The Range of Approximate Unitary Equivalence Classes of Homomorphisms from AH-algebras

The Range of Appro ximate Unitary Equiv alence Classes of Homomorphisms from AH-algebras Huaxin Lin Abstract Let C be a unital AH-algebra and A b e a unital simple C ∗ -algebra with tracia l rank zero. It has been shown t hat t wo unital monomorphisms φ, ψ : C → A are approximately unitarily equiv alent if a nd only if [ φ ] = [ ψ ] in K L ( C, A ) and τ ◦ φ = τ ◦ ψ for a ll τ ∈ T ( A ) , where T ( A ) is the tracial state spac e o f A. In this paper we prov e the following: Given κ ∈ K L ( C, A ) with κ ( K 0 ( C ) + \{ 0 } ) ⊂ K 0 ( A ) + \{ 0 } a nd with κ ([1 C ]) = [1 A ] and a co n tin uous affine map λ : T ( A ) → T f ( C ) which is co mpatible with κ, wher e T f ( C ) is the con vex set of all faithful tracial states, there exists a unital mono morphism φ : C → A suc h that [ φ ] = κ and τ ◦ φ ( c ) = λ ( τ )( c ) for all c ∈ C s.a. and τ ∈ T ( A ) . Denote by Mo n e au ( C, A ) the set o f approximate unitary equiv alenc e classes of unital mo no morphisms. W e pro vide a bij ective map Λ : Mon e au ( C, A ) → K LT ( C, A ) ++ , where K L T ( C, A ) ++ is the set of compatible pa ir s of elements in K L ( C , A ) ++ and contin uous affine maps from T ( A ) to T f ( C ) . Moreov er, we r ealized that there ar e compa ct metric spaces X , unital s imple AF-algebras A and κ ∈ K L ( C ( X ) , A ) with κ ( K 0 ( C ( X )) + \ { 0 } ) ⊂ K 0 ( A ) + \ { 0 } for which there is no homomorphism h : C ( X ) → A so that [ h ] = κ. 1 In tro duction Recall that an AH-alg ebra is a C ∗ -algebra which is an ind uctiv e limit of C ∗ -algebras C n , w here C n = P n M r ( n ) ( C ( X n )) P n for some finite CW complex X n and pro jections P n ∈ M r ( n ) ( C ( X n )) . Note that ev ery un ital separable commutativ e C ∗ -algebra is an AH-alge bra and ev ery AF- algebra is an AH-algebra. It was shown in [11] (see also T heorem 3.6 of [13]) that t w o unital monomorphisms φ, ψ : C → A, wher e A is a unital simp le C ∗ -algebra with tracial rank zero, are approximat ely u nitarily equiv alen t if and only if [ φ ] = [ ψ ] and τ ◦ φ ( c ) = τ ◦ ψ ( c ) for all c ∈ C s.a. and τ ∈ T ( A ) . This result pla ys a role in the study of classificat ion of amenable C ∗ -algebras, or otherwise kno wn as the Elliot t program. I t also has app licati ons in the study of dynamical systems b oth classical and non-comm utativ e ones (see [11]). It is desirable to kno w the range of the appro ximately unitary equiv alence classes of monomorphisms from a unital AH-alge bra C into a unital simple C ∗ -algebra with tracial ran k zero. F or example, on e may ask if giv en an y κ ∈ K L ( C, A ) and any con tin u ous affine map λ : T ( A ) → T ( C ) there exists a monomorphism φ su c h that [ φ ] = κ and τ ◦ h ( c ) = λ ( τ )( c ) for all c ∈ C s.a. and τ ∈ T ( A ) . 1 When C is a finite CW complex, it w as sho wn (see also a p r evious r esult of L. Li [6]) in [10] that, for any κ ∈ K K ( C , A ) with κ ( K 0 ( C ) + \ { 0 } ) ⊂ K 0 ( A ) + \ { 0 } and with κ ([1 C ]) = [1 A ] , there exists a unital monomorphism φ : C → A suc h that [ φ ] = κ. It sh ould b e noted that b oth conditions t hat κ ([1 C ]) = [1 A ] and κ ( K 0 ( C ) \ { 0 } ) ⊂ K 0 ( A ) \ { 0 } are n ecessary for the existence of such φ. One of the earlie st suc h results (concerning monomorphisms from C ( T 2 ) into a u nital simple AF-algebra) of this kin d app eared in a pap er of Elliott and Loring ([3] see also [2]). It wa s sho wn in [1 0] th at the sa me resu lt holds fo r the case that C is a unital simple AH-algebra whic h has real rank zero, stable rank one and we akly u np erforated K 0 ( C ) . Th erefore, it is n atural to exp ect that it h olds for general unital AH-algebras. Let C b e the u nitizatio n of K , the alg ebra of compact op erators on l 2 . Then it do es not ha ve a faithfu l tracial state. Consequen tly , it can not b e emb edded into an y unital UHF-algebra, or an y unital simp le C ∗ -algebra wh ic h h as at least one tracial state (It has b een shown that a unital AH-alge bra C can b e emb edded in to a unital simple AF-algebra if and only if C admits a faithful tracial state –see [13]) . This example at lea st suggests that for general unital AH-algebras, the problem is sligh tly more complicated than the first though t. Moreo ve r, w e note that to pro vide the range of appro ximately u nitary equiv alence classes of unital monomorph isms from C, w e also n eed to consider the map λ : T ( A ) → T ( C ) . Let X b e a compact metric space and let C = C ( X ) . Su pp ose that h : C → A is a unital monomorphism and supp ose that τ ∈ T ( A ) . Then τ ◦ h ind uces a Borel probabilit y measure on X . Sup p ose that κ ∈ K L ( C , A ) is giv en. It is clea r that not ev ery measure µ can b e ind uced b y those h for whic h [ h ] = κ. T h us, w e s hould consider a compatible pair ( κ, γ ) which giv es a more complete information on K -theory th an either κ or γ alone. The main resu lt of this pap er is to sho w that if C is a unital AH-algebra, A is an y unital simple C ∗ -algebra with tracial rank zero, κ ∈ K L ( C , A ) ++ (see 2.3 b elo w) with κ ([1 C ]) = [1 A ] and λ : T ( A ) → T f ( C ) , where T f ( C ) is the co n v ex set of faithfu l tracial sta tes, whic h is a con tin uous affine map and is compatible with κ, there is indeed a u nital monomorphism φ : C → A suc h that [ φ ] = κ in K L ( C, A ) and φ T = λ. W e also s h o w that the existence of λ is essen tial to provide homomorphisms φ. In fact, w e fin d out th at there are compact metric spaces X, u nital simple AF-algebras A and κ ∈ K L ( C ( X ) , A ) ++ with κ ([1 C ]) = [1 A ]) for whic h there are no λ : T ( A ) → T f ( C ( X )) w hic h is compatible with κ. Moreo v er, w e d isco vered that th ere are no homomorph ism h : C → A (not just monomorph isms) suc h that [ h ] = κ. This further demonstrates that tracial information is an in tegral part of K -theoretical in formation. 2 Notation 2.1. Let A b e a unital C ∗ -algebra. Denote b y T ( A ) th e tr acial state sp ace of A. Denote by Aff ( T ( A )) th e space of all real affin e conti nuous fun ctions on T ( A ) . If τ ∈ T ( A ) , w e will also use τ for the tracial state τ ⊗ T r on M k ( A ) for all i nt eger k ≥ 1 , where T r is the standard trace on M k . If a ∈ A s.a. , denote by ˇ a a real affine fun ction in Aff ( T ( A )) defined by ˇ a ( τ ) = τ ( a ) f or all τ ∈ T ( A ) . Let C b e another unital C ∗ -algebra. S upp ose that γ : Aff ( T ( C )) → Aff ( T ( A )) is a p ositiv e linear map. W e say it is unital if γ (1 C )( τ ) = 1 . W e sa y it is strictly positive , if a ∈ Aff ( T ( A )) + \ { 0 } , then γ ( a )( τ ) > 0 for all τ ∈ T ( A ) . Supp ose that φ : C → A is a unital homomorphism. Denote b y h T : T ( A ) → T ( C ) the affine con tin uous map ind uced b y h, i.e., h T ( τ )( c ) = τ ◦ h ( c ) for all c ∈ C. 2 It also ind u ces a p ositiv e linear map h ♯ : Aff ( T ( C )) → Aff ( T ( A )) defined by h ♯ (ˇ a )( τ ) = τ ◦ h ( a ) for al l a ∈ C s.a and τ ∈ T ( A ) , where ˇ a ( τ ) = τ ( a ) for τ ∈ T ( A ) . If λ : T ( A ) → T ( C ) is an affine con tin uous map, then it giv es a un ital p ositiv e linear map λ ♯ : Aff ( T ( C )) → Aff ( T ( A )) by λ ♯ ( f )( τ ) = f ( λ ( τ )) for all f ∈ Aff ( T ( C )) and for all τ ∈ T ( A ) . Con v ersely , a unital p ositiv e linear map γ : Aff ( T ( C )) → Aff ( T ( A )) give s an affine cont in uous map γ T : T ( A ) → T ( C ) b y f ( γ T ( τ )) = γ ( f )( τ ) for all f ∈ Aff ( T ( C )) and τ ∈ T ( C ) . Supp ose that A is a un ital simple C ∗ -algebra. Then γ is strictly p ositiv e if and only if γ T maps T ( A ) into T f ( C ) . Denote by ρ A : K 0 ( A ) → Aff ( T ( A )) the p ositive homomorphism in d uced by ρ A ([ p ])( τ ) = τ ( p ) for all pr o jections p ∈ M ∞ ( A ) and τ ∈ T ( A ) . Let A and C b e t w o un ital C ∗ -algebras and let κ 0 : K 0 ( C ) → K 0 ( A ) b e a unital p ositiv e homomorphism ( κ 0 ([1 C ]) = [1 A ]). Supp ose that λ : T ( A ) → T ( C ) is a con tin uous affin e map. W e sa y that λ is compatible with κ 0 , if τ ( κ ([ p ])) = λ ( τ )( p ) for all p r o jections p in M ∞ ( A ) . Similarly , a unital p ositiv e linear map γ : Aff ( T ( C )) → Aff ( T ( A )) is sai d to be compatible with κ 0 , if γ ( ˇ p )( τ ) = τ ( κ ([ p ]) for all pro jections p in M ∞ ( C ) . γ is compatible w ith κ 0 if and only if γ T is so. Tw o pro jections in A are equiv alen t if there exists a p artial isometry w ∈ A such that w ∗ w = p and w w ∗ = q . 2.2. L et A b e a unital C ∗ -algebra and let C b e a separable C ∗ -algebra which satisfies the unive rsal co efficien t theorem. By a result of Dadarlat and Loring ([1]), K L ( C , A ) = H om Λ ( K ( C ) , K ( A )) , (e 2.1) where, for any C ∗ -algebra B , K ( B ) = ⊕ i =0 , 1 K i ( B ) ∞ M n =2 ⊕ i =0 , 1 K i ( B , Z /n Z ) . W e will identify t w o ob jects in (e 2.1). Denote b y K F ,k ( C ) = M i =0 , 1 K i ( B ) M n | k ⊕ i =0 , 1 K i ( B , Z /n Z ) . If K i ( C ) is finitely generated ( i = 0 , 1), then there is k 0 ≥ 1 suc h that H om Λ ( K ( C ) , K ( A )) ∼ = H om Λ ( F k 0 K ( C ) , F k 0 K ( A )) (see [1]). Definition 2.3. Denote by K L ( C, A ) ++ the set of those κ ∈ H om Λ ( K ( C ) , K ( A )) such that κ ( K 0 ( C ) + \ { 0 } ) ⊂ K 0 ( A ) \ { 0 } . Denote by K L e ( C, A ) ++ the set of those κ ∈ K L ( C, A ) ++ suc h that κ ([1 C ]) = [1 A ] . 3 Definition 2.4. Let κ ∈ K L e ( C, A ) ++ and let λ : T ( A ) → T ( C ) b e a contin uou s affine m ap. W e sa y that λ is compatible with κ if λ is compatible with κ | K 0 ( C ) . Let γ : Aff ( T ( C )) → Aff ( T ( A )) b e a p ositiv e linear map . W e s ay γ is compatible with κ if γ is compatible with κ | K 0 ( C ) , i.e., τ ◦ κ ([ p ]) = γ ( ˇ p )( τ ) for all p ro j ections p ∈ M ∞ ( C ) . 2.5. Let C = C ( X ) for some compact metric sp ace X . On e has the follo wing short exact sequence: 0 → k er ρ C → K 0 ( C ) → C ( X , Z ) → 0 . It is then easy to see that, for ev ery p r o jection p ∈ M ∞ ( C ) , there is a pro jection q ∈ C and an in teger n suc h that ρ A ([ p ]) = nρ A ([ q ]) . It follo ws that if C is a unital A H-algebra, th en for ev ery pro jection p ∈ M ∞ ( C ) , th ere is a pro jection q ∈ C and an in teger n ≥ 1 suc h that ρ A ([ p ]) = nρ A ([ q ]) . Note also that in this case Aff ( T ( C )) = C s.a . Therefore, in th is note, instead of considering a unital p ositiv e linear maps γ : Aff ( T ( C )) → Aff ( T ( A )) , we ma y consider u nital p ositiv e linear map s γ : C s.a → Aff ( T ( A )) . Moreo v er, γ is compatible w ith some κ ∈ K L ( C, A ) ++ , if γ ( p )( τ ) = τ ( κ ([ p ])) f or all pro jections p ∈ C and τ ∈ T ( A ) . 2.6. Let φ, ψ : C → A b e t w o maps b et w een C ∗ -algebras. Let ǫ > 0 and F ⊂ C b e a subset. W e wr ite φ ≈ ǫ ψ on F , if k φ ( c ) − ψ ( c ) k < ǫ for all c ∈ F . 2.7. Let L : C → A b e a linear map. L et δ > 0 and G ⊂ C b e a (fin ite) sub set. W e s a y L is δ - G -multiplicat iv e if k L ( ab ) − L ( a ) L ( b ) k < δ for all a, b ∈ G . Definition 2.8. Let A b e a unital C ∗ -algebra. Denote b y U ( A ) the un itary group of A. Let B ⊂ A b e another C ∗ -algebra and φ : B → A b e a map. W e wr ite φ = ad u for s ome u ∈ U ( A ) if φ ( b ) = u ∗ bu for all b ∈ B . Let φ, ψ : C → A b e tw o maps. W e sa y that φ and ψ are appro ximately u nitarily equiv alent if there exists a sequence of unitaries { u n } ⊂ A suc h that lim n →∞ ad u n ◦ φ ( c ) = ψ ( c ) for all c ∈ C. 3 Appro ximate unitary equiv alence W e b egin with the follo win g theorem Theorem 3.1. (Th eorem 3.6 of [13] and see also Theorem 3.4 of [11]) L et C b e a unital AH- algebr a and let A b e a u nital simple C ∗ -algebr a with tr acial r ank zer o. Supp ose that φ, ψ : C → A ar e two unital monomorphism s. Then ther e exi sts a se quenc e of unitaries { u n } ⊂ A such that lim n →∞ ad u n ◦ ψ ( c ) = φ ( c ) for al l c ∈ C, if and only if [ φ ] = [ ψ ] in K L ( C, A ) and τ ◦ φ = τ ◦ ψ for al l τ ∈ T ( A ) . W e need th e follo wing v ariation of results in [11]. 4 Theorem 3.2. L et C b e a unital AH -algebr a, let A b e a unital s imple C ∗ -algebr a with T R ( A ) = 0 and let γ : C s.a. → Aff ( T ( A )) b e a unital strictly p ositive line ar map. F or any ǫ > 0 and any finite subset F ⊂ C, ther e exist η > 0 , δ > 0 , a finite subset G ⊂ C, a finite subset H ⊂ C s.a. and a finite subset P ⊂ K ( C ) satisfying the fol lowing: Supp ose that L 1 , L 2 : C → A ar e two unital c ompletely p ositive line ar maps which ar e δ - G - multiplic ative suc h that [ L 1 ] | P = [ L 2 ] | P , (e 3.2) | τ ◦ L i ( g ) − γ ( g )( τ ) | < η for al l g ∈ H , i = 1 , 2 . (e 3.3) Then ther e is a unitary u ∈ A such that ad u ◦ L 2 ≈ ǫ L 2 on F . (e 3.4) Pr o of. W rite C = ∪ ∞ n =1 C n , w here C n = P n M r ( n ) ( C ( X n )) P n , w here X is a compact su bset of a finite CW complex and where P n ∈ M r ( n ) ( C ( X n )) is a pro j ectio n. Let ǫ > 0 and a finite sub set F ⊂ C b e fixed. Without loss of generalit y , we ma y assume that F ⊂ C 1 . Let η 0 > 0 suc h that | f ( x ) − f ( x ′ ) | < ǫ/ 8 for all f ∈ F , if d ist( x, x ′ ) < η 0 . Let { x 1 , x 2 , ..., x m } ⊂ X b e η 0 / 2-dense in X . S upp ose that O i ∩ O j = ∅ if i 6 = j, where O j = { x ∈ X : dist( x, x j ) < η 0 / 2 s } , j = 1 , 2 , ..., m for some inte ger s ≥ 1 . Cho ose non-zero element g j ∈ ( C 1 ) s.a suc h that 0 ≤ g j ≤ 1 whose su p p ort lies in O j , j = 1 , 2 , ..., m. Note su ch g j exists (by taking those in the cen ter f or example). Cho ose σ 0 = min { inf { γ ( g j )( τ ) : τ ∈ T ( A ) } : 1 ≤ j ≤ m } . Since γ is strictly p ositiv e, σ 0 > 0 . Set σ = min { σ 0 / 2 , 1 / 2 s } . T h en, b y Corollary 4.8 of [11], such δ > 0 , η > 0 , G , H and P exists. Lemma 3.3. L et X b e a c omp act metric sp ac e, let A b e a unital simple C ∗ -algebr a with T R ( A ) = 0 and let γ : C ( X ) s.a. → Aff ( T ( A )) b e a unital strictly p ositive line ar map. Then, for any ǫ > 0 and any F ⊂ C ( X ) , ther e exists δ > 0 , a finite subset G ⊂ C ( X ) s.a. , a set S 1 , S 2 , ..., S n of mutual ly disjoint clop en sub se ts with ∪ n i =1 S i = X, satisfying the fol lowing: F or any two uni tal homomorph isms φ 1 , φ 2 : C ( X ) → pAp with finite dimensional r ange for some pr oje ction p ∈ A with τ (1 − p ) < δ such that [ φ 1 ( χ S i )] = [ φ 2 ( χ S i )] in K 0 ( A ) , i = 1 , 2 , ..., n, (e 3.5) | τ ◦ φ 1 ( g ) − γ ( g )( τ ) | < δ and (e 3.6) | τ ◦ φ 2 ( g ) − γ ( g )( τ ) | < δ (e 3.7) for al l g ∈ G and for al l τ ∈ T ( A ) , ther e exist a unitary u ∈ U ( pAp ) such that ad u ◦ φ 1 ≈ ǫ φ 2 on F . (e 3.8) Pr o of. This follo ws from 3.2 immediately . There is a sequence of fin ite CW complex X n suc h that C ( X ) = lim n →∞ ( C ( X n ) , h n ) , where eac h h n is a unital homomorp h ism. Fix ǫ > 0 and a finite subset F ⊂ C ( X ) . Without loss o f generalit y , we ma y assume that F ⊂ h K ( F K ) for some in teger K ≥ 1 and a finite sub set F K . 5 Giv en an y finite subset P ⊂ K ( C ( X )) , o ne obtains a finite subset Q k ⊂ K ( C ( X k )) suc h that [ h k ]( Q k ) = P f or some k ≥ 1 . L et p 1 , p 2 , ..., p n b e m utually orth ogonal pro j ectio ns corresp ondin g to the connected comp onents of X k . T o simp lify notation, w ithout loss of generalit y , we ma y assume that k = K. There are mutually disjoin t clop en sets S 1 , S 2 , ..., S n of X with ∪ n i =1 S i = X su c h that h k ( p i ) = χ S i , i = 1 , 2 , ..., n. Since φ and ψ are homomorph isms with finite dimensional range, if [ φ ( χ S i )] = [ ψ ( χ S i )] in K 0 ( A ) , then [ φ ◦ h k ] = [ ψ ◦ h k ] in K L ( C ( X k ) , A ) . This, in particular, implies that [ φ ] | P = [ ψ ] | P . This ab o v e argument sho ws that the lemma f ollo ws from 3.2. Definition 3.4. Let X b e a compact metric spac e wh ic h is a co mpact subset of so me finite CW complex Y . Then there exists a decreasing sequence of finite CW complexes X n ⊂ Y suc h that X ⊂ X n and lim n →∞ dist( X n , X ) = 0 . Denote by s m,n : C ( X m ) → C ( X n ) (for n > m ) an d s n : C ( X n ) → C ( X ) b e the surjectiv e homomorphisms indu ced b y the inclusion X n +1 ⊂ X n and X ⊂ X n , resp ectiv ely . Lemma 3.5. L et Y b e a finite CW c omplex and X ⊂ Y b e a c omp act subset. F or any ǫ > 0 , any finite subset F ⊂ C ( X ) , ther e exists a finite subset P ⊂ K ( C ( X )) , an inte ger k ≥ 1 and an inte ger N ≥ 1 satisfying the fol lowing: F or any unital homomorph isms φ, ψ : C ( X m ) → A ( m ≥ k ) for any unital simple C ∗ -algebr a with T R ( A ) = 0 for which [ φ ] | Q = [ ψ ] | Q in K L ( C ( X m ) , A ) , wher e Q ⊂ K ( C ( X m )) is a finite subset such that [ s m ]( Q ) = P , then ther e exists a u ni tary U ∈ M N +1 ( A ) such that ad U ◦ ( φ ⊕ Φ ◦ s m ) ≈ ǫ ( ψ ⊕ Φ ◦ s m ) on s − 1 m ( F ) , wher e Φ : C ( X ) → M N ( A ) is define d by Φ( f ) = d iag( f ( x 1 ) , f ( x 2 ) , ..., f ( x N )) for al l f ∈ C ( X 1 ) , (e 3.9) wher e { x 1 , x 2 , ..., x N } is a finite subset of X . Pr o of. Assume th at the lemma we re false. Then there w ould b e a p ositiv e num b er ǫ 0 > 0 , a finite subset F 0 ⊂ C ( X ) , an increasing sequence of finite sub sets {P n } ⊂ K ( C ( X )) with ∪ n P n = K ( C ( X )) , a sequ ence of un ital C ∗ -algebras, tw o subsequences { R ( n ) } , { k ( n ) } of N and t w o sequences monomorph isms φ n , ψ n : C ( X k ( n ) ) → A n suc h that [ φ n ] | Q n = [ ψ n ] | Q n in K K ( C ( X k ( n ) ) , A n ) and (e 3.10) lim s up n { inf { max {k u ∗ n ( φ n ⊕ Φ n ◦ s n )( f )) u n − ( φ ⊕ Φ n ◦ s n )( f ) k : f ∈ s − 1 m ( F ) }}} ≥ ǫ 0 , (e 3.11) where infimum is tak en among all p ossible Φ n : C ( X ) → M R ( n ) ( A n ) with the f orm describ ed ab o v e and among all p ossible unitaries { u n } ⊂ U ( M R ( n )+1 ( A )) , and where Q n ⊂ K ( C ( X k ( n ) )) 6 is a fi nite subs et suc h that [ s k ( n ) ]( Q n ) = P n . Since K i ( C ( X n ) is finitely generated, by passing to a su bsequence, if necessary , without loss of generalit y , we ma y assume (see also the end of 2.2) that [ φ n +1 ◦ s k ( n ) ,k ( n +1) ] = [ ψ n +1 ◦ s k ( n ) ,k ( n +1) ] in K L ( C ( X k ( n ) ) , A ) , n = 1 , 2 , .... (e 3.12) Let φ ( m ) n = φ m , if n ≤ m, φ ( m ) n = φ n ◦ s m,n , ψ ( m ) n = ψ m , if n ≤ m and ψ ( m ) n = ψ n ◦ s m,n , n = 1 , 2 , .... Denote b y H ( m ) 1 , H ( m ) 2 : C ( X k ( m ) ) → Q n A n b y H ( m ) 1 ( f ) = { φ ( m ) n } and H ( m ) 2 ( f ) = { ψ ( m ) n } . Let π : Q n A n → Q n A n / L n A n b e the quotien t map. Th en π ◦ H ( m ) 1 and π ◦ H ( m ) 2 b oth hav e sp ectrum X. Moreo v er, for eac h i, all π ◦ H ( m ) i giv es the same homomorp hism F i : C ( X ) → Q n A n / L n A n , i = 1 , 2 . Since T R ( A n ) = 0 , A n has real rank zero, s table rank one, w eakly unp erforated K 0 ( A n ) , b y Corollary 2.1 of [5] and (e 3.12) [ H ( m +1) 1 ◦ s k ( m ) ,k ( m +1) ] = [ H ( m +1) 2 ◦ s k ( m ) ,k ( m +1) ] in K L ( C ( X k ( m ) ) , Y n A n ) It follo ws from C orollary 2.1 of [5] again that [ F 1 ] = [ F 2 ] in K L ( C, Y n A n / M n A n ) . It then follo ws from Theorem 1.1 and the Remark 1.1 of [5] that ther e is an in teger N ≥ 1 and a unitary W ∈ U ( M N +1 ( Q n A n / L n A n )) suc h that ad W ◦ ( F 2 ⊕ H 0 ) ≈ ǫ 0 / 2 ( F 1 ⊕ H 0 ) on F 0 , (e 3.13) where H 0 : C ( X ) → M N ( Q n A n / L n A n ) is defin ed b y H 0 ( f ) = P N i =1 f ( x i ) E i for all f ∈ C ( X ) , x i ∈ X and E i = diag( i − 1 z }| { 0 , ..., 0 , 1 , 0 , ..., 0) , i = 1 , 2 , ..., N . There is a unitary { W n } ∈ U ( Q n A n ) such that π ( { W n } ) = W . Then, for some su fficien tly large n, W ∗ n diag( φ n ( f ) , f ( x 1 ) , f ( x 2 ) , ..., f ( x N )) W n ≈ ǫ 0 ( ψ n ( f ) , f ( x 1 ) , f ( x 2 ) , ..., f ( x N )) (e 3.14) on F 0 . This con tradicts (e 3.11). Remark 3.6. There exists a p ositiv e num b er η > 0 a nd in teger N 1 > 0 whic h depend only on ǫ and F suc h that { x 1 , x 2 , ..., x N } and an integ er N can b e r ep laced b y any η -dense finite sub set { ξ 1 , ξ 2 , ..., ξ N 1 } and integ er N 1 . F rom the pr o of, w e also know that the assum p tion that A has tracial rank zero can b e replaced b y muc h wea k er conditions (see Corollary 2.1 of [5]). The main d ifference of 3.5 and results in [5] is that homomorphisms φ and ψ are n ot assum ed to b e from C ( X ) to A. 4 Monomorphisms from C ( X ) Lemma 4.1. L et X b e a finite CW c omplex and let A b e a unital simple C ∗ -algebr a with r e al r ank zer o, stable r ank one and we akly unp erfor ate d K 0 ( A ) . L e t e 1 , e 2 , ..., e m ∈ C ( X ) b e mutual ly ortho gonal pr oje c tions c orr esp onding to c onne cte d c omp onents of X . Supp ose that κ ∈ K K ( C ( X ) , A ) ++ with κ ([1 C ( X ) ]) = [1 A ] . Then, for any pr oje ction p ∈ A and any unital homomorp hism φ 0 : C ( X ) → (1 − p ) A (1 − p ) with finite dimensional r ange such 7 that φ 0 ([ e i ]) < κ ([ e i ]) , i = 1 , 2 , ..., m. Then ther e exists a unital monomor phism φ 1 : C ( X ) → pAp such that [ φ 1 + φ 0 ] = κ i n K K ( C ( X ) , A ) . (e 4.15) Pr o of. Since P m i =1 κ ([ e i ]) = [1 A ] and A has stable rank one, there are mutually orthogonal pro jections p 1 , p 2 , ..., p m ∈ A suc h that m X i =1 p i = 1 A and [ p i ] = κ ([ e i ]) , i = 1 , 2 , ..., m (e 4.16) F rom this it is clear that we ma y reduce the general case to the case that X is conn ected. So no w we assume that X is connected. Th en it is easy to see that κ − [ φ 0 ] ∈ K K ( C ( X ) , A ) ++ and ( κ − [ φ 0 ])([1 C ( X ) ]) = p. It follo w s from Theorem 4.7 of [10] that there is a monomorphism φ 1 : C ( X ) → pAp su ch that [ φ ] = κ − [ φ 0 ] . Lemma 4.2. L et X a c omp act metric sp ac e and let A b e a unital simple C ∗ -algebr a with tr acial r ank zer o. Supp ose that γ : C ( X ) s,a → Aff ( T ( A )) is a unital strictly p ositive line ar map. L et S 1 , S 2 , ..., S n b e a set of mutual ly disjoint clop e n subsets of X with ∪ n i =1 S i = X. Then for any δ > 0 and any finite subset G ⊂ C ( X ) s.a , ther e exists a pr oje ction p ∈ A with p 6 = 1 A and a unital homomo rphism h : C ( X ) → pAp with finite dimensional r ange such that | τ ◦ h ( g ) − γ ( g )( τ ) | < δ for al l g ∈ G and τ ∈ T ( A ) , and (e 4.17) τ ◦ h ( χ S i ) < γ ( ξ S i )( τ ) for al l τ ∈ T ( A ) , (e 4.18) i = 1 , 2 , ..., n . Pr o of. Put d = min { δ , min { inf { γ ( χ S i )( τ ) : τ ∈ T ( A ) } : 1 ≤ i ≤ n }} . Since γ is strictly p ositiv e, d > 0 . Let G 0 = G ∪ { χ S 1 , χ S 2 , ..., χ S n } . It f ollo ws from 4.3 of [12] that there is a u nital h omomorphism h 0 : C ( X ) → A with finite dimensional range such that | τ ◦ h ( g ) − γ ( g )( τ ) | < d/ 8 n for all g ∈ G 0 (e 4.19) and for all τ ∈ T ( A ) . In particular, | τ ◦ h ( χ S i ) − γ ( χ S i )( τ ) | < d/ 8 n for all τ ∈ T ( A ) (e 4.20) i = 1 , 2 , ..., n . Since ρ A ( K 0 ( A )) is dense in Aff ( T ( A )) , there exists a pro jection p 0 ∈ A suc h that d/ 2 n < τ ( p 0 ) < d/n for a ll τ ∈ T ( A ) . (e 4.21) Note that τ ( p 0 ) < γ ( χ S i )( τ ) for all τ ∈ T ( A ) , i = 1 , 2 , ..., n. Moreo v er , b y (e 4.20), τ ◦ h ( χ S i ) > γ ( ξ S i )( τ ) − d/ 8 n ≥ d − d/ 8 n > τ ( p 0 ) . (e 4.22) 8 for all τ ∈ T ( A ) . W rite h 0 ( f ) = P m k =1 f ( x k ) e k for all f ∈ C ( X ) , w here x k ∈ X and { e 1 , e 2 , ..., e k } is a set of m utually orthogonal p ro j ectio ns with P m k =1 e k = 1 A . Note that h 0 ( ξ S j ) = X x k ∈ S j e k . Therefore (b y (e 4.22)) [ p 0 ] ≤ [ X x k ∈ S j e k ] . (e 4.23) By Zhang’s Riesz interp olation prop ert y (see [14] ), there are pro jections e ′ k ≤ e k suc h that [ p 0 ] = [ X k ∈ S j e ′ k ] . By Zhang’s half pro jection theorem (see Theorem 1.1 of [15 ]), for eac h k , there is a pro jection e ′′ k ≤ e ′ k suc h that [ e ′′ k ] + [ e ′′ k ] ≥ [ e ′ k ] . (e 4.24) Th us 2[ X χ k ∈ S i e ′′ k ] ≥ [ p 0 ] , i = 1 , 2 , ..., n. (e 4.25) Therefore (b y (e 4.21) and (e 4.20)) τ ( X x k ∈ S i ( e k − e ′′ k )) < τ ◦ h 0 ( χ S i ) − (1 / 2) τ ( p 0 ) (e 4.26) < τ ◦ h ( χ S i ) − d/ 4 n (e 4.27) < γ ( χ S i )( τ ) − d/ 8 n for all τ ∈ T ( A ) . (e 4.28) Let p = P m k =1 ( e k − e ′′ k ) . Then clearly th at p 6 = 1 . Moreo ver, τ (1 − p ) < d/ 4 f or all τ ∈ T ( A ) . Define h ( f ) = P m k =1 f ( x k )( e k − e ′′ k ) for all f ∈ C ( X ) . Then | τ ◦ h ( f ) − τ ◦ h 0 ( f ) | < τ ( m X k =1 e ′′ k ) = τ (1 − p ) < d/ 4 < δ (e 4.29) for all τ ∈ T ( A ) . Then, by (e 4.28), τ ◦ h ( χ S i ) < γ ( χ S i )( τ ) f or all τ ∈ T ( A ) . (e 4.30) Lemma 4.3. L et X a c omp act metric sp ac e and let A b e a unital simple C ∗ -algebr a with tr acial r ank zer o. Supp ose that γ : C ( X ) s,a → Aff ( T ( A )) is a unital strictly p ositive line ar map. L et S 1 , S 2 , ..., S n b e a set of mutual ly disjoint clop e n subsets of X with ∪ n i =1 S i = X. Then for any δ > 0 , η > 0 , for any inte ge r N and any η - dense su b set { x 1 , x 2 , ..., x N } of X and any finite 9 subset G ⊂ C ( X ) s.a , ther e exists a pr oje ction p ∈ A with p 6 = 1 A and a unital homomo rphism h : C ( X ) → pAp with finite dimensional r ange suc h that | τ ◦ h ( g ) − γ ( g )( τ ) | < δ for al l g ∈ G and τ ∈ T ( A ) , and (e 4.31) τ ◦ h ( χ S i ) < γ ( χ S i )( τ ) for al l τ ∈ T ( A ) , (e 4.32) i = 1 , 2 , ..., n , h ( f ) = N X i =1 f ( x i ) e i ⊕ h 1 ( f ) for al l f ∈ C ( X ) , (e 4.33) wher e h 1 : C ( X ) → (1 − P N i =1 e i ) A (1 − P N i =1 e i ) is a uni tal homomorph ism with finite dimen- sional r ange and { e 1 , e 2 , ..., e N } is a set of mutual ly o rtho g onal p r oje ctions such th at [ e i ] = [ e 1 ] ≥ [1 − p ] , i = 1 , 2 , ..., N . Pr o of. Let N ≥ 1 and let η -dense sub set { x 1 , x 2 , ..., x N } of X b e giv en. Let η 0 > 0 suc h that | f ( x ) − f ( x ′ ) | < δ/ 4 for all f ∈ G , (e 4.34) pro vided that dist( x, x ′ ) < η 0 . Cho ose η 0 > η 1 > 0 suc h that B ( x i , η 1 ) in tersects with one and only one S i among { S 1 , S 2 , ..., S n } . Cho ose, for eac h i, a non-zero function f i ∈ C ( X ) with 0 ≤ f ≤ 1 whose supp ort is in B ( x i , η 1 / 2) . Put d 0 = min { inf { γ ( f i )( τ ) : τ ∈ T ( A ) } : 1 ≤ i ≤ N } . So d 0 > 0 . Put δ 1 = min { δ / 8 , δ 0 / 4 } and pu t G 1 = G ∪ { 1 C ( X ) } ∪ { f i : i = 1 , 2 , ..., N } . No w applying 4.2. W e obtain a pro jection p ∈ A and a unital homomorphism h 0 : C ( X ) → pAp suc h that | τ ◦ h 0 ( g ) − γ ( g )( τ ) | < δ 1 for all g ∈ G 1 and (e 4.35) τ ◦ h 0 ( χ S i ) < γ ( χ S i )( τ ) (e 4.36) for all τ ∈ T ( A ) , i = 1 , 2 , ..., n. Since 1 C ( X ) ∈ G 1 , by (e 4.35), τ (1 − p ) < δ 1 < δ 0 / 4 for all τ ∈ T ( A ) . (e 4.37) W rite h 0 ( f ) = P L j =1 f ( ξ j ) q j for all f ∈ C ( X ) , w h ere ξ j ∈ X and { q 1 , q 2 , ..., q L } is a set of m utually orthogonal p ro j ectio ns with P L j =1 q j = p. Define e ′ i = X ξ j ∈ B ( x i ,η 1 / 2) q j , i = 1 , 2 , ..., N . It follo ws from (e 4.35 ) that, for eac h i, τ ( e ′ i ) ≥ τ ◦ h 0 ( f i ) (e 4.38) > γ ( f i )( τ ) − δ 1 > 3 δ 0 / 4 ≥ τ ( p ) (e 4.39) for all τ ∈ T ( A ) . It follo ws that [ e ′ i ] ≥ [ p ] , i = 1 , 2 , ..., N . 10 There are pr o jections e i ≤ e ′ i suc h that [ e i ] = [ e 1 ] ≥ [1 − p ] , i = 1 , 2 , ..., N . (e 4.40) Define h 1 ( f ) = X ξ j 6∈∪ N i =1 B ( x i ,η 1 / 2) f ( ξ j ) q j + N X i = j f ( x j )( e ′ i − e i ) and (e 4.41) h ( f ) = N X i =1 f ( x i ) e i ⊕ h 1 ( f ) (e 4.42) for all f ∈ C ( X ) . Sin ce B ( x j , η 1 / 2) lies in one of S i , τ ◦ h ( χ S i ) = τ ◦ h 0 ( χ S i ) for all τ ∈ T ( A ) , i = 1 , 2 , ..., n . It follo ws fr om (e 4.36) that (e 4.32 ) holds. By the c hoice of η 0 , w e also ha v e k h 0 ( g ) − h ( g ) k < δ / 2 for all h ∈ G . (e 4.43) Th us, by (e 4.35), (e 4.31 ) also holds. Lemma 4.4. L et X b e a c omp act metric sp ac e and let A b e a unital simple C ∗ -algebr a with tr acial r ank zer o. Supp ose that γ : C ( X ) s.a → Aff ( T ( A )) is a unital strictly p ositive line ar map which is c omp atible with a strictly p ositive homomorphism κ 0 : K 0 ( C ( X )) → K 0 ( A ) . Fix δ > 0 , η > 0 , a finite subset F ⊂ C ( X ) s.a. , an inte ger N ≥ 1 , an η -dense subset { x 1 , x 2 , ..., x N } of X, a finitely many mutual ly disjoint clop en su bset S 1 , S 2 , ..., S n ⊂ X with ∪ n i =1 S i = X, a finite subset set { a 1 , a 2 , ..., a n } ⊂ A of mutual ly ortho gonal pr oje ctions with 0 < a i < κ 0 ([ χ S i )]) , i = 1 , 2 , ..., n, a finitely many mutual ly disjoint clop en subsets { F 1 , F 2 , ..., F n 1 } of X with ∪ n 1 i =1 F i = X , and a pr oje ction p with τ ( p ) = τ ( P n i =1 a i ) for al l τ ∈ T ( A ) . Ther e is a pr oje ction q ∈ A such tha t [ p ] ≤ [ q ] and a unital homomorp hism h : C ( X ) → q Aq with finite dimensional r ange suc h that | τ ◦ h ( g ) − γ ( g )( τ ) | < δ for al l g ∈ F and τ ∈ T ( A ) , and (e 4.44) τ ◦ h ( χ F i ) < γ ( χ F i )( τ ) for al l τ ∈ T ( A ) , (e 4.45) i = 1 , 2 , ..., n , h ( f ) = N X i =1 f ( x i ) e i ⊕ h 1 ( f ) for al l f ∈ C ( X ) , (e 4.46) wher e h 1 : C ( X ) → (1 − P N i =1 e i ) A (1 − P N i =1 e i ) is a uni tal homomorph ism with finite dimen- sional r ange and { e 1 , e 2 , ..., e N } is a set of mutual ly o rtho g onal p r oje ctions such th at [ e i ] = [ e 1 ] ≥ [1 − p ] , i = 1 , 2 , ..., N . Mor e over, ther e exists a pr oje ction p ′ ∈ q such that p ′ h ( f ) = h ( f ) p ′ for al l f ∈ C ( X ) and (e 4.47) [ h ( χ S j ) p ′ ] = [ a j ] , j = 1 , 2 , ..., n. (e 4.48) 11 Pr o of. Let d 0 = min { inf { τ ( κ 0 ([ χ S i )]) − [ a i ]) : τ ∈ T ( A ) } : 1 ≤ i ≤ n } and let d 1 = inf { τ (1 − p ) : τ ∈ T ( A ) } . Then d 0 , d 1 > 0 . Define δ 1 = min { δ / 4 , d 0 / 2 , d 1 / 2 } and G 1 = F ∪ { 1 C ( X ) , χ S i , i = 1 , 2 , ..., n } . By applying 4.3, we obtain a p ro jection q ∈ A and a u nital homomorph ism h : C ( X ) → q Aq w ith finite dimensional r ange satisfying the follo wing: | τ ◦ h ( g ) − γ ( g )( τ ) | < δ 1 for all g ∈ G 1 , (e 4.49) τ ◦ h ( χ F j ) < γ ( χ F j )( τ ) (e 4.50) for all τ ∈ T ( A ) , j = 1 , 2 , ..., n 1 , h ( f ) = N X k =1 f ( x k ) e i ⊕ h 1 ( f ) for all f ∈ C ( X ) , (e 4.51) where { e 1 , e 2 , ..., e N } is a set of m utually orthogonal and mutually equiv alent pro jections su ch that [ e 1 ] ≥ [1 − q ] , and wh ere h 1 : C ( X ) → ( q − P N k =1 e k ) A ( q − P N k =1 e k ) is a unital h omomorp hism with finite dimensional range. Since 1 C ( X ) ∈ G 1 , b y the c hoice of δ 1 , w e conclude that [ p ] ≤ [ q ] . Moreo ver, by (e 4.49 ), τ ◦ h ( χ S i ) > τ ( a i ) f or all τ ∈ T ( A ) , i = 1 , 2 , ..., n. (e 4.52) W rite h ( f ) = L X s =1 f ( ξ s ) E s for all f ∈ C ( X ) , where ξ s ∈ X and { E 1 , E 2 , ..., E L } is a set of mutually orth ogonal p ro j ections s u c h that P L s =1 E s = q . By (e 4.52), one has X ξ s ∈ S i E s ≥ a i , i = 1 , 2 , ..., n. (e 4.53) F or eac h i, b y the Riesz Inte rp olation Prop erty ([14]), th ere is a pro jection E ′ s ≤ E s for which x s ∈ S i suc h that [ X ξ s ∈ S i E ′ s ] = [ a i ] . (e 4.54) Put p ′ = P L s =1 E ′ s Then p ′ h ( f ) = h ( f ) p ′ for all f ∈ C ( X ) and (e 4.55) [ h ( χ S i ) p ′ ] = [ a i ] , i = 1 , 2 , ..., n. (e 4.56) Theorem 4.5. L et X b e a c omp act subset of a finite CW c omplex and let A b e a unital simple C ∗ -algebr a with T R ( A ) = 0 . Supp ose that κ ∈ K L e ( C ( X ) , A ) ++ and supp ose that ther e exists a unital strictly p ositive line ar map γ : C ( X ) s.a → Aff ( T ( A )) which is c omp atible with κ. Then ther e exists a unital monomo rphism φ : C ( X ) → A such that [ φ ] = κ in K L ( C , A ) . 12 Pr o of. Supp ose that X ⊂ Y , wh ere Y is a fin ite CW complex. Let X n ⊂ Y be a decreasing sequence of fi nite C W complexes for w hic h 3.4 holds. Supp ose that p n, 1 , p n, 2 , ..., p n,r ( n ) are m utually orthogonal p ro j ections of C ( X n ) wh ic h corresp ond to the conn ected comp onen ts of X n . It is clear that w e ma y assume that eac h connected comp onent of X n con tains at least one p oin t of X . This implies th at [ s n ] ∈ K K ( C ( X n ) , C ( X )) ++ . It follo ws that [ s n ] × κ ∈ K K ( C ( X n ) , A ) ++ . (e 4.57) Let {F n } b e an increasing sequence of finite subsets of C ( X ) whose union is dense i n C ( X ) . Let { η n } b e a decreasing sequence of p ositiv e num b ers with lim n →∞ η n = 0 , {P n } b e an in - creasing sequence of fi nite subsets of K ( C ) whose union is K ( C ) , let { k ( n ) } , { N ( n ) } ⊂ N b e t w o sequences of integ ers suc h that k ( n ) , N ( n ) ր ∞ , and { x n, 1 , x ( n, 2 , ..., x n,N ( n ) } b e η n -dense subsets of X w hic h satisfy the requirements of 3.5 and 3.6 for corresp onding ǫ n = 1 / 2 n +2 and F n . By passing to a su bsequence if necessary , we ma y assume that there is a fi nite subset F ′ n ⊂ C ( X k ( n +1) ) suc h that s k ( n +1) ( F ′ n ) = F n and a finite subset Q k ( n ) ⊂ K ( C ( X k ( n ) )) suc h that [ s k ( n ) ]( Q k ( n ) ) = P n , n = 1 , 2 , .... W e may assume that 1 C ( X k ( n ) ) ∈ F ′ n , without loss o f generalit y . Set κ n = [ s k ( n ) ] × κ. Note that κ n ([1 C ( X k ( n ) ) ]) = [1 A ] . Let δ n (in place of δ ), G ′ n ⊂ C ( X ) s.a ( in place of G ), S 1 ,n , S 2 ,n , ..., S m ( n ) ,n (in place of { S 1 , S 2 , ..., } ) b e a set of disj oin t clop en subsets of X with ∪ m ( n ) i =1 S i = X r equired b y 3.3 for ǫ n and F n , n = 1 , 2 , .... W e ma y assum e that 1 C ( X ) ∈ G ′ n , n = 1 , 2 , .... By taking a refinement of the clo p en partition of X, we ma y assume that s n ( p n,i ) is a finite sum of fun ctions in { χ S j,n : 1 ≤ j ≤ m ( n ) } , i = 1 , 2 , ..., r ( n ) . Let G n ⊂ C ( X k ( n ) ) s.a. b e a finite su bsets for whic h s k ( n ) ( G n ) = G ′ n , n = 1 , 2 , .... By app lying 4.3, we obtain a pro jection P 1 ∈ A and a u nital homomorph ism Φ ′ 1 : C ( X ) → P 1 AP 1 suc h that | τ ◦ Φ ′ 1 ( g ) − γ ( g )( τ ) | < δ 1 / 2 for all g ∈ G ′ 1 , (e 4.58) τ ◦ Φ ′ 1 ( χ S j, 1 ) < γ ( χ S j, 1 )( τ ) (e 4.59) for all τ ∈ T ( A ) , i = 1 , 2 , ..., m (1) , and Φ ′ 1 ( f ) = N (1) X i =1 f ( x 1 ,i ) e (1) i ⊕ Φ ′ 0 , 1 ( f ) for all f ∈ C ( X ) , (e 4.60) where { e (1) 1 , e (1) 2 , ..., e (1) N (1) } is a set of m utually orthogonal and m utually equiv alent pr o jections with [ e 1 ] ≥ [(1 − P 1 )] and where Φ ′ 0 , 1 : C ( X ) → ( P 1 − P N (1) i =1 e (1) i ) A (( P 1 − P N (1) i =1 e (1) i ) is a unital homomorphism w ith finite dimensional range. Note also, since 1 C ( X ) ∈ G ′ 1 , τ (1 − P 1 ) < δ 1 / 2 for all τ ∈ T ( A ) . It follo ws from 4.1 th at there is a u n ital monomorphism φ ′ 1 : C ( X k (1) ) → (1 − P 1 ) A (1 − P 1 ) suc h that [ φ ′ 1 ] + [Φ ′ 1 ◦ s 1 ] = κ 1 in K K ( C ( X k (1) ) , A ) . (e 4.61) Define φ 1 = φ ′ 1 + Φ ′ 1 ◦ s 1 . Supp ose that, for 1 ≤ m ≤ n, there are unital homomorph isms φ ′ m : C ( X k ( m ) ) → (1 − P m ) A (1 − P m ) and Φ ′ m : C ( X ) → P m AP m and a unital (inj ectiv e) homomorphism φ m = φ ′ m + Φ ′ m ◦ s k ( m ) suc h that 13 (1) there are m utually orthogonal and mutually equiv alen t pro jections e ( m ) 1 , e ( m ) 2 , ..., e ( m ) N ( m ) ∈ P m AP m for whic h [ e ( m ) 1 ] ≥ [1 − P m ] , and Φ ′ m ( f ) = N ( m ) X i =1 f ( x m,i ) e ( m ) i ⊕ Φ (0) m ( f ) for all f ∈ C ( X ) where Φ (0) m : C ( X ) → ( P m − P N ( m ) i =1 e ( m ) i ) A ( P m − P N ( m ) i =1 e ( m ) i ) is a u nital homomorph ism with finite d imensional range; (2) τ ◦ Φ ′ ( χ S j,m ) < γ ( χ S j,m )( τ ) f or all τ ∈ T ( A ) , j = 1 , 2 , ..., m ( m ); (3) | τ ◦ Φ ′ m ( g ) − γ ( g )( τ ) | < δ m / 2 for all g ∈ G ′ m and for all τ ∈ T ( A ); (4) [ P m +1 ] ≥ [ P m ] in K 0 ( A ) and τ (1 − P m ) < δ m / 2 for all τ ∈ T ( A ); (5) there is a pr o jection P ′ m +1 ≤ P m +1 suc h that P ′ m +1 Φ m +1 = Φ ′ m +1 P m +1 and [Φ ′ m +1 ( χ S j,m ) P ′ m +1 ] = [Φ ′ m ( χ S j,m )] in K 0 ( A ) , j = 1 , 2 , ..., m ( m ); (6) φ ′ m is a unital m on omorp hism; (7) [ φ m ] = [ φ ′ m ] + [Φ ′ m ◦ s k ( m ) ] = κ m ; (8) there exists a u nitary u m ∈ A such that ad u m ◦ φ m +1 ◦ s k ( m ) ,k ( m +1) ≈ 1 / 2 m +1 φ m on s − 1 k ( m ) ( F m ) , m = 1 , 2 , ..., n − 1 . It f ollo ws from 4.4 th at there is a pro jection P n +1 ∈ A and a unital homomorphism Φ ′ n +1 : C ( X ) → P n +1 AP n +1 satisfying the follo wing: (1) there are mutually orthogonal and mutually equiv alent p ro jections e ( n +1) 1 , e ( n +1) 2 , ..., e ( n +1) N ( n +1) ∈ P n +1 AP n +1 for whic h [ e ( n +1) 1 ] ≥ [1 − P n +1 ] , and Φ ′ n +1 ( f ) = N ( n +1) X i =1 f ( x n +1 ,i ) e ( n +1) i ⊕ Φ (0) n +1 ( f ) for all f ∈ C ( X ) where Φ (0) n +1 : C ( X ) → ( P n +1 − P N ( n +1) i =1 e ( n +1) i ) A ( P n +1 − P N ( n +1) i =1 e ( n +1) i ) is a unital homomorphism with fi nite dimensional range; (2) τ ◦ Φ ′ n +1 ( χ S j,n +1 ) < γ ( χ S j,n +1 )( τ ) for all τ ∈ T ( A ) , j = 1 , 2 , ..., m ( n + 1); (3) | τ ◦ Φ ′ n +1 ( g ) − γ ( g )( τ ) | < δ n +1 / 2 for all g ∈ G ′ n +1 and for all τ ∈ T ( A ); (4) [ P n +1 ] ≥ [ P n ] in K 0 ( A ) and τ (1 − P n +1 ) < δ n +1 / 2 for all τ ∈ T ( A ); (5) there is a pr o jection P ′ n +1 ≤ P n +1 suc h that P ′ n +1 Φ ′ n +1 = Φ ′ n +1 P n +1 and [Φ ′ n +1 ( χ S n,j ) P ′ n +1 ] = [Φ ′ n ( χ S n,j )] in K 0 ( A ) , j = 1 , 2 , ..., m ( n ) . 14 It follo w s from 4.1 that there is a unital m on omorp hism φ ′ n +1 : C ( X k ( n +1) ) → (1 − P n +1 ) A (1 − P n +1 ) such that [ φ ′ n +1 ] = κ n +1 − [Φ ′ n +1 ◦ s k ( n +1) ] in K K ( C ( X k ( n +1) , A ) (e 4.62) Define φ n +1 = φ ′ n +1 + Φ ′ n +1 ◦ s k ( n +1) . Th us φ ′ n +1 , φ ′ n +1 and φ n +1 satisfy (1), (2), (3), (4), (5), (6) and (7). T o complete the indu ction, define Φ ′′ n +1 : C ( X ) → P ′ n +1 AP ′ n +1 b y Φ ′′ n +1 ( f ) = P ′ n +1 Φ ′ n +1 ( f ) P ′ n +1 for all f ∈ C ( X ) . By (3) and (4), | τ ◦ Φ ′′ n +1 ( g ) − γ ( g )( τ ) | < δ n +1 / 2 for all g ∈ G n for all τ ∈ T ( A ) . Note that, by ( 5), [ P ′ n +1 ] = [ P n ] . There is a unitary w n ∈ U ( A ) suc h that w ∗ n P ′ n +1 w n = P n . Th us, by (5) and (3), and by applying 3.3, there exists a u nitary v n ∈ U ( P n AP n ) such that ad v n ◦ ad w n ◦ Φ ′′ n +1 ≈ ǫ n Φ ′ n on F n . (e 4.63) Denote Ψ ′ n +1 = P ′ n +1 Φ n +1 P ′ n +1 and Ψ n +1 = ad w n ◦ Ψ ′ n +1 . Let φ ′′ n +1 = ad w n ◦ φ ′ n +1 ⊕ Ψ n +1 . No w consider φ ′ n and φ ′′ n +1 ◦ s k ( n ) ,k ( n +1) . By (7) and (e 4.62 ), w e ha v e [ φ ′′ n +1 ◦ s k ( n ) ,k ( n +1) ] | Q k ( n ) = [ φ ′ n ] | Q k ( n ) . It follo w s fr om (1) and 3.5 that th ere exists a unitary V n ∈ U ( A ) su c h that ad V n ◦ ( φ ′′ n +1 ◦ s k ( n ) ,k ( n +1) ⊕ Φ ′ n ◦ s k ( n ) ) ≈ ǫ n φ ′ n ⊕ Φ ′ n ◦ s k ( n ) on s − 1 k ( n ) ( F n ) . (e 4.64) Define u n = w n ( v n + (1 − P n )) V n . Then, by (e 4.63) and (e 4.64), ad u n ◦ φ n +1 ≈ 2 ǫ n φ n on s − 1 k ( n ) ( F n ) . (e 4.65) Note 2 ǫ n = 1 / 2 n +1 . This concludes the induction. Define ψ 1 = φ 1 and ψ n +1 = ad u n ◦ φ n +1 , n = 1 , 2 , .... Then, by (8 ) ab o ve, k ψ n ( c ) − ψ n +1 ◦ s k ( n ) ,k ( n +1) ( c ) k < 1 / 2 n +2 for all c ∈ s − 1 k ( n ) ( F n ) , (e 4.66) n = 1 , 2 , .... Fix m and f ∈ F m , let g ∈ s − 1 k ( m ) ( F m ) suc h that s k ( m ) ( g ) = f . It follo ws that { ψ n ◦ s m,n ( g ) } n ≥ m is a Cauc hy sequence b y (e 4.66 ). Note th at if g ′ ∈ s − 1 k ( m ) ( F m ) su c h that s k ( m ) ( g ) = s k ( m ) ( g ′ ) , then , for any ǫ > 0 , there exists n ≥ m suc h that k s k ( m ) ,k ( n ) ( g ) − s k ( m ) ,k ( n ) ( g ′ ) k < ǫ. Th us h ( f ) = lim n →∞ ψ n ◦ s m,n ( g ) is we ll-defined. It is then easy to ve rify that h d efines a unital homomorphism from C ( X ) int o A. Sin ce eac h φ n is injectiv e, it is easy to c hec k th at h is also injectiv e. If x ∈ Q m , then by (7) ab o ve, [ h ] ◦ [ s k ( m ) ]( x ) = κ n ◦ [ s k ( m ) ,k ( n ) ]( x ) = κ ◦ [ s k ( m ) ]( x ) . Therefore [ h ] = κ in K L ( C, A ) . It is also easy to c h ec k fr om (3) and (4) that τ ◦ h ( g ) = γ ( ˇ g )( τ ) for all g ∈ C ( X ) s.a (e 4.67) and for all τ ∈ T ( A ) . 15 5 AH-algebras Lemma 5.1. L et X b e a c omp act subset of a finite CW c omplex, let C = P M k ( C ( X )) P, wher e P ∈ M k ( C ( X )) is a pr oje ction, and let A b e a unital simple C ∗ -algebr a with tr acial r ank zer o. Supp ose tha t κ ∈ K L e ( C, A ) ++ and sup p ose that γ : C s.a. → Aff ( T ( A )) is a unital p ositive line ar map which is c omp atible with κ. Then, for any ǫ > 0 and finite subset F ⊂ C ( X ) , ther e is a unital monomor phism φ : C → A such that [ φ ] = κ and τ ◦ φ ( f ) = γ ( f )( τ ) for al l τ ∈ T ( A ) . (e 5.68) Pr o of. It is clear th at the case th at C = M k ( C ( X )) follo ws from 4.5 immediately . F or the general case, there is an integ er d ≥ 1 and a p ro j ection p ∈ M d ( C ) such that pM d ( C ) p ∼ = M m ( C ( X )) for some inte ger m ≥ 1 . Th us the general case is reduced to the case that C = M m ( C ( X )) . Theorem 5.2. L et C b e a unital AH-algebr a and let A b e a unital simple C ∗ -algebr a with T R ( A ) = 0 . Supp ose that κ ∈ KL e ( C, A ) ++ . Supp ose also that ther e is a unital strictly p ositive line ar map γ : C s,a → Aff ( T ( A )) which is c omp atible with κ. Then ther e is a monom orphism φ : C → A such that [ φ ] = κ in KL( C, A ) an d (e 5.69) τ ◦ φ ( c ) = γ ( c )( τ ) (e 5.70) for al l c ∈ C s.a. and τ ∈ T ( A ) . Pr o of. W e may write C = ∪ ∞ n =1 C n , where C n = P n M k ( C ( X n )) P n , where X n is a compact subset of a finite CW complex and P n ∈ M k ( C ( X n )) is a pr o jection. W e ma y also assume that 1 C n = 1 C for all n. Denote by ı n : C n → C the em b edding, n = 1 , 2 , .... Define κ n = κ ◦ [ ı n ] and γ n = γ ◦ ( ı n ) ♯ n = 1 , 2 , .... Sin ce ı n is injectiv e κ n ∈ K L e ( C n , A ) ++ and γ n is unital strictly p ositiv e. It is also clear that γ n is compatible with κ n , since γ is compatible with κ. It follo ws from 5.1 that there is a sequence of u nital monomorphisms φ n : C n → A suc h that [ φ n ] = κ n and τ ◦ φ n ( c ) = γ n ( c )( τ ) (e 5.71) for all c ∈ C s.a. and τ ∈ T ( A ) . Let {F n } b e an in cr easing sequence of fi nite subsets of C wh ose u nion is dense in C . By passing to a subsequence, if necessary , without loss of generalit y , we ma y assume that F n ⊂ C n . It follo ws (fr om 2.3.13 of [9], for example) that there is, for eac h n, a u nital completely p ositiv e linear map L n : C → A su c h that L n ≈ 1 / 2 n +1 φ n ◦ ı n on F n . (e 5.72) It follo ws fr om L emm a 5.1, b y passing to a s ubsequence again and b y applying (e 5.71), ther e is a sequence of u nitaries u n and a su b sequence of { k ( n ) } such that ad u n ◦ L k ( n +1) ≈ 1 / 2 n L k ( n ) on F n , (e 5.73) n = 1 , 2 , .... Define ψ 1 = L 1 , ψ n +1 = ad u n ◦ L n +1 , n = 1 , 2 , .... Note that { ψ n ( c ) } is a Cauc h y sequence in A for eac h c ∈ F m . Defin e h ( c ) = lim n →∞ ψ n ( c ) . I t is easy to see that h giv es a unital homomorph ism from C into A. Moreo ver, for eac h x ∈ ∪ ∞ n =1 F n , h ( x ) = lim n →∞ ad u n ◦ φ k ( n ) ◦ ı k ( n ) ◦ · · · ◦ ı n ( x ) . (e 5.74) 16 Since eac h φ n is injectiv e, it follo ws that h is a m onomorphism. F rom (e 5.74) and (e 5.71 ), w e ha v e [ h ] = κ as w ell as τ ◦ h ( c ) = γ ( c )( τ ) for all c ∈ C s.a. and τ ∈ T ( A ) . Corollary 5.3. L et X b e a c omp act metric sp ac e and let A b e a unital simple C ∗ -algebr a with tr acial r ank zer o. Supp ose that κ ∈ K L e ( C ( X ) , A ) ++ . Supp ose also that ther e i s a unital strictly p ositive line ar map γ : C s,a → Aff ( T ( A )) which is c omp atible with κ. Then ther e is a monomor phism α : C → A such that [ α ] = κ in KL( C ( X ) , A ) and (e 5.75) τ ◦ φ ( c ) = γ ( c )( τ ) (e 5.76) for al l c ∈ C ( X ) s.a. and τ ∈ T ( A ) . Example 5.4. Let X = { − 1 n : n ∈ N } ∪ [0 , 1] ∪ { 1 + 1 n : n ∈ N } ⊂ [ − 1 , 2] . Put C = C ( X ) . Then K 0 ( C ( X )) = C ( X , Z ) . T ak e t w o sequences of p ositive rational n umbers { a n } and { b n } suc h that P ∞ n =1 a n = 1 − √ 2 / 2 and P ∞ n =1 = √ 2 / 2 . Define a unital p ositiv e linear fun ctional F : C ( X ) → R as follo ws: F ( f ) = X n ∈ N a n f ( − 1 n ) + X n ∈ N b n f ( 1 n ) for all f ∈ C ( X ) . Let D 0 = F ( C ( X, Z )) . Note th at, if S is a clop en su bset whic h do es n ot con tain [0 , 1] , then F ( S ) ∈ Q . If S ⊃ [0 , 1] , Then F ( S ) = 1 − F ( S 1 ) for some clop en sub set S 1 ⊂ X whic h do es not intersect with [0 , 1] . It follo ws that D 0 ⊂ Q . This gives a unital p ositiv e lin ear map F ∗ : C ( X, Z ) → Q . Let p ∈ C ( X ) b e a p ro jection whose supp ort Ω has a n on -emp t y in tersection with [0 , 1] . S ince Ω is clop en, Ω ⊃ [0 , 1] . It follo ws that there is N ≥ 1 s uc h that 1 k ∈ Ω for | k | ≥ N . It follo w s that F ( p ) ≥ X | k |≥ N 1 2 | k | +1 > 0 . F rom this one sees that F ∗ is strictly p ositiv e. Let A b e a unital simple AF-algebra with ( K 0 ( A ) , K 0 ( A ) , [1 A ]) = ( Q , Q + , 1) . There is an element κ ∈ K L ( C ( X ) , A ) such that κ | K 0 ( C ( X )) = F ∗ . Th us κ ( K 0 ( C ( X )) + \ { 0 } ) ⊂ K 0 ( A ) + \ { 0 } . I n other w ords, κ ∈ K L e ( C, A ) ++ . Supp ose t hat γ : C s.a → Aff ( T ( A )) = R is unital and p ositiv e such that γ ( ˇ p )( τ ) = τ ( κ ([ p ])) 17 for all pr o jections p ∈ C and τ ∈ T ( A ) . Consid er a p ositiv e contin uou s function f ∈ C ( X ) with 0 ≤ f ≤ 1 wh ose su p p ort S is an op en subset of (0 , 1) . C onsider pro jection p n ( t ) = 0 if t 6∈ [ − 1 /n, 1 + 1 /n ] ∩ X and p n ( t ) = 1 if t ∈ [ − 1 /n, 1 + 1 /n ] ∩ X . Th en f ≤ p n , n = 1 , 2 , .... It follo w s that, for all τ ∈ T ( A ) , γ ( ˇ f )( τ ) ≤ γ ( ˇ p n )( τ ) (e 5.77) < X | k |≥ n ( a k + b k ) → 0 (e 5.78) as | n | → ∞ . It follo ws that γ ( ˇ f )( τ ) = 0 for all τ ∈ T ( A ) . This shows that γ is not strictly p ositiv e. In particular, th er e is no unital monomorph ism φ : C ( X ) → A such that [ φ ] = κ. Ho w ab ou t homomorphisms? Supp ose that there exists a unital h omomorp hism h : C ( X ) → A such that [ h ] = κ. Let f ∈ C ( X ) + b e so that its sup p ort is con tained in [0 , 1] . Then, as sho wn ab o v e, τ ( h ( f )) = 0 for τ ∈ T ( A ) . Since A is simple, this implies that h ( f ) = 0 . It is then easy to see that k er h = { f ∈ C ( X ) : f | X \ (0 , 1) = 0 } . Th us C / ker h ∼ = C ( Y ) , where Y = X \ (0 , 1) . Let φ : C ( Y ) → A b e the unital h omomorp hism induced by h. Then φ is a monomorp hism. Let Y 1 = { 1 + 1 /n : n ∈ N } ∪ { 1 } and Y 2 = {− 1 /n : n ∈ N } ∪ { 0 } . Then Y 1 and Y 2 are clop en sub sets of Y . Let p i b e the pro jection corresp ondin g to Y i , i = 1 , 2 . Then τ ( p 1 ) ≥ ∞ X n =1 b n = 1 − √ 2 / 2 and τ ( p 2 ) ≥ ∞ X n =1 a n = √ 2 / 2 for τ ∈ T ( A ) . Since τ ( p 1 ) + τ ( p 2 ) = 1 , it follo ws that τ ( p 1 ) = 1 − √ 2 / 2 and τ ( p 2 ) = √ 2 / 2 . This is imp ossib le since K 0 ( A ) = Q . F rom this we arriv e at the follo wing conclusion: Prop osition 5.5. Ther e ar e c omp act metric sp ac es X with dimension one, unital simple AF- algebr as A with u nique tr acial states and κ ∈ K L e ( C, A ) ++ which has no strictly p ositive affine map fr om Aff ( T ( C ( X )) to Aff ( T ( A )) c omp atible with κ. F urthermor e , ther e is no u nital h omomorp hism φ : C ( X ) → A such th at [ φ ] = κ in K L ( C , A ) . Definition 5.6. Let C b e a unital AH-algebra w hic h admits a faithful tracial state and let A b e a unital simple C ∗ -algebra with T ( A ) 6 = ∅ . Denote by K LT ( C, A ) ++ the set of pairs ( κ, λ ) where κ ∈ K L ( C, A ) ++ with κ ([1 C ]) = [1 A ] and λ : T ( A ) → T f ( C ) w hic h is compatible with κ, i.e., λ ( τ )( p ) = τ ( κ ([ p ]) for all pro jections p ∈ M ∞ ( C ) and f or all τ ∈ T ( A ) . Denote by Mon e au ( C, A ) th e set of app r o ximate ly unitary equiv alen t classes of u nital mon omor- phisms from C into A. 18 T o conclude this note, com bing the previous result in ?? (see 3.1) and 5.2, we s tate the follo w ing: Theorem 5.7. L et C b e a unital AH-algebr a which admits a faithful tr acial state and let A b e a unital sep ar able simple C ∗ -algebr a with T R ( A ) = 0 . Then map Λ : Mon e au ( C, A ) → K L T ( C , A ) ++ define d by φ 7→ ([ φ ] , φ T ) is bije ctive. References [1] M. D˘ ad˘ arlat and T. A. Loring, A universal multic o efficient the or e m for the Kasp ar ov gr oups , Duke Math. J. 84 (1996), 355–377. [2] M. D˘ ad˘ arlat and T. A. Loring, The K -the ory of ab e lian sub algebr as of AF algebr as , J. Reine Angew. Math. 432 (1992), 39–55. [3] G. A. Elliott and T. Loring, AF emb e ddings of C ( T 2 ) with a p r escrib e d K -the ory , J. F un ct. Anal. 103 (1992) , 1–25. [4] G. Gong and H. Lin , Classific ation of homomorphisms fr om C ( X ) to simple C ∗ -algebr as of r e al r ank zer o , Acta Math. Sin. (En gl. Ser.) 16 (2000), 181–206 . [5] G. Gong an d H. Lin, Almost multiplic ative morphisms and K -the ory , Internat. J. 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