show that the following properties of unital C ∗ -algebras in a class P are inherited by simple unital C ∗ -algebras in the class ATA P : (1) stably finite, (2) qua-sidiagonal C ∗ -algebras, (3) purely infinite simple C ∗ -algebras, (4) tracial Z -absorption, (5) the Cuntz semigroup is almost unperforated, and (6) strict comparison property.In this paper, we show that the following comparison properties of unital C ∗ -algebrasin a class P are inherited by simple unital C ∗ -algebras in the class of asymptotically tra-cially in P : (1) β -comparison ( in the sense of Kirchberg and Rørdam; see [16]), (2) n -comparison (in the sense of Winter; see [32]).2. D EFINITIONS AND PRELIMINARIES
Let M n ( A ) + denote the positive elements of M n ( A ) . Given a, b ∈ M n ( A ) + , we saythat a is Cuntz subequivalent to b (written a . b ) if there is a sequence ( v n ) ∞ n =1 of elementsof M n ( A ) such that lim n →∞ k v n bv ∗ n − a k = 0 . We say that a and b are Cuntz equivalent (written a ∼ b ) if a . b and b . a . We write h a i for the equivalence class of a .The object W( A ) := M ∞ ( A ) + / ∼ will be called the Cuntz semigroup of A . (See [2].) W( A ) becomes an ordered semigroup when equipped with the addition operation h a i + h b i = h a ⊕ b i , and the order relation h a i ≤ h b i ⇔ a . b. Given a in M ∞ ( A ) + and ε > , we denote by ( a − ε ) + the element of C ∗ ( a ) corre-sponding (via the functional calculus) to the function f ( t ) = max(0 , t − ε ) , t ∈ σ ( a ) .We shall say that a separable exact C ∗ -algebra A has strict comparison if for a, b ∈ M k ( A ) + , with d τ ( a ) < d τ ( b ) for any τ ∈ T( A ) , then we have a . b , where T( A ) is theset of tracial states of A . Definition 2.1. ( [32] ) Let A be a unital C ∗ -algebra. We say A has n -comparison, if,whenever x, y , y , y , · · · , y n are elements in W( A ) such that x < s y j for all j =0 , , · · · , n, then x ≤ y + y + · · · + y n . Here, x < s y means ( k + 1) x ≤ ky for somenatural number k . Definition 2.2. ( [16] ) Let A be a unital C ∗ -algebra and let ≤ β < ∞ . We say that A has β -comparison if for all x, y ∈ W( A ) and all integers k, l ≥ with k > βl, the inequality kx ≤ ly implies x ≤ y .It follows immediately from the definitions that W( A ) is almost unperforated if and onlyif A has 1-comparison in the sense of Kirchberg and Rørdam. SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 3 Theorem 2.3. ( [1] , [15] .) Let A be a stably finite C ∗ -algebra. (1) Let a, b ∈ A + and δ > be such that k a − b k < δ . Then we have ( a − δ ) + . b . (2) Let a be a purely positive element of A (i.e., a is not Cuntz equivalent to a projec-tion). Let η > , and let f ∈ C (0 , be a non-negative function with f = 0 on ( η, ,f > on (0 , η ) , and k f k = 1 . We have f ( a ) = 0 and ( a − η ) + + f ( a ) . a. (3) The following conditions are equivalent: (1) ′ a . b, (2) ′ for any δ > , ( a − δ ) + . b, and (3) ′ for any δ > , there is ε > , such that ( a − δ ) + . ( b − ε ) + . Lemma 2.4. ( [10] .) If the class P is closed under tensoring with matrix algebras andunder passing to unital hereditary C ∗ -subalgebras, then the class ATA P is closed undertensoring with matrix algebras and under passing to unital hereditary C ∗ -subalgebras. The following lemma is obvious, and we omit the proof.
Lemma 2.5.
The n -comparison (or β -comparison) is preserved under tensoring with ma-trix algebras and under passing to unital hereditary C ∗ -subalgebras.
3. T
HE MAIN RESULTS
Theorem 3.1.
Let P be a class of stably finite unital C ∗ -algebras which have β -comparison(in the sense of Kirchberg and Rørdam), for some ≤ β < ∞ . Then A has β -comparisonfor any simple unital C ∗ -algebra A ∈ ATA P . Proof.
By Lemma 2.5 and Lemma 2.4, enlarging the class P , we may suppose it is closedunder passing to matrix algebras and unital hereditary C ∗ -subalgebras (i.e., Morita equiv-alent C ∗ -algebras).Let a, b ∈ M ∞ ( A ) + . By Theorem 2.3 (3), we need to show that h ( a − ε ) + i ≤ h b i forany ε > , and any integers k, l ≥ such that k > βl, k h a i ≤ l h b i . We may assume that a, b ∈ M n ( A ) + for some integer n . By Lemma 2.4 and Lemma2.5, we may assume that a, b ∈ A + and k a k ≤ , k b k ≤ .We divide the proof into three cases. ( I ) , we suppose that b is not Cuntz equivalent to a projection.Given δ > , and k, l ≥ as above, with in particular k > l , since k h a i ≤ l h b i . Hence,by Theorem 2.3 (1), there exists v = ( v i,j ) ∈ M k ( A ) , ≤ i ≤ k, ≤ j ≤ k such that k v (diag( b ⊗ l , ⊗ k − l )) v ∗ − a ⊗ k k < δ. By Theorem 2.3 (2), there is a non-zero positive element d orthogonal to b such that ( b − δ/ + + d . b. With F = { a, b, v i,j : 1 ≤ i ≤ k, ≤ j ≤ k } , and ε ′ > , since A ∈ ATA P , thereexist a C ∗ -algebra B in P and completely positive contractive linear maps α : A → B and β n : B → A , and γ n : A → A ∩ β n ( B ) ⊥ such that (1) the map α is unital completely positive linear map, β n (1 B ) and γ n (1 A ) are allprojections, and β n (1 B ) + γ n (1 A ) = 1 A for all n ∈ N , (2) k x − γ n ( x ) − β n ( α ( x )) k < ε ′ for all x ∈ F and for all n ∈ N , (3) α is an F - ε ′ approximate embedding, (4) lim n →∞ k β n ( xy ) − β n ( x ) β n ( y ) k = 0 and lim n →∞ k β n ( xy ) k = k x k for all x, y ∈ B , and (5) γ n (1) . d for all n ∈ N .Since k v diag( b ⊗ l , ⊗ k − l ) v ∗ − a ⊗ k k < δ , by (1) , we have k α ⊗ id M k ( v (diag( b ⊗ l , ⊗ k − l )) v ∗ ) − α ( a ) ⊗ k k < δ. QINGZHAI FAN AND XIAOCHUN FANG By (3) , we have k α ⊗ id M k ( v )(diag( α ( b ) ⊗ l , ⊗ k − l )) α ⊗ id M k ( v ∗ ) − α ( a ) ⊗ k k < δ. By Theorem 2.3 (1), we have k h ( α ( a ) − δ ) + i ≤ l h α ( b ) i . Since B ∈ Ω , we have h ( α ( a ) − δ ) + i ≤ h α ( b ) i . Since h ( α ( a ) − δ ) + i ≤ h α ( b ) i , there exist w ∈ B such that k wα ( b ) w ∗ − ( α ( a ) − δ ) + k < δ. Since k wα ( b ) w ∗ − ( α ( a ) − δ ) + k < δ, we have k β n ( wα ( b ) w ∗ ) − β n (( α ( a ) − δ ) + ) k < δ. By (4) we have k β n ( w ) β n ( α ( b )) β n ( w ∗ ) − β n (( α ( a ) − δ ) + ) k < δ. By Theorem 2.3 (1), we have h ( β n (( α ( a ) − δ ) + ) − δ ) + i ≤ h ( β n α ( b ) − δ ) + i . Therefore, we have h ( a − ε ) + i≤ h γ n ( a ) i + h ( β n α ( a ) − δ ) + i≤ h γ n (1 A ) i + h ( β n α ( a ) − δ ) + ) i≤ h d i + h (( β n ( α ( a ) − δ ) + ) − δ ) + i≤ h ( b − δ/ + i + h d i ≤ h b i . ( II ) , we suppose that b is Cuntz equivalent to a projection and a is not Cuntz equivalentto a projection. Choose a projection p such that b is Cuntz equivalent to p . We mayassume that b = p . By Theorem 2.3 (2), let δ > (with δ < ε ), there exists a non-zero positive element d orthogonal to a such that h ( a − δ ) + + d i ≤ h a i . Since k h a i ≤ l h p i , we have k h ( a − δ ) + + d i ≤ l h p i . Hence, as above, by Theorem 2.3 (1), on increasing δ slightly, and decreasing d slightly, there exists v = ( v i,j ) ∈ M k ( A ) , ≤ i ≤ k, ≤ j ≤ k , such that k v diag( p ⊗ l , ⊗ k − l ) v ∗ − (( a − δ ) + + d ) ⊗ k k < δ. Since A ∈ ATA P , for F = { a, p , d, v i,j : 1 ≤ i ≤ k, ≤ j ≤ k } , and ε ′ > , thereexist a C ∗ -algebra B in P and completely positive contractive linear maps α : A → B and β n : B → A , and γ n : A → A ∩ β n ( B ) ⊥ such that (1) the map α is unital completely positive linear map, β n (1 B ) and γ n (1 A ) are allprojections, β n (1 B ) + γ n (1 A ) = 1 A for all n ∈ N , (2) k x − γ n ( x ) − β n ( α ( x )) k < ε ′ for all x ∈ F and for all n ∈ N . (3) α is an F − ε ′ -approximate embedding, (4) lim n →∞ k β n ( xy ) − β n ( x ) β n ( y ) k = 0 and lim n →∞ k β n ( xy ) k = k x k for all x, y ∈ B . Since k v diag( p ⊗ l , ⊗ k − l ) v ∗ − (( a − δ ) + + d ) ⊗ k k < δ , by (1) , we have k α ⊗ id M k ( v (diag( p ⊗ l , ⊗ k − l )) v ∗ ) − α (( a − δ ) + + d ) ⊗ k k < δ. By (3) , we have k α ⊗ id M k ( v )(diag( α ( p ) ⊗ l , ⊗ k − l )) α ⊗ id M k ( v ∗ ) − α (( a − δ ) + + d ) ⊗ k k < δ. SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 5 By Theorem 2.3 (1), we have k h ( α (( a − δ ) + + d ) − δ ) + i ≤ l h α ( b ) i . Since B ∈ Ω , we have h ( α (( a − δ ) + + d ) − δ ) + i ≤ h α ( b ) i . Since h ( α (( a − δ ) + + d ) − δ ) + i ≤ h α ( b ) i , there exist w ∈ B such that k wα ( b ) w ∗ − ( α (( a − δ ) + + d ) − δ ) + k < δ. Since k wα ( b ) w ∗ − ( α (( a − δ ) + + d ) − δ ) + k < δ , we have k β n ( wα ( b ) w ∗ ) − β n (( α (( a − δ ) + + d ) − δ ) + ) k < δ. By (4) we have k β n ( w ) β n α ( b ) β n ( w ∗ ) − β n (( α (( a − δ ) + + d ) − δ ) + ) k < δ. By Theorem 2.3 (1), we have h ( β n ( α (( a − δ ) + + d )) − δ ) + i ≤ h β n α ( b ) i . Since ( a − δ ) + orthogonal to d , by (3) and (4) , we may assume that β n α (( a − δ ) + ) orthogonal to β n α ( d ) .With G = { γ n ( a ) , γ n ( p ) , γ n ( v i,j ) : 1 ≤ i ≤ k, ≤ j ≤ k } , and any ε ′′ > , let E = γ n (1) Aγ n (1) , since E is asymptotically tracially in Ω , there exist a C ∗ -algebra D in Ω and completely positive contractive linear maps α ′ : E → D and β ′ n : D → E , and γ ′ n : E → E ∩ β ′ n ( D ) ⊥ such that (1) ′ the map α ′ is unital completely positive linear map, β ′ n (1 D ) and γ ′ n (1 E ) are allprojections, β ′ n (1 D ) + γ ′ n (1 E ) = 1 E for all n ∈ N , (2) ′ k x − γ ′ n ( x ) − β ′ n ( α ′ ( x )) k < ε ′′ for all x ∈ G and for all n ∈ N , (3) ′ α ′ is an F - ε ′′ approximate embedding, (4) ′ lim n →∞ k β ′ n ( xy ) − β ′ n ( x ) β n ( y ) k = 0 and lim n →∞ k β ′ n ( xy ) k = k x k for all x, y ∈ D , and (5) ′ γ ′ n (1) . β n α ( d ) for all n ∈ N .By (2) , we have k ( γ n ⊗ id M k ( v ))(diag( γ n ( p ) ⊗ l , ⊗ k − l ))( γ n ⊗ id M k ( v ∗ )) − γ n (( a − δ ) + + d ) ⊗ k k < δ. Since k ( γ n ⊗ id M k ( v ))(diag( γ n ( p ) ⊗ l , ⊗ k − l ))( γ n ⊗ id M k ( v ∗ )) − γ n (( a − δ ) + + d ) ⊗ k k < δ , by (1) ′ and (3) ′ , we have k α ′ ⊗ id M k γ n ⊗ id M k ( v )diag( α ′ γ n ( p ) ⊗ l , ⊗ k − l )) α ′ ⊗ id M k γ n ⊗ id M k ( v ∗ ) − α ′ γ n (( a − δ ) + + d ) ⊗ k k < δ. By Theorem 2.3 (1), we have k h ( α ′ γ n (( a − δ ) + + d ) − δ ) + i ≤ l h ( α ′ γ n ( p ) − δ ) + i . Since B ∈ Ω , this implies h ( α ′ γ n (( a − δ ) + + d ) − δ ) + i ≤ h ( α ′ γ n ( p ) − δ ) + i . Since h ( α ′ γ n (( a − δ ) + + d ) − δ ) + i ≤ h ( α ′ γ n ( p ) − δ ) + i , there exist w ∈ D such that k w ( α ′ γ n ( p ) − δ ) + w ∗ − α ′ γ n (( a − δ ) + + d ) − δ ) + k < δ. By (4) ′ , we have k β ′ n ( w ) β ′ n α ′ γ n ( p ) − δ ) + β n ( w ∗ ) − β n α ′ γ n (( a − δ ) + + d ) − δ ) + k < δ. QINGZHAI FAN AND XIAOCHUN FANG
By Theorem 2.3 (1), we have h ( β ′ n ( α ′ γ n (( a − δ ) + + d ) − δ ) + − δ ) + i ≤ h β ′ n ( α ′ γ n ( p ) − δ ) + ) i . Therefore, we have h ( a − ε ) + i≤ h ( γ n ( a ) − δ ) + i + h ( β n α (( a − δ ) + ) − δ ) + i≤ h γ ′ n γ n ( a ) i + h β ′ n α ′ γ n (( a − δ ) + − δ ) + )) i + h ( β n α (( a − δ ) + ) − δ ) + i ≤ h γ ′ n γ n (1 A ) i + h ( β n α (( a − δ ) + ) − δ ) + i + h ( β ′ n α ′ γ n (( a − δ ) + ) − δ ) + i≤ h β n α ( d ) i + h ( β n α (( a − δ ) + ) − δ ) + i + h β ′ n α ′ γ n (( a − δ ) + ) − δ ) + i≤ h ( β ′ n α ′ γ n ( p ) − δ ) + i + h β n α ( p ) i ≤ h p i . ( III ) , we suppose that both a and b are Cuntz equivalent to projections.Choose projections p, q such that a is Cuntz equivalent to p and b is Cuntz equivalentto q. We may assume that a = p, b = q. Since k h p i ≤ l h q i . Hence, by Theorem 2.3 (1),there exists v = ( v i,j ) ∈ M k ( A ) , ≤ i ≤ k, ≤ j ≤ k such that v diag( q ⊗ l , ⊗ k − l ) v ∗ = p ⊗ k . Since A ∈ ATA P , for F = { p, q, γ n ( v i,j ) : 1 ≤ i ≤ k, ≤ j ≤ k } , for any ε ′ > , thereexist a C ∗ -algebra B in P and completely positive contractive linear maps α : A → B and β n : B → A , and γ n : A → A ∩ β n ( B ) ⊥ such that (1) the map α is unital completely positive linear map, β n (1 B ) and γ n (1 A ) are allprojections β n (1 B ) + γ n (1 A ) = 1 A for all n ∈ N , (2) k x − γ n ( x ) − β n ( α ( x )) k < ε ′ for all x ∈ F and for all n ∈ N . (3) α is an F - ε ′ approximate embedding, (4) lim n →∞ k β n ( xy ) − β n ( x ) β n ( y ) k = 0 and lim n →∞ k β n ( xy ) k = k x k for all x, y ∈ B . Since v diag( q ⊗ l , ⊗ k − l ) v ∗ = p ⊗ k , by (2) , we have k ( γ n ⊗ id M k ( v ) + β n ⊗ id M k ( α ⊗ id M k ( v )))(diag( γ n ( q ) ⊗ l , ⊗ k − l )+diag( β n α ( q ) ⊗ l , ⊗ k − l ))( γ n ⊗ id M k ( v ∗ ) + β n ⊗ id M k ( α ⊗ id M k ( v ∗ ))) − ( γ n ( p ) ⊗ k , + β n α ( p ) ⊗ k ) k < δ. By (1) , we have k ( γ n ⊗ id M k ( v ))diag( γ n ( q ) ⊗ l , ⊗ k − l )( γ n ⊗ id M k ( v ∗ )) − γ n ( p ) ⊗ k k < δ. Since v diag( q ⊗ l , ⊗ k − l ) v ∗ = p ⊗ k , by (1) and (3) , we have k α ⊗ id M k ( v ))diag( α ( q ) ⊗ l , ⊗ k − l ) α ⊗ id M k ( v ∗ ) − α ( p ) ⊗ k k < δ. Since k α ⊗ id M k ( v ))diag( α ( q ) ⊗ l , ⊗ k − l ) α ⊗ id M k ( v ∗ ) − α ( p ) ⊗ k k < δ, byTheorem 2.3 (1), we have k h ( α ( p ) − δ ) + i ≤ l h α ( q ) i . Since B ∈ Ω , we have h ( α ( p ) − δ ) + i ≤ h α ( q ) i . Since h ( α ( p ) − δ ) + i ≤ h α ( q ) i , there exists w ∈ B such that k wα ( q ) w ∗ − ( α ( p ) − δ ) + k < δ. SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 7 By (4) , we have k β n ( w ) β n α ( q ) β n ( w ∗ ) − β n (( α ( p ) − δ ) + ) k < δ. By Theorem 2.3 (1), we have h ( β n α ( p ) − δ ) + i ≤ h β n α ( q ) i . ( III.I ) , if ( β n α ( p ) − δ ) + and β n α ( q ) are not Cuntz equivalent to a pure positive ele-ment, then there exist projections p , q such that ( β n α ( p ) − δ ) + ∼ p and β n α ( q ) ∼ q ,then we have k h p i ≤ l h q i = l h p i ≤ k h p i . So L kn =1 p is equivalent to a proper sub-projection of itself, and this contradicts the stable finiteness of A (since C ∗ -algebras in P are stably finite (cf. proposition 4.2 in [10]). So we may assume that there exist a nonzeroprojection s ∈ A and orthogonal to ( β n α ( p ) − δ ) + such that ( β n α ( p ) − δ ) + + s . β n α ( q ) .With G = { γ n ( p ) , γ n ( q ) , γ n ( v i,j ) : 1 ≤ i ≤ k, ≤ j ≤ k } , and any ε ′′ > , E = γ n (1) Aγ n (1) , since E is asymptotically tracially in Ω , there exist a C ∗ -algebra D in Ω and completely positive contractive linear maps α ′ : E → D and β ′ n : D → E , and γ ′ n : E → E ∩ β ′ n ( D ) ⊥ such that (1) ′ the map α ′ is unital completely positive linear map, β ′ n (1 D ) and γ ′ n (1 E ) are allprojections, β ′ n (1 D ) + γ ′ n (1 E ) = 1 E for all n ∈ N , (2) ′ k x − γ ′ n ( x ) − β ′ n ( α ′ ( x )) k < ε ′′ for all x ∈ G and for all n ∈ N , (3) ′ α ′ is an F - ε ′′ approximate embedding, (4) ′ lim n →∞ k β ′ n ( xy ) − β ′ n ( x ) β n ( y ) k = 0 and lim n →∞ k β ′ n ( xy ) k = k x k for all x, y ∈ D , and (5) ′ γ ′ n γ n (1) . s for all n ∈ N .Since k ( γ n ⊗ id M k ( v ))diag( γ n ( q ) ⊗ l , ⊗ k − l )( γ n ⊗ id M k ( v ∗ )) − γ n ( p ) ⊗ k k < δ. By (1) ′ , we have k α ′ ⊗ id M k γ n ⊗ id M k ( v ))diag( α ′ γ n ( q ) ⊗ l , ⊗ k − l ) α ′ ⊗ id M k γ n ⊗ id M k ( v ∗ ) − α ′ γ n ( p ) ⊗ k k < δ. By Theorem 2.3 (1), we have k h ( α ′ γ n ( p ) − δ ) + i ≤ l h ( α ′ γ n ( q ) − δ ) + i . Since D ∈ Ω , we have h ( α ′ γ n ( p ) − δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i . There exists w ∈ D such that k w ( α ′ γ n ( q ) − δ ) + w ∗ − ( α ′ γ n ( p ) − δ ) + k < δ. By (4 ′ ) , we have k β ′ n ( w ) β ′ n ( α ′ γ n ( q ) − δ ) + β n ( w ∗ ) − β n (( α ′ γ n ( p ) − δ ) + ) k < δ. We have h ( β ′ n α ′ γ n ( p ) − δ ) + i ≤ h ( β ′ n α ′ γ n ( q ) − δ ) + i . QINGZHAI FAN AND XIAOCHUN FANG
Therefore, if ε ′ , are small enough, then h ( p − ε ) + i≤ h γ n ( p ) − δ ) + i + h ( β n α ( p ) − δ ) + i≤ h ( γ ′ n γ n ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) i + h ( β n α ( p ) − δ ) + i ≤ h γ ′ n γ n (1 A ) i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − δ ) + i≤ h s i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − δ ) + i ≤ h q i . ( III.II ) , We suppose that ( β n α ( p ) − δ ) + is a purely positive element, then, by Theorem2.3 (2), there is a non-zero positive element d orthogonal to ( β n α ( p ) − δ ) + such that (( β n α ( p ) − δ ) + − δ ) + + d . ( β n α ( p ) − δ ) + .With G = { γ n ( p ) , γ n ( q ) , γ n ( v i,j ) : 1 ≤ i ≤ k, ≤ j ≤ k } , and ε ′′ > , E = γ n (1) Aγ n (1) , since E is asymptotically tracially in Ω , there exist a C ∗ -algebra D in Ω and completely positive contractive linear maps α ′ : E → D and β ′ n : D → E , and γ ′ n : E → E ∩ β ′ n ( D ) ⊥ such that (1) ′ the map α ′ is unital completely positive linear map, β ′ n (1 D ) and γ ′ n (1 E ) are allprojections, β ′ n (1 D ) + γ ′ n (1 E ) = 1 E for all n ∈ N , (2) ′ k x − γ ′ n ( x ) − β ′ n ( α ′ ( x )) k < ε ′′ for all x ∈ G and for all n ∈ N , (3) ′ α ′ is an F - ε ′′ approximate embedding, (4) ′ lim n →∞ k β ′ n ( xy ) − β ′ n ( x ) β n ( y ) k = 0 and lim n →∞ k β ′ n ( xy ) k = k x k for all x, y ∈ D , and (5) ′ γ ′ n γ n (1) . d for all n ∈ N .Since k ( γ n ⊗ id M k ( v ))diag( γ n ( q ) ⊗ l , ⊗ k − l )( γ n ⊗ id M k ( v ∗ )) − γ n ( p ) ⊗ k k < δ. By (1) ′ , we have k α ′ ⊗ id M k γ n ⊗ id M k ( v )diag( α ′ γ n ( q ) ⊗ l , ⊗ k − l ) α ′ ⊗ id M k γ n ⊗ id M k ( v ∗ ) − α ′ γ n ( p ) ⊗ k k < δ. By Theorem 2.3 (1), we have k h ( α ′ γ n ( p ) − δ ) + i ≤ l h ( α ′ γ n ( q ) − δ ) + i . Since D ∈ Ω , we have h ( α ′ γ n ( p ) − δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i . There exists w ∈ D such that k w ( α ′ γ n ( q ) − δ ) + w ∗ − ( α ′ γ n ( p ) − δ ) + k < δ. By (4 ′ ) , we have k β ′ n ( w ) β ′ n ( α ′ γ n ( q ) − δ ) + β n ( w ∗ ) − β n (( α ′ γ n ( p ) − δ ) + ) k < δ. We have h ( β ′ n α ′ γ n ( p ) − δ ) + i ≤ h ( β ′ n α ′ γ n ( q ) − δ ) + i . SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 9 Therefore, if ε ′ , are small enough, then h ( p − ε ) + i≤ h γ n ( p ) − δ ) + i + h ( β n α ( p ) − δ ) + i≤ h ( γ ′ n γ n ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) i + h ( β n α ( p ) − δ ) + i ≤ h γ ′ n γ n (1 A ) i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − δ ) + i≤ h s i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − δ ) + i≤ h q i . ( III.III ) , we suppose that ( β n α ( p ) − δ ) + is Cuntz equivalent to a projection and β n α ( q ) is not Cuntz equivalent to a projection. Choose a projection p such that ( β n α ( p ) − δ ) + is Cuntz equivalent to p . We may assume that ( β n α ( p ) − δ ) + = p . By Theorem 2.3 (2), there exists a non-zero positive element d orthogonal to β n α ( q ) such that h ( β n α ( q ) − δ ) + + d i ≤ h β n α ( q ) i . With G = { γ n ( p ) , γ n ( q ) , γ n ( v i,j ) : 1 ≤ i ≤ k, ≤ j ≤ k } , and any sufficientlysmall ε ′′ > , E = γ n (1) Aγ n (1) , since E is asymptotically tracially in Ω , there exista C ∗ -algebra D in Ω and completely positive contractive linear maps α ′ : E → D and β ′ n : D → E , and γ ′ n : E → E ∩ β ′ n ( D ) ⊥ such that (1) ′ the map α ′ is unital completely positive linear map, β ′ n (1 D ) and γ ′ n (1 E ) are allprojections, β ′ n (1 D ) + γ ′ n (1 E ) = 1 E for all n ∈ N , (2) ′ k x − γ ′ n ( x ) − β ′ n ( α ′ ( x )) k < ε ′′ for all x ∈ G and for all n ∈ N , (3) ′ α ′ is an F - ε ′′ approximate embedding, (4) ′ lim n →∞ k β ′ n ( xy ) − β ′ n ( x ) β n ( y ) k = 0 and lim n →∞ k β ′ n ( xy ) k = k x k for all x, y ∈ D , and (5) ′ γ ′ n γ n (1) . d for all n ∈ N .Since k ( γ n ⊗ id M k ( v ))diag( γ n ( q ) ⊗ l , ⊗ k − l )( γ n ⊗ id M k ( v ∗ )) − γ n ( p ) ⊗ k k < δ. By (1) ′ , we have k α ′ ⊗ id M k γ n ⊗ id M k ( v )diag( α ′ γ n ( q ) ⊗ l , ⊗ k − l ) α ′ ⊗ id M k γ n ⊗ id M k ( v ∗ ) − α ′ γ n ( p ) ⊗ k k < δ. By Theorem 2.3 (1), we have k h ( α ′ γ n ( p ) − δ ) + i ≤ l h ( α ′ γ n ( q ) − δ ) + i . Since D ∈ Ω , we have h ( α ′ γ n ( p ) − δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i . There exists w ∈ D such that k w ( α ′ γ n ( q ) − δ ) + w ∗ − ( α ′ γ n ( p ) − δ ) + k < δ. By (4 ′ ) , we have k β ′ n ( w ) β ′ n ( α ′ γ n ( q ) − δ ) + β n ( w ∗ ) − β n (( α ′ γ n ( p ) − δ ) + ) k < δ. We have h ( β ′ n α ′ γ n ( p ) − δ ) + i ≤ h ( β ′ n α ′ γ n ( q ) − δ ) + i . Therefore, we have h ( p − ε ) + i≤ h γ n ( p ) − δ ) + i + h ( β n α ( p ) − δ ) + i≤ h ( γ ′ n γ n ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) i + h ( β n α ( p ) − δ ) + i ≤ h γ ′ n γ n (1 A ) i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − δ ) + i≤ h s i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − δ ) + i≤ h q i . (cid:3) Theorem 3.2.
Let P be a class of stably finite unital C ∗ -algebras which have Winter’s n -comparison. Then A has Winter’s n -comparison for any simple unital C ∗ -algebra A ∈ ATA P . Proof.
As in the proof of Theorem 3.1, we may suppose that P is closed under Moritaequivalence.Let a, b , b , · · · , b n ∈ M ∞ ( A ) + . By Theorem 2.3 (3), we need only to show that h ( a − ε ) + i ≤ h b i + h b i + · · · + h b n i for any ε > , if ( k i +1) h a i ≤ k i h b i i , ≤ i ≤ n. Notethat k i can be chosen to be the same for all b i , as, with k = ( k +1)( k +1) · · · ( k n +1) − , one has ( k + 1) h a i ≤ k h b i i for all ≤ i ≤ n. We may assume that a, b , b , · · · , b n ∈ M k ( A ) + for some sufficiently large integer k .By Lemma 2.4, and Lemma 2.5, we may assume that a, b , b , · · · , b n ∈ A + .We divide the proof into three cases. ( I ) , let us suppose that b i is not Cuntz equivalent to a projection for some ≤ i ≤ n. We may assume that b is a purely positive element. By Theorem 2.3 (2), for δ > , thereis a non-zero positive element d orthogonal to b such that ( b − δ/ + + d . b . Hence,as the proof of Theorem 3.1, by Theorem 2.3 (1), there exist v k = ( v ki,j ) , ≤ k ≤ n, ≤ i ≤ k + 1 , ≤ j ≤ k + 1 such that k v i diag( b i ⊗ k , v ∗ i − a ⊗ k +1 k < δ, where ≤ i ≤ n. Since A ∈ ATA P , for F = { a, b , b , · · · , b n , d, v ki,j : 1 ≤ i ≤ k + 1 , ≤ j ≤ k + 1 , ≤ k ≤ n } , and ε ′ > , there exist a C ∗ -algebra B in P and completely positivecontractive linear maps α : A → B and β n : B → A , and γ n : A → A ∩ β n ( B ) ⊥ suchthat (1) the map α is unital completely positive linear map, β n (1 B ) and γ n (1 A ) are allprojections β n (1 B ) + γ n (1 A ) = 1 A for all n ∈ N , (2) k x − γ n ( x ) − β n ( α ( x )) k < ε ′ for all x ∈ F and for all n ∈ N , (3) α is an F - ε ′ approximate embedding, (4) lim n →∞ k β n ( xy ) − β n ( x ) β n ( y ) k = 0 and lim n →∞ k β n ( xy ) k = k x k for all x, y ∈ B , and (5) γ n (1) . d for all n ∈ N .Since k v i diag( b i ⊗ k , v ∗ i − a ⊗ k +1 k < δ, for all ≤ i ≤ n. By (3) , we have k α ⊗ id M k +1 ( v i )diag( α ( b i ) ⊗ k , α ⊗ id M k +1 ( v i ∗ ) − α ( a ) ⊗ k +1 k < δ, SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 11 for all ≤ i ≤ n. By Theorem 2.3 (1), we have ( k + 1) h ( α ( a ) − δ ) + i ≤ h α ( b i ) i , for all ≤ i ≤ n. Since B ∈ Ω , we have h ( α ( a ) − δ ) + i ≤ h α ( b ) i + h α ( b ) i + · · · + h α ( b n ) i . Since h ( α ( a ) − δ ) + i ≤ h α ( b ) i + h α ( b ) i + · · · + h α ( b n ) i , There exists w ∈ M k +1 ( D ) such that k w diag( α ( b ) , α ( b ) , · · · , α ( b n )) w ∗ − ( α ( a ) − δ ) + k < δ. By (4) , we have k β n ⊗ id M k +1 ( w )diag( β n α ( b ) , β n α ( b ) , · · · , β n α ( b n )) β n ⊗ id M k +1 ( w ∗ ) − ( β n α ( a ) − δ ) + k < δ. By Theorem 2.3 (1), we have h ( β n α ( a ) − δ ) + i ≤ h β n α ( b − δ ) + i + h β n α ( b ) i + · · · + h β n α ( b n ) i , Therefore, we have h ( a − ε ) + i≤ h γ n ( a ) − δ ) + i + h ( β n α ( a ) − δ ) + i≤ h γ n ( a ) i + h β n α ( b − δ ) + ) i + h β n α ( b ) i + · · · + h β n α ( b n ) i ≤ h γ n (1 A ) i + h β n α ( b − δ ) + i + h β n α ( b ) i + · · · + h β n α ( b n ) i≤ h d i + h β n α ( b − δ ) + i + h β n α ( b ) i + · · · + h β n α ( b n ) i≤ h ( b − δ/ + i + h d i + h b i + · · · + h b n i ≤ h b i + h b i + · · · + h b n i . ( II ) , let us suppose that each b i (0 ≤ i ≤ n ) is Cuntz equivalent to a projection and a is not Cuntz equivalent to a projection. Choose projections p , p , · · · , p n such that b i isCuntz equivalent to p i for all ≤ i ≤ n. We may assume that b i = p i for all ≤ i ≤ n. By Theorem 2.3 (2), let δ > (with δ < ε ), there is a non-zero positive element d orthogonal to a such that h ( a − δ ) + + d i ≤ h a i . Since ( k + 1) h a i ≤ k h p i i , we have ( k + 1) h ( a − δ ) + + d i ≤ k h p i i for all ≤ i ≤ n. Hence, by Theorem 2.3 (1), as earlier, there exist v k = ( v ki,j ) ∈ M k +1 ( A ) , ≤ k ≤ n, ≤ i ≤ k + 1 , ≤ j ≤ k + 1 , such that k v i diag( p i ⊗ k , v ∗ i − (( a − δ ) + + d ) ⊗ k +1 k < δ, ≤ i ≤ n. Since A ∈ ATA P , for F = { a, b , b , · · · , b n , d, v ki,j : 1 ≤ i ≤ k + 1 , ≤ j ≤ k + 1 , ≤ k ≤ n } , for any ε ′ > , there exist a C ∗ -algebra B in P and completelypositive contractive linear maps α : A → B and β n : B → A , and γ n : A → A ∩ β n ( B ) ⊥ such that (1) the map α is unital completely positive linear map, β n (1 B ) and γ n (1 A ) are allprojections β n (1 B ) + γ n (1 A ) = 1 A for all n ∈ N , (2) k x − γ n ( x ) − β n ( α ( x )) k < ε ′ for all x ∈ F and for all n ∈ N . (3) α is an F - ε ′ approximate embedding, (4) lim n →∞ k β n ( xy ) − β n ( x ) β n ( y ) k = 0 and lim n →∞ k β n ( xy ) k = k x k for all x, y ∈ B . Since k v i diag( p i ⊗ k , v ∗ i − (( a − δ ) + + d ) ⊗ k +1 k < δ, ≤ i ≤ n , By (2) , (3) and (4) , we have k ( γ n ⊗ id M k +1 ( v i ) + β n ⊗ id M k +1 ( α ⊗ id M k ( v i )))(diag( γ n ( p i ) ⊗ k , β n α ( p i ) ⊗ k , γ n ⊗ id M k +1 ( v i ∗ ) + β n ⊗ id M k +1 ( α ⊗ id M k +1 ( v i ∗ ))) − ( γ n (( a − δ ) + + d ) ⊗ k + , + β n α (( a − δ ) + + d ) ⊗ k +1 ) k < δ. By (1) , we have k ( γ n ⊗ id M k +1 ( v i ))(diag( γ n ( p i ) ⊗ k , γ n ⊗ id M k +1 ( v i ∗ )) − γ n (( a − δ ) + + d ) ⊗ k +1 k < δ. Since k v i diag( p i ⊗ k , v ∗ i − (( a − δ ) + + d ) ⊗ k +1 k < δ, ≤ i ≤ n , by (1) and (3) , we have k α ⊗ id M k +1 ( v ))diag( α ( p i ) ⊗ l , α ⊗ id M k +1 ( v ∗ ) − α (( a − δ ) + + d ) ⊗ k +1 k < δ, for ≤ i ≤ n .By Theorem 2.3 (1), we have ( k + 1) h ( α (( a − δ ) + + d ) − δ ) + i ≤ k h α ( p i ) i for ≤ i ≤ n .Since B ∈ Ω , we have h ( α (( a − δ ) + + d ) − δ ) + i ≤ h α ( p ) i + h α ( p ) i + · · · + h α ( p n ) i . Since h ( α (( a − δ ) + + d ) − δ ) + i ≤ h α ( p ) i + h α ( p ) i + · · · + h α ( p n ) i , there exists w ∈ M k +1 ( B ) such that k w diag( α ( p ) , α ( p ) , · · · , α ( p n )) w ∗ − ( α (( a − δ ) + + d ) − δ ) + k < δ. By (4) , we have k β n ⊗ id M k +1 ( w )diag( β n α ( p ) , β n α ( p ) , · · · , β n α ( p n )) β n ⊗ id M k +1 ( w ∗ ) − ( β n ( α (( a − δ ) + + d ) − δ ) + ) k < δ. By Theorem 2.3 (1), we have h ( β n ( α (( a − δ ) + + d ) − δ ) + ) i ≤ h ( β n α ( p ) − δ ) + i + h β n α ( p ) i + · · · + h β n α ( p n ) i . Since ( a − δ ) + is orthogonal to d , by (3) and (4) , we may assume that β n α (( a − δ ) + ) is orthogonal to β n α ( d ) .With G = { γ n ( a ) , γ n ( p i ) , γ n ( v ki,j ) : 1 ≤ i ≤ k + 1 , ≤ j ≤ k + 1 , ≤ k ≤ n } , and any ε ′′ > , E = γ n (1) Aγ n (1) , since E is asymptotically tracially in P , there exista C ∗ -algebra D in P and completely positive contractive linear maps α ′ : E → D and β ′ n : D → E , and γ ′ n : E → E ∩ β ′ n ( D ) ⊥ such that (1 ′ ) the map α ′ is unital completely positive linear map, β ′ n (1 D ) and γ ′ n (1 A ) are allprojections, and β ′ n (1 D ) + γ ′ n (1 A ) = 1 A for all n ∈ N , (2 ′ ) k x − γ ′ n ( x ) − β ′ n ( α ′ ( x )) k < ε ′′ for all x ∈ G and for all n ∈ N , (3 ′ ) α ′ is an F - ε ′′ approximate embedding, (4 ′ ) lim n →∞ k β ′ n ( xy ) − β ′ n ( x ) β n ( y ) k = 0 and lim n →∞ k β ′ n ( xy ) k = k x k for all x, y ∈ D , and (5 ′ ) γ ′ n γ n (1) . β n α ( d ) for all n ∈ N . SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 13 Since k ( γ n ⊗ id M k +1 ( v i ))diag( γ n ( p i ) ⊗ k , γ n ⊗ id M k ( v ∗ )) − γ n ( a ) ⊗ k +1 k < δ, for ≤ i ≤ n , by (1) ′ , we have k α ′ ⊗ id M k +1 γ n ⊗ id M k +1 ( v i )diag( α ′ γ n ( p i ) ⊗ k , α ′ ⊗ id M k +1 ( γ n ⊗ id M k +1 ( v ∗ i )) − α ′ γ n ( a ) ⊗ k +1 k < δ. By Theorem 2.3 (1), we have ( k + 1) h ( α ′ γ n ( a ) − δ ) + i ≤ k h α ′ γ n ( p i ) i . Since D ∈ Ω , we have h ( α ′ γ n ( a ) − δ ) + i ≤ h α ′ γ n ( p ) i + α ′ γ n ( p ) i + · · · + α ′ γ n ( p n ) i . Since h ( α ′ γ n ( a ) − δ ) + i ≤ h ( α ′ γ n ( p ) − δ ) + i + ( α ′ γ n ( p ) − δ ) + i + · · · + ( α ′ γ n ( p n ) − δ ) + i , there exists w ∈ M k +1 ( D ) such that k w diag( α ′ γ n ( p ) , α ′ γ n ( p ) , · · · , α ′ γ n ( p n )) w ∗ − ( α ′ γ n (( a − δ ) + + d ) − δ ) + k < δ. By (4) , we have k β ′ n ⊗ id M k +1 ( w )diag( β ′ n α ′ γ n ( p ) , β ′ n α ′ β n α ( p ) , · · · , β ′ n α ′ β n α ( p n )) β n ⊗ id M k +1 ( w ∗ ) − ( β ′ n α ′ β n ( α (( a − δ − δ ) + ) k < δ. By Theorem 2.3 (1), we have kh ( β ′ n ( α ′ (( a − δ ) + − δ ) + ) i ≤ h β ′ n α ′ γ n ( p ) i + h β ′ n α ′ γ n ( p ) i + · · · + h β ′ n α ′ γ n ( p n ) i . Therefore, we have h ( a − ε ) + i≤ h ( γ n ( a ) − δ ) + i + h ( β n α ( a − δ ) + ) − δ ) + i≤ h ( γ ′ n γ n ( a ) − δ ) + i + h ( β ′ n α ′ γ n (( a − δ ) + ) − δ ) + i + h ( β n α (( a − δ ) + ) − δ ) + i ≤ h γ ′ n (1 A ) i + h ( β n α (( a − δ ) + ) − δ ) + i + h ( β ′ n α ′ γ n (( a − δ ) + ) − δ ) + i≤ h β n α ( d ) i + h ( β n α (( a − δ ) + ) − δ ) + i + h ( β ′ n α ′ γ n (( a − δ ) + ) − δ ) + i≤ h ( β n α ( a ) − δ ) + i + h ( β n α ( p ) i + · · · + h ( β n α ( p n ) i + h ( β ′ n α ′ γ n ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − δ ) + i + · · · + h ( β ′ n α ′ γ n ( p n ) − δ ) + i≤ h ( β n α ( p ) − δ ) + i + h ( β n α ( p ) − δ ) + i + · · · + h ( β n α ( p n ) − δ ) + i≤ h p i + h p i + · · · + h p n i . ( III ) , we suppose that both a and b i ( ≤ i ≤ n ) are all Cuntz equivalent to projections.Choose projections p, q i such that a is Cuntz equivalent to p and b i is Cuntz equivalentto q i . We may assume that a = p, b i = q i . Since ( k + 1) h p i ≤ k h q i i for all ≤ i ≤ n, so for any δ > , by Theorem 2.3 (1), thereexist v k = ( v ki,j ) , ≤ k ≤ n, ≤ i ≤ k + 1 , ≤ j ≤ k + 1 such that k v i diag( q i ⊗ k , v ∗ i − p ⊗ k +1 k < δ. Since A ∈ ATA P , for F = { p, q k , ( v ki,j ) , ≤ k ≤ n, ≤ i ≤ k + 1 , ≤ j ≤ k + 1 } , for any ε ′ > , there exist a a C ∗ -algebra B in P and completely positive contractive linearmaps α : A → B and β n : B → A , and γ n : A → A ∩ β n ( B ) ⊥ such that (1) the map α is unital completely positive linear map, β n (1 B ) and γ n (1 A ) are allprojections β n (1 B ) + γ n (1 A ) = 1 A for all n ∈ N , (2) k x − γ n ( x ) − β n ( α ( x )) k < ε ′ for all x ∈ F and for all n ∈ N . (3) α is an F - ε ′ approximate embedding, (4) lim n →∞ k β n ( xy ) − β n ( x ) β n ( y ) k = 0 and lim n →∞ k β n ( xy ) k = k x k for all x, y ∈ B . Since k v i diag( q i ⊗ k , v ∗ i − p ⊗ k +1 k < δ for ≤ i ≤ n , by (2) , we have k ( γ n ⊗ id M k +1 ( v i ) + β n ⊗ id M k +1 ( α ⊗ id M k ( v i )))(diag( γ n ( q i ) ⊗ k , β n α ( q i ) ⊗ k , γ n ⊗ id M k +1 ( v i ∗ ) + β n ⊗ id M k +1 ( α ⊗ id M k +1 ( v i ∗ ))) − ( γ n ( p ) ⊗ k +1 + β n α ( p ) ⊗ k +1 ) k < δ. By (1) , we have k ( γ n ⊗ id M k +1 ( v i ))(diag( γ n ( q i ) ⊗ k , γ n ⊗ id M k +1 ( v i ∗ )) − γ n ( p ) ⊗ k +1 k < δ. Since k v i diag( q i ⊗ k , v ∗ i − p ⊗ k +1 k < δ for ≤ i ≤ n , by (1) and (3) , we have k α ⊗ id M k +1 ( v i )diag( α ( q i ) ⊗ k , α ⊗ id M k +1 ( v i ∗ ) − α ( p ) ⊗ k +1 k < δ. By Theorem 2.3 (1), we have ( k + 1) h ( α ( p ) − δ ) + i ≤ k h α ( q i ) i . Since B ∈ Ω , we have h ( α ( p ) − δ ) + i ≤ h α ( q ) i + h α ( q ) i + · · · h α ( q n ) i . Since h ( α ( p ) − δ ) + i ≤ h α ( q ) i + h α ( q ) i + · · · h α ( q n ) i , there exists w ∈ M k +1 ( D ) suchthat k w diag( α ( p ) , α ( p ) , · · · , α ( p n )) w ∗ − ( α (( a − δ ) + ) k < δ. By (4) , we have k β n ⊗ id M k +1 ( w )diag( β n α ( p ) , β n α ( p ) , · · · , β n α ( p n )) β n ⊗ id M k +1 ( w ∗ ) − ( β n ( α (( a − δ ) + ) − δ ) + ) k < δ. By Theorem 2.3 (1), we have h ( β n ( α (( a − δ ) + ) − δ ) + ) i ≤ h β n α ( p ) i + h β n α ( p ) i + · · · + h β n α ( p n ) i . ( III.I ) , if ( β n α ( p ) − δ ) + and β n α ( q i ) are not Cuntz equivalent to a pure positiveelement, then there exist projections p ′ , q ′ i such that ( β n α ( p ) − δ ) + ∼ p ′ and β n α ( q i ) ∼ q ′ i , and we suppose that p ′ ∼ q ′ + q ′ + · · · + q ′ n , then we have ( k + 1) h p ′ i = ( k + 1) h q ′ i +( k + 1) h q ′ i + · · · + ( k + 1) h q ′ n i ≤ k h q ′ i + k h q ′ i + · · · + k h q ′ n i , and this contradicts thestable finiteness of A (since C ∗ -algebras in P are stably finite (cf. proposition 4.2 in [10]).So there exist a nonzero projection s ∈ A , orthogonal to p ′ such that ( β n α ( p ) − δ ) + + s . q ′ + q ′ + · · · + q ′ n . With G = { γ n ( a ) , γ n ( p i ) , γ n ( v ki,j ) : 1 ≤ i ≤ k + 1 , ≤ j ≤ k + 1 , ≤ k ≤ n } , and any ε ′′ > , E = γ n (1) Aγ n (1) , since E is asymptotically tracially in P , there exista C ∗ -algebra D in Ω and completely positive contractive linear maps α ′ : E → D and β ′ n : D → E , and γ ′ n : E → E ∩ β ′ n ( D ) ⊥ such that (1) ′ the map α ′ is unital completely positive linear map, β ′ n (1 D ) and γ ′ n (1 E ) are allprojections, β ′ n (1 D ) + γ ′ n (1 E ) = 1 E for all n ∈ N , (2) ′ k x − γ ′ n ( x ) − β ′ n ( α ′ ( x )) k < ε ′′ for all x ∈ G and for all n ∈ N , (3) ′ α ′ is an F - ε ′′ approximate embedding, SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 15 (4) ′ lim n →∞ k β ′ n ( xy ) − β ′ n ( x ) β n ( y ) k = 0 and lim n →∞ k β ′ n ( xy ) k = k x k for all x, y ∈ D , and (5) ′ γ ′ n γ n (1) . s for all n ∈ N .Since k ( γ n ⊗ id M k +1 ( v i ))diag( γ n ( q i ) ⊗ k , γ n ⊗ id M k ( v ∗ )) − γ n ( p ) ⊗ k +1 k < δ, for ≤ i ≤ n , by (1) ′ , we have k α ′ ⊗ id M k +1 γ n ⊗ id M k +1 ( v i )diag( α ′ γ n ( q i ) ⊗ k , α ′ ⊗ id M k +1 ( γ n ⊗ id M k +1 ( v ∗ i )) − α ′ γ n ( p ) ⊗ k +1 k < δ. By Theorem 2.3 (1), we have ( k + 1) h ( α ′ γ n ( p ) − ( n + 6) δ ) + i ≤ k h ( α ′ γ n ( q i ) − δ ) + i . Since D ∈ Ω , we have h ( α ′ γ n ( p ) − ( n +6) δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i + h ( α ′ γ n ( q ) − δ ) + i + · · · + h ( α ′ γ n ( q n ) − δ ) + i . Since h ( α ′ γ n ( p ) − ( n + 6) δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i + h ( α ′ γ n ( q ) − δ ) + i + · · · + h ( α ′ γ n ( q n ) − δ ) + i , there exists w ∈ M k +1 ( D ) such that k w diag(( α ′ γ n ( q ) − δ ) + , ( α ′ γ n ( q ) − δ ) + , · · · , ( α ′ γ n ( q n ) − δ ) + ) w ∗ − ( α ′ γ n ( p ) − ( n +3) δ ) + k < δ. By (4) , we have k β ′ n ⊗ id M k +1 ( w )diag(( β ′ n α ′ γ n ( q ) − δ ) + , ( β ′ n α ′ γ n ( q ) − δ ) + , · · · , ( β ′ n α ′ γ n ( q n ) − δ ) + ) β ′ n ⊗ id M k +1 ( w ∗ ) − ( β ′ n ( α ′ γ n ( p ) − ( n + 3) δ ) + k < δ. By Theorem 2.3 (1), we have h β ′ n ( α ′ γ n ( p ) − ( n +10) δ ) + i ≤ h ( β ′ n α ′ γ n ( q ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + · · · + h ( β ′ n α ′ γ n ( q n ) − δ ) + i . Therefore, we have h p i ≤ h ( γ n ( p ) − ( n + 10) δ ) + ) i + h ( β n α ( p ) − ( n + 10) δ ) + i≤ h γ ′ n γ n ( p ) i + h ( β ′ n α ′ γ n ( p ) − ( n + 12) δ ) + i + h ( β n α ( p ) − δ ) + i ≤ h γ ′ n γ n (1 A ) i + h ( β n α (( a − δ ) + ) − δ ) + i + h ( β ′ n α ′ γ n (( a − ( n + 8) δ ) + ) − δ ) + i≤ h s i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − ( n + 10) δ ) + i≤ h ( β n α ( q ) − δ ) + i + h ( β n α ( q ) − δ ) + i + · · · + h ( β n α ( q n ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + · · · + h ( β ′ n α ′ γ n ( q n ) − δ ) + i≤ h ( β n α ( q ) − δ ) + i + h ( β n α ( q ) − δ ) + i + · · · + h ( β n α ( q n ) − δ ) + i≤ h q i + h q i + · · · + h q n i . ( III.II ) , We suppose that ( β n α ( p ) − δ ) + is a purely positive element, then By Theorem2.3 (2), there is a non-zero positive element d orthogonal to ( β n α ( p ) − δ ) + such that ( β n α ( p ) − δ ) + + d . ( β n α ( p ) − δ ) + . With G = { γ n ( a ) , γ n ( p i ) , γ n ( v ki,j ) : 1 ≤ i ≤ k + 1 , ≤ j ≤ k + 1 , ≤ k ≤ n } , and ε ′′ > , E = γ n (1) Aγ n (1) , since E is asymptotically tracially in P , there exist a C ∗ -algebra D in P and completely positive contractive linear maps α ′ : E → D and β ′ n : D → E , and γ ′ n : E → E ∩ β ′ n ( D ) ⊥ such that (1) ′ the map α ′′ is unital completely positive linear map, β ′ n (1 D ) and γ ′ n (1 A ) are allprojections, β ′ n (1 D ) + γ ′ n (1 A ) = 1 A for all n ∈ N , (2) ′ k x − γ ′ n ( x ) − β ′ n ( α ′ ( x )) k < ε ′′ for all x ∈ G and for all n ∈ N , (3) ′ α ′ is an F - ε ′′ approximate embedding, (4) ′ lim n →∞ k β ′ n ( xy ) − β ′ n ( x ) β n ( y ) k = 0 and lim n →∞ k β ′ n ( xy ) k = k x k for all x, y ∈ D , and (5) ′ γ ′ n γ n (1) . d for all n ∈ N .Since k ( γ n ⊗ id M k +1 ( v i ))diag( γ n ( q i ) ⊗ k , γ n ⊗ id M k ( v ∗ )) − γ n ( p ) ⊗ k +1 k < δ, for ≤ i ≤ n , by (1) ′ , we have k α ′ ⊗ id M k +1 γ n ⊗ id M k +1 ( v i )diag( α ′ γ n ( q i ) ⊗ k , α ′ ⊗ id M k +1 ( γ n ⊗ id M k +1 ( v ∗ i )) − α ′ γ n ( p ) ⊗ k +1 k < δ. By Theorem 2.3 (1), we have ( k + 1) h ( α ′ γ n ( p ) − ( n + 6) δ ) + i ≤ k h ( α ′ γ n ( q i ) − δ ) + i . Since D ∈ Ω , we have h ( α ′ γ n ( p ) − ( n +6) δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i + h ( α ′ γ n ( q ) − δ ) + i + · · · + h ( α ′ γ n ( q n ) − δ ) + i . Since h ( α ′ γ n ( p ) − ( n + 6) δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i + h ( α ′ γ n ( q ) − δ ) + i + · · · + h ( α ′ γ n ( q n ) − δ ) + i , there exists w ∈ M k +1 ( D ) such that k w diag(( α ′ γ n ( q ) − δ ) + , ( α ′ γ n ( q ) − δ ) + , · · · , ( α ′ γ n ( q n ) − δ ) + ) w ∗ − ( α ′ γ n ( p ) − ( n +6) δ ) + k < δ. By (4) , we have k β ′ n ⊗ id M k +1 ( w )diag(( β ′ n α ′ γ n ( q ) − δ ) + , ( β ′ n α ′ γ n ( q ) − δ ) + , · · · , ( β ′ n α ′ γ n ( q n ) − δ ) + ) β ′ n ⊗ id M k +1 ( w ∗ ) − ( β ′ n ( α ′ γ n ( p ) − ( n + 6) δ ) + k < δ. By Theorem 2.3 (1), we have h β ′ n ( α ′ γ n ( p ) − ( n +10) δ ) + i ≤ h ( β ′ n α ′ γ n ( q ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + · · · + h ( β ′ n α ′ γ n ( q n ) − δ ) + i . Therefore, we have h p i ≤ h ( γ n ( p ) − ( n + 10) δ ) + ) i + h ( β n α ( p ) − ( n + 10) δ ) + i≤ h γ ′ n γ n ( p ) i + h ( β ′ n α ′ γ n ( p ) − ( n + 12) δ ) + i + h ( β n α ( p ) − δ ) + i ≤ h γ ′ n γ n (1 A ) i + h ( β n α (( a − δ ) + ) − δ ) + i + h ( β ′ n α ′ γ n (( a − ( n + 8) δ ) + ) − δ ) + i≤ h d i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − ( n + 10) δ ) + i≤ h ( β n α ( q ) − δ ) + i + h ( β n α ( q ) − δ ) + i + · · · + h ( β n α ( q n ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + · · · + h ( β ′ n α ′ γ n ( q n ) − δ ) + i≤ h ( β n α ( q ) − δ ) + i + h ( β n α ( q ) − δ ) + i + · · · + h ( β n α ( q n ) − δ ) + i≤ h q i + h q i + · · · + h q n i . ( III.III ) , we suppose that ( β n α ( p ) − δ ) + is Cuntz equivalent to a projection and thereexist some q i such that β n α ( q i ) is not Cuntz equivalent to a projection. Choose a projection p such that ( β n α ( p ) − δ ) + is Cuntz equivalent to p . We may assume that ( β n α ( p ) − δ ) + = p . We may also assume that q such that β n α ( q ) is a pure positive. By Theorem 2.3 (2),there exists a non-zero positive element d orthogonal to β n α ( q ) such that h ( β n α ( q ) − δ ) + + d i ≤ h β n α ( q ) i . With G = { γ n ( a ) , γ n ( p i ) , γ n ( v ki,j ) : 1 ≤ i ≤ k + 1 , ≤ j ≤ k + 1 , ≤ k ≤ n } , and any sufficiently small ε ′′ > , E = γ n (1) Aγ n (1) , since E is asymptotically tracially SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 17 in P , there exist a C ∗ -algebra D in P and completely positive contractive linear maps α ′ : E → D and β ′ n : D → E , and γ ′ n : E → E ∩ β ′ n ( D ) ⊥ such that (1 ′ ) the map α ′ is unital completely positive linear map, β ′ n (1 D ) and γ ′ n (1 A ) are allprojections, β ′ n (1 D ) + γ ′ n (1 A ) = 1 A for all n ∈ N , (2 ′ ) k x − γ ′ n ( x ) − β ′ n ( α ′ ( x )) k < ε ′′ for all x ∈ G and for all n ∈ N , (3 ′ ) α ′ is an F - ε ′′ approximate embedding, (4 ′ ) lim n →∞ k β ′ n ( xy ) − β ′ n ( x ) β n ( y ) k = 0 and lim n →∞ k β ′ n ( xy ) k = k x k for all x, y ∈ D , and (5 ′ ) γ ′ n (1) γ n . d for all n ∈ N .Since k ( γ n ⊗ id M k +1 ( v i ))diag( γ n ( q i ) ⊗ k , γ n ⊗ id M k ( v ∗ )) − γ n ( p ) ⊗ k +1 k < δ, for ≤ i ≤ n , by (1) ′ , we have k α ′ ⊗ id M k +1 γ n ⊗ id M k +1 ( v i )diag( α ′ γ n ( q i ) ⊗ k , α ′ ⊗ id M k +1 ( γ n ⊗ id M k +1 ( v ∗ i )) − α ′ γ n ( p ) ⊗ k +1 k < δ. By Theorem 2.3 (1), we have ( k + 1) h ( α ′ γ n ( p ) − ( n + 6) δ ) + i ≤ k h ( α ′ γ n ( q i ) − δ ) + i . Since D ∈ Ω , we have h ( α ′ γ n ( p ) − ( n +6) δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i + h ( α ′ γ n ( q ) − δ ) + i + · · · + h ( α ′ γ n ( q n ) − δ ) + i . Since h ( α ′ γ n ( p ) − ( n + 6) δ ) + i ≤ h ( α ′ γ n ( q ) − δ ) + i + h ( α ′ γ n ( q ) − δ ) + i + · · · + h ( α ′ γ n ( q n ) − δ ) + i , there exists w ∈ M k +1 ( D ) such that k w diag(( α ′ γ n ( q ) − δ ) + , ( α ′ γ n ( q ) − δ ) + , · · · , ( α ′ γ n ( q n ) − δ ) + ) w ∗ − ( α ′ γ n ( p ) − ( n +3) δ ) + k < δ. By (4) , we have k β ′ n ⊗ id M k +1 ( w )diag(( β ′ n α ′ γ n ( q ) − δ ) + , ( β ′ n α ′ γ n ( q ) − δ ) + , · · · , ( β ′ n α ′ γ n ( q n ) − δ ) + ) β ′ n ⊗ id M k +1 ( w ∗ ) − ( β ′ n ( α ′ γ n ( p ) − ( n + 3) δ ) + k < δ. By Theorem 2.3 (1), we have h β ′ n ( α ′ γ n ( p ) − ( n +10) δ ) + i ≤ h ( β ′ n α ′ γ n ( q ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + · · · + h ( β ′ n α ′ γ n ( q n ) − δ ) + i . Therefore, we have h p i ≤ h ( γ n ( p ) − ( n + 10) δ ) + ) i + h ( β n α ( p ) − ( n + 10) δ ) + i≤ h γ ′ n γ n ( p ) i + h ( β ′ n α ′ γ n ( p ) − ( n + 12) δ ) + i + h ( β n α ( p ) − δ ) + i ≤ h γ ′ n γ n (1 A ) i + h ( β n α (( a − δ ) + ) − δ ) + i + h ( β ′ n α ′ γ n (( a − ( n + 8) δ ) + ) − δ ) + i≤ h d i + h ( β n α ( p ) − δ ) + i + h ( β ′ n α ′ γ n ( p ) − ( n + 10) δ ) + i≤ h ( β n α ( q ) − δ ) + i + h ( β n α ( q ) − δ ) + i + · · · + h ( β n α ( q n ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + h ( β ′ n α ′ γ n ( q ) − δ ) + i + · · · + h ( β ′ n α ′ γ n ( q n ) − δ ) + i≤ h ( β n α ( q ) − δ ) + i + h ( β n α ( q ) − δ ) + i + · · · + h ( β n α ( q n ) − δ ) + i≤ h q i + h q i + · · · + h q n i . (cid:3) Corollary 3.3.
Let P be a class of stably finite unital C ∗ -algebras such that for any B ∈ P , W( B ) is almost unperforated. Then W( A ) is almost unperforated for any simple unital C ∗ -algebra A ∈ TA P . Proof.
This is a special case of Theorem 3.1 and also of Theorem 3.2. (cid:3) R EFERENCES[1] P. Ara, F. Perera, and A. Toms, K -theory for operator algebras. Classification of C ∗ -algebras , Aspectsof operator algebras and applications, 1–71, Contemp. Math., 534, American Mathematical Society, Provi-dence, RI, 2011.[2] K. T. Coward, G. A. Elliott, and C. Ivanescu, The Cuntz semigroup as an invariant for C ∗ -algebras , J.Reine Angew. Math., (2008), 161–193.[3] G. A. Elliott and G. Gong, On the classification of C ∗ -algebras of real rank zero, II , Ann. of Math., (1996), 497–610.[4] G. A. Elliott, G. Gong, and L. Li, On the classification of simple inductive limit C ∗ -algebras II: The iso-morphism theorem , Invent. Math., (2007), 249–320.[5] G. A. Elliott, G. Gong, H. Lin, and Z. Niu, On the classification of simple amenable C ∗ -algebras with finitedecomposition rank, II , arXiv: 1507.03437.[6] G. A. Elliott and Z. Niu, On tracial approximation , J. Funct. Anal., (2008), 396–440.[7] G. A. Elliott and A. Toms,
Regularity properties in the classification program for separable amenable C ∗ -algebras , Bull. Amer. Math. Soc., (2008), 229–245.[8] Q. Fan, Some C ∗ -algebra properties preserved by tracial approximation , Israel J. Math., (2013), 545–563.[9] Q. Fan, K -monoid properties preserved by tracial approximation , J. Operator Theory, (2013), 535–543.[10] X. Fu and H. Lin, Tracial approximation in simple C ∗ -algebras , arXiv:2004. 10901v1.[11] G. Gong, On the classification of simple inductive limit C ∗ -algebras. I: The reduction theorem , Doc. Math., (2002), 255–461.[12] G. Gong, H. Lin, and Z. Niu, Classification of finite simple amenable Z -stable C ∗ -algebras , arXiv:1501.00135.[13] G. Gong, H. Lin, and Z. Niu, A classification of finite simple amenable Z -stable C ∗ -algebras, I: C ∗ -algebras with generalized tracial rank one , arXiv:1812.11590.[14] G. Gong, H. Lin, and Z. Niu, A classification of finite simple amenable Z -stable C ∗ -algebras, , II, C ∗ -algebras with rational generalized tracial rank one , arXiv:1909.13382.[15] I. Hirshberg and J. Orovitz, Tracially Z -absorbing C ∗ -algebras , J. Funct. Anal., (2013), 765–785.[16] E. Kirchberg and M. Rørdam, Central sequence C ∗ -algebras and tensorial absorption of the Jiang-Sualgebra , J. Reine Angew. Math., (2014), 175–214.[17] H. Lin, Tracially AF C ∗ -algebras , Trans. Amer. Math. Soc., (2001), 683–722.[18] H. Lin, The tracial topological rank of C ∗ -algebras , Proc. London Math. Soc., (2001), 199–234.[19] H. Lin, An Introduction to the Classification of Amenable C ∗ -Algebras , World Scientific, New Jersey,London, Singapore, Hong Kong, 2001.[20] H. Lin, Classification of simple C ∗ -algebras with tracial topological rank zero , Duke. Math. J., (2005),91–119.[21] H. Lin, Classification of simple C ∗ -algebras with tracial rank one , J. Funct. Anal., (2008), 396–440.[22] H. Lin, Asymptotic unitary equivalence and classification of simple amenable C ∗ -algebras , Invent. Math., (2011), 385–450.[23] E. Ortega, M. Rørdam, and H. Thiel, The Cuntz semigroup and comparison of open projections , J. Funct.Anal., (2011), 3474–3493.[24] F. Perera and A. Toms,
Recasting the Elliott conjecture , Math. Ann., (2007), 669–702.[25] M. Rørdam,
On the structure of simple C ∗ -algebras tensored with a UHF algebra , J. Funct. Anal., (1992), 255–269.[26] M. Rørdam, A simple C ∗ -algebra with a finite and an infinite projection , Acta Math. (2003), 109–142.[27] M. Rørdam, The stable and the real rank of Z -absorbing C ∗ -algebras , Internat. J. Math. (2004), 1065–1084.[28] A. Tikuisis, S. White, and W. Winter, Quasidiagonality of nuclear C ∗ -algebras , Ann. of Math., (2017),229–284.[29] A. Toms, On the classification problem for nuclear C ∗ -algebras , Ann. of Math., (2008), 1059–1074.[30] A. Toms, W. White, and W. Winter, Z -stability and finite dimensional tracial boundaries , Int. Math. Res.Not. IMRN, (2015), 2702–2727.[31] J. Villadsen, Simple C ∗ -algebras with perforation , J. Funct. Anal., (1998), 110–116.[32] W. Winter, Nuclear dimension and Z -stability of pure C ∗ -algebras , Invent. Math., (2012), 259–342. SYMPTOTICALLY TRACIALLY APPROXIMATION C ∗ -ALGEBRAS 19 Q INGZHAI F AN , D EPARTMENT OF M ATHEMATICS , S
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