Some properties for certain generalized tracial approximated {\rm C^*}-algebras
aa r X i v : . [ m a t h . OA ] J a n SOME PROPERTIES FOR CERTAIN GENERALIZED TRACIALAPPROXIMATED C ∗ -ALGEBRAS QINGZHAI FAN AND XIAOCHUN FANGA
BSTRACT . In this paper, we introduce a class of generalized tracial approximation C ∗ -algebras. Let P be a class of unital C ∗ -algebras which have tracially Z -absorbing (tracialnuclear dimension at most n , SP property, m -almost divisible, weakly ( m, n ) -divisible).Then A has tracially Z -absorbing (tracial nuclear dimension at most n , SP property,weakly m -almost divisible, secondly weakly ( m, n ) -divisible) for any simple unital C ∗ -algebra A in the class of this generalized tracial approximation C ∗ -algebras. As an appli-cation, Let A be an infinite dimensional unital simple C ∗ -algebra, and let B be a centrallylarge subalgebra of A . If B is tracially Z -absorbing, then A is tracially Z -absorbing. Thisresult was obtained by Archey, Buck and Phillips in [2].
1. I
NTRODUCTION
The Elliott program for the classification of amenable C ∗ -algebras might be said to havebegun with the K -theoretical classification of AF algebras in [11]. Since then, many classesof C ∗ -algebras have been classified by the Elliott invariant. Among them, one importantclass is the class of simple unital AH algebras without dimension growth (in the real rankzero case see [14], and in the general case see [15]). To axiomatize Elliott-Gong’s de-composition theorem for AH algebras of real rank zero (classified by Elliott-Gong in [14])and Gong’s decomposition theorem ([22]) for simple AH algebras (classified by Elliott-Gong-Li in [15]), Huaxin Lin introduced the concepts of TAF and TAI ([30] and [31]).Instead of assuming inductive limit structure, he started with a certain abstract approxima-tion property, and showed that C ∗ -algebras with this abstract approximation property andcertain additional properties are AH algebras without dimension growth. More precisely,Lin introduced the class of tracially approximate interval algebras (also called C ∗ -algebrasof tracial topological rank one). This axiomatization has proved to be very important inthe classification of simple amenable C ∗ -algebras. For example, it led to the classifica-tion of unital simple separable amenable C ∗ -algebras with finite nuclear dimension in theUCT class (see [23], [17], [58]). The isomorphism theorem was established first for thoseseparable amenable C ∗ -algebras with generalized tracial rank at most one (see [23]). Sim-ple C ∗ -algebras with generalized tracial topological rank at most one have good regularityproperties. There are three regularity properties of particular interest: tensorial absorptionof the Jiang-Su algebra Z , also called Z -stability; finite nuclear dimension; and strict com-parison of positive elements. The last property can be reformulated as an algebraic propertyof the Cuntz semigroup, called almost unperforation. Toms and Winter have conjectured(see e.g. [18]) that these three fundamental properties are equivalent for all separable, sim-ple, unital, amenable C ∗ -algebras (and this has now almost completely been proved (see[5], [27], [54], [57] and [7]). Key words C ∗ -algebras, tracial approximation, Cuntz semigroup.2000 Mathematics Subject Classification.
Inspired by Lin’s tracial approximation by interval algebras in [31], Elliott and Niu in[16] considered the natural notion of tracial approximation by other classes of C ∗ -algebras.Let P be a class of unital C ∗ -algebras. Then the class of simple separable C ∗ -algebraswhich can be tracially approximated by C ∗ -algebras in P , denoted by TA P , is defined asfollows. A simple unital C ∗ -algebra A is said to belong to the class TA P if, for any ε > , any finite subset F ⊆ A, and any non-zero element a ≥ , there are a projection p ∈ A and a C ∗ -subalgebra B of A with B = p and B ∈ P such that (1) k xp − px k < ε for all x ∈ F , (2) pxp ∈ ε B for all x ∈ F , and (3) 1 − p is Murray-von Neumann equivalent to a projection in aAa .The question of which properties pass from a class P to the class TA P is interesting andsometimes important. In fact, the property of being of stable rank one, and the property thatthe strict order on projections is determined by traces, are important in the classificationtheorem in [23].In [16], Elliott and Niu showed that the following properties of C ∗ -algebras in a class P are inherited by a simple unital C ∗ -algebras in the class TA P : (1) being stable finite, (2) having stable rank one, (3) having at least one tracial state, (4) the strict order on projections determined by traces, (5) any state of the order-unit K -group comes from a tracial state of the algebra, (6) if the restriction of a tracial state to the order-unit K -group is the average of twodistinct states on the K -group, then it is the average of two distinct tracial states, (7) the property of being K -injective.In [13], Elliott, Fan and Fang showed that some regularity properties of C ∗ -algebras ina class P are inherited by a simple unital C ∗ -algebras in the class TA P .Large and stably large subalgebra were introduced in [46] by Phillips, as an abstractionof Putnam’s orbit breaking subalgebra, of the crossed product algebra C ∗ ( X, Z , σ ) of theCantor set by a minimal homeomorphism in [48]. The Putnam subalgebra played a keyrole in [36], in which it is proved that this abstraction C ∗ ( X, Z , σ ) { y } has tracial rank zerowhenever this property is consistent with it’s K-theory and dim( X ) < ∞ .Let A be an infinite dimensional simple unital C ∗ -algebra and B be a stably large sub-algebra of A . In [46], Phillips showed that the following results: (1) B is simple and infinite dimensional, (2) If B is stably finite then so is A , and if B is purely infinite then so is A , (3) The restriction maps T( A ) → T( B ) and QT( A ) → QT( B ) (on tracial states andquasitraces) are bijective, (4) when A is stably finite, the inclusion of B in A induces an isomorphism on thesemgroups that remain after deleting from Cu( B ) and Cu( A ) all the classes of nonzeroprojections, (5) when A is stably finite, B and A have the same radius of comparison.In [3], Archey and Phillips, define centrally large subalgebras, and they proved that if B is centrally large in A and B has stable rank one, then so does A , In [2], Archey, Buckand Phillips proved that if A is a simple infinite dimensional stably finite unital C ∗ -algebraand B ⊂ A is a centrally large subalgebra, then A is tracially Z -absorbing in the sense of[25] if and only if B is tracially Z -absorbing.Inspired by centrally large subalgebra and tracal approximation C ∗ -algebras. We in-troduce a class of generalized tracial approximation C ∗ -algebra. This generalized tracial ENERALIZED TRACIAL APPROXIMATION C ∗ -ALGEBRAS 3 approximation C ∗ -algebra both generalizes Phillips’s centrally large subalgebras and par-tially generalizes Lin’s notion of tracial approximation. In Theorem 3.9 of [40], Niu showthat if ( X, Γ) be a dynamical system with the (URP), then the crossed product C ∗ -algebrais in our class of generalized tracial approximation C ∗ -algebras, however, the crossed prod-uct C ∗ -algebra is not in the class of tracial approximation C ∗ -algebras and not a centrallylarge subalgebras.Let P be a class of unital C ∗ -algebras. Then the class of C ∗ -algebras which can begeneralized tracial approximated by C ∗ -algebras in P is denoted by TGA P . Definition 1.1.
A simple unital C ∗ -algebra A is said to belong to the class TGA P , if forany ε > , any finite subset F ⊆ A , and any nonzero element a ≥ , there exist a nonzeroprojection p ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra B of A with g ∈ B, B = p and B ∈ P , such that (1) ( p − g ) x ∈ ε B, x ( p − g ) ∈ ε B for all x ∈ F , (2) k ( p − g ) x − x ( p − g ) k < ε for all x ∈ F , (3) 1 − ( p − g ) . a , and (4) k ( p − g ) a ( p − g ) k ≥ k a k − ε . We know that if tsr ( A ) = 1 and A ∈ TA P , then A ∈ TGA P , and if B ∈ P , B ⊂ A is a centrally large subalgebra of A , then A ∈ TGA P . Let P be a class of the form L Ss =1 M K s ( C ( Z s )) , where Z s is compact space, then in Theorem 3.9 of [40], Niu showthat if the topological dynamical system ( X, Γ) with the (URP), the the crossed product C ∗ -algebra belongs to TGA P .In this paper, we show that the following theorems: Theorem 1.2.
Let P be a class of stably finite unital C ∗ -algebras which have tracially Z -absorbing (tracial nuclear dimension at most n , SP property, m -almost divisible, weakly ( m, n ) -divisible). Then A has tracially Z -absorbing (tracial nuclear dimension at most n , SP property, weakly m -almost divisible, secondly weakly ( m, n ) -divisible) for any simpleunital C ∗ -algebra A ∈ TGA P . As applications, Let A be a simple infinite dimensional unital C ∗ -algebra, and let B be a centrally large subalgebra of A . If B is tracially Z -absorbing, then A is tracially Z -absorbing. This result was obtained by Archey, Buck and Phillips in [2]. Let P be a classof stably finite unital C ∗ -algebras which are tracially Z -absorbing. Then A is tracially Z -absorbing for any simple unital C ∗ -algebra A ∈ TA P . This result was obtain by Elliott,Fan and Fang in [13]. 2. P RELIMINARIES AND DEFINITIONS
Recall that a C ∗ -algebra A has SP property, if every nonzero hereditary C ∗ -subalgebraof A contains a nonzero projection.Let a and b be positive elements of a C ∗ -algebra A . We write [ a ] ≤ [ b ] if there is apartial isometry v ∈ A ∗∗ with vv ∗ = P a such that, for every ≤ c ∈ Her ( a ) , cv ∈ A and v ∗ cv ∈ Her ( b ) . ( [ a ] ≤ [ b ] implies that a is Cuntz subequivalent to b, i.e. a . b . If A has stable rank one then, by [9], [ a ] ≤ [ b ] if a . b but even in this case the preorderrelation [ a ] ≤ [ b ] is not necessarily an order relation.) We write [ a ] = [ b ] if, for some v as above, v ∗ Her ( a ) v = Her ( b ) . Let n be a positive integer. We write n [ a ] ≤ [ b ] if inaddition there are n mutually orthogonal positive elements b , b , · · · , b n ∈ Her ( b ) suchthat [ a ] ≤ [ b i ] , i = 1 , , · · · , n (cf. Definition 1.1 in [44], Definition 3.2 in [43], orDefinition 3.5.2 in [32]). QINGZHAI FAN AND XIAOCHUN FANG
Let A be a C ∗ -algebra, and let M n ( A ) denote the C ∗ -algebra of n × n matrices withentries elements of A . Let M ∞ ( A ) denote the algebraic inductive limit of the sequence (M n ( A ) , φ n ) , where φ n : M n ( A ) → M n +1 ( A ) is the canonical embedding as the upperleft-hand corner block. Let M ∞ ( A ) + (respectively, M n ( A ) + ) denote the positive elementsof M ∞ ( A ) (respectively, M n ( A ) ). Given a, b ∈ M ∞ ( A ) + , we say that a is Cuntz sube-quivalent to b (written a . b ) if there is a sequence ( v n ) ∞ n =1 of elements of M ∞ ( A ) suchthat 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 [9].)Observe that any a, b ∈ M ∞ ( A ) + are Cuntz equivalent to orthogonal elements a ′ , b ′ ∈ M ∞ ( A ) + (i.e., a ′ b ′ = 0 ), and so W( A ) becomes an ordered semigroup when equippedwith the addition operation h a i + h b i = h a + b i whenever ab = 0 , and the order relation h a i ≤ h b i ⇔ a . b. Let A be a stably finite unital C ∗ -algebra. Recall that a positive element a ∈ A is calledpurely positive if a is not Cuntz equivalent to a projection. This is equivalent to saying that is an accumulation point of σ ( a ) (recall that σ ( a ) denotes the spectrum of a ).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 ) . Bythe functional calculus, it follows in a straightforward manner that (( a − ε ) + − ε ) + =( a − ( ε + ε )) + . The following Theorem is well known.
Theorem 2.1. ( [1] , [25] , [53] .) Let A be a stably finite C ∗ -algebra. (1) Let a, b ∈ A + and ε > be such that k a − b k < ε . Then there is a contraction d in A with ( a − ε ) + = dbd ∗ . (2) Let a, p be positive elements in M ∞ ( A ) with p a projection. If p . a, then there is b in M ∞ ( A ) + such that bp = 0 and b + p ∼ a . (3) The following conditions are equivalent: (1) ′ a . b, (2) ′ for any ε > , ( a − ε ) + . b, and (3) ′ for any ε > , there is δ > , such that ( a − ε ) + . ( b − δ ) + . (4) Let a be a purely positive element of A (i.e., a is not Cuntz equivalent to a projec-tion). Let δ > , and let f ∈ C (0 , be a non-negative function with f = 0 on ( δ, ,f > on (0 , δ ) , and k f k = 1 . We have f ( a ) = 0 and ( a − δ ) + + f ( a ) . a. Winter and Zacharias introduced the nuclear dimension for C ∗ -algebras in [62]. Definition 2.2. ( [62] .) Let A be a C ∗ -algebra, m ∈ N . A complete positive compression ϕ : F → A is m -decomposable (where F is finite dimensional C ∗ -algebra), if there is adecomposition F = F (0) ⊕ F (1) ⊕ · · · ⊕ F ( m ) such that the restriction ϕ ( i ) of ϕ to F ( i ) hasorder zero (which means preserves orthogonality, i.e., ψ ( e ) ψ ( f ) = 0 for all e, f ∈ M n with ef = 0 ), for each i ∈ { , · · · ,m } , we say ϕ is m -decomposable with respect to F = F (0) ⊕ F (1) ⊕ · · · ⊕ F ( m ) . A has nuclear dimension m , write dim nuc ( A ) = m , if m is the least integer such that the following holds: For any finite subset G ⊆ A and ε > ,there is a finite dimension complete positive compression approximation ( F, ϕ, ψ ) for G towithin ε (i.e., F is finite-dimensional ψ : A → F and ϕ : F → A are complete positiveand k ϕψ ( b ) − b k < ε for any b ∈ G ) such that ψ is complete positive compression, and ϕ ENERALIZED TRACIAL APPROXIMATION C ∗ -ALGEBRAS 5 is m -decomposable with complete positive compression order zero components ϕ i . If nosuch m exists, we write dim nuc ( A ) = ∞ . Hirshberg and Orovitz’s introduced tracial Z -absorbing in [25]. Definition 2.3. ( [25] .) We say a unital C ∗ -algebra A is tracially Z -absorbing, if A = C and for any finite set F ⊆ A, ε > , non-zero positive element a ∈ A , and n ∈ N , thereis a completely positive order zero contraction ψ : M n → A , where completely positiveorder zero map means preserving orthogonality, i.e., ψ ( e ) ψ ( f ) = 0 for all e, f ∈ M n with ef = 0 , such that the following properties hold: (1) 1 − ψ (1) - a, and (2) for any normalized element x ∈ M n (i.e., with k x k = 1 ), and any y ∈ F we have k ψ ( x ) y − yψ ( x ) k < ε . Inspired by Hirshberg and Orovitz’s tracial Z -absorbing in [25], Fu introduced the finitetracial nuclear dimension in his doctoral dissertation in [21], and he show that finite tracialnuclear dimension implies tracially Z -absorbing for separable, exact, simple C ∗ -algebrawith nonempty tracial state space. Definition 2.4. ( [21] .) A unital C ∗ -algebra A is said to have tracial nuclear dimension atmost m , denote Tdim nuc ( A ) ≤ m , if for any ε > , any finite subset F ⊆ A , any nonzeropositive element a of A , there exist a contractive completely positive linear map ϕ : A → A and a contractive completely positive linear map ψ : A → B with dim nuc ( B ) ≤ m suchthat (1) ϕ (1) . a , and (2) k x − ϕ ( x ) − ψ ( x ) k < ε , for any x ∈ F . Centrally large and stably centrally large subalgebra were introduced in [3] by Archeyand Phillips.
Definition 2.5. ( [3] .) Let A be an infinite dimensional simple unital C ∗ -algebra. A unitalsubalgebra B ⊂ A is said to be centrally large in A if for every m ∈ N , a , a , . . . , a m ∈ A , ε > , x ∈ A + with k x k = 1 , and y ∈ B + \ { } , there are c , c , . . . , c m ∈ A and g ∈ B such that: (1) 0 ≤ g ≤ . (2) For j = 1 , , . . . , m we have k c j − a j k < ε . (3) For j = 1 , , . . . , m we have (1 − g ) c j ∈ B . (4) g - B y and g - A x . (5) k (1 − g ) x (1 − g ) k > − ε . (6) For j = 1 , , . . . , m we have k ga j − a j g k < ε . The property of m -almost divisible was introduced by Robert and Tikuisis in [49]. Definition 2.6. ( [49] .) Let m ∈ N . We say that A is m -almost divisible if for each a ∈ M ∞ ( A ) + , k ∈ N and ε > , there exists b ∈ M ∞ ( A ) + such that k h b i ≤ h a i and h ( a − ε ) + i ≤ ( k + 1)( m + 1) h b i . Definition 2.7.
Let m ∈ N . We say that A is weakly m -almost divisible if for each a ∈ M ∞ ( A ) + , k ∈ N and ε > , there exists b ∈ M ∞ ( A ) + such that k h b i ≤ h a i + h a i and h ( a − ε ) + i ≤ ( k + 1)( m + 1) h b i . The property of weakly ( m, n ) -divisible was introduced by Kirchberg and Rørdam in[26]. QINGZHAI FAN AND XIAOCHUN FANG
Definition 2.8. ( [26] .) Let A be unital C ∗ -algebra. Let m, n ≥ be integers. A issaid has weakly ( m, n ) -divisible, if for every u in W( A ) , any ε > , there exist elements x , x , · · · , x n ∈ W( A ) , such that mx j ≤ u for all j = 1 , , · · · , n and ( u − ε ) + ≤ x + x + · · · + x n . Definition 2.9.
Let A be unital C ∗ -algebra. Let m, n ≥ be integers. A is said hassecond weakly ( m, n ) -divisible if for every u in W( A ) , any ε > , there exist elements x , x , · · · , x n ∈ W( A ) , such that mx j ≤ u + u for all j = 1 , , · · · , n and ( u − ε ) + ≤ x + x + · · · + x n . The following two Theorem are Lemma 1.7 and Lemma 1.8 of [26]
Theorem 2.10.
For every ε > there is δ > such that the following holds. Let A be a C ∗ -algebra, let B ⊂ A be a subalgebra, let n be a nonzero integer, let ϕ : M n → A be acompletely positive contractive order zero map, and let x ∈ B satisfy: (1) 0 ≤ x ≤ , (2) with ( e j,k ) , j, k = 1 , , · · · , n be the standard system of matrix units for M n , wehave k ϕ ( e j,k ) x − xϕ ( e j,k ) k < ε for j, k = 1 , , · · · , n , (3) ϕ ( e j,k ) x ∈ ε B .Then there is a completely positive contractive order zero map ϕ : M n → B such thatfor all z ∈ M n with k z k ≤ , we have k ϕ ( z ) x − ϕ ( z ) k < ε . Theorem 2.11.
For every ε > and n be a nonzero integer, there is δ > such thatthe following holds. Whenever A, B , ϕ : M n → A and x ∈ B satisfy the conditions inTheorem 2.10, and in addition A is unital and B conditions the identity of A , there existsa completely positive contractive order zero map ϕ : M n → A such that: (1) k ϕ ( z ) − ϕ ( z ) x k < ε , for all z ∈ M n with k z k ≤ , (2) 1 − ϕ (1) . (1 − x ) ⊕ (1 − ϕ (1) .
3. T
HE MAIN RESULTS
Theorem 3.1.
Let P be a class of stably finite unital C ∗ -algebras which have SP property.Then A has SP property for any simple unital C ∗ -algebra A ∈ TGA P .Proof. Let B be any hereditary C ∗ -algebra of A , If B is finite dimensional C ∗ -algebra.Then B contains a nonzero projection. So we may assume that B is infinite dimensional.Then B contains a nonzero element a with k a k = 1 and with infinite spectrum. Choose ε with ≤ ε ≤ / , put a = f ε ( a ) , a = f ε ( a ) then ≤ a i ≤ , i = 1 , and a a = a .For F = { a , a , a / , a / } , any ε > , since A ∈ TGA P there exist a nonzeroprojection p ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with B = p and D has SP property such that (1) ( p − g ) x ∈ ε D, x ( p − g ) ∈ ε D for all x ∈ F , (2) k ( p − g ) x − x ( p − g ) k < ε for all x ∈ F .By (1) , there exist positive elements b , b ∈ D such that k ( p − g ) a i ( p − g ) − b i k < ε for i = 1 , . Since a a = a , we have k b b − b k < ε , i.e., k ( p − b ) b k < ε . Byperturbation, if ε is sufficiently small, there exists δ > and exist c ′ , c ∈ D such that k c ′ − ( p − b ) k < δ and k c − b k < δ and c ′ c = 0 . Let c = p − c ′ . Then we have k c − b k < δ and c c = c .Since D has SP property, there exists a nonzero projection q ∈ c Dc .Since c c = c , we have c q = q .Therefore, we have ENERALIZED TRACIAL APPROXIMATION C ∗ -ALGEBRAS 7 k a / ( p − g ) a / − c k ≤ k a / ( p − g ) a / − b k + k b − c k < ε + δ and k a / ( p − g ) a / qa / ( p − g ) a / − q k≤ k a / ( p − g ) a / qa / ( p − g ) a / − c qc k + k c qc − q k < ε + δ ) . When ε is small enough, then δ is small enough. By Lemma 2.3 in [32], there exists anon-zero projection e ∈ Her( a ) , therefore, A has the SP property. (cid:3) Corollary 3.2.
Let A be a simple infinite dimensional unital C ∗ -algebra, and let B be acentrally large subalgebra of A . If B has SP property, then A has SP property. Corollary 3.3.
Let P be a class of stably finite unital C ∗ -algebras which have SP property.Then A has SP property for any simple unital C ∗ -algebra A ∈ TA P . Theorem 3.4.
Let P be a class of nuclear unital C ∗ -algebras such that Tdim nuc ( B ) ≤ m for any B ∈ Ω . Then we have Tdim nuc ( A ) ≤ m for any infinite dimension simple unital C ∗ -algebra A ∈ TGA P .Proof. We need to show that for any ε > , any finite subset F = { a , a , · · · , a n } of A ,any nonzero positive element b of A , there exist a contractive completely positive linearmap ϕ : A → A and a contractive completely positive linear map ψ : A → D with dim nuc ( D ) ≤ m such that (1) ϕ (1) . b , and (2) k x − ϕ ( x ) − ψ ( x ) k < ε , for any x ∈ F .By Lemma 2.3 in [46], there exist nonzero positive elements b , b ∈ A such that b b = 0 and b + b . b .For any sufficiently small ε ′ > , since A ∈ TGA P there exist a nonzero projection p ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra B of A with B = p and Tdim nuc ( B ) ≤ m such that (1) ′ ( p − g ) b ∈ ε ′ B, b ( p − g ) ∈ ε ′ B , ( p − g ) a i ∈ ε ′ B, a i ( p − g ) ∈ ε ′ B for ≤ i ≤ n , (2) ′ k ( p − g ) b − b ( p − g ) k < ε ′ , k ( p − g ) a i − a i ( p − g ) k < ε ′ for ≤ i ≤ n , (3) ′ − ( p − g ) . b , and (4) ′ k ( p − g ) / b ( p − g ) / k > − ε ′ .By (1) ′ and (2) ′ , there exist positive elements a ′ , a ′ , · · · , a ′ n , b ′ ∈ B such that k ( p − g ) / a i ( p − g ) / − a ′ i k < ε , k ( p − g ) / b ( p − g ) / − b ′ k < ε for ≤ i ≤ n .We have k b − b ′ − (1 − ( p − g )) / b (1 − ( p − g )) / k < ε .Since k ( p − g ) / b ( p − g ) / − b ′ k < ε , we have ( b ′ − ε ) + . (( p − g ) / b ( p − g ) / − ε ) + .Since k ( p − g ) / b i ( p − g ) / k > − ε , we have ( b ′ − ε ) + = 0 .We define ϕ ′′ : A → A by ϕ ′′ ( a ) = (1 − ( p − g )) / a (1 − ( p − g )) / , then ϕ ′′ is acontractive completely positive linear map. Since B is a nuclear C ∗ -algebra, by Theorem2.3.13 of [32], there exist a contractive completely positive linear map ψ ′′ : A → B suchthat k ψ ′′ ( a ′ i ) − a ′ i k < ε for all ≤ i ≤ n .Since Tdim nuc ( B ) ≤ m , there exist a contractive completely positive linear map ϕ ′ : B → B and a contractive completely positive linear map ψ ′ : B → D with dim nuc ( D ) ≤ m such that QINGZHAI FAN AND XIAOCHUN FANG (1) ′′ ϕ ′ ( p ) . ( b ′ − ε ) + , and (2) ′′ k a ′ i − ϕ ′ ( a ′ i ) − ψ ′ ( a ′ i ) k < ε for all ≤ i ≤ n .Write ϕ : A → A by ϕ ( a ) = ϕ ′′ ( a ) + ϕ ′ ( ψ ′′ (( p − g ) / a ( p − g ) / )) and ψ : A → D by ψ ( a ) = ψ ′ ( ψ ′′ (( p − g ) / a ( p − g ) / ))) , then ϕ and ψ are contractive completelypositive linear maps. We have ϕ (1) = ϕ ′′ (1) + ϕ ′ ( ψ ′′ ( p − g )) . − ( p − g ) ⊕ ϕ ′ ( p − g ) . − ( p − g ) ⊕ ϕ ′ ( p ) . b ⊕ ( b ′ − ε ) + . b ⊕ ((1 − ( p − g )) / b (1 − ( p − g )) / + ( p − g ) / b ( p − g ) / ) − ε ) + . b ⊕ b . b, and k a i − ϕ ( a i ) − ψ ( a i ) k≤ k a i − (1 − ( p − g )) / a i (1 − ( p − g )) / − a ′ i k≤ ε + k a ′ i − ϕ ′ ( ψ ′′ (( p − g ) / a i ( p − g ) / ))) k + k a ′ i − ϕ ′ ( a ′ i ) − ψ ′ ( a ′ i ) k + k ϕ ′ ( a ′ i ) − ϕ ′ ( ψ ′′ (( p − g ) / a i ( p − g ) / )) k + k ψ ′ ( a ′ i ) − ψ ′ ( ψ ′′ (( p − g ) / a i ( p − g ) / )) k + k ψ ′′ (( p − g ) / a i ( p − g ) / )) − ψ ′′ ( a ′ i ) k + k ψ ′′ ( a ′ i ) − a ′ i k < ε + 2 ε + 2 ε + ε + ε < ε, for all ≤ i ≤ n . (cid:3) Corollary 3.5.
Let A be a simple infinite dimensional unital C ∗ -algebra, and let B be annuclear centrally large subalgebra of A . If Tdim nuc ( B ) ≤ m , then Tdim nuc ( A ) ≤ m . Corollary 3.6.
Let P be a class of stably finite unital C ∗ -algebras which have tracialnuclear dimension at most n . Then A has tracial nuclear dimension at most n for anysimple unital C ∗ -algebra A ∈ TA P . Theorem 3.7.
Let P be a class of stably finite unital C ∗ -algebras which are tracially Z -absorbing. Then A is tracially Z -absorbing for any infinite dimensional unital simple C ∗ -algebra A ∈ TGA P .Proof. We need to show that for any finite set F = { a , a , · · · , a k } ⊆ A, any ε > , any non-zero positive element b ∈ A and n ∈ N , there is an order zero contraction ψ :M n → A such that the following conditions hold: (1) 1 − ψ (1) - b , and (2) for any normalized element z ∈ M n and any y ∈ F, we have k ψ ( z ) y − yψ ( z ) k < ε. Since A is a simple C ∗ -algebra there exist b ′ , b ′′ ∈ A such that b ′ b ′′ = 0 and b ′ + b ′′ . b .Since A ∈ TGA P , for any ε ′ > , there exist a nonzero projection p ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra B of A with B = p and B is tracially Z -absorbing such that (1) ′ ( p − g ) b ′′ , b ′′ ( p − g ) ∈ ε ′ B , ( p − g ) a i , a i ( p − g ) ∈ ε ′ B for ≤ i ≤ n , (2) ′ k ( p − g ) b ′′ − b ′′ ( p − g ) k < ε ′ , k ( p − g ) a i − a i ( p − g ) k < ε ′ for ≤ i ≤ n , (3) ′ − ( p − g ) . b ′ , and (4) ′ k ( p − g ) / b ′′ ( p − g ) / k > − ε ′ .By (1) ′ and (2) ′ , there exist positive elements a ′ , a ′ , · · · , a ′ n , b ′′′ ∈ B such that k ( p − g ) / a i ( p − g ) / − a ′ i k < ε, ENERALIZED TRACIAL APPROXIMATION C ∗ -ALGEBRAS 9 k ( p − g ) / b ′′ ( p − g ) / − b ′′′ k < ε for ≤ i ≤ n .By (4) ′ , we have k ( p − g ) / b ′′ ( p − g ) / k > − ε ′ , Since k ( p − g ) / b ′′ ( p − g ) / − b ′′′ k < ε, therefore we have k ( b ′′′ − ε ) + k + 2 ε ≥ k ( p − g ) / b ′′ ( p − g ) / k > − ε So we have k ( b ′′′ − ε ) + k ≥ k ( p − g ) / b ′′ ( p − g ) / k > − ε > . Since B ∈ P , for G = { a ′ , a ′ , · · · , a ′ k , p − g, ( p − g ) / , ( p − g ) a i } ⊆ B, ε ′′ > asspecified, there is an order zero contraction ψ : M n → B with the following properties: (1) ′′ p − ψ (1) - ( b ′′′ − ε ) + , and (2) ′′ for any normalized element z ∈ M n and any a ′ i ∈ G, we have k ψ ( z ) a ′ i − a ′ i ψ ( z ) k < ε ′′ , k ψ ( z )( p − g ) − ( p − g ) ψ ( z ) k < ε ′′ and k ψ ( z )( p − g ) / − ( p − g ) / ψ ( z ) k < ε ′′ , k ψ ( z )( p − g ) a i − ( p − g ) a i ψ ( z ) k < ε ′′ . By Theorem 2.11, there exist ψ : M n → B ⊆ A such that (1) ′′′ k ψ ( z ) − ψ ( z )( p − g ) k < ε , (2) ′′′ − ψ (1) . − ( p − g ) ⊕ ( p − ψ (1)) . b ′ ⊕ ( b ′′′ − ε ) + . b ′ ⊕ ( p − g ) / b ′′ ( p − g ) / . b ′ + b ′′ . b .For any normalized element z ∈ M n , we have k ψ ( z ) a ′ i − a ′ i ψ ( z ) k≤ k ψ ( z ) a ′ i − ψ ( z )( p − g ) a ′ i k + k ψ ( z )( p − g ) a ′ i − ( p − g ) a ′ i ψ ( z ) k + k ( p − g ) a ′ i ψ ( z ) − ( p − g )( p − g ) / a i ( p − g ) / ψ ( z ) k + k ( p − g )( p − g ) / a i ( p − g ) / ψ ( z ) − ( p − g ) a i ( p − g ) ψ ( z ) k + k ( p − g ) a i ( p − g ) ψ ( z ) − ( p − g ) / a i ( p − g ) / )( p − g ) ψ ( z ) k + k ( p − g ) / a i ( p − g ) / )( p − g ) ψ ( z ) − a ′ i ( p − g ) ψ ( z ) k + k a ′ i ( p − g ) ψ ( z ) − a ′ i ψ ( z ) k≤ ε + 2 ε + ε + ε + 3 ε + ε + ε + ε = 11 ε. We also have k ψ ( z )(1 − ( p − g )) / a i (1 − ( p − g )) / − (1 − ( p − g )) / a i (1 − ( p − g )) / ψ ( z ) k≤ k ψ ( z )(1 − ( p − g )) / a i ((1 − ( p − g )) / − ψ ( z )( p − g )(1 − ( p − g )) / a i (1 − ( p − g )) / k + k ψ ( z )( p − g )(1 − ( p − g )) / a i (1 − ( p − g )) / − ψ ( z )( p − g )(1 − ( p − g )) a i k + k ψ ( z )( p − g )(1 − ( p − g )) a i − ψ ( z )(1 − ( p − g ))( p − g ) a i k + k ψ ( z )(1 − ( p − g ))( p − g ) a i − (1 − ( p − g )) ψ ( z )( p − g ) a i k + k (1 − ( p − g )) ψ ( z )( p − g ) a i − (1 − ( p − g ))( p − g ) a i ψ ( z ) k + k ((1 − ( p − g ))( p − g ) a i ψ ( z ) − (1 − ( p − g )) a i ( p − g ) ψ ( z ) k + k (1 − ( p − g )) a i ( p − g ) ψ ( z ) k − (1 − ( p − g )) / a i ((1 − ( p − g )) / )( p − g ) ψ ( z ) k + k (1 − ( p − g )) / a i (1 − ( p − g )) / )( p − g ) ψ ( z ) − (1 − ( p − g )) / a i ((1 − ( p − g )) / ) ψ ( z )( p − g ) k + k (1 − ( p − g )) / a i ((1 − ( p − g )) / ) ψ ( z )( p − g ) − (1 − ( p − g )) / a i ((1 − ( p − g )) / ) ψ ( z ) k≤ ε + 2 ε + ε + 2 ε + 2 ε + ε + 2 ε + ε + 2 ε + 2 ε ≤ ε. Therefore, we have k ψ ( z ) a i − a i ψ ( z ) k≤ k ψ ( z ) a i − ψ ( z )( a ′ i + (1 − ( p − g )) / a i (1 − ( p − g )) / ) k + k ψ ( z )( a ′ i + (1 − ( p − g )) / a i (1 − ( p − g )) / ) − ( a ′ i + (1 − ( p − g )) / a i (1 − ( p − g )) / ) ψ ( z ) k + k ( a ′ i + (1 − ( p − g )) / a i (1 − ( p − g )) / ) ψ ( z ) − a i ψ ( z ) k≤ ε + 2 ε + k ψ ( z ) a ′ i − a ′ i ψ ( z ) k + k ψ ( z )( p − g )) / a i ((1 − ( p − g )) / − ( p − g )) / a i ((1 − ( p − g )) / ψ ( z ) k≤ ε + 11 ε + 13 ε = 27 ε. (cid:3) The following two Corollaries are well-known.
Corollary 3.8. ( [2] ) Let A be a simple infinite dimensional unital C ∗ -algebra, and let B be a centrally large subalgebra of A . If B is tracially Z -absorbing, then A is tracially Z -absorbing. Corollary 3.9. ( [13] ) Let P be a class of stably finite unital C ∗ -algebras which are tra-cially Z -absorbing. Then A is tracially Z -absorbing for any simple unital C ∗ -algebra A ∈ TA P . Theorem 3.10.
Let P be a class of stably finite unital C ∗ -algebras which have m -almostdivisible. Then A has weakly m -almost divisible for any simple unital C ∗ -algebra A ∈ TGA P . Proof.
We need to show that there is b ∈ M ∞ ( A ) + such that kb . a ⊕ a and ( a − ε ) + . ( k + 1)( m + 1) b for any a ∈ A + , ε > and k ∈ N . We may assume that k a k = 1 . With F = { a } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra B of A with B = p and B ∈ P , such that (1) ( p − g ) x ∈ ε B, x ( p − g ) ∈ ε B for all x ∈ F , and (2) k ( p − g ) x − x ( p − g ) k < ε for all x ∈ F . ENERALIZED TRACIAL APPROXIMATION C ∗ -ALGEBRAS 11 By (1) and (2) there exist positive elements a ′ ∈ B and a ′′ ∈ A such that k a − a ′ − a ′′ k < ε, k ( p − g ) / a ( p − g ) / − a ′ k < ε/ , and k (1 − ( p − g ) / ) a (1 − ( p − g ) / ) − a ′′ k <ε/ . Since B has m -almost divisible, and ( a ′ − ε ) + ∈ B , there exists b ∈ B such that kb . ( a ′ − ε ) + and ( a ′ − ε ) + . ( k + 1)( m + 1) b .Since B has m -almost divisible, and ( a ′ − ε ) + ∈ B , there exists b ′ ∈ B such that kb ′ . ( a ′ − ε ) + and ( a ′ − ε ) + . ( k + 1)( m + 1) b ′ .We divide the proof into two cases. Case (1) , we assume that ( a ′ − ε ) + is Cuntz equivalent to a projection. (1.1) , we assume that ( a ′ − ε ) + is Cuntz equivalent to a projection. (1.1.1) , If ( a ′ − ε ) + is not Cuntz equivalent to ( k + 1)( m + 1) b . We may assume thatthere exist non-zero c ∈ A + such that ( a ′ − ε ) + ⊕ c . ( k + 1)( m + 1) b .With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x, x ( p ′ − g ) ∈ ε D for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ B and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . Since D has m -almost divisible, and ( a ′′′ − ε ) + ∈ B , there exists b ∈ D + such that kb . ( a ′′′ − ε ) + and ( a ′′′ − ε ) + . ( k + 1)( m + 1) b . Therefore we have k ( b ⊕ b ) ∼ kb ⊕ kb . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, and we also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ c . ( k + 1)( m + 1) b ⊕ ( k + 1)( m + 1) b ∼ ( k + 1)( m + 1)( b ⊕ b ) . (1.1.2) , If ( a ′ − ε ) + is Cuntz equivalent to ( k + 1)( m + 1) b , then kb . ( a ′ − ε ) + and ( k +1)( m +1) b . ( a ′ − ε ) + , we have k ( b ⊕ b ) . ( a ′ − ε ) + , so ( a ′ − ε ) + ⊕ b . ( k + 1)( m + 1)( b ⊕ b ) .With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra B of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x, x ( p ′ − g ) ∈ ε B for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ B and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . Since D has m -almost divisible, and ( a ′′′ − ε ) + ∈ D , there exists b ∈ D such that kb . ( a ′′′ − ε ) + and ( a ′′′ − ε ) + . ( k + 1)( m + 1) b .Therefore we have k ( b ⊕ b ⊕ b ) ∼ kb ⊕ b ⊕ kb . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, and we also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ b . ( k + 1)( m + 1) b ⊕ ( k + 1)( m + 1)( b ⊕ b ) ∼ ( k + 1)( m + 1)( b ⊕ b ⊕ b ) . ∼ ( k + 1)( m + 1)( b ⊕ b ⊕ b ) . (1.2) , we assume that ( a ′ − ε ) + is not Cuntz equivalent to a projection. By Theorem2.1, there is a non-zero positive element d such that ( a ′ − ε ) + + d . ( a ′ − ε ) + .With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with B = p ′ and B ∈ P , such that (1) ( p ′ − g ) x, x ( p ′ − g ) ∈ ε D for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ B and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . Since D has m -almost divisible, and ( a ′′′ − ε ) + ∈ B , there exists b ∈ D + such that kb . ( a ′′′ − ε ) + and ( a ′′′ − ε ) + . ( k + 1)( m + 1) b .Therefore we have k ( b ⊕ b ) ∼ kb ⊕ kb . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, and we also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ d . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . ( k + 1)( m + 1) b ⊕ ( k + 1)( m + 1)( b ) ∼ ( k + 1)( m + 1)( b ⊕ b ) . Case (2) , we suppose that we assume that ( a ′ − ε ) + is not Cuntz equivalent to aprojection.By Theorem 2.1, there is a non-zero positive element d such that ( a ′ − ε ) + + d . ( a ′ − ε ) + . ENERALIZED TRACIAL APPROXIMATION C ∗ -ALGEBRAS 13 With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x, x ( p ′ − g ) ∈ ε D for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ B and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . Since D has m -almost divisible, and ( a ′′′ − ε ) + ∈ D , there exists b ∈ D + such that kb . ( a ′′′ − ε ) + and ( a ′′′ − ε ) + . ( k + 1)( m + 1) b .Therefore we have k ( b ′ ⊕ b ) ∼ kb ′ ⊕ kb . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, and we also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ d . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . ( k + 1)( m + 1) b ⊕ ( k + 1)( m + 1) b ′ ∼ ( k + 1)( m + 1)( b ⊕ b ′ ) . (cid:3) Theorem 3.11.
Let P be a class of stably finite unital C ∗ -algebras such that for any B ∈ P , B has weakly ( m, n ) -divisible ( m = n ). Then A has second weakly ( m, n ) -divisible for any simple unital C ∗ -algebra A ∈ TGA P .Proof. We need to show that for any a ∈ M ∞ ( A ) + , any ε > , there exist x , x , · · · , x n ∈ M ∞ ( A ) + such that x j ⊕ x j ⊕ · · · ⊕ x j . a ⊕ a for all ≤ j ≤ n , where x j repeat m times, and ( a − ε ) + . ⊕ ni =1 x i .We may assume a ∈ A + .With F = { a } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra B of A with B = p and B ∈ P , such that (1) ( p − g ) x ∈ ε B, x ( p − g ) ∈ ε B for all x ∈ F , and (2) k ( p − g ) x − x ( p − g ) k < ε for all x ∈ F .By (2) and (3) there exist positive elements a ′ ∈ B and a ′′ ∈ A such that k a − a ′ − a ′′ k < ε, k ( p − g ) / a ( p − g ) / − a ′ k < ε/ , and k (1 − ( p − g ) / ) a (1 − ( p − g ) / ) − a ′′ k <ε/ . Since B has weakly ( m, n ) -divisible, there exist x ′ , x ′ , · · · , x ′ n ∈ M ∞ ( B ) + such that x ′ j ⊕ x ′ j ⊕ · · · ⊕ x ′ j . ( a ′ − ε ) + where x ′ j repeat m times and ( a ′ − ε ) + . ⊕ ni =1 x ′ i .Since B has weakly ( m, n ) -divisible, there exist y ′ , y ′ , · · · , y ′ n ∈ M ∞ ( B ) + such that y ′ j ⊕ y ′ j ⊕ · · · ⊕ y ′ j . ( a ′ − ε ) + where y ′ j repeat m times and ( a ′ − ε ) + . ⊕ ni =1 y ′ i .We divide the proof into two cases. Case (1) , we assume that ( a ′ − ε ) + is Cuntz equivalent to a projection. (1.1) , we assume that ( a ′ − ε ) + is Cuntz equivalent to a projection. (1.1.1) , we assume that ( a ′ − ε ) + ∼ ( a ′ − ε ) + . (1.1.1.1) , If x ′ , x ′ , · · · , x ′ n ∈ M ∞ ( B ) + are all Cuntz equivalent to projections, and ( a ′ − ε ) + ∼ ⊕ ni =1 x ′ i . Then there exist some j and a nonzero projection r such that ( x ′ j ⊕ r ) ⊕ ( x ′ j ⊕ r ) ⊕ · · · ⊕ ( x ′ j ⊕ r ) . ( a ′ − ε ) + where x ′ j ⊕ r repeat m times, otherwise,this contradicts the stable finiteness of A (since m = n and C ∗ -algebra A is stably finite).With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x ∈ ε D, x ( p ′ − g ) ∈ ε D for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ D and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . Since D has weakly ( m, n ) -divisible, there exist x ′′ , x ′′ , · · · , x ′′ n ∈ M ∞ ( D ) + such that x ′′ j ⊕ x ′′ j ⊕ · · · ⊕ x ′′ j . ( a ′′′ − ε ) + where x ′′ j repeat m times and ( a ′′′ − ε ) + . ⊕ ni =1 x ′′ i .Therefore we have (( x ′ j ⊕ r ) ⊕ x ′′ j )) ⊕ (( x ′ j ⊕ r ) ⊕ x ′′ j )) ⊕ · · · ⊕ (( x ′ j ⊕ r ) ⊕ x ′′ j )) . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, and ( x ′ i ⊕ x ′′ i ) ⊕ ( x ′ i ⊕ x ′′ i ) ⊕ · · · ⊕ ( x ′ i ⊕ x ′′ i ) . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, for all i = j and ≤ i ≤ n where ( x ′ i ⊕ x ′′ i ) repeat m times.We also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ r . ⊕ ni =1 ,i = j ( x ′ i ⊕ x ′′ i ) ⊕ (( x ′ j ⊕ r ) ⊕ x ′′ j ) . (1.1.1.2) , If x ′ , x ′ , · · · , x ′ k ∈ M ∞ ( B ) + are all projections, and ( a ′ − ε ) + < ⊕ ki =1 x ′ i .Then there exists a nonzero projection s such that ( a ′ − ε ) + ⊕ s . ⊕ ki =1 x ′ i .With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x ∈ ε D, x ( p ′ − g ) ∈ ε D for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ D and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . ENERALIZED TRACIAL APPROXIMATION C ∗ -ALGEBRAS 15 Since D has weakly ( m, n ) -divisible, there exist x ′′ , x ′′ , · · · , x ′′ n ∈ M ∞ ( D ) + such that x ′′ j ⊕ x ′′ j ⊕ · · · ⊕ x ′′ j . ( a ′′′ − ε ) + where x ′′ j repeat m times and ( a ′′′ − ε ) + . ⊕ ni =1 x ′′ i .Therefore we have ( x ′ i ⊕ x ′′ i ) ⊕ ( x ′ i ⊕ x ′′ i ) ⊕ · · · ⊕ ( x ′ i ⊕ x ′′ i ) . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, for ≤ i ≤ n where ( x ′ i ⊕ x ′′ i ) repeat m times.We also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ s . ⊕ ni =1 ( x ′ i ⊕ x ′′ i ) . (1.1.1.3) , we assume that there is a purely positive element x ′ . Since ( a ′ − ε ) + . ⊕ ni =1 x ′ i , for any ε > , there exists δ > , such that ( a ′ − ε ) + . ( x ′ − δ ) + ⊕ ni =2 x ′ i ,By Theorem 2.1, there exists a nonzero positive element d such that ( x ′ − δ ) + + d . x ′ .With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x ∈ ε B, x ( p ′ − g ) ∈ ε B for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ D and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . Since D has weakly ( m, n ) -divisible, there exist x ′′ , x ′′ , · · · , x ′′ n ∈ M ∞ ( D ) + such that x ′′ j ⊕ x ′′ j ⊕ · · · ⊕ x ′′ j . ( a ′′′ − ε ) + where x ′′ j repeat m times and ( a ′′′ − ε ) + . ⊕ ni =1 x ′′ i .Therefore we have ( x ′ i ⊕ x ′′ i ) ⊕ ( x ′ i ⊕ x ′′ i ) ⊕ · · · ⊕ ( x ′ i ⊕ x ′′ i ) . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, for ≤ i ≤ n where ( x ′ i ⊕ x ′′ i ) repeat m times.We also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ d . ⊕ ni =1 ( x ′ i ⊕ x ′′ i ) . (1.1.2) We assume that there exists nonzero projection r such that ( a ′ − ε ) + + r . ( a ′ − ε ) + . With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x ∈ ε D, x ( p ′ − g ) ∈ ε D for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ D and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g )) / a ′′ − ( p ′ − g ) / − a k < ε/ . Since D has weakly ( m, n ) -divisible, there exist x ′′ , x ′′ , · · · , x ′′ n ∈ M ∞ ( D ) + such that x ′′ j ⊕ x ′′ j ⊕ · · · ⊕ x ′′ j . ( a ′′′ − ε ) + where x ′′ j repeat m times and ( a ′′′ − ε ) + . ⊕ ni =1 x ′′ i .Therefore we have ( y ′ i ⊕ x ′′ i ) ⊕ ( y ′ i ⊕ x ′′ i ) ⊕ · · · ⊕ ( y ′ i ⊕ x ′′ i ) . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, for ≤ i ≤ n where ( x ′ i ⊕ x ′′ i ) repeat m times.We also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ r . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . ⊕ ni =1 ( y ′ i ⊕ x ′′ i ) . (1.2) If ( a ′ − ε ) + is not Cuntz equivalent to a projection.By Theorem 2.1, there is a non-zero positive element d such that ( a ′ − ε ) + + d . ( a ′ − ε ) + .With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x ∈ ε D, x ( p ′ − g ) ∈ ε D for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ D and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . Since D has weakly ( m, n ) -divisible, there exist x ′′ , x ′′ , · · · , x ′′ n ∈ M ∞ ( D ) + such that x ′′ j ⊕ x ′′ j ⊕ · · · ⊕ x ′′ j . ( a ′′′ − ε ) + where x ′′ j repeat m times and ( a ′′′ − ε ) + . ⊕ ni =1 x ′′ i .Therefore we have ( x ′ j ⊕ x ′′ j )) ⊕ ( x ′ j ⊕ x ′′ j )) ⊕ · · · ⊕ ( x ′ j ⊕ x ′′ j )) . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, for ≤ j ≤ n where ( x ′ j ⊕ x ′′ j ) repeat m times. ENERALIZED TRACIAL APPROXIMATION C ∗ -ALGEBRAS 17 We also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ d . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . ⊕ ni =1 ( x ′ i ⊕ x ′′ i ) . Case (2) , If ( a ′ − ε ) + is not Cuntz equivalent to a projection.By Theorem 2.1, there is a non-zero positive element d such that ( a ′ − ε ) + + d . ( a ′ − ε ) + .With F = { a ′′ } , any ε ′ > with ε ′ sufficiently small, since A ∈ TGA P there exist anonzero projection p ′ ∈ A and element g ∈ A with ≤ g ≤ and a C ∗ -subalgebra D of A with D = p ′ and D ∈ P , such that (1) ( p ′ − g ) x ∈ ε D, x ( p ′ − g ) ∈ ε D for all x ∈ F , and (2) k ( p ′ − g ) x − x ( p ′ − g ) k < ε for all x ∈ F . (3) 1 − ( p ′ − g ) . c .By (2) ′ and (3) ′ , there exist positive elements a ′′′ ∈ B and a ∈ A such that k a − a ′′′ − a k < ε, k ( p ′ − g ) / a ′′ ( p ′ − g ) / − a ′′′ k < ε/ , and k (1 − ( p ′ − g ) / ) a ′′ (1 − ( p ′ − g ) / ) − a k < ε/ . Since D has weakly ( m, n ) -divisible, there exist x ′′ , x ′′ , · · · , x ′′ n ∈ M ∞ ( D ) + such that x ′′ j ⊕ x ′′ j ⊕ · · · ⊕ x ′′ j . ( a ′′′ − ε ) + where x ′′ j repeat m times and ( a ′′′ − ε ) + . ⊕ ni =1 x ′′ i .Therefore we have ( y ′ i ⊕ x ′′ i ) ⊕ ( y ′ i ⊕ x ′′ i ) ⊕ · · · ⊕ ( y ′ i ⊕ x ′′ i ) . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . a ⊕ a, for ≤ i ≤ n where ( x ′ i ⊕ x ′′ i ) repeat m times.We also have ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ( a − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ ((1 − ( p ′ − g )) − ε ) + . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + ⊕ d . ( a ′ − ε ) + ⊕ ( a ′′′ − ε ) + . ⊕ ni =1 ( y ′ i ⊕ x ′′ i ) . (cid:3) Acknowledgement:
The research of the first author was supported by grant from theNational Natural Sciences Foundation of China (No.11571008). The research of the sec-ond author was supported by a grant from the National Natural Sciences Foundation ofChina (No.11871375). R
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