A classification of finite simple amenable Z-stable C*-algebras, II, --C*-algebras with rational generalized tracial rank one
aa r X i v : . [ m a t h . OA ] S e p A classification of finite simple amenable Z -stable C*-algebras, II.C*-algebras with rational generalized tracial rank one. Guihua Gong, Huaxin Lin and Zhuang Niu
Abstract
A classification theorem is obtained for a class of unital simple separable amenable Z -stable C ∗ -algebras which exhausts all possible values of the Elliott invariant for unital stablyfinite simple separable amenable Z -stable C ∗ -algebras. Moreover, it contains all unital simpleseparable amenable C ∗ -algebras which satisfy the UCT and have finite rational tracial rank. Contents
22 Construction of maps 223 A pair of almost commuting unitaries 2124 More existence theorems for Bott elements 2625 Another Basic Homotopy Lemma 3126 Stable results 3927 Asymptotic unitary equivalence 4528 Rotation maps and strong asymptotic equivalence 5929 The general classification theorem 64
This is the second part of the paper entitled “A classification of finite simple amenable Z -stable C*-algebras” (see [21]).The main theorem of this part is the following isomorphism theorem: Theorem (see Theorem 29.8). Let A and B be two unital separable simple amenable Z -stable C ∗ -algebras which satisfy the UCT. Suppose that gT R ( A ⊗ Q ) ≤ gT R ( B ⊗ Q ) ≤ . Then A ∼ = B if and only if Ell( A ) ∼ = Ell( B ) . See Section 29 for a brief explanation. We also refer to the first part [21], in particular,Section 2 of [21], for the notations and definitions.
Acknowledgements : A large part of this article was written during the summers of 2012,2013, and 2014 when all three authors visited The Research Center for Operator Algebras inEast China Normal University. They were partially supported by the Center. Both the firstnamed author and the second named author have been supported partially by NSF grants. Thework of the third named author has been partially supported by a NSERC Discovery Grant,1 Start-Up Grant from the University of Wyoming, and a Simons Foundation CollaborationGrant.
22 Construction of maps
In this section, we will introduce some technical results on the existence of certain maps.
Lemma 22.1.
Let X be a finite CW complex, let C = P M k ( C ( X )) P, and let A ∈ B be a unitalsimple C ∗ -algebra. Assume that A = A ⊗ U for an infinite dimensional UHF-algebra U. Let α ∈ KK e ( C, A ) ++ (see Definition 2.10 of [21]). Then there exists a unital monomorphism ϕ : C → A such that [ ϕ ] = α. Moreover we may write ϕ = ϕ ′ n ⊕ ϕ ′′ n , where ϕ ′ n : C → (1 − p n ) A (1 − p n ) is a unital monomorphism, ϕ ′′ n : C → p n Ap n is a unital homomorphism with [ ϕ ′′ n ] = [Φ] in KK ( C, p n Ap n ) for some homomorphism Φ with finite dimensional range, and lim n →∞ max { τ (1 − p n ) : τ ∈ T ( A ) } = 0 for all τ ∈ T ( A ) , where p n ∈ A is a sequence of projections.Proof. To simplify the matter, we may assume that X is connected. Suppose that the lemmaholds for the case C = M k ( C ( X )) for some integer k ≥ . Consider the case C = P M k ( C ( X )) P. Note C ⊗ K ∼ = C ( X ) ⊗ K . Let q ∈ M m ( A ) be a projection (for some integer m ≥
1) suchthat [ q ] = α ([1 M k ( C ( X )) ]) . Put A = qM m ( A ) q. Then α ∈ KK e ( M k ( C ( X )) , A ) ++ . Let ψ : M k ( C ( X )) → qM m ( A ) q be the map given by the lemma for the case that C = M k ( C ( X )) . Notenow C = P M k ( C ( X )) P. Let ψ ′ = ψ | C . Since P ≤ M k ( C ( X )) , ψ ′ (1 C ) = ψ ′ ( P ) ≤ q. Moreover,[ ψ ′ (1 C )] = [1 A ] . Since A = A ⊗ U, there is a unitary v ∈ A such that v ∗ ψ ′ (1 C ) v = 1 A . Define ϕ = Ad v ◦ ψ ′ . We see that the general case reduces to the case C = M k ( C ( X )) . This case thenreduces to the case C = C ( X ) . Since quasitraces of C and A are traces (see 9.10 of [21]) by Corollary 3.4 of [3], α (ker ρ C ) ⊂ ker ρ A .Since K i ( C ) is finitely generated, i = 0 , , KK ( C, A ) = KL ( C, A ) . Let α ∈ KL e ( C, A ) ++ . We may identify α with an element in Hom Λ ( K ( C ) , K ( A )) by a result in ([6]).Write A = lim n →∞ ( A ⊗ M r n , ı n,n +1 ) , where r n | r n +1 , r n +1 = m n r n and ı n,n +1 ( a ) = a ⊗ M mn ,n = 1 , , .... Since K ∗ ( C ) is finitely generated and consequently, K ( C ) is finitely generatedmodulo Bockstein Λ operations, there is an element α ∈ KK ( C, A ⊗ M r n ) such that α = α × [ ı n ] , where [ ı n ] ∈ KK ( A ⊗ M r n , A ) is induced by the inclusion ı n : A ⊗ M r n → A . Increasing n , we may assume that α (ker ρ C ) ⊂ ker ρ A ⊗ M rn and further that α ∈ KK e ( C, A ⊗ M r n ) ++ .Replacing A by A ⊗ M r n , we may assume that α = α × [ ı ], where α ∈ KK e ( C, A ) ++ and ı : A → A is the inclusion.It induces an element ˜ α ∈ KL ( C ⊗ U, A ⊗ U ) . Let K ( U ) = D , a dense subgroup of Q . Notethat K i ( C ⊗ U ) = K i ( C ) ⊗ D , i = 0 , , by the K¨unneth formula.We verify that ˜ α ( K ( C ⊗ U ) + \ { } ) ⊂ K ( A ⊗ U ) + \ { } . Consider x = P mi =1 x i ⊗ d i ∈ K ( C ⊗ U ) + \ { } with x i ∈ K ( C ) and d i ∈ D , i = 1 , , ..., m. There is a projection p ∈ M r ( C )for some r ≥ p ] = x. Let t ∈ T ( C ); then m X i =1 t ( x i ) d i > . (e 22.1)It should be noted that, since C = C ( X ) and X is connected, t ( x i ) ∈ Z and t ( x i ) = t ′ ( x i ) forall t, t ′ ∈ T ( C ) . Since α ([1 C ]) = [1 A ] , τ ◦ α ( x i ) = t ( x i ) for any τ ∈ T ( A ) and t ∈ T ( C ) . By2e 22.1), τ ( ˜ α ( x )) = m X i =1 τ ◦ α ( x i ) d i = m X i =1 t ( x i ) d i > τ ∈ T ( A ) . This shows that ˜ α is strictly positive. For any C ∗ -algebra A ′ , in this proof,we will use j A ′ : A ′ → A ′ ⊗ U for the homomorphism j A ′ ( a ) = a ⊗ U for all a ∈ A ′ . Evidently, α = ˜ α ◦ j C = j A ◦ α . (e 22.3)Write U = lim n →∞ ( M r n , ı n ) , where r n | r n +1 , r n +1 = m n r n and ı n ( a ) = a ⊗ M mn , n = 1 , , .... We may assume that r = 1 . The UHF algebra U corresponds to the supernatural number Q ∞ i =1 m i . Evidently, we can choose m i carefully so that we can write the supernatural number Q ∞ i =1 m i in another way as Q ∞ i =1 m i = Q ∞ i =1 l i with l i | m i and lim i →∞ m i l i = ∞ .Let { x n } be a sequence of points in X such that { x k , x k +1 , ..., x n , ... } is dense in X for each k and each point in { x n } repeated infinitely many times. Let B = lim n →∞ ( B n = M r n ( C ) , ψ n ) , where ψ n ( f ) = diag( f, f..., f | {z } l n , f ( x ) , f ( x ) , ..., f ( x m n − l n )) for all f ∈ M r n ( C ) ,n = 1 , , .... Note that ψ n is injective. Set e n = diag(1 M rn · ln , , ..., ∈ M r n +1 ( C ) , n = 1 , , ... It is standard that B has tracial rank zero (see [23] and also, 3.77 and 3.79 of [31]) and K ∗ ( B ) = K ∗ ( C ⊗ U ). Note that B is a unital simple AH-algebra with no dimension growth,with real rank zero, and with a unique tracial state. Note that( K ( B ) , K ( B ) + , [1 B ] , K ( B )) = ( K ( C ⊗ U ) , K ( C ⊗ U ) + , [1 C ⊗ U ] , K ( C ⊗ U )) . Thus we obtain a KK -equivalence κ ∈ KL e ( B, C ⊗ U ) ++ (by the UCT). It is standard toconstruct a unital homomorphism h : C → B such that KK ( h ) = κ − ◦ j C (see [28]). Inparticular, if N is large enough, we can choose a homomorphism h ′ : C → B N such that h = ψ N, ∞ ◦ h ′ . Note that 1 − ψ n, ∞ ( e n ) commutes with the image of h for all n ≥ N. Moreover,(1 − ψ n, ∞ ( e n )) h ( c )(1 − ψ n, ∞ ( e n )) = ψ n, ∞ ((1 − e n ) ψ N,n ◦ h ′ ( c )(1 − e n )) for all c ∈ C. Thereforethe map (1 − ψ n, ∞ ( e n )) h ( C )(1 − ψ n, ∞ ( e n )) has finite dimensional range.We also have ˜ α ◦ κ ∈ KL e ( B, A ) ++ , where recall A = A ⊗ U. We also note that B hasa unique tracial state. Let γ : T ( A ) → T ( B ) be defined by γ ( τ ) = t where t ∈ T ( B ) is theunique tracial state. It follows that ˜ α ◦ κ and γ is compatible. By Corollary 21.11 of [21],there is a unital homomorphism H : B → A such that [ H ] = ˜ α ◦ κ. Define ϕ : C → A by ϕ = H ◦ h = H ◦ ψ N, ∞ ◦ h ′ . Then, ϕ is injective, and, by (e 22.3) and [ h ] = κ − ◦ j C , we have[ ϕ ] = α. To show the last part, define q n = ψ n +1 , ∞ ( e n ) ∈ B, n = N + 1 , N + 2 , .... Define p n =1 − H ( q n ) , n = N + 1 , N + 2 , .... One checks thatlim n →∞ max { τ (1 − p n ) : τ ∈ T ( A ) } = lim l n m n =0 (e 22.4)Note that for n > N , q n commutes with the image of h and the homomorphism (1 − q n ) h (1 − q n ) : C → (1 − q n ) B (1 − q n ) has finite dimensional range. Define ϕ ′ n : C → (1 − p n ) A (1 − p n ) by ϕ ′ n ( f ) = H ( q n ) H ◦ h ( f ) H ( q n ) for f ∈ C. Define ϕ ′′ n ( f ) = (1 − p n ) H ◦ h ( f )(1 − p n ) , which is apoint-evaluation map. The lemma follows.We also have the following: 3 emma 22.2. Let C = M k ( C ( T )) and let A be a unital infinite dimensional simple C ∗ -algebrawith stable rank one and with the property (SP). Then the conclusion of 22.1 also holds for agiven α ∈ KK e ( C, A ) ++ . Proof.
Let p ∈ C be a minimal rank one projection. Since kα ([ p ]) = α ([1 C ]) = [1 A ] , A containsmutually equivalent and mutually orthogonal projections e , e , ..., e k such that P ki =1 e i = 1 A . Thus A = M k ( A ′ ) , where A ′ ∼ = e Ae . Since e i Ae i are unital infinite dimensional simple C ∗ -algebras with stable rank one and with (SP), the general case can be reduced to the case that k = 1 . Fix 1 > δ > . Choose a non-zero projection p ∈ A such that τ ( p ) < δ for all τ ∈ T ( A ) . Note K ( pAp ) = K ( A ) , since A is simple. Let α : K ( C ( T )) → K ( pAp ) be the homomorphismgiven by α. Let z ∈ C ( T ) be the standard unitary generator. Let x = α ([ z ]) ∈ K ( pAp ) . Since pAp has stable rank one, there is a unitary u ∈ pAp such that [ u ] = x in K ( pAp ) = K ( A ) . Define ϕ ′ : C ( T ) → pAp by ϕ ′ ( f ) = f ( u ) for all f ∈ C ( T ) . Define ϕ ′′ : C ( T ) → (1 − p ) A (1 − p )by ϕ ′′ ( f ) = f (1)(1 − p ) for all f ∈ C ( T ) (where f (1) is the point evaluation at 1 on the unitcircle). Define ϕ = ϕ ′ ⊕ ϕ ′′ : C ( T ) → A. The map ϕ verifies the conclusion of lemma follows. Corollary 22.3.
Let X be a connected finite CW complex, let C = P M m ( C ( X )) P, where P ∈ M m ( C ( X )) is a projection, let A ∈ B be a unital separable simple C ∗ -algebra whichsatisfies the UCT, and let A = A ⊗ U, where U is a UHF-algebra of infinite type. Suppose that α ∈ KK ( C, A ) ++ and γ : T ( A ) → T f ( C ( X )) is a continuous affine map. Then there exists asequence of contractive completely positive linear maps h n : C → A such that (1) lim n →∞ k h n ( ab ) − h n ( a ) h n ( b ) k = 0 , for any a, b ∈ C , (2) for each h n , the map [ h n ] is well defined and [ h n ] = α , and (3) lim n →∞ max {| τ ◦ h n ( f ) − γ ( τ )( f ) | : τ ∈ T ( A ) } = 0 for any f ∈ C .Proof. By Theorem 21.10 of [21], one may assume that A is a unital C ∗ -algebra as describedin Theorem 14.10 of [21]. It follows from Lemma 22.1 that there is a unital homomorphism h n : C → A such that [ h n ] = α . Moreover, h n = h ′ n ⊕ h ′′ n , where h ′′ n : C → p n Ap n is a homomorphism with [ h ′′ n ] = [Φ ′ ] in KK ( C, p n Ap n ) for some pointevaluation map Φ ′ , where p n is a projection in A with τ (1 − p n ) converging to 0 uniformly as n → ∞ . We will modify the map h n = h ′ n ⊕ h ′′ n to get the homomorphism.We assert that for any finite subset H ⊂ C s.a , and ǫ >
0, and any sufficiently large n ,there is a unital homomorphism ˜ h n : C → p n Ap n such that [˜ h n ] = [Φ] in KK ( C, p n Ap n ) for ahomomorphism Φ with finite dimensional range, and | τ ◦ ˜ h n ( f ) − γ ( τ )( f ) | < ǫ for all τ ∈ T ( A )for all f ∈ H . The corollary then follows by replacing the map h ′′ n by the map ˜ h n —of course,we use the fact that lim n →∞ τ (1 − p n ) = 0.Let H , (in place of H , ) be the finite subset of Lemma 17.1 of [21] with respect to H (inplace of H ), ǫ/ σ ), and C (in place of C ). Since γ ( T ( A )) ⊂ T f ( C ( X )), there is σ , > γ ( τ )( h ) > σ , for all h ∈ H , for all τ ∈ T ( A ) . Let H , ⊂ C + (in place of H , ) be the finite subset of Lemma 17.1 of [21] with respect to σ , . Since γ ( T ( A )) ⊂ T f ( C ( X )), there is σ , > γ ( τ )( h ) > σ , for all h ∈ H , for all τ ∈ T ( A ) . M be the constant of Lemma 17.1 of [21] with respect to σ , . By Lemma 16.12 of [21](also see the proof of Lemma 16.12 of [21]) for sufficiently large n , there are a C*-subalgebra D ⊂ p n Ap n ⊂ A such that D ∈ C , and a continuous affine map γ ′ : T ( D ) → T ( C ) such that | γ ′ ( 1 τ ( p ) τ | D )( f ) − γ ( τ )( f ) | < ǫ/ τ ∈ T ( A ) for all f ∈ H , where p = 1 D , τ (1 − p ) < ǫ/ (4 + ǫ ), and further (see part (2) of Lemma 16.12 of [21]) γ ′ ( τ )( h ) > σ , for all τ ∈ T ( D ) for all h ∈ H , , and (e 22.5) γ ′ ( τ )( h ) > σ , for all τ ∈ T ( D ) for all h ∈ H , . (e 22.6)Since A is simple and not elementary, one may assume that the dimensions of the irreduciblerepresentations of D are at least M . Thus, by Lemma 17.1 of [21], there is a homomorphism ϕ : C → D such that [ ϕ ] = [Φ] in KK ( C, D ) for a point evaluation map Φ, and that | τ ◦ ϕ ( f ) − γ ′ ( τ )( f ) | < ǫ/ f ∈ H for all τ ∈ T ( D ) . Pick a point x ∈ X , and define ˜ h : C → p n Ap n by f f ( x )( p n − p ) ⊕ ϕ ( f ) for all f ∈ C. Then a calculation as in the proof of Theorem 17.3 of [21] shows that the homomorphism h ′ n ⊕ ˜ h verifies the assertion. Corollary 22.4.
Let C ∈ H (see Definition 14.5 of [21]) and let A ∈ B be a unital separablesimple C ∗ -algebra which satisfies the UCT and let A = A ⊗ U for some UHF-algebra U ofinfinite type. Suppose that α ∈ KK e ( C, A ) ++ , λ : U ( C ) /CU ( C ) → U ( A ) /CU ( A ) is a continuoushomomorphism, and γ : T ( A ) → T f ( C ) is a continuous affine map such that α, λ, and γ arecompatible. Then there exists a sequence of unital completely positive linear maps h n : C → A such that (1) lim n →∞ k h n ( ab ) − h n ( a ) h n ( b ) k = 0 for any a, b ∈ C , (2) for each h n , the map [ h n ] is well defined and [ h n ] = α , (3) lim n →∞ max {| τ ◦ h n ( f ) − γ ( τ )( f ) | : τ ∈ T ( A ) } = 0 for all f ∈ C, and (4) lim n →∞ dist( h ‡ n (¯ u ) , λ (¯ u )) = 0 for any u ∈ U ( C ) . Proof.
Let ǫ >
0. Let U be a finite subset of U ( C ) such that U generates J c ( K ( C )), where J c ( K ( C )) is as in Definition 2.16 of [21]. Let σ > δ > G be the constant and finitesubset of Lemma 21.5 of [21] with respect to U , ǫ, and λ (in the place of α ). Without loss ofgenerality, one may assume that δ < ǫ .Let F be a finite subset such that F ⊃ G . Let
H ⊂ C be a finite subset of self-adjointelements with norm at most one. By Corollary 22.3, there is a completely positive linear map h ′ : C → A such that h ′ is F - δ -multiplicative, [ h ′ ] is well defined and [ h ′ ] = α , and | τ ( h ′ ( f )) − γ ( τ )( f ) | < ǫ, τ ∈ T( A ) , f ∈ H . (e 22.7)By Theorem 21.9 of [21], the C*-algebra A is isomorphic to one of the model algebrasconstructed in Theorem 14.10 of [21], and therefore there is an inductive limit decomposition A = lim −→ ( A i , ϕ i ), where A i and ϕ i are as described in Theorem 14.10 of [21]. Without loss of5enerality, one may assume that h ′ ( C ) ⊂ A i . Therefore, by Theorem 14.10 of [21], the map ϕ , ∞ ◦ h ′ has a decomposition ϕ , ∞ ◦ h ′ = ψ ⊕ ψ such that ψ , ψ satisfy (1)–(4) of Lemma 21.5 of [21] with the σ and δ above.It then follows from Lemma 21.5 of [21] that there is a homomorphism Φ : C → e Ae ,where e = ψ (1 C ), such that(i) Φ is homotopic to a homomorphism with finite dimensional range and[Φ] ∗ = [ ψ ] , and (e 22.8)(ii) for each w ∈ U , there is g w ∈ U ( B ) with cel( g w ) < ǫ such that λ ( ¯ w ) − (Φ ⊕ ψ ) ‡ ( ¯ w ) = ¯ g w . (e 22.9)Consider the map h := Φ ⊕ ψ . Then h is F - ǫ -multiplicative. By (e 22.8), one has[ h ] = [ ψ ] ⊕ [ ψ ] = [ h ′ ] = α. By (e 22.7) and Condition (4) of Lemma 21.5 of [21], one has, for all f ∈ H , | τ ( h ( f )) − γ ( τ )( f ) | ≤ | τ ( h ′ ( f )) − γ ( τ )( f ) | + δ < ǫ + δ < ǫ. It follows from (e 22.9) that, for all u ∈ U , dist( h ( u ) , λ ( u )) < ǫ. Since F , H , and ǫ are arbitrary, this proves the corollary. Corollary 22.5.
Let C ∈ H and let A ∈ B be a unital separable simple C ∗ -algebra whichsatisfies the UCT, and let A = A ⊗ U for some UHF-algebra U of infinite type. Suppose that α ∈ KL e ( C, A ) ++ and λ : U ( C ) /CU ( C ) → U ( A ) /CU ( A ) is a continuous homomorphism, and γ : T ( A ) → T f ( C ) is a continuous affine map such that α, λ, and γ are compatible. Then thereexists a unital homomorphism h : C → A such that (1) [ h ] = α, (2) τ ◦ h ( f ) = γ ( τ )( f ) for any f ∈ C, and (3) h ‡ n = λ. Proof.
Let us construct a sequence of unital completely positive linear maps h n : C → A whichsatisfies (1)–(4) of Corollary 22.4, and moreover, is such that the sequence { h n ( f ) } is Cauchyfor any f ∈ C . Then the limit map h = lim n →∞ h n is the desired homomorphism.Let {F n } be an increasing sequence in the unit ball of C with union dense in the unit ballof C. Define ∆( a ) = min { γ ( τ )( a ) : τ ∈ T( A ) } . Since γ is continuous and T( A ) is compact,the map ∆ is an order preserving map from C ,q + \ { } to (0 , G ( n ) , H ( n ) , H ( n ) ⊂ C , U ( n ) ⊂ U ∞ ( C ), P ( n ) ⊂ K ( C ), γ ( n ), γ ( n ), and δ ( n ) be the finite subsets and constants ofTheorem 12.7 of [21] with respect to F n , 1 / n +1 , and ∆ /
2. We may assume that δ ( n ) decreasesto 0 if n → ∞ , P ( n ) ⊂ P ( n + 1) , n = 1 , , ..., and S ∞ n =1 P ( n ) = K ( C ) . Let G ⊂ G ⊂ · · · be an increasing sequence of finite subsets of C such that S G n is dense in C , and let U ⊂ U ⊂ · · · be an increasing sequence of finite subsets of U ( C ) such that S U n isdense in U ( C ). One may assume that G n ⊃ G ( n ) ∪ G ( n − G n ⊃ H ( n ) ∪ H ( n + 1) ∪ H ( n ) ∪H ( n − U n ⊃ U ( n ) ∪ U ( n − G - δ (1)-multiplicative map h ′ : C → A such that64) the map [ h ′ ] is well defined and [ h ] = α ,(5) | τ ◦ h n ( f ) − γ ( τ )( f ) | < min { γ (1) , ∆( f ) : f ∈ H } for any f ∈ G , and(6) dist( h ‡ n (¯ u ) , λ (¯ u )) < γ (1) for any u ∈ U n . Define h = h ′ . Assume that h , h , ..., h n : C → A are constructed such that(7) h i is G i - δ ( i )-multiplicative, i = 1 , ..., n ,(8) the map [ h i ] is well defined and [ h i ] = α , i = 1 , ..., n ,(9) | τ ◦ h i ( f ) − γ ( τ )( f ) | < min { γ ( i ) , ∆( f ) : f ∈ H ( i ) } for any f ∈ G i , i = 1 , ..., n ,(10) dist( h ‡ i (¯ u ) , λ (¯ u )) < γ ( i ) for any u ∈ U i , i = 1 , ..., n , and(11) k h i − ( g ) − h i ( g ) k < i − for all g ∈ G i − , i = 2 , , ..., n .Let us construct h n +1 : C → A such that(12) h n +1 is G n +1 - δ ( n + 1)-multiplicative,(13) the map [ h n +1 ] is well defined and [ h n +1 ] = α ,(14) | τ ◦ h n +1 ( f ) − γ ( τ )( f ) | < min { γ ( n + 1) , ∆( f ) : f ∈ H ( n + 1) } for any f ∈ G n +1 ,(15) dist( h ‡ n +1 (¯ u ) , λ (¯ u )) < γ ( n + 1) for any u ∈ U , i = 1 , ..., n , and(16) k h n ( g ) − h n +1 ( g ) k < n for all g ∈ F n .Then the statement follows.By Corollary 22.4, there is G ( n + 1)- δ ( n + 1)-multiplicative map h ′ n +1 : C → A such that h ′ n +1 is G n +1 - δ ( n + 1)-multiplicative, the map [ h ′ n +1 ] is well defined and [ h ′ n +1 ] = α , | τ ◦ h ′ n +1 ( f ) − γ ( τ )( f ) | < min { γ ( n + 1) ,
12 ∆( f ) : f ∈ H ( n + 1) } (e 22.10)for any f ∈ G n +1 , and dist(( h ′ n +1 ) ‡ (¯ u ) , λ (¯ u )) < γ ( n + 1)for any u ∈ U , i = 1 , ..., n . In particular, this implies that[ h ′ n +1 ] | P = [ h n ] | P , and for any f ∈ H ( n ) (note that H ( n ) ⊂ G n ), | τ ◦ h n ( f ) − τ ◦ h ′ n +1 ( f ) | < γ ( n ) + | γ ( τ )( f ) − τ ◦ h ′ n +1 ( f ) | < γ ( n ) / γ ( n + 1) / < γ ( n ) . Also by (e 22.10), for any f ∈ H ( n ), one has τ ( h ′ n +1 ( f )) ≥ γ ( τ )( f ) −
12 ∆( f ) >
12 ∆( f ) . By the inductive hypothesis, one also has τ ( h n ( f )) ≥ γ ( τ )( f ) −
12 ∆( f ) >
12 ∆( f ) for all f ∈ H ( n ) . u ∈ U ( n ), one hasdist( h ′ n +1 ( u ) , h n ( u )) < γ ( n + 1) + dist( γ ( u ) , h n ( u )) < γ ( n + 1) + 12 γ ( n ) < γ ( n ) . Note that both h ′ n +1 and h n are G ( n )- δ ( n )-multiplicative, and so, by Theorem 12.7 of [21],there is a unitary W ∈ A such that k W ∗ h ′ n +1 ( g ) W − h n ( g ) k < / n for all g ∈ F n . Then the map h n +1 := AdW ◦ h ′ n +1 satisfies the desired conditions, and the statement is proved. Lemma 22.6.
Let C ∈ C . Let ε > , F ⊂ C be any finite subset. Suppose that B is a unitalseparable simple C ∗ -algebra in B , A = B ⊗ U for some UHF-algebra of infinite type, and α ∈ KK e ( C ⊗ C ( T ) , A ) ++ . Then there is a unital ε - F -multiplicative completely positive linearmap ϕ : C ⊗ C ( T ) → A such that [ ϕ ] = α. (e 22.11) Proof.
Denote by α and α the induced maps induced by α on K -groups and K -groups.By Theorem 18.2 of [21], there exist an F - ǫ -multiplicative map ϕ : C ⊗ C ( T ) → A ⊗ K anda homomorphism ϕ : C ⊗ C ( T ) → A ⊗ K with finite dimensional range such that[ ϕ ] = α + [ ϕ ] in KK ( C, A ) . In particular, one has ( ϕ ) ∗ = α . Without loss of generality, one may assume that both ϕ and ϕ map C into M r ( A ) for some integer r .Since M r ( A ) ∈ B , for any finite subset G ⊂ M r ( A ) and any ǫ ′ >
0, there are G - ǫ ′ -multiplicative maps L : M r ( A ) → (1 − p ) M r ( A )(1 − p ) and L : M r ( A ) → S ⊂ pM r ( A ) p for a C*-subalgebra S ∈ C with 1 S = p such that(1) || a − L ( a ) ⊕ L ( a ) || < ǫ ′ for any a ∈ G and(2) τ ((1 − p )) < ǫ ′ for any τ ∈ T ( M r ( A )) . Since K ( S ) = { } , choosing G sufficiently large and ǫ ′ sufficiently small, one may assume that L ◦ ϕ is F - ε -multiplicative, and[ L ◦ ϕ ] | K ( C ⊗ C ( T )) = ( ϕ ) ∗ = α . Moreover, since the positive cone of K ( C ⊗ C ( T )) is finitely generated, choosing ǫ ′ even smaller,one may assume that the map κ := α − [ L ◦ ϕ ] | K ( C ⊗ C ( T ) : K ( C ⊗ C ( T )) → K ( A )is positive. Pick a point x ∈ T , and consider the evaluation map π : C ⊗ C ( T ) ∈ f ⊗ g f · g ( x ) ∈ C. Then π ∗ : K ( C ⊗ C ( T )) → K ( C ) is an order isomorphism, since K ( C ) = 0.8hoose a projection q ∈ A with [ q ] = κ ([1]). Since qAq ∈ B , by Corollary 18.9 of [21], thereis a unital homomorphism h : C → qAq such that[ h ] = κ ◦ π − ∗ on K ( C ) , and hence one has ( h ◦ π ) ∗ = κ, on K ( C ⊗ C ( T )).Put ϕ = ( L ◦ ϕ ) ⊕ ( h ◦ π ) : C ⊗ C ( T ) → A. Then it is clear that ϕ ∗ = [ L ◦ ϕ ] | K ( C ⊗ C ( T )) + κ = [ L ◦ ϕ ] | K ( C ⊗ C ( T )) + α − [ L ◦ ϕ ] | K ( C ⊗ C ( T )) = α and [ ϕ ] = [ L ◦ ϕ ] | K ( C ⊗ C ( T )) = α . Since K ∗ ( C ⊗ C ( T )) is finitely generated and torsion free, one has that [ ϕ ] = α in KK ( C ⊗ C ( T ) , A ). Lemma 22.7.
Let C ∈ C . Let ε > , F ⊂ C ⊗ C( T ) be a finite subset, σ > , and H ⊂ ( C ⊗ C ( T )) s.a. be a finite subset. Suppose that A is a unital C ∗ -algebra in B , B = A ⊗ U forsome UHF-algebra U of infinite type, α ∈ KK e ( C ⊗ C ( T ) , B ) ++ , and γ : T ( B ) → T f ( C ⊗ C ( T )) is a continuous affine map such that α and γ are compatible. Then there is a unital F - ε -multiplicative completely positive linear map ϕ : C ⊗ C ( T ) → B such that (1) [ ϕ ] = α and (2) | τ ◦ ϕ ( h ) − γ ( τ )( h ) | < σ for any h ∈ H .Moreover, if A ∈ B , β ∈ KK e ( C, A ) ++ , γ ′ : T ( A ) → T f ( C ) is a continuous affine map which iscompatible with β, and H ′ ⊂ C s.a. is a finite subset, then there is also a unital homomorphism ψ : C → A such that [ ψ ] = β and | τ ◦ ψ ( h ) − γ ′ ( τ )( h ) | < σ for all f ∈ H ′ . (e 22.12) Proof.
Since K ∗ ( C ⊗ C ( T )) is finitely generated and torsion free, by the UCT, the element α ∈ KK ( C ⊗ C ( T ) , A ) is determined by the induced maps α ∈ Hom( K ( C ⊗ C ( T )) , K ( A ))and α ∈ Hom( K ( C ⊗ C ( T )) , K ( A )). We may assume that projections in M r ( C ⊗ C ( T )) (forsome fixed integer r >
0) generate K ( C ⊗ C ( T )) . We may also assume that k h k ≤ h ∈ H . Fix a finite generating set G of K ( C ⊗ C ( T )).Since γ ( τ ) ∈ T f ( C ⊗ C( T )) for all τ ∈ T ( B ) and τ ( B ) is compact, one is able to define ∆ :( C ⊗ C ( T )) q, \ { } → (0 ,
1) by∆(ˆ h ) = 12 inf { γ ( τ )( h ) : τ ∈ T ( B ) } . Fix a finite generating set G of K ( C ⊗ C ( T )). Let H ⊂ C ⊗ C ( T ), δ >
0, and K ∈ N be thefinite subset and the constants of Lemma 16.10 of [21] with respect to F , H , ǫ , σ/ σ ), and ∆.Since A ∈ B and U is of infinite type, for any finite subset G ′ ⊂ B and any ǫ ′ >
0, thereare unital G ′ - ǫ ′ -multiplicative completely positive linear maps L : B → (1 − p ) B (1 − p ) and L : B → D ⊗ M K ⊂ D ⊗ M K ⊂ pBp for a C*-subalgebra D ∈ C with 1 D ⊗ M K = p such that(3) || a − L ( a ) ⊕ L ( a ) || < ǫ ′ for any a ∈ G ′ , and(4) τ ((1 − p )) < min { ǫ ′ , σ/ } for any τ ∈ T ( B ) . S = D ⊗ M K . Since K i ( C ⊗ C ( T )) is finite generated, Hom Λ ( K ( C ⊗ C ( T )) , K ( C ⊗ C ( T )))is determined on a finitely generated subgroup G K of K ( C ⊗ C ( T )) (see Corollary 2.12 of [6]).Choosing G ′ large enough and ǫ ′ small enough, one may assume [ L ] and [ L ] are well definedon α ( G K ) , and α = [ L ] ◦ α + [ j ] ◦ [ L ] ◦ α, (e 22.13)where j : S → A is the embedding. Note that since K ( S ) = { } , one has α = [ L ] ◦ α | K ( C ⊗ C ( T )) . Define κ ′ = [ L ] ◦ α | K ( C ⊗ C ( T )) , which is a homomorphism from K ( C ⊗ C ( T )) to K ( D ) whichmaps [1 C ⊗ C ( T ) ] to [1 D ] . Let { e i,j : 1 ≤ i, j ≤ K } be a system of matrix units for M K . View e i,j ∈ D ⊗ M K . Then e i,j commutes with the image of L . Define L ′ : B → D ⊗ e , by L ′ ( a ) = L ( a ) ⊗ e , for all a ∈ B , where we view L as a map from B to D (rather than to D ⊗ M K ).Put κ = [ L ′ ] ◦ α | K ( C ⊗ C ( T )) . Put D ′ = D ⊗ e , . Choosing G ′ larger and ǫ ′ smaller, if necessary, one has a continuous affine map γ ′ : T ( D ′ ) → T ( C ⊗ C ( T )) such that, for all τ ∈ T ( A ) , (5) | γ ′ ( τ ( e , ) τ | D ′ )( f ) − γ ( τ )( f ) | < σ/ f ∈ H ,(6) γ ′ ( τ )( h ) > ∆(ˆ h ) for any h ∈ H , and(7) | γ ′ ( τ ( e , ) τ | D ′ )( p ) − τ ( κ ([ p ])) | < δ for all projections p ∈ M r ( C ⊗ C ( T )) . Then it follows from Theorem 16.10 of [21] that there is an F - ǫ -multiplicative contractive com-pletely positive linear map ϕ : C ⊗ C ( T ) → M K ( D ) = S such that( ϕ ) ∗ = Kκ = κ ′ and | (1 /K ) t ◦ ϕ ( h ) − γ ′ ( t )( h ) | < σ/ , h ∈ H , t ∈ T ( D ′ ) . On the other hand, since (1 − p ) A (1 − p ) ∈ B , by Lemma 22.6, there is a unital F - ǫ -multiplicative completely positive linear map ϕ : C ⊗ C ( T ) → (1 − p ) A (1 − p ) such that[ ϕ ] = [ L ] ◦ α in KK ( C ⊗ C ( T ) , A ) . Define ϕ = ϕ ⊕ j ◦ ϕ : C ⊗ C ( T ) → (1 − p ) A (1 − p ) ⊕ S ⊂ A . Then, by (e 22.13), one has ϕ ∗ = ( ϕ ) ∗ + ( j ◦ ϕ ) ∗ = ([ L ] ◦ α ) | K ( C ⊗ C ( T )) + ([ j ◦ L ] ◦ α ) | K ( C ⊗ C ( T )) = α and ϕ ∗ = ( ϕ ) ∗ + ( j ◦ ϕ ) ∗ = ([ L ] ◦ α | K ( C ⊗ C ( T )) = α . Hence [ ϕ ] = α in KK ( C ⊗ C ( T )).For any h ∈ H and any τ ∈ T ( A ), one has (note that k h k ≤ h ∈ H , and τ (1 − p ) <δ/ | τ ◦ ϕ ( h ) − γ ( τ )( h ) | < | τ ◦ ϕ ( h ) − τ ◦ j ◦ ϕ ( h ) | + | τ ◦ j ◦ ϕ ( h ) − γ ( τ )( h ) | < σ/ | τ ◦ j ◦ ϕ ( h ) − γ ′ ( 1 τ ( e , ) τ | D ′ )( h ) | + | γ ′ ( 1 τ ( e , ) τ | D ′ )( h ) − γ ( τ )( h ) | < σ/ | τ ◦ j ◦ ϕ ( h ) − γ ′ ( 1 τ ( p ) τ | S )( h ) | + | γ ′ ( 1 τ ( p ) τ | S )( h ) − γ ( τ )( h ) | < σ/ σ/ σ/ < σ, T ( D ′ ) with T ( S ) in a standard way for S = D ′ ⊗ M K . Hence the map ϕ satisfies the requirements of the lemma.To see the last part of the lemma holds, we note that, when C ⊗ C ( T ) is replaced by C and A is assumed to be in B , the only difference is that we cannot use 22.6. But then we can appealto Theorem 18.7 of [21] to obtain ϕ . The semiprojectivity of C allows us actually to obtain aunital homomorphism (see Corollary 18.9 of [21]). Corollary 22.8.
Let C ∈ C . Suppose that A is a unital separable simple C ∗ -algebra in B , B = A ⊗ U for some UHF-algebra of infinite type, α ∈ KK e ( C, B ) ++ , and γ : T ( B ) → T f ( C )) is a continuous affine map. Suppose that ( α, λ, γ ) is a compatible triple. Then there is a unitalhomomorphism ϕ : C → B such that [ ϕ ] = α and ϕ T = γ. In particular, ϕ is a monomorphism.Proof. The proof is exactly the same as the argument employed in 22.5 but using the secondpart of Lemma 22.7 instead of 22.4. The reason ϕ is a monomorphism is because γ ( τ ) is faithfulfor each τ ∈ T ( A ) . Lemma 22.9.
Let C be a unital C*-algebra. Let p ∈ C be a full projection. Then, for any u ∈ U ( C ) , there is a unitary v ∈ pCp such that u = v ⊕ (1 − p ) in U ( C ) /CU ( C ) . If, furthermore, C is separable and has stable rank one, then, for any u ∈ U ( C ) , there is aunitary v ∈ pCp such that u = v ⊕ (1 − p ) in U ( C ) /CU ( C ) . Proof.
It suffices to prove the first part of the statement. This is essentially contained in theproof of 4.5 and 4.6 of [22]. As in the proof of 4.5 of [22], for any b ∈ C s.a. , there is c ∈ pCp such that b − c ∈ C , where C is the closed subspace of A s.a. consisting of elements of the form x − y, where x = P ∞ n =1 c ∗ n c n and y = P ∞ n =1 c n c ∗ n (convergence in norm) for some sequence { c n } in C. Now let u = Q nk =1 exp( ib k ) for some b k ∈ C s.a. , k = 1 , , ..., n. Then there are c k ∈ pCp such that b k − c k ∈ C , k = 1 , , ..., n. Put v = p ( Q nk =1 exp( ic k )) p. Then v ∈ U ( pCp ) and v + (1 − p ) = Q nk =1 exp( ic k ) . By 3.1 of [52], u ∗ ( v + (1 − p )) ∈ CU ( C ) . Lemma 22.10.
Let D be the family of unital separable residually finite dimensional C ∗ -algebrasand let A be a unital simple separable C ∗ -algebra which has the property ( L D ) (see 9.4 of [21])and the property (SP). Then A satisfies the Popa condition: Let ε > and let F ⊂ A be a finitesubset. There exists a finite dimensional C ∗ -subalgebra F ⊂ A with P = 1 F such that k [ P, x ] k < ε, P xP ∈ ε F and k P xP k ≥ k x k − ε (e 22.14) for all x ∈ F . In particular, if A ∈ B and A has the property (SP), then A satisfies the Popacondition.Proof. We may assume that
F ⊂ A and 0 < ε < / . Without loss of generality, we mayassume that d = min {k x k : x ∈ F } > . A has property ( L D ), there are a projection p ∈ A and a C ∗ -subalgebra D ⊂ A with D ∈ D and p = 1 D such that k px − xp k < dǫ/ , pxp ∈ dǫ/ D, and k pxp k ≥ (1 − ε/ k x k (e 22.15)for all x ∈ F (see 9.5 of [21]).Let F ′ ⊂ D be a finite subset such that, for each x ∈ F , there exists x ′ ∈ F ′ such that k pxp − x ′ k < dε/ . Since D ∈ D , there is a unital surjective homomorphism π : D → D/ ker π such that F := D/ ker π is a finite dimensional C ∗ -algebra and k π ( x ′ ) k ≥ (1 − ε/ k x ′ k for all x ′ ∈ F ′ . (e 22.16)Let B = (ker π ) A (ker π ) . B is a hereditary C ∗ -subalgebra of A. Let C be the closure of D + B. Note that 1 C = 1 D = p. As in the proof of 5.2 of [29], B is an ideal of C and C/B ∼ = D/ ker π = F . The lemma then follows from Lemma 2.1 of [46]. In fact, since pAp has property (SP), by Lemma2.1 of [46], there are a projection P ∈ pAp and a monomorphism h : F → P AP such that h (1 F ) = P, k P x ′ − x ′ P k < ε/
16 and (e 22.17) k h ◦ π ( x ′ ) − P x ′ P k < ε · d/
16 (e 22.18)for all x ′ ∈ F ′ . Put F = h ( F ) . Then, one estimates that, for all x ∈ F , k P x − xP k ≤ k P px − P x ′ k + k P x ′ + x ′ P k + k x ′ P − xpP k (e 22.19) < ε/
16 + ε/
16 + ε/ < ε, (e 22.20) P xP ≈ ε/ P x ′ P ∈ ε/ F , and (e 22.21) k P xP k = k P pxpP k ≥ k
P x ′ P k − dε/ ≥ k h ◦ π ( x ′ ) k − dε/ k π ( x ′ ) k − dε/ ≥ k x ′ k − dε/ − dε/ ≥ k pxp k − dε/ ≥ (1 − ε/ k x k − dε/ ≥ k x k − ε. (e 22.24) Lemma 22.11.
Let C ∈ C . Let ε > , F ⊂ C be a finite subset, > σ > , > σ > , U ⊂ J c ( K ( C ⊗ C ( T ))) ⊂ U ( C ⊗ C ( T )) /CU ( C ⊗ C ( T )) be a finite subset (see Definition 2.16 of[21]) and H ⊂ ( C ⊗ C ( T )) s.a. be a finite subset. Suppose that A is a unital separable simple C ∗ -algebra in B , B = A ⊗ U for some UHF-algebra U of infinite type, α ∈ KK e ( C ⊗ C ( T ) , B ) ++ ,λ : J c ( K ( C ⊗ C ( T ))) → U ( B ) /CU ( B ) is a homomorphism, and γ : T ( B ) → T f ( C ⊗ C ( T )) isa continuous affine map. Suppose that ( α, λ, γ ) is a compatible triple. Then there is a unital F - ε -multiplicative completely positive linear map ϕ : C ⊗ C ( T ) → B such that (1) [ ϕ ] = α , (2) dist( ϕ ‡ ( x ) , λ ( x )) < σ , for any x ∈ U , and (3) | τ ◦ ϕ ( h ) − γ ( τ )( h ) | < σ , for any h ∈ H .Proof. Note that K ( C ⊗ C ( T )) is finitely generated modulo Bockstein operations and K ( C ⊗ C ( T )) + is a finitely generated semigroup. Using the inductive limit B = lim n →∞ ( A ⊗ M r n , ı n,n +1 ) , one can find, for n large enough, α n ∈ KK e ( C ⊗ C ( T ) , A ⊗ M r n ) ++ such that α = α n × [ ı n ] where[ ı n ] ∈ KK ( A ⊗ M r n , B ) is induced by the inclusion ı n : A ⊗ M r n → B . Replacing A by A ⊗ M r n ,we may assume that α = α × [ ı ], where α ∈ KK e ( C ⊗ C ( T ) , A ) ++ and ı : A → A ⊗ U = B is the inclusion. Note that λ : J c ( K ( C ⊗ C ( T ))) → U ( A ⊗ U ) /CU ( A ⊗ U ) . By the same12rgument as above, we know that if the integer n above is large enough, then there is a map λ n : J c ( K ( C ⊗ C ( T ))) → U ( A ⊗ M r n ) /CU ( A ⊗ M r n ) such that | ı ‡ n ◦ λ n ( u ) − λ ( u ) | is arbitrarilysmall (e.g smaller than σ /
4) for all u ∈ U . Replacing A by A ⊗ M r n , we may assume λ = ı ‡ ◦ λ with λ : J c ( K ( C ⊗ C ( T ))) → U ( A ) /CU ( A ) and ı ‡ : U ( A ) /CU ( A ) → U ( B ) /CU ( B ) inducedby the inclusion map. Furthermore, we may assume that λ is compatible with α .Without loss of generality, we may assume that k h k ≤ h ∈ H . Let p i , q i ∈ M k ( C ) beprojections such that { [ p ] − [ q ] , ..., [ p d ] − [ q d ] } forms a set of independent generators of K ( C )(as an abelian group) for some integer k ≥
1. Choosing a specific J c , one may assume that U = { (( k − p i ) + p i ⊗ z )(( k − q i ) + q i ⊗ z ∗ ) : 1 ≤ i ≤ d } , where z ∈ C ( T ) is the identity function on the unit circle. Put u ′ i = ( k − p i ) + p i ⊗ z )(( k − q i ) + q i ⊗ z ∗ ) . Hence, { [ u ′ ] , ..., [ u ′ d ] } is a set of standard generators of K ( C ⊗ C ( T )) ∼ = K ( C ) ∼ = Z d . Then λ is a homomorphism from Z d to U ( B ) /CU ( B ).Let π e : C → F = L li =1 M n i be the standard evaluation map defined in Definition 3.1 of [21].By Proposition 3.5 of [21], the map ( π e ) ∗ induces an embedding of K ( C ) in Z l , and the map( π e ⊗ id) ∗ induces an embedding of K ( C ⊗ C ( T )) ∼ = Z d in K ( L li =1 M n i ⊗ C ( T )) ∼ = Z l . Define J c ( K ( L li =1 M n i ⊗ C ( T ))) to be the subgroup generated by { e i ⊗ z i ⊕ (1 − e i ); i = 1 , ..., l } , where e i is a rank one projection of M n i and z i is the standard unitary generator of the i -th copy of C ( T ) . Note that the image of J c ( K ( C ⊗ C ( T ))) under π e is contained in J c ( K ( L li =1 M n i ⊗ C ( T ))). Write w j = e j ⊗ z j ⊕ (1 − e j ), 1 ≤ j ≤ l .Let U be as in the lemma. We write B = B ⊗ U , and B = A ⊗ U , with U = U ⊗ U ,and both U and U UHF algebras of infinite type. Denote by ı : A → B , ı : B → B, and ı = ı ◦ ı : A → B the inclusion maps. Recall α = α × [ ı ] ∈ KK ( C, B ).Applying Lemma 22.7, one obtains a unital F ′ - ε ′ -multiplicative completely positive linearmap ψ : C ⊗ C( T ) → B such that [ ψ ] = α × [ ı ] and (e 22.25) | τ ◦ ψ ( h ) − γ ( τ )( h ) | < min { σ , σ } / h ∈ H , and for all τ ∈ T( B ) , (e 22.26)where ε/ > ε ′ > F ⊃ F . (Note that T ( B ) = T ( A ) = T ( B ), and the map γ : T ( B ) → T f ( C ⊗ C ( T )) can be regarded as a map with domain T ( B )). We may assume that ε ′ issufficiently small and F is sufficiently large that not only (e 22.25) and (e 22.26) make sensebut also that ψ ‡ can be defined on ¯ U , and induces a homomorphism from J c ( K ( C ⊗ C ( T ))) to U ( B ) /CU ( B ) (see 2.17 of [21]).Let M be the integer of Corollary 15.3 of [21] for K ( C ) ⊂ Z l (in place of G ⊂ Z l ).For any ε ′′ > F ′′ ⊂ B , since B has the Popa condition andhas the property (SP) (see 22.10), there exist a non-zero projection e ∈ B and a unital F ′′ - ε ′′ -multiplicative completely positive linear map L : B → F ⊂ eB e, where F is a finitedimensional and 1 F = e, and a unital F ′′ - ε ′′ -multiplicative completely positive linear map L : B → (1 − e ) B (1 − e ) such that k b − ı ◦ L ( b ) ⊕ L ( b ) k < ε ′′ for all b ∈ F ′′ , (e 22.27) k L ( b ) k ≥ k b k / b ∈ F ′′ , and (e 22.28) τ ( e ) < min { σ / , σ / } for all τ ∈ T ( B ) , (e 22.29)where ı : F → eB e is the embedding and L ( b ) = (1 − p ) b (1 − p ) for all b ∈ B . Since the positive cone of K ( C ⊗ C ( T )) is finitely generated, with sufficiently small ε ′′ and sufficiently large F ′′ , one may assume that [ L ◦ ψ ] | K ( C ⊗ C ( T )) is positive. Moreover, onemay assume that ( L ◦ ψ ) ‡ and ( L ◦ ψ ) ‡ are well defined and induce homomorphisms from13 c ( K ( C ⊗ C ( T ))) to U ( B ) /CU ( B ). One may also assume that [ L ◦ ψ ] is well defined.Moreover, we may assume that L i ◦ ψ is F - ε -multiplicative for i = 0 , . There is a projection E c ∈ U such that E c is a direct sum of M copies of some non-zeroprojections E c, ∈ U . Put E = 1 U − E c . Define ϕ : C ⊗ C ( T ) → F ⊗ EU E → eB e ⊗ EU E by ϕ ( c ) = L ◦ ψ ( c ) ⊗ E ( ∈ B ) for all c ∈ C ⊗ C ( T ) and define ϕ ′ : C → F ⊗ E c U E c by ϕ ′ ( c ) = L ◦ ψ ( c ) ⊗ E c for all c ∈ C. Notethat ϕ is also F - ε -multiplicative and ϕ ‡ is also well defined as ( L ◦ ψ ) ‡ is. Moreover [ ϕ ′ ] iswell defined. Define L = ı ◦ L ◦ ψ + ϕ : C ⊗ C ( T ) → (cid:0) (1 − e ) B (1 − e ) ⊗ U (cid:1) ⊕ (cid:0) eB e ⊗ EU E (cid:1) ( ⊂ B ) . Denote by λ = λ − L ‡ = λ − ϕ ‡ − ( ı ◦ L ◦ ψ ) ‡ : J c ( K ( C ⊗ C ( T ))) → U ( B ) /CU ( B ) . Note that L factors through the finite dimensional algebra F and therefore [ L ] = 0 on K ( B ) . Consequently [ ϕ ] | K ( C ⊗ C ( T )) = 0 and [ L ◦ ψ ] = [ ı ] ◦ [ α ] on K ( C ⊗ C ( T )). Hence, [ ı ◦ L ◦ ψ ] = α on K . Furthermore, α is compatible with λ . We know that the image of λ is in U ( B ) /CU ( B ) . Note that, by Lemma 11.5 of [21], the group U ( B ) /CU ( B ) is divisible. It is an injectiveabelian group. Therefore there is a homomorphism ˜ λ : J c ( L li =1 M n i ⊗ C ( T )) → U ( B ) /CU ( B )such that ˜ λ ◦ ( π e ) ‡ = λ − L ‡ . (e 22.30)Let β = [ L ◦ ψ ] | K ( C ) : K ( C ) → K ( F ) = Z n . Let R ≥ β : K ( C ) → Z n (in place of κ : G → Z n ; note that that [ E c ] is divisible by M implies that every element in β ( K ( C )) is divisible by M ). There is a unital C ∗ -subalgebra M MK ⊂ E c U E c such that K ≥ R and such that E c U E c can be written as M MK ⊗ U . Itfollows from Corollary 15.3 of [21] that there is a positive homomorphism β : K ( F ) → K ( F )such that β ◦ ( π e ) ∗ = M Kβ.
Let h : F → F ⊗ M MK be the unital homomorphism such that h ∗ = β . Put ϕ ′′ = h ◦ π e : C → F ⊗ M MK , and then one has ( ϕ ′′ ) ∗ = M Kβ.
Let J : M MK → E c U E c be the embedding. One verifies that( ı F ⊗ J ) ∗ ◦ ( ϕ ′′ ) ∗ = ( ı F ⊗ J ) ∗ ◦ M Kβ = ˜ ı ∗ ◦ ( ϕ ′ ) ∗ , (e 22.31)where ı F : F → eB e and ˜ ı : F ⊗ E c U E c → eB e ⊗ E c U E c is the unital embedding.Choose a unitary y i ∈ ( ı F ⊗ J ◦ h )( e j ) B ( ı F ⊗ J ◦ h )( e j ) such that¯ y j = ˜ λ ( w j ) , j = 1 , , ..., l, where we recall that w j = e j ⊗ z j ⊕ (1 − e j ) ∈ F ⊗ C ( T ) = ⊕ j M n j ⊗ C ( T ) is one of the chosengenerator of K ( M n j ⊗ C ( T )). Let 1 j be the unit of M n j ⊂ F ; then 1 j = e j ⊕ e j ⊕ · · · ⊕ e j | {z } n j .Define ˜ y j = diag( n j z }| { y j , y j , ..., y j ) ∈ ( ı F ⊗ J ◦ h )(1 j ) B ( ı F ⊗ J ◦ h )(1 j ) , j = 1 , , ..., l. Then ˜ y j commutes with ( ı F ⊗ J )( F ).Define ˜ ϕ : F ⊗ C ( T ) → (cid:0) ( ı F ⊗ J ) ◦ ϕ ′′ (cid:1) (1 C ) B (cid:0) ( ı F ⊗ J ) ◦ ϕ ′′ (cid:1) (1 C ) by˜ ϕ ( c j ⊗ f ) = (cid:0) ( ı F ⊗ J ) ◦ ϕ ′′ (cid:1) ( c j ) f (˜ y j ) for all c j ∈ M n j and f ∈ C ( T ) . Define ϕ = ˜ ϕ ◦ ( π e ⊗ id C ( T ) ) . Then, by identifying K ( C ⊗ C ( T )) with K ( C ) , one has( ϕ ) ∗ = ˜ ı ∗ ◦ ( ϕ ′ ) ∗ and ( ϕ ) ‡ = ˜ λ. (e 22.32)14efine ϕ = ϕ ⊕ ϕ ⊕ ı ◦ L ◦ ψ. By (e 22.26) and (e 22.29), | τ ◦ ϕ ( h ) − γ ( τ )( h ) | < σ / σ / σ / h ∈ H . It is ready to verify that ϕ ∗ = α | K ( C ⊗ C ( T )) and ϕ ‡ = λ. Thus, since λ is compatible with α,ϕ ∗ = α | K ( C ⊗ C ( T )) . (e 22.33)Since K ∗ i ( C ⊗ C ( T )) ∼ = K ( C ) is free and finitely generated, one concludes that[ ϕ ] = α. Corollary 22.12.
Let C ∈ C and C = C ⊗ C ( T ) . Suppose that A is a unital separable simple C ∗ -algebra in B , B = A ⊗ U for some UHF-algebra U of infinite type, α ∈ KK e ( C , B ) ++ ,λ : J c ( K ( C )) → U ( B ) /CU ( B ) is a homomorphism, and γ : T ( B ) → T f ( C )) is a continuousaffine map. Suppose that ( α, λ, γ ) is a compatible triple. Then there is a unital homomorphism ϕ : C → B such that [ ϕ ] = α, ϕ ‡ | J c ( K ( C )) = λ and ϕ T = γ. In particular, ϕ is a monomorphism.Proof. The proof is exactly the same as the argument employed in 22.5 using 22.11.
Corollary 22.13.
Let C ∈ C and let C = C or C = C ⊗ C ( T ) . Suppose that A is aunital separable simple C ∗ -algebra in B , B = A ⊗ U for some UHF-algebra of infinite type, α ∈ KK e ( C , B ) ++ , and γ : T ( B ) → T f ( C ) is a continuous affine map. Suppose that ( α, γ ) iscompatible. Then there is a unital homomorphism ϕ : C → B such that [ ϕ ] = α and ϕ T = γ. In particular, ϕ is a monomorphism.Proof. To apply 22.12, one needs a map λ. Note that J c ( K ( C )) is isomorphic to K ( C ) which isfinitely generated. Let J (1) c : K ( B ) → U ( B ) /CU ( B ) be the splitting map defined in Definition2.16 of [21], Define λ = J (1) c ◦ α | K ( C ) ◦ π | J c ( K ( C )) , where π : U ( M ( C )) /CU ( M ( C )) → K ( C ) is the quotient map (note that C has stable rank one and C = C ⊗ C ( T ) has stablerank two). Then ( α, λ, γ ) is compatible. The corollary then follows from the previous one. Lemma 22.14.
Let B ∈ B which satisfies the UCT, let A ∈ B , let C = B ⊗ U , and let A = A ⊗ U , where U and U are UHF-algebras of infinite type. Suppose that κ ∈ KL e ( C, A ) ++ ,γ : T ( A ) → T ( C ) is a continuous affine map and α : U ( C ) /CU ( C ) → U ( A ) /CU ( A ) is acontinuous homomorphism for which γ, α, and κ are compatible. Then there exists a unitalmonomorphism ϕ : C → A such that (1) [ ϕ ] = κ in KL e ( C, A ) ++ , (2) ϕ T = γ and ϕ ‡ = α. Proof.
The proof follows the same lines as that of Lemma 8.5 of [40]. By the classificationtheorem (Theorem 21.9 and Theorem 14.10 of [21]), one can write C = lim −→ ( C n , ϕ n,n +1 )15here C n is a direct sum of C ∗ -algebras in C or in H . Let κ n = κ ◦ [ ϕ n, ∞ ] , α n = α ◦ ϕ ‡ n, ∞ , and γ n = ( ϕ n, ∞ ) T ◦ γ . Write C n = C n ⊕ C n with C n ∈ H and C n ∈ C . By Corollary 22.5 applyingto C n and Corollary 22.12 applying to to C n , there are unital monomorphisms ψ n : C n → A such that [ ψ n ] = κ n ψ ‡ n = α n , and ( ψ n ) T = γ n . (Note that K ( C n ) = 0, Consequently, ( ψ n | C n ) T = ( ı C n ,C n ) T ◦ γ n implies ( ψ n | C n ) ‡ = α n | U ( C n ) /CU ( C n ) .)In particular, the sequence of monomorphisms ψ n satisfies[ ψ n +1 ◦ ϕ n,n +1 ] = [ ψ n ] , ψ ‡ n +1 ◦ ϕ n,n +1 = ψ ‡ n , and ( ψ n +1 ◦ ϕ n,n +1 ) T = ( ψ n ) T . Let F n ⊂ C n be a finite subset such that ϕ n,n +1 ( F n ) ⊂ F n +1 and S ϕ n, ∞ ( F n ) is dense in C .Applying Theorem 12.7 of [21] with ∆( h ) = inf { γ ( τ )( ϕ n, ∞ ( h )) : τ ∈ T ( A ) } , h ∈ C + n \ { } , wehave a sequence of unitaries u n ∈ A such thatAd u n +1 ◦ ψ n +1 ◦ ϕ n,n +1 ≈ / n Ad u n ◦ ψ n on F n . The maps { Ad u n ◦ ψ n : n = 1 , , ... } then converge to a unital homomorphism ϕ : C → A which satisfies the lemma. Theorem 22.15.
Let X be a finite CW complex and let C = P M n ( C ( X )) P, where n ≥ isan integer and P ∈ M n ( C ( X )) is a projection. Let A ∈ B and let A = A ⊗ U for a UHF-algebra U of infinite type. Suppose α ∈ KL e ( C, A ) ++ , λ : U ∞ ( C ) /CU ∞ ( C ) → U ( A ) /CU ( A ) is a continuous homomorphism, and γ : T ( A ) → T f ( C ) is a continuous affine map such that ( α, λ, γ ) is compatible. Then there exists a unital homomorphism h : C → A such that [ h ] = α, h ‡ = λ and h T = γ. (e 22.34) Proof.
The proof is similar to that of 6.6 of [40]. To simplify the notation, without loss of general-ity, let us assume that X is connected. Furthermore, a standard argument shows that the generalcase can be reduced to the case C = C ( X ) . We may assume that U ( M N ( C )) /U ( M N ( C )) = K ( C ) for some integer N (see [48]). Therefore, in this case, U ( M N ( C )) /CU ( M N ( C )) = U ∞ ( C ) /CU ∞ ( C ) . Write K ( C ) = G ⊕ Tor( K ( C )) , where G is the torsion free part of K ( C ) . Fix a point ξ ∈ X and let C = C ( X \ { ξ } ) . Note that C is an ideal of C and C/C ∼ = C . Write K ( C ) = Z · [1 C ] ⊕ K ( C ) . (e 22.35)Let B ∈ B be a unital separable simple C ∗ -algebra as constructed in Corollary 14.14 of [21]such that ( K ( B ) , K ( B ) + , [1 B ] , T ( B ) , r B ) = ( K ( A ) , K ( A ) , [1 A ] , T ( A ) , r A ) (e 22.36)and K ( B ) = G ⊕ Tor( K ( A )) . Put∆(ˆ g ) = inf { γ ( τ )( g ) : τ ∈ T ( A ) } . (e 22.37)For each g ∈ C + \ { } , since γ ( τ ) ∈ T f ( C ) , γ is continuous and T ( A ) is compact, ∆(ˆ g ) > . Let ε > , F ⊂ C be a finite subset, 1 > σ , σ > , H ⊂ C s.a. be a finite subset, and U ⊂ U ( M N ( C )) /CU ( M N ( C )) be a finite subset. Without loss of generality, we may assume that U = U ∪U , where U ⊂ U ( M N ( C )) /CU ( M N ( C )) and U ⊂ J c ( K ( C )) ⊂ U ( M N ( C )) /CU ( M N ( C )) . u ∈ U , write u = Q n ( u ) j =1 exp( √− a i ( u )) , where a i ( u ) ∈ M N ( C ) s.a. . Write a i ( u ) = ( a ( k,j ) i ( u )) N × N , i = 1 , , ..., n ( u ) . (e 22.38)Write c i,k,j ( u ) = a ( k,j ) i ( u ) + ( a ( k,j ) i ) ∗ d i,k,j = a ( k,j ) i ( u ) − ( a ( k,j ) i ) ∗ i . (e 22.39)Put M = max {k c k , k c i,k,j ( u ) k , k d i,k,j ( u ) k : c ∈ H , u ∈ U } . (e 22.40)Choose a non-zero projection e ∈ B such that τ ( e ) < min { σ , σ } N ( M + 1) max { n ( u ) : u ∈ U } for all τ ∈ T ( B ) . Let B = (1 − e ) B (1 − e ) . In what follows we will use the identification (e 22.36). Define κ ∈ Hom( K ( C ) , K ( B ))as follows. Define κ ( m [1 C ]) = m [1 − e ] for m ∈ Z and κ | K ( C ) = α | K ( C ) . Note that K ( B ) = G ⊕ Tor( K ( A )) and that α induces a map α | Tor( K ( C )) : Tor( K ( C )) → Tor( K ( A )).Using the given decomposition K ( C ) = G ⊕ Tor( K ( C ), we can define κ : K ( C ) → K ( B )by κ | G = id and κ | Tor( K ( C )) = [ α ] | Tor( K ( C )) .By the Universal Coefficient Theorem, there is κ ∈ KL ( C, B ) which gives rise to the twohomomorphisms κ , κ above.. Note that κ ∈ KL e ( C, B ) ++ , since K ( C ) = ker ρ C ( K ( C )).Choose H = H ∪ { c i,k,j ( u ) , d i,k,j ( u ) : u ∈ U } . Every tracial state τ ′ of B has the form τ ′ ( b ) = τ ( b ) /τ (1 − e ) for all b ∈ B for some τ ∈ T ( B ). Let γ ′ : T ( B ) → T ( C ) be defined as follows. For τ ′ ∈ T ( B ) as above, define γ ′ ( τ ′ )( f ) = γ ( τ )( f ) for f ∈ C .It follows from 22.3 that there exists a sequence of unital completely positive linear maps h n : C → B such that lim n →∞ k h n ( ab ) − h n ( a ) h n ( b ) k = 0 for all a, b ∈ C, [ h n ] = κ ( − K ∗ ( C ) is finitely generated ) , andlim n →∞ max {| τ ◦ h n ( c ) − γ ′ ( τ )( c ) | : τ ∈ T ( B ) } = 0 for all c ∈ C. Here we may assume that [ h n ] is well defined for all n and | τ ◦ h n ( c ) − γ ( τ )( c ) | < min { σ , σ } N , n = 1 , , ... (e 22.41)for all c ∈ H and for all τ ∈ T ( B ) . Choose θ ∈ KL ( B, A ) such that it gives the identification of(e 22.36), and, θ | G = α | G and θ | Tor( K ( A )) = id Tor( K ( A )) . Let e ′ ∈ A be a projection such that[ e ′ ] ∈ K ( A ) corresponds to [ e ] ∈ K ( B ) under the identification (e 22.36). Let β = α − θ ◦ κ. Then β ([1 C ]) = [ e ′ ] , β K ( C ) = 0 , and β K ( C ) = 0 . (e 22.42)Then β ∈ KL e ( C, e ′ Ae ′ ) . It follows from 22.3 that there exists a sequence of unital completelypositive linear maps ϕ ,n : C → e ′ Ae ′ such thatlim n →∞ k ϕ ,n ( ab ) − ϕ ,n ( a ) ϕ ,n ( b ) k = 0 and [ ϕ ,n ] = β. (e 22.43)17ote that, for each u ∈ U ( M N ( C )) with ¯ u ∈ U ,D C ( u ) = n ( u ) X i =1 [ a j ( u ) , (e 22.44)where b c ( τ ) = τ ( c ) for all c ∈ C s.a. and τ ∈ T ( C ) . Since κ and λ are compatible, we compute, for¯ u ∈ U , dist(( h n ) ‡ (¯ u ) , λ (¯ u )) < σ / . (e 22.45)Fix a pair of large integers n, m, and define χ n,m : J c ( G )( ⊂ U ( C ) /CU ( C )) → Aff( T ( A )) /ρ A ( K ( A ))to be λ | J c ( G ) − ( h n ) ‡ | J c ( G ) − ϕ ‡ ,m | J c ( G ) . (e 22.46)We may may also view J c ( G ) as subgroup of J c ( K ( B ))= J c ( K ( B )) . Write J c ( K ( B )) = J c ( G ) ⊕ J c (Tor( K ( B ))) and define χ n,m to be zero on Tor( K ( B )), we obtain a homomor-phism χ n,m : J c ( K ( B )) → Aff( T ( A )) /ρ A ( K ( A )) . It follows from Lemma 22.14 that there isa unital homomorphism ψ : B → (1 − e ′ ) A (1 − e ′ ) such that[ ψ ] = θ, ψ T = id T ( A ) and (e 22.47) ψ ‡ | J c ( K ( B )) = χ n,m | J c ( K ( B )) + J c ◦ θ | K ( B ) , (e 22.48)where we identify K ( B ) with K ( B ) . By (e 22.47), ψ ‡ | Aff( T ( B )) /ρ B ( K ( B )) = id . (e 22.49)Define L ( c ) = ϕ ,m ( c ) ⊕ ψ ◦ h n ( c ) for all c ∈ C. It follows, on choosing sufficiently large m and n, that L is ε - F -multiplicative,[ L ] = α, (e 22.50)max {| τ ◦ ψ ( f ) − γ ( τ )( f ) | : τ ∈ T ( A ) } < σ for all f ∈ H , and (e 22.51)dist( L ‡ (¯ u ) , λ (¯ u )) < σ . (e 22.52)This implies that that there is a sequence of contractive completely positive linear maps ψ n : C → A such thatlim n →∞ k ψ n ( ab ) − ψ n ( a ) ψ n ( b ) k = 0 for all a, b ∈ C, (e 22.53)[ ψ n ] = α, (e 22.54)lim n →∞ max {| τ ◦ ψ n ( c ) − γ ( τ )( c ) | : τ ∈ T ( A ) } = 0 for all c ∈ C s.a. , and (e 22.55)lim n →∞ dist( ψ ‡ n (¯ u ) , λ (¯ u )) = 0 for all u ∈ U ( M N ( C )) /CU ( M N ( C )) . (e 22.56)Finally, applying Theorem 12.7 of [21], as in the proof of 22.5, using ∆ / h : C → A such that[ h ] = α, h T = γ, and h ‡ = λ, (e 22.57)as desired. 18 heorem 22.16. Let C ∈ C and let G = K ( C ) . Write G = Z k with Z k generated by { x = [ p ] − [ q ] , x = [ p ] − [ q ] , ..., x k = [ p k ] − [ q k ] } , where p i , q i ∈ M n ( C ) (for some integer n ≥ ) are projections, i = 1 , ..., k .Let A be a simple C*-algebra in B , and let B = A ⊗ U for a UHF algebra U of infinitetype. Suppose that ϕ : C → B is a monomorphism. Then, for any finite subsets F ⊂ C and P ⊂ K ( C ) , any ε > and σ > , and any homomorphism Γ : Z k → U ( B ) /CU ( B ) , there is a unitary w ∈ B such that (1) k [ ϕ ( f ) , w ] k < ε , for any f ∈ F , (2) Bott( ϕ, w ) | P = 0 , and (3) dist( h (( n − ϕ ( p i )) + ϕ ( p i ) ˜ w )(( n − ϕ ( q i )) + ϕ ( q i ) ˜ w ∗ ) i , Γ( x i ))) < σ , for any ≤ i ≤ k, where ˜ w = diag( n z }| { w, ..., w ) .Proof. Write B = lim n →∞ ( A ⊗ M r n , ı n,n +1 ). Using the fact that C is semiprojective (see [9]),one can construct a sequence of homomorphisms ϕ n : C → A ⊗ M r n such that ı n ◦ ϕ n ( c ) → ϕ ( c )for all c ∈ C . Without loss of generality, we may assume ϕ = ı ◦ ϕ for a homomorphism ϕ : C → A (replacing A by A ⊗ M r n ), where ı : A → A ⊗ U = B is the standard inclusion.We may assume that || f || ≤ f ∈ F .For any non-zero positive element h ∈ C with norm at most 1, define∆( h ) = inf { τ ( ϕ ( h )); τ ∈ T ( B ) } . Since B is simple, one has that ∆( h ) ∈ (0 , H ⊂ C + \ { } , G ⊂ C , δ > P ⊂ K ( C ), H ⊂ C s.a. , and γ > C, F , ǫ/ , and ∆ / K ( C ) = { } ,one does not need U and γ ).Note that B = A ⊗ U. Pick a unitary z ∈ U with sp( u ) = T and consider the map ϕ ′ : C ⊗ C ( T ) → B = A ⊗ U defined by a ⊗ f ϕ ( a ) ⊗ f ( z ) . (Recall that ϕ ( a ) = ϕ ( a ) ⊗ U .) Set γ = ( ϕ ′ ) T : T ( B ) → T f ( C ⊗ C ( T )) . Also define α := [ ϕ ′ ] ∈ KK ( C ⊗ C ( T ) , B ) . Note that K ( C ⊗ C ( T )) = K ( C ) = Z k . Identifying J c ( K ( C ⊗ C ( T ))) with Z k , define amap λ : J c ( K ( U ( C ⊗ C ( T )))) → U ( B ) /CU ( B ) by λ ( a ) = Γ( a ) for any a ∈ Z k .Set U = { (1 n − p i + p i ˜ z ′ )(1 n − q i + q i ˜ z ′∗ ) : i = 1 , ..., k } ⊂ J c ( U ( C ⊗ C ( T ))) , where z ′ is the standard generator of C ( T ), and set δ = min { ∆( h ) / h ∈ H } . F - ǫ/ C ⊗ C ( T ) → B such that[Φ] = α, dist(Φ ‡ ( x ) , λ ( x )) < σ for all x ∈ U , and (e 22.58) | τ ◦ Φ( h ⊗ − γ ( τ )( h ⊗ | < min { γ , δ } for all h ∈ H ∪ H . (e 22.59)Let ψ denote the restriction of Φ to C ⊗
1. Then one has[ ψ ] | P = [ ϕ ] | P . By (e 22.59), one has that, for any h ∈ H , τ ( ψ ( h )) > γ ( τ )( h ) − δ = τ ( ϕ ′ ( h ⊗ − δ = τ ( ϕ ( h )) − δ > ∆( h ) / , and it is also clear that τ ( ϕ ( h )) > ∆( h ) / h ∈ H . Moreover, for any h ∈ H , one has | τ ◦ ψ ( h ) − τ ◦ ϕ ( h ) | = | τ ◦ Φ( h ⊗ − τ ◦ ϕ ′ ( h ⊗ | = | τ ◦ Φ( h ⊗ − γ ( τ )( h ⊗ | < γ . Therefore, by Theorem 12.7 of [21], there is a unitary W ∈ B such that || W ∗ ψ ( f ) W − ϕ ( f ) || < ǫ/ f ∈ F . Then the element w = W ∗ Φ(1 ⊗ z ′ ) W is the desired unitary. Theorem 22.17.
Let C be a unital C ∗ -algebra which is a finite direct sum of C ∗ -algebras in C and C ∗ -algebras of the form P M n ( C ( X )) P, where X is a finite CW complex and P is aprojection, and let G = K ( C ) . Write G = Z k L Tor( G ) with a basis for Z k the set { x = [ p ] − [ q ] , x = [ p ] − [ q ] , ..., x k = [ p k ] − [ q k ] } , where p i , q i ∈ M n ( C ) (for some integer n ≥ ) are projections, i = 1 , ..., k .Let A be a simple C*-algebra in the class B , and let B = A ⊗ U for a UHF algebra U ofinfinite type. Suppose that ϕ : C → B is a monomorphism. Then, for any finite subsets F ⊂ C and P ⊂ K ( C ) , any ε > and σ > , and any homomorphism Γ : Z k → U ( M n ( B )) /CU ( M n ( B )) , there is a unitary w ∈ B such that (1) k [ ϕ ( f ) , w ] k < ε , for any f ∈ F , (2) Bott( ϕ, w ) | P = 0 , and (3) dist( h (( n − ϕ ( p i )) + ϕ ( p i ) ˜ w )(( n − ϕ ( q i )) + ϕ ( q i ) ˜ w ∗ ) i , Γ( x i ))) < σ , for any ≤ i ≤ k, where ˜ w = diag( n z }| { w, ..., w ) .Proof. By Theorem 22.16, it suffices to prove the case that C = P M n ( C ( X )) P, where X is afinite CW complex, n ≥ P ∈ M n ( C ( X )) is a projection. The proof followsthe same lines as that of Theorem 22.16 but using Lemma 22.15 instead of Lemma 22.11.20 Lemma 23.1.
Let C ∈ C . There exists a constant M C > satisfying the following condition:For any ε > , any x ∈ K ( C ) , and any n ≥ M C /ε, if | ρ C ( x )( τ ) | < ε for all τ ∈ T ( C ⊗ M n ) , (e 23.1) then there are mutually inequivalent and mutually orthogonal minimal projections p , p , ..., p k and q , q , ..., q k in C ⊗ M n and positive integers l , l , ..., l k , m , m , ..., m k such that x = [ k X i =1 l i p i ] − [ k X j =1 m j q j ] and (e 23.2) τ ( k X i =1 l i p i ) < ε and τ ( k X j =1 m j q j ) < ε (e 23.3) for all τ ∈ T ( C ⊗ M n ) . Proof.
Let C = C ( F , F , ϕ , ϕ ) and F = L li =1 M r ( i ) . By Theorem 3.15 of [21], there is aninteger N ( C ) > C ⊗ K is equivalent to a finite direct sum ofprojections from a set of finitely many mutually inequivalent minimal projections (some of themmay repeat in the direct sum) in M N ( C ) ( C ) . Choosing a larger N ( C ), we may assume this setof mutually inequivalent projections is sitting in M N ( C ) ( C ), orthogonally. We also assume that,as in Definition 3.1 of [21], C is minimal. Let M C = N ( C ) + 2( r (1) · r (2) · · · r ( l ))Suppose that n ≥ M C /ε. With the canonical embedding of K ( C ) into K ( F ) ∼ = Z l , write x = x x ... x l ∈ Z l . (e 23.4)By (e 23.1), for any irreducible representation π of C and any tracial state t on M n ( π ( C )) , | t ◦ π ( x ) | < ε. (e 23.5)It follows that | x s | /r ( s ) n < ε, s = 1 , , ..., l. (e 23.6)Let T = max {| x s | /r ( s ) : 1 ≤ s ≤ l } . (e 23.7)Define y = x + T r (1) r (2)... r ( l ) and z = T r (1) r (2)... r ( l ) . (e 23.8)21t is clear that z ∈ K ( C ) + (see Proposition 3.5 of [21]). It follows that y ∈ K ( C ) . One alsocomputes that y ∈ K ( C ) + . It follows that there are projections p, q ∈ M L ( C ) for some integer L ≥ p ] = y and [ q ] = z. Moreover, x = [ p ] − [ q ] . One also computes that τ ( q ) < T /n < ε for all τ ∈ T ( C ⊗ M n ) . (e 23.9)One also has τ ( p ) < ε for all τ ∈ T ( C ⊗ M n ) . (e 23.10)There are two sets of mutually inequivalent and mutually orthogonal minimal projections { p , p , ..., p k } and { q , q , ..., q k } in C ⊗ M n (since n > N ( C )) such that[ p ] = k X i =1 l i [ p i ] and [ q ] = k X j =1 m j [ q j ] . (e 23.11)Therefore x = k X i =1 l i [ p i ] − k X j =1 m j [ q j ] . (e 23.12) Lemma 23.2.
Let C ∈ C . There is an integer M C > satisfying the following condition: Forany ε > and for any x ∈ K ( C ) with | τ ( ρ C ( x )) | < ε/ π for all τ ∈ T ( C ⊗ M n ) , where n ≥ M C π/ε, there exists a pair of unitaries u and v ∈ C ⊗ M n such that k uv − vu k < ε and τ (bott ( u, v )) = τ ( x ) . (e 23.13) Proof.
We may assume that C is minimal. Applying Lemma 23.1, we obtain mutually orthogonaland mutually inequivalent minimal projections p , p , ..., p k , q , q , ..., q k ∈ C ⊗ M n such that k X i =1 l i [ p i ] − k X j =1 m j [ q j ] = x, where l , l , ..., l k , m , m , ..., m k are positive integers. Moreover, k X i =1 l i τ ( p i ) < ε/ π and k X j =1 m j τ ( q j ) < ε/ π (e 23.14)for all τ ∈ T ( C ⊗ M n ) . Choose N ≤ n such that N = [2 π/ε ] + 1 . By (e 23.14), k X i =1 N l i τ ( p i ) + k X j =1 N m j τ ( q j ) < / τ ∈ T ( C ⊗ M n ) . (e 23.15)It follows that there are mutually orthogonal projections d i,k , d ′ j,k ∈ C ⊗ M n , k = 1 , , ..., N,i = 1 , , ..., k , and j = 1 , , ..., k such that[ d i,k ] = l i [ p i ] and [ d ′ j,k ] = m j [ q j ] , i = 1 , , ..., k , j = 1 , , ..., k (e 23.16)22nd k = 1 , , ..., N. Let D i = P Nk =1 d i,k and D ′ j = P Nk =1 d ′ j,k , i = 1 , , ..., k and j = 1 , , ..., k . There are partial isometries s i,k , s ′ j,k ∈ C ⊗ M n such that s ∗ i,k d i,k s i,k = d i,k +1 , ( s ′ j,k ) ∗ d ′ j,k s ′ j,k = d ′ j,k +1 , k = 1 , , ..., N − , (e 23.17) s ∗ i,N d i,N s i,N = d i, , and ( s ′ j,N ) ∗ d ′ j,N s ′ j,N = d ′ j, , (e 23.18) i = 1 , , ..., k and j = 1 , , ..., k . Thus, we obtain unitaries u i ∈ D i ( C ⊗ M n ) D i and u ′ j = D ′ j ( C ⊗ M n ) D ′ j such that u ∗ i d i,k u i = d i,k +1 , u ∗ i d i,N u i = d i, , ( u ′ j ) ∗ d ′ j,k u ′ j = d ′ j,k +1 , and ( u ′ j ) ∗ d ′ j,N u ′ j = d ′ j, , (e 23.19) i = 1 , , ..., k , j = 1 , , ..., k . Define v i = N X k =1 e √− kπ/N ) d i,k and v ′ j = N X k =1 e √− kπ/N ) d ′ j,k . We compute that k u i v i − v i u i k < ε and k u ′ j v ′ j − v ′ j u ′ j k < ε, (e 23.20)12 π √− τ (log v i u i v ∗ i u ∗ i ) = l i τ ( p i ) , and (e 23.21)12 π √− τ (log v ′ j u ′ j ( v ′ j ) ∗ ( u ′ j ) ∗ ) = m j τ ( q j ) , (e 23.22)for τ ∈ T ( C ⊗ M n ) , i = 1 , , ..., k and j = 1 , , ..., k . Now define u = k X i =1 u i + k X j =1 u ′ j + (1 C ⊗ M n − k X i =1 D i − k X j =1 D ′ j ) and (e 23.23) v = k X i =1 v i + k X j =1 ( v ′ j ) ∗ + (1 C ⊗ M n − k X i =1 D i − k X j =1 D ′ j ) . (e 23.24)We then compute that τ (bott ( u, v )) = k X i =1 π √− τ (log( v i u i v ∗ i u ∗ i )) − k X j =1 π √− τ (log v ′ j u ′ j ( v ′ j ) ∗ ( u ′ j ) ∗ ) (e 23.25)= k X i =1 l i τ ( p i ) − k X j =1 m j τ ( q j ) = τ ( x ) (e 23.26)for all τ ∈ T ( C ⊗ M n ) . Lemma 23.3.
Let ε > . There exists σ > satisfying the following condition: Let A = A ⊗ U, where U is a UHF-algebra of infinite type and A ∈ B , let u ∈ U ( A ) be a unitary with sp ( u ) = T , and let x ∈ K ( A ) with | τ ( ρ A ( x )) | < σ for all τ ∈ T ( A ) and y ∈ K ( A ) . Then there exists aunitary v ∈ U ( A ) such that k uv − vu k < ε, bott ( u, v ) = x, and [ v ] = y. (e 23.27)23 roof. Let ϕ : C ( T ) → A be the unital monomorphism defined by ϕ ( f ) = f ( u ) for all f ∈ C ( T ) . Let ∆ : C ( T ) q, + \ { } → (0 ,
1) be defined by ∆ ( ˆ f ) = inf { τ ( ϕ ( f )) : τ ∈ T ( A ) } . Let ε > < ε < ε such thatbott ( z , z ) = bott ( z ′ , z ′ )if k z − z ′ k < ε and k z − z ′ k < ε for any two pairs of unitaries z , z and z ′ , z ′ which alsohave the property that k z z − z z k < ε and k z ′ z ′ − z ′ z ′ k < ε . Let H ⊂ C ( T ) + \ { } be a finite subset, γ > , γ > , and H ⊂ C ( T ) s.a. be a finite subsetas provided by Corollary 12.9 of [21] (for ε / / H ⊂ C ( T ) . Let δ = min { γ / , γ / , min { ∆ ( ˆ f ) : f ∈ H } / } . Let σ = min { δ / , ( δ / ε / π ) } . Let e ∈ ⊗ U ⊂ A be a non-zero projection such that τ ( e ) < σ for all τ ∈ T ( A ) . Let B = eAe (then B ∼ = A ⊗ U ′ for some UHF-algebra U ′ ). It follows from Corollary 18.10 of [21] that thereis a unital simple C ∗ -algebra C ′ = lim n →∞ ( C n , ψ n ) , where C n ∈ C and C = C ′ ⊗ U such that( K ( C ) , K ( C ) + , [1 C ] , T ( C ) , r C ) = ( ρ A ( K ( A )) , ( ρ A ( K ( A ))) + , [ e ] , T ( A ) , r A ) . Moreover, we may assume that all ψ n are unital.Now suppose that x ∈ K ( A ) with | τ ( ρ A ( x )) | < σ for all τ ∈ T ( A ) and suppose that y ∈ K ( A ) . Let z = ρ A ( x ) ∈ K ( C ) . We identify z with the element in K ( C ) in the identificationabove. We claim that, there is n ≥ x ′ ∈ K ( C n ⊗ U ) such that z =( ψ n , ∞ ) ∗ ( x ′ ) ∈ K ( C ) and | t ( ρ C n ⊗ U )( x ′ ) | < σ for all t ∈ T ( C n ⊗ U ) . Otherwise, there is an increasing sequence n k , x k ∈ K ( C n k ⊗ U ) such that( ψ n k , ∞ ) ∗ ( x k ) = z ∈ K ( C ) and | t k ( ρ C nk ⊗ U )( x k ) | ≥ σ (e 23.28)for some t k ∈ T ( C n k ⊗ U ) , k = 1 , , .... Let L k : C → C n k ⊗ U be such thatlim n →∞ k ψ n, ∞ ◦ L n ( c ) − c k = 0for all c ∈ ψ k, ∞ ( C n k ⊗ U ) , k = 1 , , .... It follows that any limit point of t k ◦ L k is a tracial stateof C. Let t be one such limit. Then, by (e 23.28), t ( ρ C ( z )) ≥ σ. This proves the claim.Write U = lim n →∞ ( M r ( m ) , ı m ) , where ı m : M r ( m ) → M r ( m +1) is a unital embedding. Re-peating the argument above, we obtain m ≥ y ′ ∈ K ( C n ⊗ M r ( m ) ) = K ( C n ) such that( ı m , ∞ ) ∗ ( y ′ ) = x ′ and | t ( ρ C n ( y ′ )) | < σ for all t ∈ T ( C n ⊗ M r ( m ) ) . Let M C N be the constantgiven by Lemma 23.2. Choose r ( m ) ≥ max { M C n /σ, r ( m ) } and let y ′′ = ( ı m ,m ) ∗ ( y ′ ) . Then, we compute that | t ( ρ C n ( y ′′ )) | < σ for all t ∈ T ( C n ⊗ M r ( m ) ) . It follows from 23.2 that there exists a pair of unitaries u ′ , v ′ ∈ C n ⊗ M r ( m ) such that k u ′ v ′ − v ′ u ′ k < ε / ( u ′ , v ′ ) = y ′′ . (e 23.29)Put u = ı m , ∞ ( u ′ ) and v = ı m , ∞ ( v ′ ) . Then (e 23.29) implies that k u v − v u k < ε / ( u , v ) = x ′ . (e 23.30)24et h : C n ⊗ U → eAe be a unital homomorphism as given by Corollary 18.10 of [21] suchthat ρ A ◦ ( h ) ∗ = ( ψ n , ∞ ) ∗ . (e 23.31)It follows that ρ A (( h ) ∗ ( x ′ ) − x ) = 0 . (e 23.32)Let u = h ( u ) and v = h ( v ) . We have ρ A (bott ( u , v ) − x ) = 0 . (e 23.33)Choose another non-zero projection e ∈ A such that e e = ee = 0 and τ ( e ) < δ /
16 for all τ ∈ T ( A ) . It follows from 22.1 that there is a unital homomorphism H : C ( T ) → e Ae suchthat H ∗ ( b ) = x − bott ( u , v ) , (e 23.34)where b is the Bott element in K ( C ( T )). (In fact, we can also apply 22.15 here.) Thus weobtain a pair of unitaries u , v ∈ e Ae such that u v = v u and bott ( u , v ) = x − bott ( u , v ) . (e 23.35)Let e , e ∈ (1 − e − e ) A (1 − e − e ) be a pair of non-zero mutually orthogonal projections suchthat τ ( e ) < δ /
32 and τ ( e ) < δ /
32 for all τ ∈ T ( A ) . Thus τ ( e + e + e + e ) < δ /
16 for all τ ∈ T ( A ) . Then, together with Theorem 17.3 of [21], (applied to X = T ), we obtain a unitary u ∈ (1 − e − e − e − e ) A (1 − e − e − e − e ) such that | τ ( f ( u )) − τ ( f ( u )) k < δ / f ∈ H ∪ H and for all τ ∈ T ( A ) . (e 23.36)Let w = u + u + u + (1 − e − e − e − e ) . It follows from Theorem 3.10 of [22] that thereexists u ∈ U ( e Ae ) such that u = ¯ u ¯ w ∗ ∈ U ( A ) /CU ( A ) . (e 23.37)Since A is simple and has stable rank one, there exists a unitary v ∈ e Ae such that [ v ] = y − [ v + v + ( e + e )] ∈ K ( A ) . Now define u = u + u + u + u + e and v = v + v + (1 − e − e − e − e ) + e + v . Then k u v − v u k < ε / , bott ( u , v ) = x, and [ v ] = y. (e 23.38)Moreover, τ ( f ( u )) ≥ ∆( ˆ f ) / f ∈ H , (e 23.39) | τ ( f ( u )) − τ ( f ( u )) | < γ and ¯ u = ¯ u. (e 23.40)It follows from Corollary 12.7 of [21] that there exists a unitary W ∈ A such that k W ∗ u W − u k < ε / . (e 23.41)Now let v = W ∗ v W. We compute that k uv − vu k < ε, bott ( u, v ) = bott ( u , v ) = x, and [ v ] = y. (e 23.42)25 orollary 23.4. Let ε > , C = L ki =1 C i = L ki =1 M m ( i ) ( C ( T )) . Let P ⊂ K ( C ) and P ⊂ K ( C ) be finite sets generating K ( C ) and K ( C ) . There exists σ > satisfying the followingcondition: Let A = A ⊗ U be as in Lemma 23.3, let ι : C → A be an embedding, and let α ∈ KL ( A ⊗ C ( T ) , A ) be such that | τ ( ρ A ( α ( β ( w )))) | < σ min { τ ′ ( ι (1 C i )) /m ( i ) , ≤ i ≤ k, τ ′ ∈ T ( A ) } , for all w ∈ P and τ ∈ T ( A ) . Then there exists a unitary v ∈ ι (1 C ) Aι (1 C ) such that Bott( ι, v ) | P ∪P = α ◦ β | P ∪P . Proof.
Let e i ∈ C i = M m ( i ) ( C ( T )) be the rank one projection of the upper left corner of C i and u i ∈ K ( C i ) be the standard generator given by ze i + (1 C i − e i ), where z ∈ C ( T )is the identity function from T to T ⊂ C . Without loss of generality, we may assume that P = { [ e i ] , ≤ i ≤ k } and P = { u i , ≤ i ≤ k } . Let σ be as in Lemma 23.3. For each i ∈ { , , · · · , k } , applying Lemma 23.3 to ι ( e i ) Aι ( e i ) (in place of A ), ι ( ze i ) ∈ ι ( e i ) Aι ( e i )(in place of u ) with x = α ( β ( u i )), y = α ( β ([ e i ])), one obtains a unitary v i ∈ ι ( e i ) Aι ( e i ) inplace of v . Identifying ι (1 C i ) Aι (1 C i ) ∼ = (cid:0) ι ( e i ) Aι ( e i ) (cid:1) ⊗ M m ( i ) ( C ), we define v i = v i ⊗ m ( i ) .Finally, choose v = v ⊕ v ⊕ · · · ⊕ v k ∈ ι (1 C ) Aι (1 C ) to finish the proof.
24 More existence theorems for Bott elements
Using Lemma 23.3, 22.1, Corollary 21.11 of [21], Lemma 18.11 of [21], and Theorem 12.11 of[21], we can show the following result:
Lemma 24.1.
Let A = A ⊗ U , where A is as in Theorem 14.10 of [21] and B = B ⊗ U , where B ∈ B and U , U are two UHF-algebras of infinite type. For any ε > , any finite subset F ⊂ A, and any finite subset P ⊂ K ( A ) , there exist δ > and a finite subset Q ⊂ K ( A ) satisfyingthe following condition: Let a unital homomorphism ϕ : A → B and α ∈ KL ( A ⊗ C ( T ) , B ) besuch that | τ ◦ ρ B ( α ( β ( x ))) | < δ for all x ∈ Q and for all τ ∈ T ( B ) . (e 24.1) Then there exists a unitary u ∈ B such that k [ ϕ ( x ) , u ] k < ε for all x ∈ F and (e 24.2)Bott( ϕ, u ) | P = α ( β ) | P . (e 24.3) Proof.
Let ε > F ⊂ A be a finite subset satisfying the following condition: If L, L ′ : A ⊗ C ( T ) → B are two unital F ′ - ε -multiplicative completely positive linear maps such that k L ( f ) − L ′ ( f ) k < ε for all f ∈ F ′ , (e 24.4)where F ′ = { a ⊗ g : a ∈ F and g ∈ { z, z ∗ , C ( T ) }} , then [ L ] | β ( P ) = [ L ′ ] | β ( P ) . (e 24.5)26et B ,n = M m (1 ,n ) ( C ( T )) ⊕ M m (2 ,n ) ( C ( T )) ⊕· · ·⊕ M m ( k (1) ,n ) ( C ( T )) , B ,n = P M r ( n ) ( C ( X n )) P, where X n is a finite disjoint union of copies of S , T ,k , and T ,k (for various k ≥ B ,n be a finite direct sum of C ∗ -algebras in C (with trivial K and ker ρ B ,n = { } —see Proposi-tion 3.5 of [21]), n = 1 , , .... Put C n = B ,n ⊕ B ,n ⊕ B ,n , n = 1 , , .... We may write that A = lim n →∞ ( C n , ı n ) as in Theorem 14.10 of [21]. with the maps ı n injective (applying Theorem14.10 of [21] to A ), ker ρ A ⊂ ( ı n, ∞ ) ∗ (ker ρ C n ) , and (e 24.6)lim n →∞ sup { τ (1 B ,n ⊕ B ,n ) : τ ∈ T ( B ) } = 0 . (e 24.7)Let ε = min { ε / , ε/ } and let F = F ∪ F . Let P , ⊂ K ( B ,n ) , P , ⊂ K ( B ,n ) and P , ⊂ K ( B ,n ) be finite subsets such that P ⊂ [ ı n , ∞ ]( P , ) ∪ [ ı n , ∞ ]( P , ) ∪ [ ı n , ∞ ])( P , )for some n ≥ . Let Q ′ be a finite set of generators of K ( C n ) and let Q = [ ı n , ∞ ]( Q ′ ) . Since K i ( B ,n ) , i = 0 , , are finitely generated free abelian groups, without loss of generality, we mayassume that P , ⊂ K ( B ,n ) ∪ K ( B ,n ) and generates K ( B ,n ) ⊕ K ( B ,n ).Without loss of generality, we may assume that F ∪ F ⊂ ı n , ∞ ( C n ) . Let F , ⊂ B ,n , F , ⊂ B ,n , and F , ⊂ B ,n be finite subsets such that F ∪ F ⊂ ı n , ∞ ( F , ∪ F , ∪ F , ) . (e 24.8)Let e = ı n , ∞ (1 B ,n ) , e = ı n , ∞ (1 B ,n ) , and e = 1 − e − e . Note that B ,n = ⊕ s ( n ) i =1 B i ,n ,where s ( n ) is an integer depending on n and B i ,n = M m ( i,n ) ( C ( T )). We may write e = ⊕ s ( n ) i =1 e i with e i = ı n , ∞ (1 iB ,n ). Let ∆ : ( B ,n ) q, + \ { } → (0 ,
1) be defined by∆ (ˆ h ) = (1 /
2) inf { τ ( ϕ ( ı n , ∞ ( h )) : τ ∈ T ( B ) } for all h ∈ ( B ,n ) + \ { } . Let ∆ : B q, ,n \ { } → (0 ,
1) be defined by∆ (ˆ h ) = (1 /
2) inf { τ ( ϕ ( ı n , ∞ ( h )) : τ ∈ T ( B ) } for all h ∈ ( B ,n ) + \ { } . Note that B ,n has the form C of Theorem 12.7 of [21]. So we will apply Theorem 12.7 of [21].Let H , ⊂ ( B ,n ) + \ { } (in place of H ), γ , > γ ), δ , > δ ), G , ⊂ B ,n (in place of G ), P , ⊂ K ( B ,n ) (in place of P ), and H , ⊂ ( B ,n ) s.a. (in place of H ) be the constants and finite subsets provided by Theorem 12.7 of [21] for ε / , F , , and∆ (we do not need the set U in Theorem 12.7 of [21] since K ( B ,n ) is torsion or zero; seeCorollary 12.8 of [21]).Recall that B ,n = L s ( n ) i =1 B i ,n with B i ,n = M m ( i,n ) ( C ( T )), and e = L s ( n ) i =1 e i with e i = ı n , ∞ (1 iB ,n ). Now let σ > P , and ε / ε ). Let δ = σ · inf { τ ( e i ) /m ( i, n ) : 1 ≤ i ≤ s ( n ) , τ ∈ T ( A ) } . It follows from23.4 that if | τ ◦ ρ B ( α ( β ( x ))) | < δ for all x ∈ [ ı n , ∞ ])( P , ) then there is a unitary v ∈ e Be such that Bott( ϕ ◦ ı n , ∞ , v ) | P , = α ◦ β ◦ [ ı n , ∞ ] | ( P , ) . (e 24.9)Note that K ( B ,n ) is a finite group. Therefore, α ( β ([ ı n , ∞ ])( K ( B ,n )) ⊂ ker ρ B . (e 24.10)27efine κ ∈ KK ( B ,n ⊗ C ( T ) , A ) by κ | K ( B ,n ) = [ ϕ ◦ ı n , ∞ | B ,n ] and κ | β ( K ( B ,n ) ) = α | β ( K ( B ,n )) . Since ı n , ∞ is injective, by (e 24.10), κ ∈ KK e ( B ,n ⊗ C ( T ) , e Be ) ++ . Let σ = min { γ , / , min { ∆ (ˆ h ) : h ∈ H , } · inf { τ ( e ) : τ ∈ T ( A ) } . Define γ : T ( e Ae ) → T f ( B ,n ⊗ C ( T )) by γ ( τ )( f ⊗ C ( T ) ) = τ ◦ ϕ ◦ ı n , ∞ ( f ) for all f ∈ B ,n and γ (1 ⊗ g ) = R T g ( t ) dt for all g ∈ C ( T ) . It follows from 22.15, applied to the space X n × T ,that there is a unital monomorphism Φ : B ,n ⊗ C ( T ) → e Ae such that [Φ] = κ and Φ T = γ . Put L = Φ | B ,n (identifying B ,n with B ,n ⊗ C ( T ) ) and v ′ = Φ(1 ⊗ z ) , where z ∈ C ( T ) is theidentity function on the unit circle. Then L is a unital monomorphism from B ,n to e Ae . We also have the following facts:[ L ] = [ ϕ ◦ ı n , ∞ ] , k [ L ( f ) , v ′ ] k = 0 , (e 24.11)Bott( L , v ′ ) | P , = α ( β ([ ı n , ∞ ])) | P , , and (e 24.12) | τ ◦ L ( f ) − τ ◦ ϕ ◦ ı n , ∞ ( f ) | = 0 for all f ∈ H , ∪ H , (e 24.13)and for all τ ∈ T ( e Ae ) . It follows from (e 24.13) that τ ( L ( f )) ≥ ∆ ( ˆ f ) · τ ( e ) for all f ∈ H , and τ ∈ T ( A ) . (e 24.14)By Theorem 12.7 of [21] (see also Corollary 12.8 of [21]), there exists a unitary w ∈ e Ae suchthat k Ad w ◦ L ( f ) − ϕ ◦ ı n , ∞ ( f ) k < ε /
16 for all f ∈ F , . (e 24.15)Define v = w ∗ v ′ w. Then, for all f ∈ F , , k [ ϕ ◦ ı n , ∞ ( f ) , v ] k < ε / ϕ ◦ ı n , ∞ , v ) | P , = α ( β ([ ı n , ∞ ])) | P , . (e 24.16)Note that B ,n has the form C of Theorem 12.7 of [21]. Let H , ⊂ ( B ,n ) + \ { } (inplace of H ), γ , > γ ), δ , > δ ), G , ⊂ B ,N (in place of G ), P , ⊂ K ( B ,n ) (in place of P ), and H , ⊂ ( B ,n ) s.a. (note that K ( B ,n ) = { } ) be constantsand finite subsets as provided by Theorem 12.7 of [21] for ε / , F , , and ∆ (see also Corollary12.8 of [21]).Let σ = ( γ , /
2) min { τ ( e ) : τ ∈ T ( A ) } · min { ∆ ( ˆ f ) : f ∈ H , } . Note that ker ρ B ,n = { } and K ( B ,n ) = { } (see Proposition 3.5 of [21]) Therefore,ker ρ B ,n ⊗ C ( T ) = ker ρ B ,n = { } . Define κ ∈ KK ( B ,n ⊗ C ( T )) as follows: κ | K ( B ,n ) = [ ϕ ◦ ı n , ∞ ] | B ,n and κ | β ( K ( B ,n ) = α ( β ( ı n , ∞ ) | K ( B ,n ) . Thus κ ∈ KK e ( B ,n ⊗ C ( T ) , e Ae ) ++ . It follows from 22.7 that there is a unital G , -min { ε / , δ , / } -multiplicative completely positive linear map L : B ,n → e Ae and aunitary v ′ ∈ e Ae such that[ L ] = [ ϕ ] , k [ L ( f ) , v ′ ] k < ε /
16 for all f ∈ G , , (e 24.17)Bott( L , v ′ ) | P , = κ | β ( P , ) , and (e 24.18) | τ ◦ L ( f ) − τ ◦ ϕ ◦ ı n , ∞ ( f ) | < σ for all f ∈ H , ∪ H , (e 24.19)and for all τ ∈ T ( e Ae ) . It follows that (e 24.19) that τ ( L ( f )) ≥ ∆ ( ˆ f ) τ ( e ) for all f ∈ H , and for all τ ∈ T ( A ) . (e 24.20)28t follows from Theorem 12.7 of [21], and its corollary (see part (2) of Corollary 12.8 of [21]),that there exists a unitary w ∈ e Ae such that k Ad w ◦ L ( f ) − ϕ ◦ ı n , ∞ ( f ) k < ε /
16 for all f ∈ F , . (e 24.21)Define v = w ∗ v ′ w . Then k [ ϕ ◦ ı n , ∞ ( f ) , v ] k < ε / ϕ ◦ ı n , ∞ , v ) | P , = Bott( L , v ′ ) | P , . (e 24.22)Let v = v + v + v . Then k [ ϕ ( f ) , v ] k < ε for all f ∈ F . (e 24.23)Moreover, we compute that Bott( ϕ, v ) | P = α | β ( P ) . (e 24.24)We have actually proved the following result: Lemma 24.2.
Let A = A ⊗ U , where A is as in Theorem 14.10 of [21] and B = B ⊗ U , where B ∈ B is a unital simple C ∗ -algebra and where U , U are two UHF-algebras of infinitetype. Write A = lim n →∞ ( C n , ı n ) as described in Theorem 14.10 of [21]. For any ε > , anyfinite subset F ⊂ A, and any finite subset P ⊂ K ( A ) , there exists an integer n ≥ such that P ⊂ [ ı n, ∞ ]( K ( C n )) and there is a finite subset Q ⊂ K ( C n ) which generates K ( C n ) and thereexists δ > satisfying the following condition: Let ϕ : A → B be a unital homomorphism andlet α ∈ KK ( C n ⊗ C ( T ) , B ) such that | τ ◦ ρ B ( α ( β ( x ))) | < δ for all x ∈ Q and for all τ ∈ T ( B ) . Then there exists a unitary u ∈ B such that k [ ϕ ( x ) , u ] k < ε for all x ∈ F and Bott( ϕ ◦ [ ı n, ∞ ] , u ) = α ( β ) . Proof.
Note that, in the proof of Lemma 24.1, K i ( B j, ) is finitely generated, i = 0 , , and j =1 , , . Then KK ( B j, , A ) = KL ( B j, , B ) (for any unital C ∗ -algebra B ), j = 1 , . Moreover (see[6]), there exists an integer N > Λ ( K ( B j, ) , K ( B )) are determinedby their restrictions to K i ( B j, ) and K i ( B j, , Z /m Z ) , m = 2 , , ..., N . In particular, we mayassume, in the proof of 24.1, that P j, generates K i ( B j, ) ⊕ L N m =2 K i ( B j, , Z /m Z ) , j = 1 , , . Remark 24.3.
Note that, in the statement above, if an integer n works, any integer m ≥ n also works. In the terminology of Definition 3.6 of [44], the statement above also implies that B has properties (B1) and (B2) associated with C. Corollary 24.4.
Let B ∈ B , let A ∈ B , let C = B ⊗ U , and let A = A ⊗ U , where U and U are UHF-algebras of infinite type. Suppose that B satisfies the UCT and supposethat κ ∈ KK e ( C, A ) ++ , γ : T ( A ) → T ( C ) is a continuous affine map, and α : U ( C ) /CU ( C ) → U ( A ) /CU ( A ) is a continuous homomorphism for which γ, α, and κ are compatible. Then, thereexists a unital monomorphism h : C → A such that (1) [ h ] = κ in KK e ( C, A ) ++ , (2) h T = γ and h ‡ = α. roof. The proof follows the same lines as that of Theorem 8.6 of [40], following the proof ofTheorem 3.17 of [44]. Denote by κ ∈ KL ( C, A ) the image of κ . It follows from Lemma 22.14that there is a unital monomorphism ϕ : C → A such that[ ϕ ] = κ, ϕ ‡ = α, and ( ϕ ) T = γ. Note that it follows from the UCT that (as an element of KK ( C, A )) κ − [ ϕ ] ∈ Pext( K ∗ ( C ) , K ∗ +1 ( A )) . By Lemmas 24.2 and 23.3, the C ∗ -algebra A has Property (B1) and Property (B2) associatedwith C in the sense of [44]. Since A contains a unital copy of U , it is infinite dimensional, simpleand antiliminal. It follows from a result in [1] that A contains an element b with sp ( b ) = [0 , . Moreover, A is approximately divisible. It follows from Theorem 3.17 of [44] that there is aunital monomorphism ψ : A → A which is approximately inner and such that[ ψ ◦ ϕ ] − [ ϕ ] = κ − [ ϕ ] in KK ( C, A ) . Then the map h := ψ ◦ ϕ satisfies the requirements of the corollary. Lemma 24.5.
Let A = A ⊗ U , where A is as in Theorem 14.10 of [21] and B = B ⊗ U , where B ∈ B is a unital simple C ∗ -algebra and where U , U are two UHF-algebras of infinite type.Let A = lim n →∞ ( C n , ı n ) be as described in Theorem 14.10 of [21], For any ε > , any σ > , anyfinite subset F ⊂ A, any finite subset P ⊂ K ( A ) , and any projections p , p , ..., p k , q , q , ..., q k ∈ A such that { x , x , ..., x k } generates a free abelian subgroup G of K ( A ) , where x i = [ p i ] − [ q i ] , i =1 , , ..., k, there exists an integer n ≥ such that P ⊂ [ ı n, ∞ ]( K ( C n )) and there is a finite subset Q ⊂ K ( C n ) which generates K ( C n ) and there exists δ > satisfying the following condition:Let ϕ : A → B be a unital homomorphism, let Γ : G → U ( B ) /CU ( B ) be a homomorphism andlet α ∈ KK ( C n ⊗ C ( T ) , B ) such that | τ ◦ ρ B ( α ( β ( x ))) | < δ for all x ∈ Q and for all τ ∈ T ( B ) . Then there exists a unitary u ∈ B such that k [ ϕ ( x ) , u ] k < ε for all x ∈ F , Bott( ϕ ◦ [ ı n, ∞ ] , u ) = α ( β ) , and dist( h ((1 − ϕ ( p i )) + ϕ ( p i ) u )((1 − ϕ ( q i )) + ϕ ( q i ) u ∗ ) i , Γ( x i )) < σ, i = 1 , , ..., k. Proof.
This follows from Lemma 24.2 and Theorem 22.17. In fact, for any 0 < ε < ε/ F ⊃ F , by 24.2, there exists an integer n ≥ , a finite subset Q ⊂ K ( C n ) , and δ > u ∈ U ( B ) , such that k [ ϕ ( x ) , u ] k < ε for all x ∈ F and Bott( ϕ ◦ ı n, ∞ , u ) = α ( β ) | P . Choosing a smaller ε and a larger F , if necessary, we may assume that the class h ((1 − ϕ ( p i )) + ϕ ( p i ) u )((1 − ϕ ( q i )) + ϕ ( q i ) u ∗ ) i ∈ U ( B ) /CU ( B )30s well defined for all 1 ≤ i ≤ k. Define a map Γ : G → U ( B ) /CU ( B ) byΓ ( x i ) = h ((1 − ϕ ( p i )) + ϕ ( p i ) u )(1 − ϕ ( q i )) + ϕ ( q i ) u ∗ ) i , i = 1 , , ..., k. (e 24.25)Choosing a large enough n, without loss of generality, we may assume that there are pro-jections p ′ , p ′ , ..., p ′ k , q ′ , q , ′ ..., q ′ k ∈ C n such that ı n, ∞ ( p ′ i ) = p i and ı n, ∞ ( q ′ i ) = q i , i = 1 , , ..., k. Moreover, we may assume that F ⊂ ı n, ∞ ( C n ) . Let Γ : G → U ( B ) /CU ( B ) be defined by Γ ( x i ) = Γ ( x i ) ∗ Γ( x i ) , i = 1 , , ..., k. It followsby Theorem 22.17 that is a unitary v ∈ U ( B ) such that k [ ϕ ( x ) , v ] k < ε/ x ∈ F , (e 24.26)Bott( ϕ ◦ ı n, ∞ , v ) = 0 , and (e 24.27)dist( h ((1 − ϕ ( p i )) + ϕ ( p i ) v )((1 − ϕ ( q i )) + ϕ ( q i ) v ∗ ) i , Γ ( x i )) < σ, (e 24.28) i = 1 , , ..., k. Define u = u v,X i = h ((1 − ϕ ( p i )) + ϕ ( p i ) u )((1 − ϕ ( q i )) + ϕ ( q i ) u ∗ ) i , and (e 24.29) Y i = h ((1 − ϕ ( p i )) + ϕ ( p i ) v )((1 − ϕ ( q i )) + ϕ ( q i ) v ∗ ) i , (e 24.30) i = 1 , , ..., k. We then compute that k [ ϕ ( x ) , u ] k < ε + ε/ < ε for all x ∈ F , (e 24.31)Bott( ϕ ◦ ı n, ∞ , u ) = Bott( ϕ ◦ ı n, ∞ , u ) = α ( β ) , anddist( h ((1 − ϕ ( p i )) + ϕ ( p i ) u )((1 − ϕ ( q i )) + ϕ ( q i ) u ∗ ) i , Γ( x i )) ≤ dist( X i Y i , Γ ( x i ) Y i ) + dist(Γ ( x i ) Y i , Γ( x i ))= dist( X i , Γ ( x i )) + dist( Y i , Γ ( x i )) < σ, for i = 1 , , ..., k .
25 Another Basic Homotopy Lemma
Lemma 25.1.
Let A be a unital C*-algebra and let U be an infinite dimensional UHF-algebra.Then there is a unitary w ∈ U such that for any unitary u ∈ A , one has τ ( f ( u ⊗ w )) = τ ( f (1 A ⊗ w )) = Z T f dm, f ∈ C ( T ) , τ ∈ T ( A ⊗ U ) , (e 25.1) where m is normalized Lebesgue measure on T . Furthermore, for any a ∈ A and τ ∈ T ( A ⊗ U ) , τ ( a ⊗ w j ) = 0 if j = 0 .Proof. Denote by τ U the unique trace of U . Then any trace τ ∈ T ( A ⊗ U ) is a product trace,i.e., τ ( a ⊗ b ) = τ ( a ⊗ ⊗ τ U ( b ) , a ∈ A, b ∈ U. Pick a unitary w ∈ U such that the spectral measure of w is Lebesgue measure (a Haarunitary). Such a unitary always exists. (It can be constructed directly; or, one can consider astrictly ergodic Cantor system (Ω , σ ) such that K (C(Ω) ⋊ σ Z ) ∼ = K ( U ) . One then notes thatthe canonical unitary in C(Ω) ⋊ σ Z is a Haar unitary. Embedding C(Ω) ⋊ σ Z into U , one obtainsa Haar unitary in U. ) Then one has, for each n ∈ Z ,τ U ( w n ) = (cid:26) , if n = 0 , , otherwise . τ ∈ T ( A ⊗ U ), one has, for each n ∈ Z ,τ (( u ⊗ w ) n ) = τ ( u n ⊗ w n ) = τ ( u n ⊗ τ U ( w n ) = (cid:26) , if n = 0 , , otherwise;and therefore, τ ( P ( u ⊗ w )) = τ ( P (1 ⊗ w )) = Z T P ( z ) dm for any polynomial P . Similarly, τ ( P ( u ⊗ w ) ∗ ) = R T P (¯ z ) dm for any polynomial P. Sincepolynomials in z and z − are dense in C( T ), one has τ ( f ( u ⊗ w )) = τ ( f (1 ⊗ w )) = Z T f dm, f ∈ C( T ) , as desired. Lemma 25.2.
Let A be a unital separable amenable C ∗ -algebra and let L : A ⊗ C ( T ) → B be aunital completely positive linear map, where B is another unital amenable C ∗ -algebra. Supposethat C is a unital C ∗ -algebra and u ∈ C is a unitary. Then, there is a unique pair of unitalcompletely positive linear maps Φ , Φ : A ⊗ C ( T ) → B ⊗ C such that Φ i | A ⊗ C ( T ) = ı ◦ L | A ⊗ C ( T ) ( i = 0 , and Φ ( a ⊗ z j ) = L ( a ⊗ z j ) ⊗ u j and (e 25.2)Φ ( a ⊗ z j ) = L ( a ⊗ C ( T ) ) ⊗ u j (e 25.3) for any a ∈ A and any integer j, where ı : B → B ⊗ C is the standard inclusion.Furthermore, if δ > and G ⊂ A ⊗ C ( T ) is a finite subset, there are δ > and finite set G ⊂ A ⊗ C ( T ) (which do not depend on L ) such that if L is G - δ -multiplicative, then one canchoose u such that the unique Φ i for u is G - δ -multiplicative.Proof. Considering the map L ′ : A ⊗ C ( T ) → B by L ′ ( a ⊗ f ) = L ( a ⊗ f (1)) for all a ∈ A and f ∈ C ( T ) , where f (1) is the evaluation of f at 1 (a point on the unit circle), we see that itsuffices to prove the statement for Φ only.Denote by C the unital C*-subalgebra of C generated by u. Then the tensor product map L ⊗ id C : A ⊗ C ( T ) ⊗ C → B ⊗ C is unital and completely positive (see, for example, Theorem 3.5.3 of [4]). Define the homomor-phism ψ : C ( T ) → C ( T ) ⊗ C by ψ ( z ) = z ⊗ u. By Theorem 3.5.3 of [4] again, the tensor product mapid A ⊗ ψ : A ⊗ C ( T ) → A ⊗ C ( T ) ⊗ C is unital and completely positive. Then, the mapΦ := ( L ⊗ id C ) ◦ (id A ⊗ ψ )satisfies the requirement of the first part of the lemma.Let us consider the second part of the lemma. Let δ > G ⊂ A ⊗ C ( T ) be a finite subset.Without loss of generality, we may assume that elements in G have the form P − n ≤ i ≤ n a i ⊗ z i . Let N = max { n : P − n ≤ i ≤ n a i ⊗ z i ∈ G} , let δ = δ/ N , and let G ⊃ { a i ⊗ z i : − n ≤ i ≤ n : P − n ≤ i ≤ n a i ⊗ z i ∈ G} . X − n ≤ i ≤ n a i ⊗ z i )( X − n ≤ i ≤ n b i ⊗ z i )) = X i,j Φ( a i b j ⊗ z i + j )= X i,j L ( a i b j ⊗ z i + j ) ⊗ u i + j ≈ δ ( X − n ≤ i ≤ n L ( a i ⊗ z i ) ⊗ u i )( X − n ≤ i ≤ n L ( b i ⊗ z i ) ⊗ u i )= Φ( X − n ≤ i ≤ n a i ⊗ z i )Φ( X − n ≤ i ≤ n b i ⊗ z i ) , if P − n ≤ i ≤ n a i ⊗ z i , P − n ≤ i ≤ n b i ⊗ z i ∈ G . It follows that Φ is G - δ -multiplicative.Let P ( T ) = { P ni = − n c i z i , c i ∈ C } denote the algebra of Laurent polynomials. Uniqueness of Φfollows from the fact that A ⊗ C ( T ) is the closure of the algebraic tensor product A ⊗ alg P ( T ).The following corollary follows immediately from 25.2 and 25.1. Corollary 25.3.
Let C be a unital C ∗ -algebra and let U be an infinite dimensional UHF-algebra. For any δ > and any finite subset G ⊂ C ⊗ C ( T ) , there exist δ > and a finite subset G ⊂ C ⊗ C ( T ) satisfying the following condition: For any > σ , σ > , any finite subset H ⊂ C ( T ) + \ { } , any finite subset H ⊂ ( C ⊗ C ( T )) s.a. , and any unital G - δ -multiplicativecompletely positive linear map L : C ⊗ C ( T ) → A, where A is another unital C ∗ -algebra, thereexists a unitary w ∈ U satisfying the following conditions: | τ ( L ( f )) − τ ( L ( f )) | < σ for all f ∈ H , τ ∈ T ( B ) , and (e 25.4) τ ( g (1 A ⊗ w )) ≥ σ ( Z gdm ) for all g ∈ H , τ ∈ T ( B ) , (e 25.5) where B = A ⊗ U and m is normalized Lebesgue measure on T , and L , L : C ⊗ C ( T ) → A ⊗ U are G - δ -multiplicative contractive completely positive linear maps as given by Lemma (as Φ , Φ ) such that L i ( c ⊗ C ( T ) ) = L ( c ⊗ C ( T ) ) ⊗ U ( i = 1 , ), L ( c ⊗ z j ) = L ( c ⊗ z j ) ⊗ w j , and L ( c ⊗ z j ) = L ( c ⊗ C ( T ) ) ⊗ w j for all c ∈ C and all integers j. Proof.
Fix a δ > G . Let δ > G ⊂ C ⊗ C ( T ) be as given by Lemma25.2 for A (in place of B ).Let 0 < σ , σ < , H , and H be as given in the statement. There are a finite subset F C ⊂ C and an integer N > h ∈ H , k h − N X j = − N a h,i ⊗ z i k < σ / , (e 25.6)where a h,i ∈ F C ∪ { } . Set H ′ = { P Ni = − N a h,i ⊗ z j : h ∈ H } . Now assume that L is as stated for G and δ mentioned above.Choose w ∈ U as in Lemma 25.1. Let L , L : C ⊗ C ( T ) → A ⊗ U be as described in thecorollary. In other words, let L : C ⊗ C ( T ) → A ⊗ U be the map as Φ given by Lemma 25.2(with C in place of A, A in place of
B, U in place of C, and w ∈ U in place u ∈ U ), and let L : C ⊗ C ( T ) → A ⊗ U be defined by L ( c ⊗ f ) = L ( c ⊗ C ( T ) ) ⊗ f ( w ) for all c ∈ C and f ∈ C ( T )(as Φ in Lemma 25.2). By the choice of G and δ , L and L are G - δ -multiplicative (as in25.2). 33y Lemma 25.1 (and by Lemma 25.2), for h ∈ H ′ ,τ ( L ( h )) = τ ( N X i = − N L ( a h,i ⊗ z i )) = N X i = − N τ ( L ( a h,i ⊗ z i ) ⊗ w i ) (e 25.7)= τ ( L ( a h, ⊗ C ( T ) )) = τ ( L ( a h, ⊗ C ( T ) )) (e 25.8)= τ ( N X i = − N L ( a h,i ⊗ C ( T ) ) ⊗ w i ) = τ ( L ( h )) . (e 25.9)Thus, combining (e 25.6), inequality (e 25.4) holds. By (e 25.1), (e 25.5) also holds. Lemma 25.4.
Let A = A ⊗ U , where A ∈ B and satisfies the UCT and U is a UHF-algebraof infinite type. For any > ε > and any finite subset F ⊂ A, there exist δ > , σ > ,a finite subset G ⊂ A, a finite subset { p , p , ..., p k , q , q , ..., q k } of projections of A such that { [ p ] − [ q ] , [ p ] − [ q ] , ..., [ p k ] − [ q k ] } generates a free abelian subgroup G u of K ( A ) , and a finitesubset P ⊂ K ( A ) , satisfying the following condition:Let B = B ⊗ U , where B is in B and satisfies the UCT and U is a UHF-algebra ofinfinite type. Suppose that ϕ : A → B is a unital homomorphism.If u ∈ U ( B ) is a unitary such that k [ ϕ ( x ) , u ] k < δ for all x ∈ G , (e 25.10)Bott( ϕ, u ) | P = 0 , (e 25.11)dist( h ((1 − ϕ ( p i )) + ϕ ( p i ) u )(1 − ϕ ( q i )) + ϕ ( q i ) u ∗ ) i , ¯1) < σ, and (e 25.12)dist(¯ u, ¯1) < σ, (e 25.13) then there exists a continuous path of unitaries { u ( t ) : t ∈ [0 , } ⊂ U ( B ) such that u (0) = u, u (1) = 1 B , (e 25.14)dist( u ( t ) , CU ( B )) < ε for all t ∈ [0 , , (e 25.15) k [ ϕ ( a ) , u ( t )] k < ε for all a ∈ F and f or all t ∈ [0 , , and (e 25.16)length( { u ( t ) } ) ≤ π + ε. (e 25.17) Proof.
It is enough to prove the statement under the assumption that u ∈ CU ( B ).Recall that every C*-algebra in B has stable rank one (see Theorem 9.7 of [21]). Define∆( f ) = (1 / Z f dm for all f ∈ C ( T ) + \ { } , where m is normalized Lebesgue measure on the unit circle T . Let A = A ⊗ C ( T ) . Let F = { x ⊗ f : x ∈ F , f = 1 , z, z ∗ } . We may assume that F is a subset of the unit ball of A. Let1 > δ > δ ), G ⊂ A (in place of G ), 1 / > σ > , / > σ > , e P ⊂ K ( A ∈ ) , H ⊂ C ( T ) + \ { } , H ⊂ ( A ) s.a. , and U ⊂ U ( M ( A )) /CU ( M ( A )) be the constants and finitesubsets provided by Theorem 12.11 part (b) of [21] for ε/ ε ), F (in place of F ),∆ , and A (in place of A ).We may assume that G = { a ⊗ f : a ∈ G and f = 1 , z, z ∗ } , where G ⊂ A is a finite subset, and e P = P ∞ ∪ β ( P ∈ ) , where P , P ⊂ K ( A ) are finite subsets.Define P = P ∪ P . 34e may assume that (2 δ , P , G ) is a KL -triple for A , (2 δ , P , G ) is a KL -triple for A ,and 1 A ⊗ H ⊂ H .We may choose σ and σ such thatmax { σ , σ } < (1 /
4) inf { ∆( f ) : f ∈ H } . (e 25.18)Let δ (in place of δ ) and a finite subset G (in place of G ) be as provided by 25.3 for A (in place of C ), δ / δ ), and G (in place of G ). Choosing smaller δ , without loss ofgenerality, we may assume that G = { a ⊗ f : g ∈ G ′ and f = 1 , z, z ∗ } for a large finite subset G ′ ⊃ G . We may assume that δ < δ . We may further assume that U = U ∪ { ⊗ z } ∪ U , (e 25.19)where U = { a ⊗ a ∈ U ′ ⊂ U ( A ) } , U ′ is a finite subset, and U ⊂ U ( A ) /CU ( A ) is afinite subset whose elements represent a finite subset of β ( K ( A )) . So we may assume that U ∈ J c ( β ( K ( A ))) . As in Remark 12.12 of [21], we may assume that the subgroup of J c ( β ( K ( A )))generated by U is free abelian. Let U ′ be a finite subset of unitaries such that { ¯ x : x ∈ U ′ } = U . We may also assume that the unitaries in U ′ have the form((1 − p i ) + p i ⊗ z )((1 − q i ) + q i ⊗ z ∗ ) , i = 1 , , ..., k. (e 25.20)We may further assume that p i ⊗ z, q i ⊗ z ∈ G , i = 1 , , ..., k. Choose δ > G ′ ⊂ A (and write G := { g ⊗ f : g ∈ G ′ , f = 1 , z, z ∗ } ) such that, for any two unital G - δ -multiplicative completely positive linear maps Ψ , Ψ : A ⊗ C ( T ) → C (any unital C ∗ -algebra C ), any G ′ - δ -multiplicative contractive completely positive linear map Ψ : A → C and unitary V ∈ C (1 ≤ i ≤ k ), if k Ψ ( g ) − Ψ ( g ⊗ k < δ for all g ∈ G ′ , (e 25.21) k Ψ ( z ) − V k < δ , and k Ψ ( g ) − Ψ ( g ) k < δ for all g ∈ G , (e 25.22)then h (1 − Ψ ( p i ) + Ψ ( p i ) V )(1 − Ψ ( q i ) + Ψ ( q i ) V ∗ i (e 25.23) ≈ σ / h Ψ (((1 − p i ) + p i ⊗ z )((1 − q i ) + q i ⊗ z ∗ ) i , (e 25.24) kh Ψ ( x ) i − h Ψ ( x ) ik < σ / for all x ∈ U ′ , and (e 25.25)Ψ (((1 − p i ) + p i ⊗ z )(1 − q i ) + q i ⊗ z ∗ )) (e 25.26) ≈ σ / Ψ (((1 − p i ) + p i ⊗ z ))Ψ ((1 − q i ) + q i ⊗ z ∗ )) , (e 25.27)and, furthermore, for d (1) i = p i , d (2) i = q i , , there are projections ¯ d ( j ) i ∈ C and unitaries ¯ z ( j ) i ∈ ¯ d ( j ) i C ¯ d ( j ) i such thatΨ (((1 − d ( j ) i ) + d ( j ) i ⊗ z )) ≈ σ (1 − ¯ d ( j ) i ) + ¯ z ( j ) i , (e 25.28)¯ d ( j ) i ≈ σ Ψ ( d ( j ) i ) , ¯ z (1) i ≈ σ Ψ ( p i ⊗ z ) , and ¯ z (2) i ≈ σ Ψ ( q i ⊗ z ∗ ) , (e 25.29)where 1 ≤ i ≤ k, j = 1 , . Choose σ > { σ / , ε/ , σ / , δ / , δ / } . Choose δ > G ⊂ A satisfying the following condition: there is a unital G - σ/ L : A ⊗ C ( T ) → B ′ such that k L ( a ⊗ − ϕ ′ ( a ) k < σ/ a ∈ G ′ and k L (1 ⊗ z ) − u ′ k < σ/ ϕ ′ : A → B ′ and any unitary u ′ ∈ B ′ such that k ϕ ′ ( g ) u ′ − u ′ ϕ ′ ( g ) k < δ for all g ∈ G . Let δ = min { δ / , σ } and G = G ∪ G ′ ∪ G ′ . Now suppose that ϕ : A → B is a unital homomorphism and u ∈ CU ( B ) satisfies theassumptions (e 25.10) to (e 25.12) for the above mentioned δ, σ, G , P , p i , and q i . Choose anisomorphism s : U ⊗ U → U . Note that s ◦ ı (since it is unital) is approximately unitarilyequivalent to the identity map on U , where ı : U → U ⊗ U is defined by ı ( a ) = a ⊗ a ∈ U ). To simplify notation, let us assume that ϕ ( A ) ⊂ B ⊗ ⊂ B ⊗ U . Suppose that u ∈ U ( B ) ⊗ U is a unitary which satisfies the assumption of the lemma. As mentioned at thebeginning, we may assume that u ∈ CU ( B ) ⊗ U . Without loss of generality, we may furtherassume that u = Q m j =1 c j d j c ∗ j d ∗ j , where c j , d j ∈ U ( B ) ⊗ U , ≤ j ≤ m . Let F = { c j , d j : 1 ≤ j ≤ m } . Let L : A ⊗ C ( T ) → B be a unital G - δ / k L ( a ⊗ − ϕ ( a ) k < σ/ a ∈ G ′ and k L (1 ⊗ z ) − u k < σ/ . (e 25.31)Since Bott( ϕ, u ) | P = 0, we may also assume that[ L ] | P = [ ϕ ] | P and [ L ] | β ( P ) = 0 . (e 25.32)Since B is in B , there is a projection p ∈ B and a unital C ∗ -subalgebra C ∈ C with 1 C = p satisfying the following condition: k L ( g ) − [(1 − p ) L ( g )(1 − p ) + L ( g )] k < σ / m + 1) for all g ∈ G (e 25.33)and k (1 − p ) x − x (1 − p ) k < σ / m + 1) for all x ∈ F , (e 25.34)where L : A ⊗ C ( T ) → C is a unital G -min { δ / , ε/ } -multiplicative completely positive linearmap, τ (1 − p ) < min { σ / , σ / } for all τ ∈ T ( B ) , (e 25.35)and, using (e 25.12), (e 25.13), (e 25.31), and (e 25.23) to (e 25.27) we have thatdist( L ‡ ( x ) , ¯1) < σ / x ∈ { ⊗ ¯ z } ∪ U and (e 25.36)dist( L ‡ ( x ) , ϕ ( x ′ ) ⊗ C ( T ) ) < σ / x ∈ U , (e 25.37)where x ′ ⊗ C ( T ) = x and L ( a ) = (1 − p ) L ( a )(1 − p ) + L ( a ) for all a ∈ A ⊗ C ( T ) . Note thatwe also have k ϕ ( g ) − L ( g ⊗ k < σ/ g ∈ G ′ and [ L | A ] | P = [ ϕ ] | P . (e 25.38)By (e 25.34) and the choice of F , there are a unitary v ∈ CU ( C ) and a unitary v ∈ CU ((1 − p ) B (1 − p )) such that k L (1 ⊗ z ) − v k < min { δ / , ε/ } and (e 25.39) k (1 − p ) L (1 ⊗ z )(1 − p ) − v k < min { δ / , ε/ } . (e 25.40)By the choice of δ and G , applying Corollary 25.3, we obtain a unitary w ∈ U ( U ) = U ( U ) = CU ( U ) (see Theorem 4.1 [22], for example) such that | t ( L ( g ))) − t (Φ( g )) | < σ , g ∈ H , and (e 25.41) t ( g (1 ⊗ w )) ≥ Z T gdm for all g ∈ H (e 25.42)36or all t ∈ T ( B ⊗ U ) , where L : A ⊗ C ( T ) → B ⊗ U is the unital G - δ / L ( a ⊗
1) = L ( a ⊗ ⊗ U and L ( a ⊗ z j ) = L ( a ⊗ z j ) ⊗ ( w ) j (e 25.43)for all a ∈ A and all integers j as given by Lemma 25.2, w (1 − p ) = (1 − p ) w = (1 − p ), as weconsider both w , 1 − p as elements of B ⊗ U as 1 B ⊗ w and (1 − p ) ⊗ U , respectively, andΦ : A ⊗ C ( T ) → B ⊗ U is defined by Φ( a ⊗
1) = ϕ ( a ) ⊂ B ⊗ a ∈ A and Φ(1 ⊗ f ) = f ( w )for all f ∈ C ( T ) . Moreover, Φ(1 ⊗ f ) = f ( λ )((1 − p ) ⊗ U ) + f ( pw ) for all f ∈ C ( T ) and forsome λ ∈ T . Note that CU ( U ) = U ( U ) (see Theorem 4.1 of [22]). It is also known (by workingin matrices, for example) that there is a continuous path of unitaries in CU ( U ) connecting 1 U to w with length no more than π + ε/ . Therefore one obtains a continuous path of unitaries { v ( t ) : t ∈ [1 / , / } ⊂ CU ( U ) such that v (1 /
4) = 1 U , v (1 /
2) = w, and length( { v ( t ) : t ∈ [1 / , / } ) ≤ π + ε/ . (e 25.44)Note that ϕ ( a )Φ(1 ⊗ z ) = Φ(1 ⊗ z ) ϕ ( a ) for all a ∈ A. So, in particular, Φ is a unital homomor-phism and [Φ] | β ( K ( A )) = 0 . (e 25.45)Define a unital completely positive linear map L t : A → C ([2 , , B ⊗ U ) by L t ( f ⊗
1) = L ( f ⊗
1) and L t ( a ⊗ z j ) = L ( a ⊗ z j ) ⊗ ( v (( t − / / j for all a ∈ A and integers j and t ∈ [2 , L t (1 ⊗ z ) ≈ min( δ / ,ε/ ( v ⊕ v ) ⊗ v (( t − / / , and, since v ( s ) ∈ CU ( U ) , L t (1 ⊗ z ) ∈ min( δ / ,ε/ CU ( B ⊗ U ) for all t ∈ [2 , . Note also that, L t is G - δ / t = 2 , L t = L and at t = 3 , L t = L . It follows that [ L ] | P = [ L ] | P = [ ϕ ] | P , [ L ] | β ( P ) = 0 , and (e 25.46) L ‡ ( x ) = L ‡ ( x ) for all x ∈ U . (e 25.47)If v = ( e ⊗ z ) + (1 − e ) for some projection e ∈ A, then L ( v ) = L ( e ⊗ z ) ⊗ w + L ((1 − e )) . (e 25.48)Since w ∈ CU ( U ) , one computes from (e 25.27) that that, with x = ((1 − p i ) + p i ⊗ z )((1 − q i ) + q i ⊗ z ∗ ) , h L ( x ) i ≈ σ / (¯ z (1) i ⊗ w + (1 − ¯ p i ))(¯ z (2) i ⊗ w + (1 − ¯ q i )) (e 25.49)= (¯ z (1) i + (1 − ¯ p i ))(¯ p i ⊗ w + (1 − ¯ p i ) ⊗ U )(¯ z (2) i + (1 − ¯ q i ))(¯ q i ⊗ w + (1 − ¯ q i )) (e 25.50)= (¯ z (1) i + (1 − ¯ p i ))(¯ z (2) i + (1 − ¯ q i )) = h L ( x ) i , (e 25.51)where ¯ p i , ¯ q i , ¯ z (1) i , ¯ z (2) i are as above (see the lines following (e 25.27)), with Ψ replaced by L . Itfollows that dist( L ‡ ( x ) , ¯1) < σ / x ∈ { ⊗ z } ∪ U . (e 25.52)Note that, since w ∈ CU ( U ) and ϕ ( q ) ∈ B ⊗ U , Φ( q ⊗ z + (1 − q ) ⊗
1) = ϕ ( q ) ⊗ w + ϕ (1 − q ) ∈ CU ( B ⊗ U ) (e 25.53)37or any projection q ∈ A. It follows thatΦ ‡ ( x ) ∈ CU ( B ⊗ U ) for all x ∈ { ⊗ z } ∪ U . (e 25.54)Therefore (see also (e 25.45))[ L ] | P = [Φ] | P and dist(Φ ‡ ( x ) , L ‡ ( x )) < σ for all x ∈ U . (e 25.55)It follows from (e 25.42) that τ (Φ( f )) ≥ ∆( f ) , f ∈ H , τ ∈ T ( B ⊗ U ) , (e 25.56)and it follows from (e 25.41) that | τ (Φ( f )) − τ ( L ( f )) | < σ , f ∈ H , τ ∈ T ( B ⊗ U ) . (e 25.57)Applying Theorem 12. 11 of [21], we obtain a unitary w ∈ B ⊗ U such that k w ∗ Φ( f ) w − L ( f ) k < ε/ f ∈ F . (e 25.58)Since w ∈ U , there is a continuous path of unitaries { w ( t ) : t ∈ [3 / , } ⊂ CU ( U ) (recall that CU ( U ) = U ( U )) such that w (3 /
4) = Φ(1 ⊗ z ) = w, w (1) = 1 U and length( { w ( t ) : t ∈ [3 / , } ) ≤ π + ε/ . (e 25.59)Note that Φ( a ) w ( t ) = w ( t )Φ( a ) for all a ∈ A and t ∈ [3 / , . (e 25.60)It follows from (e 25.58) that there exists a continuous path of unitaries { u ( t ) : t ∈ [1 / , / } ⊂ B ⊗ U such that u (1 /
2) = ( v + ( v )) ⊗ w, u (3 /
4) = w ∗ Φ(1 ⊗ z ) w , and (e 25.61) k u ( t ) − u (1 / k < ε/ t ∈ [1 / , / . (e 25.62)It follows from (e 25.30), (e 25.39), and (e 25.40) that there exists a continuous path of unitaries { u ( t ) : t ∈ [0 , / } ⊂ B such that u (0) = u, u (1 /
4) = v + v , and (e 25.63) k u ( t ) − u k < ε/ t ∈ [0 , / . (e 25.64)Also, define u ( t ) = ( v ⊕ v ) ⊗ v ( t ) for all t ∈ [1 / , / . It follows that k ϕ ( g ) u ( t ) − u ( t ) ϕ ( g ) k < ε/ δ < ε/
16 for all g ∈ G . (e 25.65)Then define u ( t ) = w ∗ w ( t ) w for all t ∈ [3 / , . (e 25.66)Then { u ( t ) : t ∈ [0 , } ⊂ B ⊗ U is a continuous path of unitaries such that u (0) = u and u (1) = 1 . Moreover, by (e 25.64), (e 25.65), (e 25.61), (e 25.62), (e 25.58), (e 25.64), (e 25.44), and(e 25.59), k ϕ ( f ) u ( t ) − u ( t ) ϕ ( f ) k < ε for all f ∈ F and length( { u ( t ) } ) ≤ π + ε. (e 25.67)38 emark 25.5. Note, in the statement of Theorem 25.4, if [1 A ] ∈ P (as an element of K ( A )),by 2.14 of [21], condition (e 25.11) implies [ u ] = 0 in K ( B ) . In other words, by making [1 A ] ∈ P , (e 25.11) implies [ u ] = 0 . One also notices that if, for some i, p i = 1 A and q i = 0 , then (e 25.12) implies (e 25.13). Infact, (e 25.13) is redundant. To see this, let A be a unital simple separable amenable C ∗ -algebrawith stable rank one. Let G ⊂ K ( A ) be a finitely generated subgroup containing [1 A ] . Let G r = ρ A ( G ) . Then ρ A ([1 A ]) = 0 and G r is a finitely generated free abelian group. Then wemay write G = G ∩ ker ρ A ⊕ G ′ r , where ρ A ( G ′ r ) = G r and G ′ r ∼ = G r . Note that G ∩ ker ρ A is afinitely generated group. We may therefore write G ∩ ker ρ A = G ⊕ G , where G is a torsiongroup and G is free abelian. Note that G ⊕ G ′ r is free abelian. Therefore G = Tor( G ) ⊕ F, where F is a finitely generated free abelian subgroup. Note that there is an integer m ≥ m [1 A ] ∈ F. Let z ∈ C ( T ) be the standard unitary generator. Consider A ⊗ C ( T ) . Then β ( G ) ⊂ β ( K ( A )) is a subgroup of K ( A ⊗ C ( T )) . Moreover, β ([1 A ]) may be identified with[1 ⊗ z ] . If we choose U in the proof of 25.4 to generate β ( F ) , then m [1 A ] is in the subgroup generatedby { [ p i ] − [ q i ] : 1 ≤ i ≤ k } (see the last paragraph). Thus, for any σ > , we may assume thatdist( u m , ¯1) < σ (e 25.68)provided that (e 25.12) holds for a sufficiently small σ. Recall that B has stable rank one (seeTheorem 9.7 of [21]) and u ∈ U ( B ) (see the beginning of this remark). We may write u =exp( ih ) v for some h ∈ B s.a. . Recall, in this case, U ( B ) /CU ( B ) = Aff( T ( B )) /ρ B ( K ( B )) , where ρ B ( K ( B )) is a closed vector subspace of Aff( T ( B )) (see the proof of Lemma 11.5 of[21]). The image of u m in Aff( T ( B )) /ρ B ( K ( B )) is the same as m times the image of u inAff( T ( B )) /ρ B ( K ( B )) . It follows from (e 25.68) thatdist(¯ u, ¯1) < σ . (e 25.69)This implies that (with sufficiently small σ ) the condition (e 25.13) is redundant and thereforecan be omitted.
26 Stable results
Lemma 26.1.
Let C be a unital amenable separable C ∗ -algebra which is residually finite di-mensional and satisfies the UCT. For any ε > , any finite subset F ⊂ C, any finite sub-set P ⊂ K ( C ) , any unital homomorphism h : C → A, where A is any unital C ∗ -algebra,and any κ ∈ Hom Λ ( K ( SC ) , K ( A )) , there exists an integer N ≥ , a unital homomorphism h : C → M N ( C ) ⊂ M N ( A ) , and a unitary u ∈ U ( M N +1 ( A )) such that k H ( c ) , u ] k < ε for all c ∈ F and Bott(
H, u ) | P = κ ◦ β | P , (e 26.1) where H ( c ) = diag( h ( c ) , h ( c )) for all c ∈ C. Proof.
Define S = { z, C ( T ) } , where z is the identity function on the unit circle. Define x ∈ Hom Λ ( K ( C ⊗ C ( T )) , K ( A )) as follows: x | K ( C ) = [ h ] and x | β ( K ( C )) = κ. (e 26.2)Fix a finite subset P ⊂ β ( K ( C )) . Choose ε > F ⊂ C satisfying thefollowing condition: [ L ′ ] | P = [ L ′′ ] | P (e 26.3)39or any pair of ( F ⊗ S )- ε -multiplicative contractive completely positive linear maps L ′ , L ′′ : C ⊗ C ( T ) → B (for any unital C ∗ -algebra B ), whenever L ′ ≈ ε L ′′ on F ⊗ S. (e 26.4)Let a positive number ε > , a finite subset F and a finite subset P ⊂ K ( C ) be given. Wemay assume, without loss of generality, thatBott( H ′ , u ′ ) | P = Bott( H ′ , u ′′ ) | P (e 26.5)whenever k u ′ − u ′′ k < ε for any unital homomorphism H ′ from C. Put ε = min { ε/ , ε / } and F = F ∪ F (choosing P = β ( P ) above).Let δ > , a finite subset G ⊂ C , and a finite subset P ⊂ K ( C ) (in place of P ) be as providedby Lemma 4.17 of [21] for ε / ε ) and F (in place of F ). We may assume that F and G are in the unit ball of C and δ < min { / , ε / } . Fix another finite subset P ⊂ K ( C )and define P = P ∪ β ( P ) (as a subset of K ( C ⊗ C ( T ))). We may assume that P ⊂ β ( P ) . It follows from Theorem 18.2 of [21] that there are integers k , k , ..., k m and K , a homo-morphism h ′ : C ⊗ C ( T ) → L mj =1 M k j ( C ) → M K ( A ) , and a unital ( G ⊗ S )- δ/ L ′ : C ⊗ C ( T ) → M K +1 ( A ) such that[ L ′ ] | P = ( x + [ h ′ ]) | P . (e 26.6)Write h ′ = ⊕ mj =1 H ′ j , where H ′ j = ψ j ◦ π j , π j : A ⊗ C ( T ) → M k j ( C ) is a finite dimensionalrepresentation and ψ j : M k j ( C ) → M K ( A ) is a homomorphism. Let e j be a minimal projectionof M k j ( C ) and q j = ψ j ( e j ) ∈ M K ( A ) , and Q j = ψ j (1 M kj ( C ) ) ∈ M K ( A ) . Set p j = 1 M K ( A ) − q j , j =1 , , ..., m. Then M k j K ( C ) can be identified with M k j ( M K ( A )) = M k j (cid:0) ( q j ⊕ p j ) M K ( A )( q j ⊕ p j ) (cid:1) (since q j + p j = 1 M K ( A ) ) in such a way that Q j ( M K ( A )) Q j is identified with M k j ( q j M K ( A ) q j ).Define ψ ′ j : M k j ( C ) → M k j ( C · M K ( A ) ) ⊂ M k j K ( A ) by sending e j to p j . Define H ′′ j : C ⊗ C ( T ) → M k j K ( A ) by H ′′ j ( c ) = ψ j ◦ π j ( c ) ⊕ ψ ′ j ◦ π j (conjugating a unitary). Note we require H ′′ j mapsinto the scalar matrices of M k j K ( A ) . Let H ′ = ⊕ mj =1 ψ ′ j : C ⊗ C ( T ) → M k j ( p j M K ( A ) p j ) ⊂ M ( P j k j ) K ( A ) (conjugating a suitable unitary). Let N = ( P mj =1 k j ) K. Define h = h ′ ⊕ H ′ and L = L ′ ⊕ H ′ . Then h maps C ⊗ C ( T ) into M N ( C · A ) ⊂ M N ( A ) . In other words, there are an integer N ≥ , a unital homomorphism h ′ : C ⊗ C ( T ) → M N ( C ) ⊂ M N ( A ), and a unital ( G ⊗ S )- δ/ L : C ⊗ C ( T ) → M N +1 ( A ) such that[ L ] | P = ( x + [ h ]) | P . (e 26.7)We may assume that there is a unitary v ∈ M N +1 ( A ) such that k L (1 ⊗ z ) − v k < ε / . (e 26.8)Define H : C → M N +1 ( A ) by H ( c ) = h ( c ) ⊕ h ( c ⊗
1) for all c ∈ C. (e 26.9)Define L : C → M N +1 ( A ) by L ( c ) = L ( c ⊗
1) for all c ∈ C. Note that[ L ] | P = [ H ] | P . (e 26.10)It follows from Lemma 4.17 of [21] that there exists an integer N ≥ , a unital homomorphism h : C → M N ( N +1) ( C ) ⊂ M N ( N +1) ( A ) , and a unitary W ∈ M ( N +1)(1+ N ) ( A ) such that W ∗ ( L ( c ) ⊕ h ( c )) W ≈ ε/ H ( c ) ⊕ h ( c ) for all c ∈ F . (e 26.11)40ut N = N ( N + 1) + N . Now define h : C → M N ( C ) and H : C → M N +1 ( A ) by h ( c ) = h ( c ⊗ ⊕ h ( c ) and H ( c ) = h ( c ) ⊕ h ( c ) (e 26.12)for all c ∈ C. Define u = W ∗ ( v ⊕ M N N ) W. Then, by (e 26.11), and as L is ( G ⊗ S )- δ/ k [ H ( c ) , u ] k ≤ k ( H ( c ) − Ad W ◦ ( L ( c ) ⊕ h ( c ))) u ] k (e 26.13)+ k Ad W ◦ ( L ( c ) ⊕ h ( c )) , u ] k + k u ( H ( c ) − Ad W ◦ ( L ( c ) ⊕ h ( c ))) k (e 26.14) < ε/ δ/ ε/ < ε for all c ∈ F . (e 26.15)Define L : C → M N +1 ( A ) by L ( c ) = L ( c ) ⊕ h ( c ) for all c ∈ C. Then, we compute thatBott(
H, u ) | P = Bott(Ad W ◦ L , u ) | P = Bott( L , v ⊕ M N N ) | P (e 26.16)= Bott( L , v ) | P + Bott( h , M N N ) | P (e 26.17)= [ L ] | β ( P ) + 0 = ( x + [ h ]) | β ( P ) = κ | P . (e 26.18) Theorem 26.2.
Let C be a unital amenable separable C ∗ -algebra which is residually finitedimensional and satisfies the UCT. For any ε > and any finite subset F ⊂ C, there are δ > , a finite subset G ⊂ C, and a finite subset P ⊂ K ( C ) satisfying the following condition:Suppose that A is a unital C ∗ -algebra, suppose that h : C → A is a unital homomorphismand suppose that u ∈ U ( A ) is a unitary such that k [ h ( a ) , u ] k < δ for all a ∈ G and Bott( h, u ) | P = 0 . (e 26.19) Then there exist an integer N ≥ , a unital homomorphism H : C ⊗ C ( T ) → M N ( C ) ( ⊂ M N ( A ) ) (with finite dimensional range), and a continuous path of unitaries { U ( t ) : t ∈ [0 , } in M N +1 ( A ) such that U (0) = u ′ , U (1) = 1 M N +1 ( A ) , and k [ h ′ ( a ) , U ( t )] k < ε for all a ∈ F , (e 26.20) where u ′ = diag( u, H (1 ⊗ z )) and h ′ ( f ) = h ( f ) ⊕ H ( f ⊗ for f ∈ C, and z ∈ C ( T ) is the identity function on the unit circle.Moreover, Length( { U ( t ) } ) ≤ π + ε. (e 26.21) Proof.
Let ε >
F ⊂ C be given. We may assume that F is in the unit ball of C. Let δ > , G ⊂ C ⊗ C ( T ) , and P ⊂ K ( C ⊗ C ( T )) be as provided by Lemma 4.17 of[21] for ε/ F ⊗ S. We may assume that G = G ′ ⊗ S, where G ′ is in the unit ball of C and S = { C ( T ) , z } ⊂ C ( T ) . Moreover, we may assume that P = P ∪ P , where P ⊂ K ( C )and P ⊂ β ( K ( C )) . Let P = P ∪ β − ( P ) ⊂ K ( C ). Furthermore, we may assume that any δ - G -multiplicative contractive completely positive linear map L ′ from C ⊗ C ( T ) to a unital C ∗ -algebra gives rise to a well defined map [ L ′ ] | P . Let δ > G ⊂ C be as provided by 2.8 of [38] for δ / G ′ above.Let δ = min { δ / , δ / , ε/ } and G = F ∪ G . h and u satisfy the assumption with δ, G and P as above. Thus, by 2.8 of [38],there is a δ / G -multiplicative contractive completely positive linear map L : C ⊗ C ( T ) → A such that k L ( f ⊗ − h ( f ) k < δ / f ∈ G ′ and (e 26.22) k L (1 ⊗ z ) − u k < δ / . (e 26.23)Define y ∈ Hom Λ ( K ( C ⊗ C ( T )) , K ( A )) as follows: y | K ( C ) = [ h ] | K ( C ) and y | β ( K ( C )) = 0 . It follows from Bott( h, u ) | P = 0 that [ L ] | β ( P ) = 0.Then [ L ] | P = y | P . (e 26.24)Define H : C ⊗ C ( T ) → A by H ( c ⊗ g ) = h ( c ) · g (1) · A for all c ∈ C and g ∈ C ( T ) , where T refers to the unit circle (and 1 ∈ T ).It follows that [ H ] | P = y | P = [ L ] | P . (e 26.25)It follows from Lemma 4.17 of [21] that there are an integer N ≥ , a unital homomorphism H : C ⊗ C ( T ) → M N ( C ) ( ⊂ M N ( A )) with finite dimensional range, and a unitary W ∈ U ( M N ( A )) such that W ∗ ( H ( c ) ⊕ H ( c )) W ≈ ε/ L ( c ) ⊕ H ( c ) for all c ∈ F ⊗ S. (e 26.26)Since H has finite dimensional range and since H (1 ⊗ z ) is in the center of range( H ) ⊂ M N ( C ), it is easy to construct a continuous path { V ′ ( t ) : t ∈ [0 , } in a finite dimensional C ∗ -subalgebra of M N ( C ) such that V ′ (0) = H (1 ⊗ z ) , V ′ (1) = 1 M N ( A ) and (e 26.27) H ( c ⊗ V ′ ( t ) = V ′ ( t ) H ( c ⊗
1) (e 26.28)for all c ∈ C and t ∈ [0 , . Moreover, we may ensure thatLength( { V ′ ( t ) } ) ≤ π. (e 26.29)Now define U (1 / t/
4) = W ∗ diag(1 , V ′ ( t )) W for t ∈ [0 ,
1] and u ′ = u ⊕ H (1 A ⊗ z ) and h ′ ( c ) = h ( c ) ⊕ H ( c ⊗ c ∈ C for t ∈ [0 , . Then, by (e 26.26), k u ′ − U (1 / k < ε/ k [ U ( t ) , h ′ ( a )] k < ε/ a ∈ F and t ∈ [1 / , . The desired conclusion follows by connecting U (1 /
4) with u ′ witha short path as follows: There is a self-adjoint element a ∈ M N ( A ) with k a k ≤ επ/ ia ) = u ′ U (1 / ∗ (e 26.31)Then the path of unitaries U ( t ) = exp( i (1 − t ) a ) U (1 /
4) for t ∈ [0 , /
4) satisfies the requirements.42 emma 26.3.
Let C be a unital separable C ∗ -algebra whose irreducible representations havebounded dimension and let B be a unital C ∗ -algebra with T ( B ) = ∅ . Suppose that ϕ , ϕ : C → B are two unital monomorphisms such that [ ϕ ] = [ ϕ ] i n KK ( C, B ) , Let θ : K ( C ) → K ( M ϕ ,ϕ ) be the splitting map defined in Equation (e 2.46) in Definition 2.21of [21].For any / > ε > , any finite subset F ⊂ C and any finite subset P ⊂ K ( C ) , thereare integers N ≥ , a unital ε/ - F -multiplicative completely positive linear map L : C → M N ( M ϕ ,ϕ ) , a unital homomorphism h : C → M N ( C ) (later, M N ( C ) can be regardedas unital subalgebra of M N ( B ) and also of M N ( M ϕ ,ϕ ) ), and a continuous path of unitaries { V ( t ) : t ∈ [0 , − d ] } in M N ( B ) for some / > d > , such that [ L ] | P is well defined, V (0) = 1 M N ( B ) , [ L ] | P = ( θ + [ h ]) | P , (e 26.32) π t ◦ L ≈ ε Ad V ( t ) ◦ ( ϕ ⊕ h ) on F for all t ∈ (0 , − d ] , (e 26.33) π t ◦ L ≈ ε Ad V (1 − d ) ◦ ( ϕ ⊕ h ) on F for all t ∈ (1 − d, , and (e 26.34) π ◦ L ≈ ε ϕ ⊕ h on F , (e 26.35) where π t : M ϕ ,ϕ → B is the point evaluation at t ∈ (0 , . Proof.
Let ε >
F ⊂ C be a finite subset. Let δ > , a finite subset G ⊂ C , and afinite subset P ⊂ K ( C ) be as provided by 26.2 for ε/ F above. In particular, we assumethat δ < δ P (see Definition 2.14 of [21]). By Lemma 2.15 of [40], we may further assume that δ is sufficiently small thatBott(Φ , U U U ) | P = X i =1 Bott(Φ , U i ) | P (e 26.36)whenever k [Φ( a ) , U i ] k < δ for all a ∈ G , i = 1 , , . Let ε = min { δ / , ε/ } and F = F ∪ G . We may assume that F is in the unit ballof C. We may also assume that [ L ′ ] | P is well defined for any ε - F -multiplicative contractivecompletely positive linear map L ′ from C to any unital C*-algebra.Let δ > G ⊂ C , and P ⊂ K ( C ) be a constant and finite subsets as provided by Lemma4.17 of [21] for ε / F . We may assume that δ < ε / , G ⊃ F , and P ⊃ P . We alsoassume that G is in the unit ball of C. It follows from Theorem 18.2 of [21] that there exist an integer K ≥ , a unital homomor-phism h ′ : C → M K ( C ) (see also lines around (e 26.7)), and a δ / G -multiplicative contractivecompletely positive linear map L : C → M K +1 ( M ϕ ,ϕ ) such that[ L ] | P = ( θ + [ h ′ ]) | P . (e 26.37)Note that [ π ] ◦ θ = [ ϕ ] and [ π ] ◦ θ = [ ϕ ] and, for each t ∈ (0 , , [ π t ] ◦ θ = [ ϕ ] = [ ϕ ] . (e 26.38)By Lemma 4.17 of [21], we obtain an integer K , a unitary V ∈ U ( M K + K (( C ))), and aunital homomorphism h : C → M K ( C ) such thatAd V ◦ ( π e ◦ L ⊕ h ) ≈ ε / (id ⊕ h ′ ⊕ h ) on F , (e 26.39)43here π e : M ϕ ,ϕ → C is the canonical projection.(Here and below, we will identify a homomorphism mapping to M k ( C ) with a homomorphismto M k ( A ) for any unital C ∗ algebra A , without introducing new notation.)Write V = ϕ ( V ) and V ′ = ϕ ( V ) . The assumption that [ ϕ ] = [ ϕ ] implies that [ V ] =[ V ′ ] in K ( B ) . By adding another homomorphism to h in (e 26.39), replacing K by 2 K , andreplacing V by V ⊕ M K , if necessary, we may assume that V and V ′ are in the same connectedcomponent of U ( M K + K ( B )) . (Note that [ V ] = [ V ′ ].)One obtains a continuous path of unitaries { Z ( t ) : t ∈ [0 , } in M K + K ( B ) such that Z (0) = V and Z (1) = V ′ . (e 26.40)It follows that Z ∈ M K + K ( M ϕ ,ϕ ) . By replacing L by ad Z ◦ ( L ⊕ h ) and using a new h ′ , we may assume that π ◦ L ≈ ε / ϕ ⊕ h ′ on F and π ◦ L ≈ ε / ϕ ⊕ h ′ on F . (e 26.41)Define λ : C → M K + K ( C ) by λ ( c ) = diag( c, h ′ ( c )) , where we also identify M K + K ( C )with the scalar matrices in M K + K ( C ) . In particular, since ϕ i is unital, ϕ i ⊗ id M K K is theidentity on M K + K ( C ) , i = 1 , . Consequently, ( ϕ i ⊗ id M K K ) ◦ h ′ = h ′ . Therefore, one maywrite ϕ i ( c ) ⊕ h ′ ( c ) = ( ϕ i ⊗ id M K K ) ◦ λ ( c ) for all c ∈ C. There is a partition 0 = t < t < · · · < t n = 1 such that π t i ◦ L ≈ δ / π t ◦ L on G for all t i ≤ t ≤ t i +1 , i = 1 , , ..., n − . (e 26.42)Applying Lemma 4.17 of [21] again, we obtain an integer K ≥ , a unital homomorphism h : C → M K ( C ) , and a unitary V t i ∈ M K + K + K ( B ) such thatAd V t i ◦ ( ϕ ⊕ h ′ ⊕ h ) ≈ ε / ( π t i ◦ L ⊕ h ) on F . (e 26.43)Note that, by (e 26.42), (e 26.43), and (e 26.41), k [ ϕ ⊕ h ′ ⊕ h ( a ) , V t i V ∗ t i +1 ] k < δ / ε for all a ∈ F . Define η − = 0 and η k = k X i =0 Bott( ϕ ⊕ h ′ ⊕ h , V t i V ∗ t i +1 ) | P , k = 0 , , ..., n − . Now we will construct, for each i, a unital homomorphism F i : C → M J i ( C ) ⊂ M J i ( B ) anda unitary W i ∈ M K + K + K + P ik =1 J i ( B ) such that k [ H i ( a ) , W i ] k < δ / a ∈ F and Bott( H i , W i ) = η i − , (e 26.44)where H i = ϕ ⊕ h ′ ⊕ h ⊕ L ik =1 F i , i = 1 , , ..., n − . Let W = 1 M K K K . It follows from Lemma 26.1 that there are an integer J ≥ , aunital homomorphism F : C → M J ( C ), and a unitary W ∈ U ( M K + K + K + J ( B )) suchthat k [ H ( a ) , W ] k < δ / a ∈ F and Bott( H , W ) = η , (e 26.45)where H = ϕ ⊕ h ′ ⊕ h ⊕ F . F i and W i for i = 0 , , ..., k < n − . It followsfrom Lemma 26.1 that there are an integer J k +1 ≥ , a unital homomorphism F k +1 : C → M J k +1 ( C ) , and a unitary W k +1 ∈ U ( M K + K + K + P k +1 i =1 J i ( B )) such that k [ H k +1 ( a ) , W k +1 ] k < δ / a ∈ F and Bott( H k +1 , W k +1 ) = η k , (e 26.46)where H k +1 = ϕ ⊕ h ′ ⊕ h ⊕ L k +1 i =1 F i . This finished the construction of F i , W i and H i for i = 0 , , ..., n − . Now define F = h ⊕ L n − i =0 F i and define K = 1 + K + K + K + P n − i =1 J i . Define v t k = diag( W k diag( V t k , id M P ki =1 Ji ) , M P n − i = k +1 Ji ) ,k = 1 , , ..., n − v t = 1 M K K K P n − i =1 Ji . ThenAd v t i ◦ ( ϕ ⊕ h ′ ⊕ F ) ≈ δ + ε π t i ◦ ( L ⊕ F ) on F , (e 26.47) k [ ϕ ⊕ h ′ ⊕ F ( a ) , v t i v ∗ t i +1 ] k < δ / ε for all a ∈ F , and (e 26.48)Bott( ϕ ⊕ h ′ ⊕ F , v t i v ∗ t i +1 ) (e 26.49)= Bott( ϕ ′ , W ′ i ) + Bott( ϕ ′ , V ′ t i ( V ′ t i +1 ) ∗ ) + Bott( ϕ ′ , ( W ′ i +1 ) ∗ ) (e 26.50)= η i − + Bott( ϕ ′ , V t i V ∗ t i +1 ) − η i = 0 , (e 26.51)where ϕ ′ = ϕ ⊕ h ′ ⊕ F , W ′ i = diag( W i , M P n − j = i +1 Ji ) and V ′ t i = diag( V t i , M P n − i =1 Ji ) , i =0 , , , ..., n − . It follows by Lemma 26.2 that there are an integer N ≥ , a unital homomorphism F ′ : C → M N ( C ), and a continuous path of unitaries { w i ( t ) : t ∈ [ t i − , t i ] } in M K ( B ) such that w i ( t i − ) = v ′ i − ( v ′ i ) ∗ , w i ( t i ) = 1 , and (e 26.52) k [ ϕ ⊕ h ′ ⊕ F ⊕ F ′ ( a ) , w i ( t )] k < ε/ a ∈ F , (e 26.53)where v ′ i = diag( v i , M N ( B )) , i = 1 , , ..., n − . Define V ( t ) = w i ( t ) v ′ i for t ∈ [ t i − , t i ] , i =1 , , ..., n − . Then V ( t ) ∈ C ([0 , t n − ] , M K + N ( B )) . Moreover,Ad V ( t ) ◦ ( ϕ ⊕ h ′ ⊕ F ⊕ F ′ ) ≈ ε π t ◦ L ⊕ F ⊕ F ′ on F . (e 26.54)Define h = h ′ ⊕ F ⊕ F ′ , L = L ⊕ F + F ′ , and d = 1 − t n − . Then, by (e 26.54), (e 26.33)and (e 26.34) hold. From (e 26.41), it follows that (e 26.35) also holds.
27 Asymptotic unitary equivalence
Lemma 27.1.
Let C and A be two unital separable simple C ∗ -algebras in B , let U and U betwo UHF-algebras of infinite type and consider the C ∗ -algebras C = C ⊗ U and A = A ⊗ U . Suppose that ϕ , ϕ : C → A are two unital monomorphisms. Suppose also that [ ϕ ] = [ ϕ ] i n KL ( C, A ) , (e 27.1)( ϕ ) T = ( ϕ ) T and ϕ ‡ = ϕ ‡ . (e 27.2) Then ϕ and ϕ are approximately unitarily equivalent.Proof. This follows immediately from Theorem 12.11 part (a) of [21]. Note that both A and C are in B . emma 27.2. Let B be a unital C ∗ -algebra and let u , u , ..., u n ⊂ U ( B ) be unitaries. Supposethat v , v , ..., v m ⊂ U ( B ) are also unitaries such that [ v j ] ⊂ G, j = 1 , ..., m , where G is thesubgroup of K ( B ) generated by [ u ] , [ u ] , ..., [ u n ] . There exist δ > and a finite subset F ⊂ B satisfying the following condition: For any unital C ∗ -algebra A and any unital monomorphisms ϕ , ϕ : B → A , if τ ◦ ϕ = τ ◦ ϕ for all τ ∈ T ( A ) and if there is a unitary w ∈ U ( B ) such that k w ∗ ϕ ( b ) w − ϕ ( b ) k < δ for all b ∈ F , (e 27.3) then there exists a group homomorphism α : G → Aff( T ( A )) such that πi τ (log( ϕ ( u k ) w ∗ ϕ ( u ∗ k ) w ) = α ([ u k ])( τ ) and (e 27.4)12 πi τ (log( ϕ ( v j ) w ∗ ϕ ( v ∗ j ) w ) = α ([ v j ])( τ ) , (e 27.5) for any τ ∈ T ( A ) , k = 1 , , ..., n and j = 1 , , ..., m. Proof.
The proof is essentially contained in the proofs of 6.1, 6.2, and 6.3 of [36]. Note thatthere is a typo in Lemma 6.2 and Lemma 6.3 in [36]: “ τ ( α ( a )) = a ” should be “ τ ( α ( a )) = τ ( a )”.Here the condition τ ◦ ϕ = τ ◦ ϕ plays the role of condition τ ( α ( a )) = τ ( a ) there. Lemma 27.3.
Let C be a unital simple C ∗ -algebra as in Theorem 14 .10 of [21], let A be aunital separable simple C ∗ -algebra in B , and let U and U be UHF-algebras of infinite type.Let C = C ⊗ U and A = A ⊗ U . Suppose that ϕ , ϕ : C → A are unital monomorphisms.Suppose also that [ ϕ ] = [ ϕ ] i n KL ( C, A ) , (e 27.6) ϕ ‡ = ϕ ‡ , ( ϕ ) T = ( ϕ ) T , and (e 27.7) R ϕ,ψ ( K ( M ϕ ,ϕ )) ⊂ ρ A ( K ( A )) . (e 27.8) Then, for any increasing sequence of finite subsets {F n } of C whose union is dense in C, anyincreasing sequence of finite subsets P n of K ( C ) with S ∞ n =1 P n = K ( C ) , and any decreasingsequence of positive numbers { δ n } with P ∞ n =1 δ n < ∞ , there exists a sequence of unitaries { u n } in U ( A ) such that Ad u n ◦ ϕ ≈ δ n ϕ o n F n and (e 27.9) ρ A (bott ( ϕ , u ∗ n u n +1 )( x )) = 0 for all x ∈ P n ( ⊂ K ( C )) (e 27.10) and for all sufficiently large n. Proof.
Note that A ∼ = A ⊗ U . Therefore, as U is of infinite type, there is a unital homomorphism s : A ⊗ U → A such that s ◦ ı is approximately unitarily equivalent to the identity map on A, where ı : A → A ⊗ U is defined by a → a ⊗ U for all a ∈ A . Therefore, we may assume that ϕ ( C ) , ϕ ( C ) ⊂ A ⊗ U . By Lemma 27.1, there exists a sequence of unitaries { v n } ⊂ A suchthat lim n →∞ Ad v n ◦ ϕ ( c ) = ϕ ( c ) for all c ∈ C. (e 27.11)We may assume that the set F n are in the unit ball of C, with dense union. For the nextfour paragraphs of the proof, fix n = 1 , , .... Put ε ′ n = min { / n +1 , δ n / } . Let C n ⊂ C be a unital C ∗ -subalgebra (in place of C n ) suchthat K i ( C n ) is finitely generated ( i = 0 , Q n be a finite set of generators of K ( C n ) , δ ′ n > δ ) be as in Lemma 24.2 for C (in place of A ), ε ′ n (in place of ε ), F n (inplace of F ), and [ ı n ]( Q n − ) (in place of P ), where ı n : C n → C is the embedding. Note that weassume that [ ı n +1 ]( Q n +1 ) ⊃ P n +1 ∪ [ ı n ]( Q n ) . (e 27.12)Write K ( C n ) = G n,f ⊕ Tor( K ( C n )) , where G n,f is a finitely generated free abelian group.Let z ,n , z ,n , ..., z f ( n ) ,n be independent generators of G n,f and z ′ ,n , z ′ ,n , ..., z ′ t ( n ) ,n be generatorsof Tor( K ( C n )) . We may assume that Q n = { z ,n , z ,n , ..., z f ( n ) ,n , z ′ ,n , z ′ ,n , ..., z ′ t ( n ) ,n } . Choose 1 / > ε ′′ n > ( h ′ , u ′ ) | K ( C n ) is a well defined group homomorphism,bott ( h ′ , u ′ ) | Q n is well defined, and (bott ( h ′ , u ′ ) | K ( C n ) ) | Q n = bott ( h ′ , u ′ ) | Q n for any unitalhomomorphism h ′ : C → A and any unitary u ′ ∈ A for which k [ h ′ ( c ) , u ′ ] k < ε ′′ n for all c ∈ G ′ n (e 27.13)for some finite subset G ′ n ⊂ C which contains F n . Let w ,n , w ,n , ..., w f ( n ) ,n , w ′ ,n , w ′ ,n , ..., w ′ t ( n ) ,n ∈ C be unitaries (note that, by Theorem 9.7of [21], C has stable rank one) such that [ w i,n ] = ( ı n ) ∗ ( z i,n ) and [ w ′ j,n ] = ( ı n ) ∗ ( z ′ j,n ) , i =1 , , ..., f ( n ) , j = 1 , , ..., t ( n ), and n = 1 , , .... Since we may choose larger G ′ n , without loss ofgenerality, we may assume that w i,n ∈ G ′ n . Let δ ′′ = 1 / n ≥ , let δ ′′ n > δ ) and G ′′ n (in place of F ) be as in Lemma27.2 associated with w ,n , w ,n , ..., w f ( n ) ,n , w ′ ,n , w ′ ,n , ..., w ′ t ( n ) ,n (in place of u , u , ..., u n ) and { w ,n − , w ,n − , ..., w f ( n − ,n − , w ′ ,n − , w ′ ,n − , ..., w ′ t ( n − ,n − } (in place of v , v , ..., v m ).Now consider all n = 1 , , .... Put ε n = min { ε ′′ n / , ε ′ n / , δ ′ n , δ ′′ n / } and G n = G ′ n ∪ G ′′ n . By(e 27.11), we may assume thatAd v n ◦ ϕ ≈ ε n ϕ on G n , n = 1 , , .... (e 27.14)Thus, bott ( ϕ ◦ ı n , v ∗ n v n +1 ) is well defined. Since Aff( T ( A )) is torsion free, τ (cid:0) bott ( ϕ ◦ ı n , v ∗ n v n +1 ) | Tor( K ( C n )) (cid:1) = 0 . (e 27.15)From (e 27.14), we have k ϕ ( w j,n )Ad v n ( ϕ ( w j,n ) ∗ ) − k < (1 /
4) sin(2 πε n ) < ε n , n = 1 , , .... (e 27.16)Define h j,n = 12 πi log( ϕ ( w j,n )Ad v n ( ϕ ( w j,n ) ∗ )) , j = 1 , , ..., f ( n ) , n = 1 , , .... (e 27.17)Then, for any τ ∈ T ( A ) , | τ ( h j,n ) | < ε n < δ ′ n , j = 1 , , ..., f ( n ), n = 1 , , .... Since Aff( T ( A )) istorsion free, and the classes [ w ′ j,n ] are torsion, it follows from Lemma 27.2 that τ ( 12 πi log( ϕ ( w ′ j,n )Ad v n ( ϕ ( w ′∗ j,n )))) = 0 , (e 27.18)47 = 1 , , ..., t ( n ) and n = 1 , , .... By the assumption that R ϕ ,ϕ ( K ( M ϕ ,ϕ )) ⊂ ρ A ( K ( A )) , byExel’s formula (see [24]), and by Lemma 3.5 of [37], we conclude that d h j,n ( τ ) = τ ( h j,n ) ∈ R ϕ ,ϕ ( K ( M ϕ ,ϕ )) ⊂ ρ A ( K ( A )) . Now define α ′ n : K ( C n ) → ρ A ( K ( A )) by α ′ n ( z j,n )( τ ) = d h j,n ( τ ) = τ ( h j,n ) , j = 1 , , ..., f ( n ) and α ′ n ( z ′ j,n ) = 0 , j = 1 , , ..., t ( n ) , (e 27.19) n = 1 , , .... Since α ′ n ( K ( C n )) is free abelian, it follows that there is a homomorphism α (1) n : K ( C n ) → K ( A ) such that( ρ A ◦ α (1) n ( z j,n ))( τ ) = τ ( h j,n ) , j = 1 , , ..., f ( n ) , τ ∈ T ( A ) , and (e 27.20) α (1) n ( z ′ j,n ) = 0 , j = 1 , , ..., t ( n ) . (e 27.21)Define α (0) n : K ( C n ) → K ( A ) by α (0) n = 0 . By the UCT, there is κ n ∈ KL ( SC n , A ) such that κ n | K i ( C n ) = α ( i ) n , i = 0 , , where SC n is the suspension of C n (here, we identify K i ( C n ) with K i +1 ( SC n )).By the UCT again, there is α n ∈ KL ( C n ⊗ C ( T ) , A ) such that α n ◦ β | K ( C n ) = κ n . In particular, α n ◦ β | K ( C n ) = α (1) n . It follows from Lemma 24.2 that there exists a unitary U n ∈ U ( A ) suchthat k [ ϕ ( c ) , U n ] k < ε ′′ n for all c ∈ F n and (e 27.22) ρ A (bott ( ϕ , U n )( z j,n )) = − ρ A ◦ α (1) n ( z j,n ) , (e 27.23) j = 1 , , ..., f ( n ) . We also have ρ A (bott ( ϕ , U n )( z ′ j,n )) = 0 , j = 1 , , ..., t ( n ) , (e 27.24)as the elements z j,n are torsion. By the Exel trace formula (see [24]), (e 27.20), and (e 27.23),we have τ ( h j,n ) = − ρ A (bott ( ϕ , U n )( z j,n )( τ ) = − τ ( 12 πi log( U n ϕ ( w j,n ) U ∗ n ϕ ( w ∗ j,n )))(e 27.25)for all τ ∈ T ( A ) , j = 1 , , ..., f ( n ) . Define u n = v n U n , n = 1 , , .... By 6.1 of [36], (e 27.25), and(e 27.23), we compute that τ ( 12 πi log( ϕ ( w j,n )Ad u n ( ϕ ( w ∗ j,n ))))) (e 27.26)= τ ( 12 πi log( U n ϕ ( w j,n ) U ∗ n v ∗ n ϕ ( w ∗ j,n ) v n ))) (e 27.27)= τ ( 12 πi log( U n ϕ ( w j,n ) U ∗ n ϕ ( w ∗ j,n ) ϕ ( w j,n ) v ∗ n ϕ ( w ∗ j,n ) v n ))) (e 27.28)= τ ( 12 πi log( U n ϕ ( w j,n ) U ∗ n ϕ ( w ∗ j,n )))) + τ ( 12 πi log( ϕ ( w j,n ) v ∗ n ϕ ( w ∗ j,n ) v n ))) (e 27.29)= ρ A (bott ( ϕ , U n )( z j,n ))( τ ) + τ ( h j,n ) = 0 (e 27.30)for all τ ∈ T ( A ) , j = 1 , , ..., f ( n ) and n = 1 , , .... By (e 27.18) and (e 27.24), τ ( 12 πi log( ϕ ( w ′ j,n )Ad u n ( ϕ (( w ′ j,n ) ∗ )))) = 0 , (e 27.31)48 = 1 , , ..., t ( n ) and n = 1 , , .... Let b j,n = 12 πi log( u n ϕ ( w j,n ) u ∗ n ϕ ( w ∗ j,n )) , (e 27.32) b ′ j,n = 12 πi log( ϕ ( w j,n ) u ∗ n u n +1 ϕ ( w ∗ j,n ) u ∗ n +1 u n ) , and (e 27.33) b ′′ j,n +1 = 12 πi log( u n +1 ϕ ( w j,n ) u ∗ n +1 ϕ ( w ∗ j,n )) , (e 27.34) j = 1 , , ..., f ( n ) and n = 1 , , .... We have, by (e 27.26), τ ( b j,n ) = τ ( 12 πi log( u n ϕ ( w j,n ) u ∗ n ϕ ( w ∗ j,n ))) (e 27.35)= τ ( 12 πi log( ϕ ( w j,n ) u ∗ n ϕ ( w ∗ j,n ) u n )) = 0 (e 27.36)for all τ ∈ T ( A ) , j = 1 , , ..., f ( n ), and n = 1 , , .... Note that τ ( b j,n +1 ) = 0 for all τ ∈ T ( A ) ,j = 1 , , ..., f ( n + 1) . It follows from Lemma 27.2 and (e 27.12) that τ ( b ′′ j,n +1 ) = 0 for all τ ∈ T ( A ) , j = 1 , , ..., f ( n ) , n = 1 , , .... Note that u n e πib ′ j,n u ∗ n = e πib j,n · e − πib ′′ j,n +1 , j = 1 , , ..., f ( n ) . Hence, using 6.1 of [36], we compute that τ ( b ′ j,n ) = τ ( b j,n ) − τ ( b ′′ j,n +1 ) = 0 for all τ ∈ T ( A ) . (e 27.37)By the Exel formula (see [24]) and (e 27.37), ρ A (bott ( ϕ , u ∗ n u n +1 ))( w ∗ j,n )( τ ) = τ ( 12 πi log( u ∗ n u n +1 ϕ ( w j,n ) u ∗ n +1 u n ϕ ( w ∗ j,n )))(e 27.38)= τ ( 12 πi log( ϕ ( w j,n ) u ∗ n u n +1 ϕ ( w ∗ j,n ) u ∗ n +1 u n )) = 0 (e 27.39)for all τ ∈ T ( A ) and j = 1 , , ..., f ( n ) . Thus, ρ A (bott ( ϕ , u ∗ n u n +1 )( w j,n ))( τ ) = 0 for all τ ∈ T ( A ) , (e 27.40) j = 1 , , ..., f ( n ), and n = 1 , , .... We also have ρ A (bott ( ϕ , u ∗ n u n +1 )( w ′ j,n ))( τ ) = 0 for all τ ∈ T ( A ) , (e 27.41) j = 1 , , ..., f ( n ), and n = 1 , , .... By 27.2, we have that ρ A (bott ( ϕ , u ∗ n u n +1 )( z )) = 0 for all z ∈ P n , (e 27.42) n = 1 , , .... Remark 27.4.
Let C be a unital separable amenable C*-algebra satisfying the UCT with finitelygenerated K i ( C ) ( i = 0 , A be a unital separable C*-algebra and let ϕ , ϕ : C → A betwo unital homomorphisms. In what follows, we will continue to use ϕ and ϕ for the inducedhomomorphisms from M k ( C ) to M k ( A ). Suppose that v ∈ U ( A ) and k v ∗ ϕ ( a ) v − ϕ ( a ) k < ε < / , a ∈ { z , z , ..., z n } ∪ F F ⊆ M k ( C ) and some z , z , ..., z n ∈ U ( M k ( C )) such that [ z ] , [ z ] , ..., [ z n ]generate K ( C ). Define W j ( t ) ∈ U ( M ( C ([0 , , M k ( A )))) as follows W j ( t ) = ( T t V T − t ) ∗ diag( ϕ ( z j ) , M k ) T t V T − t , where V = diag( v, M k ) and T t = (cid:18) cos( π t ) − sin( π t )sin( π t ) cos( π t ) (cid:19) . Note that W j (0) = diag( v ∗ ϕ ( z j ) v,
1) and W j (1) = diag( ϕ ( z j ) , W j (0) withdiag( ϕ ( z ) ,
1) by a path, we obtain a continuous path of unitary Z j ( t ) such that Z j (0) =diag( ϕ ( z j ) , Z (1 /
4) = W (0) and Z j (1) = diag( ϕ ( z j ) ,
1) and k Z j ( t ) − Z j (1 / k < / t ∈ [0 , / Z j ∈ M k ( M ϕ ,ϕ ). With sufficiently small ε >
0, since K ( C ) is finitelygenerated, the map K ( C ) ∋ [ z j ] [ Z j ] ∈ K ( M ϕ ,ϕ ) , j = 1 , , ..., n, induces a homomorphism.Set h j = 1 i diag(log( ϕ ( z j ) ∗ V ∗ ϕ ( z j ) V ) , , j = 1 , , ..., n. (e 27.43)We may specifically use Z j ( t ) = diag( ϕ ( z j ) ,
1) exp( i th j ) , t ∈ [0 , / . Still use ϕ and ϕ for the induced homomorphisms from M k ( C ⊗ C ′ ) to M k ( A ⊗ C ′ ), where C ′ is a commutative C*-algebra C ′ with finitely generated K i ( C ′ ) ( i = 0 , z , ..., z n ∈ M ∞ ( C ⊗ C ′ ) which generates K ( C ⊗ C ′ ). We also obtain a homomorphism K ( C ⊗ C ′ ) → K ( M ϕ ,ϕ ⊗ C ′ ) provided that ε is small.Let F ⊂ C be a finite subset and ε >
0. Suppose that there is a unitary v ∈ U ( A ) such thatad v ◦ ϕ ≈ ε ϕ on F . Let U ′ ( t ) = T t V T − t . Define L ( c )( t ) = ( U ′ ( 4 t −
13 )) ∗ diag( ϕ ( c ) , U ′ ( 4 t −
13 ) , t ∈ [1 / ,
1] (e 27.44)and L ( c )( t ) = 4 tL ( c )(1 /
4) + (1 − t )diag( ϕ ( c ) , , t ∈ [0 , / . Note that L maps C into M ( M ϕ ,ϕ ). Thus, since K i ( C ) ( i = 0 ,
1) is finitely generated, byCorollary 2.11 of [6], there is N > Λ ( K ( C ) , K ( A )) is determinedby its restriction to K i ( A, Z /n Z ), i = 0 , n = 0 , , ..., N . Hence, if ε is sufficiently small and F is sufficiently large, there is γ ϕ ,ϕ ,v ∈ Hom Λ ( K ( C ) , K ( M ϕ ,ϕ )) (e 27.45)such that [ L ] | P = γ ϕ ,ϕ ,v | P (e 27.46)for any given finite subset P ⊂ K ( C ). 50ne computes that Z τ ( dZ j ( t ) dt Z j ( t )) dt = τ ( h j ) , τ ∈ T ( A ) . Therefore, if R ϕ ,ϕ ◦ γ ϕ ,ϕ ,v ( K ( C )) = 0, then τ ( h j ) = 0 , τ ∈ T ( A ) . (e 27.47)On the other hand, for any given η > { z , z , ..., z n } of generators of K ( C ),by (e 27.43), | τ ( h i ) | < η, τ ∈ T ( A ) , (e 27.48)provided that ε is sufficiently small and F is sufficiently large.Now, assume that ϕ = ϕ . Then, with sufficiently large F and sufficiently small ε , theelement Bott( ϕ , v ) : K ( C ) → K ( SA ) is well defined.We have the following splitting short exact sequence:0 / / S A ı / / M ϕ ,ϕ π e / / C / / . Define θ : C → M ϕ ,ϕ by θ ( b ) = ϕ ( b ) as a constant element in M ϕ ,ϕ . Then θ may be identifiedwith a splitting map and K ( M ϕ ,ϕ ) may be written as K ( SA ) ⊕ K ( C ) . Let P : K ( M ϕ ,ϕ ) ∼ = K ( C ) ⊕ K ( SA ) → K ( SA ) be the standard projection map. Onecan verify that for any two elements x, y ∈ K ( C ) if Bott( ϕ , v )( x ) = Bott( ϕ , v )( y ), then P ◦ γ ϕ ,ϕ ,v ( x ) = P ◦ γ ϕ ,ϕ ,v ( y ). So we will also use Γ(Bott( ϕ , v )) to denote the map P ◦ γ ϕ ,ϕ ,v ∈ Hom Λ ( K ( C ) , K ( SA )).By shifting the index, we see Γ(Bott( ϕ , v )) | P maps P to K ( A ) . One may identify P withid M ϕ ,ϕ − [ θ ] ◦ [ π e ] . Note that π e ◦ θ = id C and π e ◦ L = id C . So Γ(Bott( ϕ , v )) | P = γ ϕ ,ϕ ,v | P − θ | P . (e 27.49)Furthermore, it is shown in 10.6 of [37] that Γ(Bott( ϕ , v )) = 0 if and only if Bott( ϕ , v ) = 0.Note that since the K-theory of C is finitely generated, by Corollary 2.11 of [6], one has thatany element of Hom Λ ( K ( C ) , K ( A )) is determined by its restriction to K i ( A, Z /n Z ), i = 0 , n = 0 , , ..., N . Fix separable commutative C*-algebras C = C , C , ..., C N , C N +1 , ..., C N +1 with K ( C n ) = Z /n Z and K ( C n ) = { } , n = 0 , , ..., N , and C N + i = SC i − , i = 1 , , ..., N +1 . For each C ⊗ C ′ , where C ′ is one of the C , C , ..., C N +1 ,fix a finite set of unitaries z ( n )1 , z ( n )2 , ..., z ( n ) k ( n ) of M N ( ^ C ⊗ C ′ ) ⊂ M N ( C ⊗ e C ′ ) (for some N ≥ K ( C ⊗ C n ) , n = 0 , , ..., N + 1 . Let C ′ i = M N ( C i ) , i = 0 , , ..., N + 1 . Let 1 / > ε > / > η > . Choose 0 < δ < ε/
F ⊆ A sufficiently large such that if u ∗ ϕ u ≈ δ ϕ on F for some unitary u ∈ A , then, for each C n ,u ∗ ˜ ϕ u ≈ ε/ ˜ ϕ on F ,n , (e 27.50)51here F ,n is a finite subset which contains { z ( n )1 , z ( n )2 , ..., z ( n ) k ( n ) } , n = 0 , , ..., N + 1 . We alsoassume that δ is sufficiently small and F is sufficiently large so that (e 27.45), (e 27.46), (e 27.48),(e 27.49) hold.Suppose that there are unitaries u , u ∈ A such that u ∗ i ϕ u i ≈ δ/ ϕ on F ,i = 1 , . Then, as in (e 27.50), for each C n ,u ∗ i ˜ ϕ u i ≈ ε/ ˜ ϕ on F ,n , (e 27.51)where F ,n is a finite subset which contains { z ( n )1 , z ( n )2 , ..., z ( n ) k ( n ) } , n = 0 , , ..., N + 1 . Let L i : C → M ϕ ,ϕ and γ ϕ ,ϕ ,u i ∈ Hom Λ ( K ( C ) , K ( M ϕ ,ϕ )) be the element defined by the pair ( ϕ , u i )( i = 1 ,
2) as above.On the other hand, one also has that u u ∗ ϕ u u ∗ ≈ δ ϕ on F . Note that π e ◦ ( L − L ) = 0 . Fix n ∈ { , , ..., N + 1 } . Consider z ∈ { z ( n )1 , z ( n )2 , ..., z ( n ) k ( n ) } and ˜ u i = u i ⊗ f C ′ n , i = 1 , . Wealso write ˜ ϕ i for ϕ i ⊗ id f C ′ n . Define ˜ T ( t ) = T t − / for t ∈ [1 / , /
4] and ˜ T t = T for t ∈ [3 / , . Let W i ( t ) =˜ T t (cid:18) ˜ u ∗ i
00 1 (cid:19) ˜ T ∗ t , t ∈ [1 / , , and W i ( t ) = diag(1 ,
1) for t ∈ [0 , / , i = 0 , . Note that k diag(˜ u ∗ ˜ ϕ ( z )˜ u ˜ u ∗ ˜ ϕ ( z ) ∗ ˜ u , − diag(1 , k < ε < / . (e 27.52)There is a continuous path d ( z )( t ) (for t ∈ [0 , / d ( z )(0) = diag(1 ,
1) and d ( z )(1 /
4) =diag(˜ u ∗ ˜ ϕ ( z )˜ u ˜ u ∗ ˜ ϕ ( z ) ∗ ˜ u,
1) and k d ( z )( t ) − diag(1 , k < ε for all t ∈ [0 , / . (e 27.53)Define, for t ∈ [1 / , ,d ( z )( t ) = (cid:0) W ( t )diag( ˜ ϕ ( z ) , W ( t ) ∗ (cid:1)(cid:0) W ( t )diag( ˜ ϕ ( z ) ∗ , W ( t ) ∗ (cid:1) = ˜ T t (cid:18) ˜ u ∗
00 1 (cid:19) ˜ T ∗ t (cid:18) ˜ ϕ ( z ) 00 1 (cid:19) ˜ T t (cid:18) ˜ u ˜ u ∗
00 1 (cid:19) ˜ T ∗ t (cid:18) ˜ ϕ ( z ) ∗
00 1 (cid:19) ˜ T t (cid:18) ˜ u
00 1 (cid:19) ˜ T ∗ t . On [3 / , , define d ( z )( t ) = diag(1 , . Define U ( t ) = diag( W ( t ) ∗ , W ( t )) for t ∈ (1 / , / , On[0 , / , there is a continuous path U ( t ) of unitaries in M ( A ⊗ C ′ n ) with U (0) = diag(1 , , , U (1 /
4) = diag( W (0) ∗ , W (0)) . On [3 / , , there is a continuous path U ( t ) of unitaries in M ( A ⊗ C ′ n ) with U (3 /
4) = diag( W (3 / ∗ , W (3 / U (1) = diag(1 , , , . For n = 0 (so z is represented by unitaries in M N ( C )), we may also assume that (see(e 27.48)), | τ ( h z ) | < η for all τ ∈ T ( A ) , (e 27.54)where h z = diag(log( ˜ ϕ ( z ) ∗ ˜ u ∗ ˜ ϕ ( z )˜ u , . Note that d ( z )(0) = d ( z )(1) = 1 and[ d ( z )] = γ ϕ ,ϕ ,u ( z ) − γ ϕ ,ϕ ,u ( z ) , (e 27.55)52here γ ϕ ,ϕ ,u i , i = 1 ,
2, are the maps defined above (see (e 27.44)).One has, on [1 / , / ,U ( t ) ∗ diag( d ( z )( t ) , , U ( t )= diag( (cid:18) ˜ T t (cid:18) ˜ u
00 1 (cid:19) ˜ T ∗ t (cid:19) (cid:18) ˜ T t (cid:18) ˜ u ∗
00 1 (cid:19) ˜ T ∗ t (cid:18) ˜ ϕ ( z ) 00 1 (cid:19) ˜ T t (cid:18) ˜ u ˜ u ∗
00 1 (cid:19) ˜ T ∗ t (cid:18) ˜ ϕ ( z ) ∗
00 1 (cid:19)(cid:19) , (cid:18) (cid:19) ) (e 27.56)= diag( ˜ T t (cid:18) ˜ u ˜ u ∗
00 1 (cid:19) ˜ T ∗ t (cid:18) ˜ ϕ ( z ) 00 1 (cid:19) ˜ T t (cid:18) ˜ u ˜ u ∗
00 1 (cid:19) ˜ T ∗ t (cid:18) ˜ ϕ ( z ) ∗
00 1 (cid:19) , (cid:18) (cid:19) ) , (e 27.57)on [0 , / , and on [3 / , , k U ( t ) ∗ diag( d ( z )( t ) , , U ( t ) − diag(1 , , , k < ε. (e 27.58)Moreover, U (0) ∗ diag( d ( z )(0) , , U (0) = U ∗ (1)diag( d ( z )(1) , , U (1) = diag(1 , , , . There-fore [ d ( z )] = [ U ∗ d ( z ) U ] in K ( SA ⊗ f C n ) . (e 27.59)Since the short exact sequence 0 → SA ⊗ C n → SA ⊗ f C n → SA → d ( z )] = [ U ∗ d ( z ) U ] in K ( SA ⊗ C n ) . (e 27.60)On the other hand, the class Γ(Bott( ϕ , u u ∗ ))( z ) is represented by the path r ( t ) = ˜ T t (cid:18) ˜ u ˜ u ∗
00 1 (cid:19) ˜ T ∗ t (cid:18) ˜ ϕ ( z ) 00 1 (cid:19) ˜ T t (cid:18) ˜ u ˜ u ∗
00 1 (cid:19) ˜ T ∗ t (cid:18) ˜ ϕ ( z ) ∗
00 1 (cid:19) for all t ∈ [1 / , / , and k r ( t ) − diag(1 , k < ε for all t ∈ [0 , / ∪ [3 / , . (e 27.61)Hence (see (e 27.57)), by (e 27.55) and (e 27.60),Γ(Bott( ϕ , u u ∗ )) = γ ϕ ,ϕ ,u − γ ϕ ,ϕ ,u . (e 27.62) Theorem 27.5.
Let C be a unital simple C ∗ -algebra as in Theorem 14.10 of [21], let A be aunital separable simple C ∗ -algebra in B , let C = C ⊗ U and let A = A ⊗ U , where U and U are UHF-algebras of infinite type. Suppose that ϕ , ϕ : C → A are two unital monomorphisms.Then ϕ and ϕ are asymptotically unitarily equivalent if and only if [ ϕ ] = [ ϕ ] i n KK ( C, A ) , (e 27.63) ϕ ‡ = ψ ‡ , ( ϕ ) T = ( ϕ ) T , and R ϕ ,ϕ = 0 . (e 27.64) Proof.
We will prove the “if ” part only. The “only if” part follows from 4.3 of [40]. Note C = C ⊗ U can be also regarded as a C ∗ -algebra as in Theorem 14.10 of [21]. Let C =lim n →∞ ( C n , ı n ) be as in Theorem 14.10 of [21], where each ı n : C n → C n +1 is an injectivehomomorphism. Let F n ⊂ C be an increasing sequence of finite subsets of C such that S ∞ n =1 F n is dense in C. Put M ϕ ,ϕ = { ( f, c ) ∈ C ([0 , , A ) ⊕ C : f (0) = ϕ ( c ) and f (1) = ϕ ( c ) } . C satisfies the UCT, the assumption that [ ϕ ] = [ ϕ ] in KK ( C, A ) implies that thefollowing exact sequence splits:0 → K ( SA ) → K ( M ϕ ,ϕ ) ⇋ π e θ K ( C ) → θ ∈ Hom( K ( C ) , K ( A )) , where π e : M ϕ ,ϕ → C is the projection to C defined inDefinition 2.20 of [21]. Furthermore, since τ ◦ ϕ = τ ◦ ϕ for all τ ∈ T ( A ), and R ϕ ,ϕ = 0 , wemay also assume that R ϕ ,ϕ ( θ ( x )) = 0 for all x ∈ K ( C ) . (e 27.66)By [6], we have lim n →∞ ( K ( C n ) , [ ı n ]) = K ( C ) . (e 27.67)Since K i ( C n ) is finitely generated, there exists K ( n ) ≥ Λ ( F K ( n ) K ( C n ) , F K ( n ) K ( A )) = Hom Λ ( K ( C n ) , K ( A )) (e 27.68)(see also [6] for the notation F m there).Let δ ′ n > δ ), σ ′ n > σ ), G ′ n ⊂ C (in place of G ), { p ′ ,n , p ′ ,n , ..., p ′ I ( n ) ,n ) , q ′ ,n , q ′ ,n , ..., q ′ I ( n ) ,n } (in place of { p , p , ..., p k , q , q , ..., q k } ) , P ′ n ⊂ K ( C )(in place of P ) corresponding to 1 / n +2 (in place of ε ), and F n (in place of F ) be as providedby Lemma 25.4 (see also Remark 25.5). Note that, by the choice as in 25.4, we may assumethat G ′ u,n , the subgroup generated by { [ p ′ i,n ] − [ q ′ i,n ] : 1 ≤ i ≤ I ( n ) } is free abelian.Without loss of generality, we may assume that G ′ n ⊂ ı n, ∞ ( G n ) and P ′ n ⊂ [ ı n, ∞ ]( P n ) forsome finite subset G n ⊂ C n , and for some finite subset P n ⊂ K ( C n ) , and we may assume that p ′ i,n = ı n, ∞ ( p i,n ) and q ′ i,n = ı n, ∞ ( q i,n ) for some projections p i,n , q i,n ∈ C n , i = 1 , , ..., I ( n ) . Wemay also assume that the subgroup G n,u generated by { [ p i,n ] − [ q i,n ] : 1 ≤ i ≤ I ( n ) } is freeabelian and p i,n , q i,n ∈ G n , n = 1 , , ..., I ( n ) . We may assume that P n contains a set of generators of F K ( n ) K ( C n ) , F n ⊂ G ′ n , and δ ′ n < / n +3 . We may also assume that Bott( h ′ , u ′ ) | P n is well defined whenever k [ h ′ ( a ) , u ′ ] k < δ ′ n forall a ∈ G ′ n and for any unital homomorphism h ′ from C n and unitary u ′ in the target algebra.Note that Bott( h ′ , u ′ ) | P n defines Bott( h ′ u ′ ) . We may further assume thatBott( h, u ) | P n = Bott( h ′ , u ) | P n (e 27.69)provided that h ≈ δ ′ n h ′ on G ′ n . We may also assume that δ ′ n is smaller than δ/
16 for the δ definedin 2.15 of [40] for C n (in place of A ) and P n (in place of P ). Let k ( n ) ≥ n (in place of n ), η ′ n > δ ), and Q k ( n ) ⊂ K ( C k ( n ) ) be as provided by Lemma 24.5 for δ ′ k ( n ) / ε ), ı n, ∞ ( G k ( n ) ) (in place of F ), P k ( n ) (in place of P ), { p i,n , q i,n , : i = 1 , , ..., k ( n ) } (in place of { p i , q i : i = 1 , , ..., k } ), and σ ′ k ( n ) /
16 (in place of σ ). We may assume that Q k ( n ) generates thegroup K ( C k ( n ) ) . Since P generates F K ( n ) K ( C k ( n +1) ) , we may assume that Q n ⊂ P k ( n ) . Since K i ( C n ) ( i = 0 ,
1) is finitely generated, by (e 27.68), we may further assume that [ ı k ( n ) , ∞ ]is injective on [ ı n,k ( n ) ]( K ( C n )) , n = 1 , , .... Passing to a subsequence, we may also assume that k ( n ) = n + 1 . Let δ n = min { η n , σ ′ n , δ ′ n / } . By Lemma 27.3, there are unitaries v n ∈ U ( A ) suchthat Ad v n ◦ ϕ ≈ δ n +1 / ϕ on ı n, ∞ ( G n +1 ) , (e 27.70) ρ A (bott ( ϕ , v ∗ n v n +1 ))( x ) = 0 for all x ∈ ( ı n, ∞ ) ∗ ( K ( C n +1 )) , and (e 27.71) k [ ϕ ( c ) , v ∗ n v n +1 ] k < δ n +1 / a ∈ ı n, ∞ ( G n +1 ) (e 27.72)54Recall that K ( C n +1 ) is finitely generated). Note that, by (e 27.69), we may also assume thatBott( ϕ , v n +1 v ∗ n ) | [ ı n, ∞ ]( P n ) = Bott( v ∗ n ϕ v n , v ∗ n v n +1 ) | [ ı n, ∞ ]( P n ) (e 27.73)= Bott( ϕ , v ∗ n v n +1 ]) | [ ı n, ∞ ]( P n ) . (e 27.74)In particular, bott ( v ∗ n ϕ v n , v ∗ n v n +1 )( x ) = bott ( ϕ , v ∗ n v n +1 )( x ) (e 27.75)for all x ∈ ( ı n, ∞ ) ∗ ( K ( C n +1 )) . Applying 10.4 and 10.5 of [37] (see also Remark 27.4), we may assume that the pair ( ϕ , ϕ )and v n define an element γ n := γ ϕ | Cn +1 ,ϕ | Cn +1 ,v n ∈ Hom Λ ( K ( C n +1 ) , K ( M ϕ ,ϕ )) and [ π e ] ◦ γ n =[id C n +1 ] (see Remark 27.4 for the definition of γ n ). Moreover, we may assume (see (e 27.54))that | τ (log( ϕ ◦ ı n, ∞ ( z ∗ j )˜ v n ϕ ◦ ı n, ∞ ( z j )˜ v n )) | < δ n +1 , (e 27.76) j = 1 , , ..., r ( n ) , where { z , z , ..., z r ( n ) } ⊂ U ( M k ( C n +1 )) , and this set generates K ( C n +1 ), andwhere ˜ v n = diag( k z }| { v n , v n , ..., v n ) . We may assume that z j ∈ Q n ⊂ P n , j = 1 , , ..., r ( n ) . Let H n = [ ı n +1 ]( K ( C n +1 )) ⊂ K ( C n +2 ) . Since S n =1 [ ı n +1 , ∞ ]( K ( C n )) = K ( C ) and [ π e ] ◦ γ n =[id C n +1 ] , we conclude that K ( M ϕ ,ϕ ) = K ( SA ) + ∞ [ n =1 γ n +1 ( H n ) . (e 27.77)Thus, passing to a subsequence, we may further assume that γ n +1 ( H n ) ⊂ K ( SA ) + γ n +2 ( H n +1 ) , n = 1 , , .... (e 27.78)Identifying H n with γ n +1 ( H n ) , let us write j n : K ( SA ) ⊕ H n → K ( SA ) ⊕ H n +1 for theinclusion in (e 27.78). By (e 27.77), the inductive limit is K ( M ϕ ,ϕ ) . From the definition of γ n , we note that γ n − γ n +1 ◦ [ ı n +1 ] maps K ( C n +1 ) into K ( SA ) . By Remark 27.4 (see (e 27.62)), themap Γ(Bott( ϕ , v n v ∗ n +1 )) | H n = ( γ n +1 − γ n +2 ◦ [ ı n +2 ]) | H n (see 27.4 for the definition of Γ(Bott( , ))) is then a homomorphism ξ n : H n → K ( SA ) . Put ζ n = γ n +1 | H n . Then j n ( x, y ) = ( x + ξ n ( y ) , [ ı n +2 ]( y )) (e 27.79)for all ( x, y ) ∈ K ( SA ) ⊕ H n . Thus we obtain the following diagram:0 → K ( SA ) → K ( SA ) ⊕ H n → H n → k k ւ ξ n ↓ [ ı n +2 , ∞ ] ↓ [ ı n +2 , ∞ ] → K ( SA ) → K ( SA ) ⊕ H n +1 → H n +1 → k k ւ ξ n +1 ↓ [ ı n +3 , ∞ ] ↓ [ ı n +3 , ∞ ] → K ( SA ) → K ( SA ) ⊕ H n +2 → H n +2 → . By the assumption that ¯ R ϕ ,ϕ = 0 , the map θ also gives the following decomposition:ker R ϕ ,ϕ = ker ρ A ⊕ K ( C ) . (e 27.80)55efine θ n = θ ◦ [ ı n +2 , ∞ ] and κ n = ζ n − θ n . Note that θ n = θ n +1 ◦ [ ı n +2 ] . (e 27.81)We also have that ζ n − ζ n +1 ◦ [ ı n +2 ] = ξ n . (e 27.82)Since [ π e ] ◦ ( ζ n − θ n ) | H n = 0 , κ n maps H n into K ( SA ) . It follows that κ n − κ n +1 ◦ [ ı n +2 ] = ζ n − θ n − ζ n +1 ◦ [ ı n +2 ] + θ n +1 ◦ [ ı n +2 ] (e 27.83)= ζ n − ζ n +1 ◦ [ ı n +2 ] = ξ n . (e 27.84)It follows from Lemma 26.3 that there are an integer N ≥ , a unital δ n +1 - ı n +1 ( G n +1 )-multiplicative completely positive linear map L n : ı n, ∞ ( C n +1 ) → M N ( M ϕ ,ϕ ) , a unital homo-morphism h : ı n +1 , ∞ ( C n +1 ) → M N ( C ) , and a continuous path of unitaries { V n ( t ) : t ∈ [0 , / } in M N ( A ) such that [ L n ] | P ′ n +1 is well defined, V n (0) = 1 M N ( A ) , [ L n ◦ ı n, ∞ ] | P n = ( θ ◦ [ ı n +1 , ∞ ] + [ h ◦ ı n +1 , ∞ ]) | P n ,π t ◦ L n ◦ ı n +1 , ∞ ≈ δ n +1 / Ad V n ( t ) ◦ (( ϕ ◦ ı n +1 , ∞ ) ⊕ ( h ◦ ψ n +1 , ∞ ))on ı n +1 , ∞ ( G n +1 ) for all t ∈ (0 , / ,π t ◦ L n ◦ ı n +1 , ∞ ≈ δ n +1 / Ad V n (3 / ◦ (( ϕ ◦ ı n +1 , ∞ ) ⊕ ( h ◦ ı n +1 , ∞ ))on ı n +1 , ∞ ( G n +1 ) for all t ∈ (3 / , , and π ◦ L n ◦ ı n +1 , ∞ ≈ δ n +1 / ϕ ◦ ı n +1 , ∞ ⊕ h ◦ ı n +1 , ∞ on ı n +1 , ∞ ( G n +1 ) , where π t : M ϕ ,ϕ → A is the point evaluation at t ∈ (0 , . Note that R ϕ ,ϕ ( θ ( x )) = 0 for all x ∈ ı n +1 , ∞ ( K ( C n +1 )) . As in (e 27.47) (see also 10.4 of[37]), τ (log(( ϕ ( x ) ⊕ h ( x ) ∗ V n (3 / ∗ ( ϕ ( x ) ⊕ h ( x )) V n (3 / x = ı n +1 , ∞ ( y ) , where y is in a set of generators of K ( C n +1 ) , and for all τ ∈ T ( A ) . Define W ′ n = diag( v n +1 , ∈ M N ( A ) . Then˜ κ n := Bott(( ϕ ⊕ h ) ◦ ı n +1 , ∞ , W ′ n ( V n (3 / ∗ ) (e 27.86)defines a homomorphism in Hom Λ ( K ( C n +1 ) , K ( SA )) . By (e 27.76) | τ (log(( ϕ ⊕ h ) ◦ ı n +1 , ∞ ( z j ) ∗ ( W ′ n ) ∗ ( ϕ ⊕ h ) ◦ ı n +1 , ∞ ( z j ) W ′ n )) | < δ n +1 , (e 27.87) j = 1 , , ..., r ( n ) . One computes (see (e 27.49)) thatΓ(Bott( ϕ ◦ ı n +1 , ∞ ⊕ h , W ′ n V (3 / ∗ ) | P n = ( γ n +1 − θ )[ ı n ] | P n . (e 27.88)Put ˜ V n = V n (3 / . Let b j,n = 12 πi log( ˜ V ∗ n ( ϕ ⊕ h ) ı n +1 , ∞ ( z j ) ˜ V n ( ϕ ⊕ h ) ◦ ı n +1 , ∞ ( z j ) ∗ ) , (e 27.89) b ′ j,n = 12 πi log(( ϕ ⊕ h ) ◦ ı n +1 , ∞ ( z j ) ˜ V n ( W ′ n ) ∗ ( ϕ ⊕ h ) ◦ ı n +1 , ∞ ( z j ) ∗ W ′ n ˜ V ∗ n ) , and (e 27.90) b ′′ j,n = 12 πi log(( ϕ ⊕ h ) ı n +1 , ∞ ( z j )( W ′ n ) ∗ ( ϕ ⊕ h ) ◦ ı n +1 , ∞ ( z j ) ∗ W ′ n ) , (e 27.91)56 = 1 , , ..., r ( n ) . By (e 27.85) and (e 27.87), τ ( b j,n ) = 0 and | τ ( b ′′ j,n ) | < δ n +1 (e 27.92)for all τ ∈ T ( A ) . Note that ˜ V ∗ n e πib ′ j,n ˜ V n = e πib j,n e πib ′′ j,n . (e 27.93)Then, by 6.1 of [36] and by (e 27.92), τ ( b ′ j,n ) = τ ( b j,n ) − τ ( b ′′ j,n ) = τ ( b ′′ j,n ) and | τ ( b ′ j,n ) | < δ n +1 (e 27.94)for all τ ∈ T ( A ) . It follows from this, (e 27.86), and (e 27.90) that | ρ A (˜ κ n ( z j ))( τ ) | < δ n +1 , j = 1 , , ..., (e 27.95)for all τ ∈ T ( A ) . It follows from 24.5 that there is a unitary w ′ n ∈ U ( A ) such that k [ ϕ ( a ) , w ′ n ] k < δ ′ n +1 / a ∈ ı n +1 , ∞ ( G n +1 ) and (e 27.96)Bott( ϕ ◦ ı n +1 , ∞ , w ′ n ) = − ˜ κ n ◦ [ ı n +1 ] . (e 27.97)By (e 27.69), Bott( ϕ ◦ ı n +1 , ∞ , v ∗ n w ′ n v n ) | P n = − ˜ κ n ◦ [ ı n +1 ] | P n . (e 27.98)It follows from (e 27.49) (see also 10.6 of [37]) and (e 27.88) thatΓ(Bott( ϕ ◦ ı n +1 , ∞ , w ′ n )) = − κ n ◦ [ ı n +1 ] and (e 27.99)Γ(Bott( ϕ ◦ ı n +2 , ∞ , w ′ n +1 )) = − κ n +1 ◦ [ ı n +2 ] . (e 27.100)We also have Γ(Bott( ϕ ◦ ı n +1 , ∞ , v n v ∗ n +1 )) | H n = ζ n − ζ n +1 ◦ [ ı n +2 ] = ξ n . (e 27.101)But, by (e 27.83) and (e 27.84),( − κ n + ξ n + κ n +1 ◦ [ ı n +2 ]) = 0 . (e 27.102)By 10.6 of [37] (see also Remark 27.4), Γ(Bott( ., . )) = 0 if and only if Bott( ., . ) = 0 . Thus, by(e 27.98), (e 27.99), and (e 27.101), − Bott( ϕ ◦ ı n +1 , ∞ , w ′ n ) + Bott( ϕ ◦ ı n +1 , ∞ , v n v ∗ n +1 ) + Bott( ϕ ◦ ı n +1 , ∞ , w ′ n +1 ) = 0 . (e 27.103)Put w n = v ∗ n ( w ′ n ) v n and u n = v n w ∗ n , n = 1 , , .... Then, by (e 27.70) and (e 27.96),Ad u n ◦ ϕ ≈ δ ′ n / ϕ for all a ∈ ı n +1 , ∞ ( G n +1 ) . (e 27.104)From (e 27.73), (e 27.69), and (e 27.103), we compute thatBott( ϕ ◦ ı n +1 , ∞ , u ∗ n u n +1 ) = Bott( ϕ ◦ ı n +1 , ∞ , w n v ∗ n v n +1 w ∗ n +1 ) (e 27.105)= Bott( ϕ ◦ ı n +1 , ∞ , w n ) + Bott( ϕ ◦ ı n +1 , ∞ , v ∗ n v n +1 ) (e 27.106)+Bott( ϕ ◦ ı n +1 , ∞ , w ∗ n +1 ) (e 27.107)= Bott( ϕ ◦ ı n +1 , ∞ , w ′ n ) + Bott( ϕ ◦ ı n +1 , ∞ , v n +1 v ∗ n ) (e 27.108)+Bott( ϕ ◦ ı n +1 , ∞ , ( w ′ n +1 ) ∗ ) (e 27.109)= − [ − Bott( ϕ ◦ ı n +1 , ∞ , w ′ n ) + Bott( ϕ ◦ ı n +1 , ∞ , v n v ∗ n +1 ) (e 27.110)+Bott( ϕ ◦ ı n +1 , ∞ , w ′ n +1 )] = 0 . (e 27.111)57et x i,n = [ p i,n ] − [ q i,n ] , ≤ i ≤ I ( n ) . Note that we assume that G u,n is a free abelian groupgenerated by { x i,n : 1 ≤ i ≤ I ( n ) } . Without loss of generality, we may assume that thesegenerators are independent. Define, for each n ≥ , a homomorphism Λ n : G u,n → U ( A ) /CU ( A )by Λ n ( x i,n ) = ( h ((1 − e i,n ) + e i,n u n )((1 − e ′ i,n ) + e ′ i,n u ∗ n ) i , (e 27.112)where e i,n = ϕ ◦ ı n +1 , ∞ ( p i,n ) , e ′ i,n = ϕ ◦ ı n +1 , ∞ ( q i,n ) , i = 1 , , ..., I ( n ) . In what follows, we willconstruct unitaries s , s , ..., s n , ... in A such that || [ ϕ ◦ ι n +1 , ∞ ( f ) , s n ] || < δ ′ n +1 / f ∈ G n +1 , (e 27.113)Bott( ϕ ◦ ι n +1 , ∞ , s n ) | P n = 0 , and (e 27.114)dist( h ((1 − e i,n ) + e i,n s n )((1 − e ′ i,n ) + e ′ i,n s ∗ n ) i , Λ n ( − x i,n )) < σ ′ n / , (e 27.115)Let s = 1, and assume that s , s , ..., s n are already constructed. Let us construct s n +1 .Note that by (e 27.105), the K class of the unitary u ∗ n u n +1 is trivial. In particular, the K classof s n u ∗ n u n +1 is trivial. Since Λ factors through G ′ u,n , applying Theorem 24.5 to ϕ ◦ ι n +2 , ∞ , oneobtains a unitary s n +1 ∈ B such that || [ ϕ ◦ ι n +2 , ∞ ( f ) , s n ] || < δ ′ n +1 / f ∈ G n +2 , (e 27.116)Bott( ϕ ◦ ι n +2 , ∞ , s n +1 ) | P n = 0 , and (e 27.117)dist( h ((1 − e i,n +1 ) + e i,n +1 s ∗ n +1 )((1 − e ′ i,n +1 ) + e ′ i,n +1 s n +1 ) i , Λ n +1 ( − x i,n +1 )) < σ ′ n / , (e 27.118) i = 1 , , ..., I ( n + 1) . Then s , s , ..., s n +1 satisfies (e 27.113), (e 27.114), and (e 27.115).Put f u n = u n s ∗ n . Then by (e 27.104) and (e 27.113), one hasad f u n ◦ ϕ ≈ δ ′ n ϕ for all a ∈ ı n +1 , ∞ ( G n +1 ) . (e 27.119)By (e 27.105) and (e 27.114), one hasBott( ϕ ◦ ι n +1 , ∞ , ( f u n ) ∗ ] u n +1 ) | P n = 0 . (e 27.120)Note that h (1 − e i,n ) + e i,n u n +1 s ∗ n +1 ih (1 − e ′ i,n +1 ) + e ′ i,n +1 s n +1 u ∗ n +1 i = c c c c = c c c c , (e 27.121)where c = h (1 − e i,n +1 ) + e i,n +1 u n +1 i , c = h (1 − e i,n +1 ) + e i,n +1 s ∗ n +1 i , (e 27.122) c = h (1 − e ′ i,n +1 ) + e ′ i,n +1 s n +1 i , c = h (1 − e ′ i,n +1 ) + e ′ i,n +1 u ∗ n +1 i . (e 27.123)Therefore, by (e 27.118) and (e 27.112), one hasdist( h ((1 − e i,n +1 ) + e i,n +1 ] u n +1 )((1 − e ′ i,n +1 ) + e ′ i,n +1 ] u n +1 ∗ ) i ) , ¯1) (e 27.124) < σ ′ n +1 /
16 + dist(Λ( − x i,n +1 )Λ( x i,n +1 ) , ¯1) = σ ′ n +1 / , (e 27.125) i = 1 , , ..., I ( n ) . Therefore, by Lemma 25.4 (and Remark 25.5), there exists a continuous andpiecewise smooth path of unitaries { z n ( t ) : t ∈ [0 , } of A such that z n (0) = 1 , z n (1) = ( f u n ) ∗ ] u n +1 and (e 27.126) k [ ϕ ( a ) , z n ( t )] k < / n +2 for all a ∈ F n and t ∈ [0 , . (e 27.127)58efine u ( t + n −
1) = f u n z n +1 ( t ) t ∈ (0 , . Note that u ( n ) = ] u n +1 for all integers n and { u ( t ) : t ∈ [0 , ∞ ) } is a continuous path of unitariesin A. One estimates that, by (e 27.104) and (e 27.127),Ad u ( t + n − ◦ ϕ ≈ δ ′ n Ad z n +1 ( t ) ◦ ϕ ≈ / n +2 ϕ on F n (e 27.128)for all t ∈ (0 , . It then follows thatlim t →∞ u ∗ ( t ) ϕ ( a ) u ( t ) = ϕ ( a ) for all a ∈ C. (e 27.129)
28 Rotation maps and strong asymptotic equivalence
Lemma 28.1.
Let A be a unital separable simple C ∗ -algebra of stable rank one. Suppose that u ∈ CU ( A ) . Then, for any continuous and piecewise smooth path { u ( t ) : t ∈ [0 , } ⊂ U ( A ) with u (0) = u and u (1) = 1 A ,D A ( { u ( t ) } ) ∈ ρ A ( K ( A )) (recall Definition 2.16 of [21] for D A ). (e 28.1) Proof.
It follows from Corollary 11.11 of [21] that the map j : u diag( u, , ...,
1) from U ( A )to U ( M n ( A )) induces an isomorphism from U ( A ) /CU ( A ) to U ( M n ( A )) /CU ( M n ( A )) . Then theconclusion follows from 3.1 and 3.2 of [52].
Lemma 28.2.
Let A be a unital separable simple C ∗ -algebra of stable rank one. Suppose that B is a unital separable C ∗ -algebra and suppose that ϕ, ψ : B → A are two unital monomorphismssuch that [ ϕ ] = [ ψ ] i n KK ( B, A ) (e 28.2) ϕ T = ψ T and ϕ ‡ = ψ ‡ . (e 28.3) Then R ϕ,ψ ∈ Hom( K ( B ) , ρ A ( K ( A ))) . (e 28.4) Proof.
Let z ∈ K ( B ) be represented by the unitary u ∈ U ( M m ( B )) for some integer m . Then,by (e 28.3), ( ϕ ⊗ id M m )( u )( ψ ⊗ id M m )( u ) ∗ ∈ CU ( M m ( A )) . Suppose that { u ( t ) : t ∈ [0 , } is a continuous and piecewise smooth path in M m ( U ( A )) suchthat u (0) = ( ϕ ⊗ id M m )( u ) and u (1) = ( ψ ⊗ id M m )( u ) . Put w ( t ) = ( ψ ⊗ id M m )( u ) ∗ u ( t ) . Then w (0) = ( ψ ⊗ id M m )( u ) ∗ ( ϕ ⊗ id M m )( u ) ∈ CU ( A ) and w (1) = 1 A . Thus, R ϕ,ψ ( z )( τ ) = 12 πi Z τ ( du ( t ) dt u ∗ ( t )) dt = 12 πi Z τ ( ψ ( u ) ∗ du ( t ) dt u ∗ ( t ) ψ ( u )) dt (e 28.5)= 12 πi Z τ ( dw ( t ) dt w ∗ ( t )) dt (e 28.6)for all τ ∈ T ( A ) . By 28.1, R ϕ,ψ ( z ) ∈ ρ A ( K ( A )) . (e 28.7)It follows that R ϕ,ψ ∈ Hom( K ( B ) , ρ A ( K ( A ))) . (e 28.8)59 heorem 28.3. Let C , C ∈ B be unital separable simple C ∗ -algebras, and A = C ⊗ U ,B = C ⊗ U , where U and U are UHF-algebras of infinite type. Suppose that B is a unital C ∗ -subalgebra of A, and denote by ı the embedding. For any λ ∈ Hom( K ( B ) , ρ A ( K ( A ))) , there exists ϕ ∈ Inn(
B, A ) (see Definition 2.8 of [21]) such that there are homomorphisms θ i : K i ( B ) → K i ( M ı,ϕ ) with ( π ) ∗ i ◦ θ i = id K i ( B ) , i = 0 , , and the rotation map R ı,ϕ : K ( M ı,ϕ ) → Aff( T ( A )) is given by R ı,ϕ ( x ) = ρ A ( x − θ ( π ) ∗ ( x )) + λ ◦ ( π ) ∗ ( x )) (e 28.9) for all x ∈ K ( M ı,ϕ ) . In other words, [ ϕ ] = [ ı ] in KK ( B, A ) (e 28.10) and the rotation map R ı,ϕ : K ( M ı,ϕ ) → Aff( T ( A )) is given by R ı,ϕ ( a, b ) = ρ A ( a ) + λ ( b ) (e 28.11) for some identification of K ( M ı,ϕ ) with K ( A ) ⊕ K ( B ) . Proof.
The proof is exactly the same as that of Theorem 4.2 of [44]. By Lemma 23.3 and Lemma24.1 (see also Lemma 24.2), we have the properties (B1) and (B2) associated with B (defined in3.6 of [44]) as in Theorem 4.2 of [44]. In 4.2 of [44], it is also assumed that ρ A ( K ( A )) is densein Aff( T ( A )), which is only used to get that ψ ( K ( B )) ⊂ ρ A ( K ( A )), which corresponds to theassumption λ ( K ( B )) ⊂ ρ A ( K ( A )) here. Definition 28.4.
Let A be a unital C ∗ -algebra and let C be a unital separable C ∗ -algebra.Denote by Mon easu ( C, A ) the set of all asymptotic unitary equivalence classes of unital monomor-phisms from C into A. Denote by K : Mon easu ( C, A ) → KK e ( C, A ) ++ the map defined by ϕ [ ϕ ] for all ϕ ∈ Mon easu ( C, A ) . Let κ ∈ KK e ( C, A ) ++ . Denote by h κ i the set of classes of all ϕ ∈ Mon easu ( C, A ) such that K ( ϕ ) = κ. Denote by
KKU T e ( A, B ) ++ the set of triples ( κ, α, γ ) for which κ ∈ KK e ( A, B ) ++ , α : U ( A ) /CU ( A ) → U ( B ) /CU ( B ) is a homomorphism , γ : T ( B ) → T ( A ) is a continuous affinemap, and both α and γ are compatible with κ. Denote by K the map from Mon easu ( C, A ) into
KKU T ( C, A ) ++ defined by ϕ ([ ϕ ] , ϕ ‡ , ϕ T ) for all ϕ ∈ Mon easu ( C, A ) . Denote by h κ, α, γ i the subset of ϕ ∈ Mon easu ( C, A ) such that K ( ϕ ) = ( κ, α, γ ) . Theorem 28.5.
Let C and A be two unital separable amenable C ∗ -algebras. Suppose that ϕ , ϕ , ϕ : C → A are three unital monomorphisms for which [ ϕ ] = [ ϕ ] = [ ϕ ] i n KK ( C, A )) and ( ϕ ) T = ( ϕ ) T = ( ϕ ) T . (e 28.12) Then R ϕ ,ϕ + R ϕ ,ϕ = R ϕ ,ϕ . (e 28.13) Proof.
The proof is exactly the same as that of Theorem 9.6 of [40].60 emma 28.6.
Let A and B be two unital separable amenable C ∗ -algebras. Suppose that ϕ , ϕ : A → B are two unital monomorphisms such that [ ϕ ] = [ ϕ ] i n KK ( A, B ) and ( ϕ ) T = ( ϕ ) T . Suppose that ( ϕ ) T : T ( B ) → T ( A ) is an affine homeomorphism. Suppose also that there is α ∈ Aut ( B ) such that [ α ] = [id B ] i n KK ( B, B ) and α T = id T . Then R ϕ ,α ◦ ϕ = R id B ,α ◦ ( ϕ ) ∗ + R ϕ ,ϕ (e 28.14) in Hom( K ( A ) , Aff( T ( B ))) / R . Proof.
Using 28.5, we compute that R ϕ ,α ◦ ϕ = R ϕ ,ϕ + R ϕ ,α ◦ ϕ = R ϕ ,ϕ + R id B ,α ◦ ( ϕ ) ∗ . Theorem 28.7.
Let B ∈ N be a unital separable simple C ∗ -algebra in B , let C = B ⊗ U , where U is a UHF-algebra of infinite type, let A be a unital separable amenable simple C ∗ -algebra in B , and let A = A ⊗ U , where U is another UHF-algebra of infinite type. Thenthe map K : Mon easu ( C, A ) → KKU T ( C, A ) ++ is surjective. Moreover, for each ( κ, α, γ ) ∈ KKU T ( C, A ) ++ , there exists a bijection η : h κ, α, γ i → Hom( K ( C ) , ρ A ( K ( A ))) / R . Proof.
It follows from Lemma 24.4 that K is surjective.Fix a triple ( κ, α, γ ) ∈ KKT ( C, A ) ++ and choose a unital monomorphism ϕ : C → A suchthat [ ϕ ] = κ , ϕ ‡ = α , and ϕ T = γ. If ϕ : C → A is another unital monomorphism such that K ( ϕ ) = K ( ϕ ) , then by Lemma 28.2, R ϕ,ϕ ∈ Hom( K ( C ) , ρ A ( K ( A ))) / R . (e 28.15)Let λ ∈ Hom( K ( C ) , ρ A ( K ( A ))) be a homomorphism. It follows from Theorem 28.3 thatthere is a unital monomorphism ψ ∈ Inn( ϕ ( C ) , A ) with [ ψ ◦ ϕ ] = [ ϕ ] in KK ( C, A ) such thatthere exists a homomorphism θ : K ( C ) → K ( M ϕ,ψ ◦ ϕ ) with ( π ) ∗ ◦ θ = id K ( C ) for which R ϕ,ψ ◦ ϕ ◦ θ = λ. Let β = ψ ◦ ϕ. Then R ϕ,β ◦ θ = λ. Note also that, since ψ ∈ Inn( ϕ ( C ) , A ) ,β ‡ = ϕ ‡ and β T = ϕ T . In particular, K ( β ) = K ( ϕ ) . Thus, for each unital monomorphism ϕ , we obtain a well-defined and surjective map η ϕ : h [ ϕ ] , ϕ ‡ , ϕ T i → Hom( K ( A ) , ρ A ( K ( A ))) / R . To see that η ϕ is injective, consider two monomorphisms ϕ , ϕ : C → A in h [ ϕ ] , ϕ ‡ , ϕ T i suchthat R ϕ,ϕ = R ϕ,ϕ . Then, by Theorem 28.5, R ϕ ,ϕ = R ϕ ,ϕ + R ϕ,ϕ = − R ϕ,ϕ + R ϕ,ϕ = 0 . (e 28.16)It follows from Theorem 27.5 that ϕ and ϕ are asymptotically unitarily equivalent. The map η ϕ is the desired bijection η as h [ ϕ ] , ϕ ‡ , ϕ T i = h κ, α, γ i .61 efinition 28.8. Denote by
KKU T − e ( A, A ) ++ the subset of those elements ( κ, α, γ ) ∈ KKU T e ( A, A ) ++ for which κ | K i ( A ) is an isomorphism ( i = 0 , α is an isomorphism, and γ is an affine homeo-morphism. Recall from the proof of Theorem 28.7 that η id A : h [id A ] , id ‡ A , (id A ) T i → Hom( K ( A ) , ρ A ( K ( A ))) / R is a bijection.Denote by h id A i the class of those automorphisms ψ which are asymptotically unitarilyequivalent to id A —this subset of Aut( A ) gives rise to a single element in Mon easu ( A, A ) whichshould not be confused with the subset h [id A ] , id ‡ A , (id A ) T i ⊂ Mon easu ( A, A ). Note that, if ψ ∈h id A i , then ψ is asymptotically inner , i.e., there exists a continuous path of unitaries { u ( t ) : t ∈ [0 , ∞ ) } ⊂ A such that ψ ( a ) = lim t →∞ u ( t ) ∗ au ( t ) for all a ∈ A. Note that h id A i is a normal subgroup of Aut( A ). Corollary 28.9.
Let A ∈ N ∩ B be a unital simple C ∗ -algebra and let A = A ⊗ U for someUHF-algebra U of infinite type. Then one has the following short exact sequence: → Hom( K ( A ) , ρ A ( K ( A ))) / R η − A → Aut( A ) / h id A i K → KKU T − e ( A, A ) ++ → . (e 28.17) In particular, if ϕ, ψ ∈ Aut( A ) are such that K ( ϕ ) = K ( ψ ) = K (id A ) , then η id A ( ϕ ◦ ψ ) = η id A ( ϕ ) + η id A ( ψ ) . Proof.
It follows from Lemma 24.4 that, for any h κ, α, γ i , there is a unital monomorphism h : A → A such that K ( h ) = h κ, α, γ i . The fact that κ ∈ KK − e ( A, A ) ++ implies that there is κ ∈ KK − e ( A, A ) ++ such that κ × κ = κ × κ = [id A ] . Using Lemma 24.4, choose h : A → A such that K ( h ) = h κ , α − , γ − i . It follows from Lemma 27.1 that h ◦ h and h ◦ h are approximately unitarily equivalent.Applying the standard approximate intertwining argument of G. A. Elliott (Theorem 2.1 of[12]), one obtains two isomorphisms ϕ and ϕ − such that there is a sequence of unitaries { u n } in A such that ϕ ( a ) = lim n →∞ Ad u n +1 ◦ h ( a ) and ϕ − ( a ) = lim n →∞ Ad u n ◦ h ( a )for all a ∈ A. Thus, [ ϕ ] = [ h ] in KL ( A, A ) and ϕ ‡ = h ‡ and ϕ T = h T . Then, as in the proof of24.4, there is ψ ∈ Inn(
A, A ) such that [ ψ ◦ ϕ ] = [id A ] in KK ( A, A ) as well as ( ψ ◦ ϕ ) ‡ = h ‡ and ( ψ ◦ ϕ ) T = h T . So we have ψ ◦ ϕ ∈ Aut(
A, A ) such that K ( ψ ◦ ϕ ) = h κ, α, γ i . This impliesthat K is surjective.Now let λ ∈ Hom( K ( C ) , Aff( T ( A ))) / R . The proof Theorem 28.7 says that there is ψ ∈ Inn(
A, A ) (in place of ψ ) such that K ( ψ ◦ id A ) = K (id A ) and R id A ,ψ = λ. Note that ψ is again an automorphism. The last part of the lemma then follows from Lemma28.6. 62 efinition 28.10 (Definition 10.2 of [37] and see also [41]) . Let A be a unital C ∗ -algebra and B be another C ∗ -algebra. Recall ([41]) that H ( K ( A ) , K ( B )) = { x ∈ K ( B ) : ϕ ([1 A ]) = x for some ϕ ∈ Hom( K ( A ) , K ( B )) } . Proposition 28.11 (Proposition 12.3 of [37])) . Let A be a unital separable C ∗ -algebra and let B be a unital C ∗ -algebra. Suppose that ϕ : A → B is a unital homomorphism and u ∈ U ( B ) isa unitary. Suppose that there is a continuous path of unitaries { u ( t ) : t ∈ [0 , ∞ ) } ⊂ B such that u (0) = 1 B and lim t →∞ Ad u ( t ) ◦ ϕ ( a ) = Ad u ◦ ϕ ( a ) (e 28.18) for all a ∈ A. Then [ u ] ∈ H ( K ( A ) , K ( B )) . Lemma 28.12.
Let C = C ′ ⊗ U for some C ′ = lim −→ ( C n , ψ n ) and a UHF algebra U of infinitetype, where each C n is a direct sum of C*-algebras in C and H . Assume that ψ n is unitaland injective. Let A ∈ B . Let ϕ , ϕ : C → A be two monomorphisms such that there isan increasing sequence of finite subsets F n ⊂ C with dense union, an increasing sequence offinite subsets P n ⊂ K ( C ) with union equal to K ( C ) , a sequence of positive numbers ( δ n ) with P δ n < and a sequence of unitaries { u n } ⊂ A such that Ad u n ◦ ϕ ≈ δ n ϕ on F n and ρ A (bott ( ϕ , u ∗ n u n +1 )) = 0 for all x ∈ P n . Suppose that H ( K ( C ) , K ( A )) = K ( A ) . Then there exists a sequence of unitaries v n ∈ U ( A ) such that Ad v n ◦ ϕ ≈ δ n ϕ on F n and (e 28.19) ρ A (bott ( ϕ , v ∗ n v n +1 )) = 0 , x ∈ P n . (e 28.20) Proof.
Let x n = [ u n ] ∈ K ( A ). Since H ( K ( C ) , K ( A )) = K ( A ), there is a homomorphism κ n, : K ( C ) → K ( A )such that κ n, ([1 C ]) = − x n . Since C satisfies the Universal Coefficient Theorem, there is κ n ∈ KL ( C ⊗ C ( T ) , A ) such that( κ n ) | β ( K ( C )) = κ n, and ( κ n ) | β ( K ( C )) = 0 . Without loss of generality, we may assume that [1 C ] ∈ P n , n = 1 , , ..... For each δ n , choose apositive number η n < δ n , such thatAd u n ◦ ϕ ≈ η n ϕ on F n . By Lemma 24.1, there is a unitary w n ∈ U ( A ) such that k [ ϕ ( a ) , w n ] k < ( δ n − η n ) / a ∈ F n and Bott( ϕ , w n ) | P n = κ n | β ( P n ) . Put v n = u n w n , n = 1 , , ... . ThenAd v n ◦ ϕ ≈ δ n ϕ on F n , ρ A (bott ( ϕ , v ∗ n v n +1 )) | P n = 0and, since [1 C ] ∈ P n , [ v n ] = [ u n ] − x n = 0 , as desired. 63 heorem 28.13. Let B ∈ B be a unital separable simple C ∗ -algebra which satisfies the UCT,let A ∈ B be a unital separable simple C ∗ -algebra, and let C = B ⊗ U and A = A ⊗ U , where U and U are unital infinite dimensional UHF-algebras. Suppose that H ( K ( C ) , K ( A )) = K ( A ) and suppose that ϕ , ϕ : C → A are two unital monomorphisms which are asymptoticallyunitarily equivalent. Then ϕ and ϕ are strongly asymptotically unitarily equivalent, that is,there exists a continuous path of unitaries { u ( t ) : t ∈ [0 , ∞ ) } ⊂ A such that u (0) = 1 and lim t →∞ Ad u ( t ) ◦ ϕ ( a ) = ϕ ( a ) for all a ∈ C. Proof.
By 4.3 of [40], one has [ ϕ ] = [ ϕ ] in KK ( C, A ) ,ϕ ‡ = ψ ‡ , ( ϕ ) T = ( ϕ ) T and R ϕ ,ϕ = 0 . Then by Lemma 28.12, one may assume that v n ∈ U ( A ) ( n = 1 , , ... ) in the proof of Theorem27.5. It follows that ξ n ([1 C ]) = 0, n = 1 , , ... , and therefore κ n ([1 C ]) = 0. This implies that γ n ◦ β ([1 C ]) = 0. Hence w n ∈ U ( A ), and also u n ∈ U ( A ). Therefore, the continuous pathof unitaries { u ( t ) } constructed in Theorem 27.5 is in U ( A ), and then one may require that u (0) = 1 A by connecting u (0) to 1 A .
29 The general classification theorem
Lemma 29.1.
Let A ∈ B be a unital separable simple C ∗ -algebra, let A = A ⊗ U for someinfinite dimensional UHF-algebra, and let p be a supernatural number of infinite type. Thenthe homomorphism ı : a a ⊗ induces an isomorphism from U ( A ) /CU ( A ) to U ( A ⊗ M p ) /CU ( A ⊗ M p ) . Proof.
There are sequences of positive integers { m ( n ) } and { k ( n ) } such that A ⊗ M p = lim n →∞ ( A ⊗ M m ( n ) , ı n ) , where ı n : M m ( n ) ( A ) → M m ( n +1) ( A )is defined by ı ( a ) = diag( k ( n ) z }| { a, a, ..., a ) for all a ∈ M m ( n ) ( A ) , n = 1 , , .... Note, M m ( n ) ( A ) = M m ( n ) ( A ) ⊗ U and M m ( n ) ( A ) ∈ B . Let j n : U ( M m ( n ) ( A )) /CU ( M m ( n ) ( A ))) → U ( M m ( n +1) ( A )) /CU ( M m ( n +1) ( A ))be defined by j n (¯ u ) = diag( u, , , ..., | {z } k ( n ) − ) for all u ∈ U (( M m ( n ) ( A )) . It follows from Corollary11.11 of [21] that j n is an isomorphism. By Corollary 11.7 of [21], theabelian group U ( M m ( n ) ( A )) /CU ( M m ( n ) ( A )) is divisible. For each n and i, there is a unitary U i ∈ M m ( n +1) ( A ) such that U ∗ i E , U i = E i,i , i = 2 , , ..., k ( n ) , where E i,i = P im ( n ) j =( i − m ( n )+1 e j,j and { e i,j } is a system of matrix units for M m ( n +1) . Then ı n ( u ) = u ′ ( U ∗ u ′ U )( U ∗ u ′ U ) · · · ( U ∗ k ( n ) u ′ U k ( n ) ) , u ′ = diag( u, z }| { , , ..., , for all u ∈ M m ( n ) ( A ) . Thus, ı ‡ n (¯ u ) = k ( n ) j n (¯ u ) . It follows that ı ‡ n | U ( M m ( n ) ( A )) /CU ( M m ( n ) ( A )) is injective, since U ( M m ( n +1) ( A )) /CU ( M m ( n +1) ( A ))is torsion free (see Lemma 11.5 of [21]) and j n is injective. For each z ∈ U ( M m ( n +1) ( A ) /CU ( M m ( n +1) ) , there is a unitary v ∈ M m ( n +1) ( A ) such that j n (¯ v ) = z, since j n is an isomorphism. By the divisibility of U ( M m ( n ) ( A ) /CU ( M m ( n ) ) , there is u ∈ M m ( n ) ( A ) such that u k ( n ) = u k ( n ) = v. As above, ı ‡ n (¯ u ) = k ( n ) j n (¯ v ) = z. So ı ‡ n | U ( M m ( n ) ( A )) /CU ( M m ( n ) ( A )) is surjective. It follows that ı ‡ n, ∞ | U ( M m ( n ) ( A )) /CU ( M m ( n ) ( A )) is anisomorphism. One then concludes that ı ‡ | U ( A ) /CU ( A ) is an isomorphism. Lemma 29.2.
Let A and B be two unital separable simple C ∗ -algebras in B , let A = A ⊗ U and let B = B ⊗ U , where U and U are two UHF-algebras of infinite type. Let ϕ : A → B bean isomorphism and let β : B ⊗ M p → B ⊗ M p be an automorphism such that β ∗ = id K ( B ⊗ M p ) for some supernatural number p of infinite type. Then ψ ‡ ( U ( A ) /CU ( A )) = ( ϕ ) ‡ ( U ( A ) /CU ( A )) = U ( B ) /CU ( B ) , where ϕ = ı ◦ ϕ, ψ = β ◦ ı ◦ ϕ and where ı : B → B ⊗ M p is defined by ı ( b ) = b ⊗ for all b ∈ B. Moreover, there is an isomorphism µ : U ( B ) /CU ( B ) → U ( B ) /CU ( B ) with µ ( U ( B ) /CU ( B )) ⊂ U ( B ) /CU ( B ) such that ı ‡ ◦ µ ◦ ϕ ‡ = ψ ‡ and q ◦ µ = q , where q : U ( B ) /CU ( B ) → K ( B ) is the quotient map.Proof. The proof is exactly the same as that of Lemma 11.3 of [40].
Lemma 29.3.
Let A and B be two unital simple amenable C ∗ -algebras in N ∩ B , let A = A ⊗ U , and let B = B ⊗ U , where U and U are UHF-algebras of infinite type. Suppose that ϕ , ϕ : A → B are two isomorphisms such that [ ϕ ] = [ ϕ ] in KK ( A, B ) . Then there existsan automorphism β : B → B such that [ β ] = [id B ] in KK ( B, B ) and β ◦ ϕ is asymptoticallyunitarily equivalent to ϕ . Moreover, if H ( K ( A ) , K ( B )) = K ( B ) , then β can be chosen sothat β ◦ ϕ and β ◦ ϕ are strongly asymptotically unitarily equivalent.Proof. It follows from Theorem 28.7 that there is an automorphism β : B → B satisfying thefollowing condition: [ β ] = [id B ] in KK ( B, B ) , (e 29.1) β ‡ = ϕ ‡ ◦ ( ϕ − ) ‡ and ( β ) T = ( ϕ ) T ◦ ( ϕ ) − T . (e 29.2)By Corollary 28.9, there is automorphism β ∈ Aut( B ) such that[ β ] = [id B ] in KK ( B, B ) , (e 29.3) β ‡ = id ‡ B , ( β ) T = (id B ) T , and (e 29.4) R id B ,β = − R ϕ ,β ◦ ϕ ◦ ( ϕ ) − ∗ . (e 29.5)65ut β = β ◦ β . It follows that[ β ◦ ϕ ] = [ ϕ ] in KK ( A, B ) , ( β ◦ ϕ ) ‡ = ϕ ‡ , and ( β ◦ ϕ ) T = ( ϕ ) T . (e 29.6)Moreover, by 28.6, R ϕ ,β ◦ ϕ = R id B ,β ◦ ( ϕ ) ∗ + R ϕ ,β ◦ ϕ (e 29.7)= ( − R ϕ ,β ◦ ϕ ◦ ( ϕ ) − ∗ ) ◦ ( ϕ ) ∗ + R ϕ ,β ◦ ϕ = 0 . (e 29.8)It follows from 28.7 that β ◦ ϕ and ϕ are asymptotically unitarily equivalent.In the case that H ( K ( A ) , K ( B )) = K ( B ) , it follows from Theorem 28.13 that β ◦ ϕ and ϕ are strongly asymptotically unitarily equivalent. Lemma 29.4.
Let A and B be two unital simple amenable C ∗ -algebras in N ∩ B and let A = A ⊗ U and B = B ⊗ U for UHF-algebras U and U of infinite type. Let ϕ : A → B bean isomorphism. Suppose that β ∈ Aut( B ⊗ M p ) is such that [ β ] = [id B ⊗ M p ] i n KK ( B ⊗ M p , B ⊗ M p ) and β T = (id B ⊗ M p ) T for some supernatural number p of infinite type.Then there exists an automorphism α ∈ Aut ( B ) with [ α ] = [id B ] in KK ( B, B ) such that ı ◦ α ◦ ϕ and β ◦ ı ◦ ϕ are asymptotically unitarily equivalent, where ı : B → B ⊗ M p is definedby ı ( b ) = b ⊗ for all b ∈ B. Proof.
It follows from Lemma 29.2 that there is an isomorphism µ : U ( B ) /CU ( B ) → U ( B ) /CU ( B )such that ı ‡ ◦ µ ◦ ϕ ‡ = ( β ◦ ı ◦ ϕ ) ‡ . Note that ı T : T ( B ⊗ M p ) → T ( B ) is an affine homeomorphism.It follows from Theorem 28.7 that there is an automorphism α : B → B such that[ α ] = [id B ] in KK ( B, B ) , (e 29.9) α ‡ = µ, α T = ( β ◦ ı ◦ ϕ ) T ◦ (( ı ◦ ϕ ) T ) − = (id B ⊗ M p ) T and (e 29.10) R id B ,α ( x )( τ ) = − R β ◦ ı ◦ ϕ, ı ◦ ϕ ( ϕ − ∗ ( x ))( ı T ( τ )) for all x ∈ K ( A ) (e 29.11)and for all τ ∈ T ( B ) . Denote by ψ = ı ◦ α ◦ ϕ. Then we have, by Lemma 28.6,[ ψ ] = [ ı ◦ ϕ ] = [ β ◦ ı ◦ ϕ ] in KK ( A, B ⊗ M p ) (e 29.12) ψ ‡ = ı ‡ ◦ µ ◦ ϕ ‡ = ( β ◦ ı ◦ ϕ ) ‡ , and (e 29.13) ψ T = ( ı ◦ α ◦ ϕ ) T = ( ı ◦ ϕ ) T = ( β ◦ ı ◦ ϕ ) T . (e 29.14)Moreover, for any x ∈ K ( A ) and τ ∈ T ( B ⊗ M p ) ,R β ◦ ı ◦ ϕ,ψ ( x )( τ ) = R β ◦ ı ◦ ϕ,ı ◦ ϕ ( x )( τ ) + R ı,ı ◦ α ◦ ϕ ∗ ( x )( τ ) (e 29.15)= R β ◦ ı ◦ ϕ,ı ◦ ϕ ( x )( τ ) + R id B ,ı ◦ α ◦ ϕ ∗ ( x )( ı − T ( τ )) (e 29.16)= R β ◦ ı ◦ ϕ,ı ◦ ϕ ( x )( τ ) − R β ◦ ı ◦ ϕ,ı ◦ ϕ ( ϕ − ∗ )( ϕ ∗ ( x ))( τ ) = 0 . (e 29.17)It follows from Theorem 27.5 that ı ◦ α ◦ ϕ and β ◦ ı ◦ ϕ are asymptotically unitarily equivalent.66 heorem 29.5. Let A and B be two unital separable simple C ∗ -algebras in N . Suppose thatthere is an isomorphism
Γ : Ell( A ) → Ell( B ) . Suppose also that, for some pair of relatively prime supernatural numbers p and q of infinite typesuch that M p ⊗ M q ∼ = Q, we have A ⊗ M p ∈ B , B ⊗ M p ∈ B , A ⊗ M q ∈ B , and B ⊗ M q ∈ B . Then, A ⊗ Z ∼ = B ⊗ Z . Proof.
The proof is almost identical to that of 11.7 of [40], with a few necessary modifications.Note that Γ induces an isomorphismΓ p : Ell( A ⊗ M p ) → Ell( B ⊗ M p ) . Since A ⊗ M p ∈ B and B ⊗ M p ∈ B , by Theorem 21.10 of [21], there is an isomorphism ϕ p : A ⊗ M p → B ⊗ M p . Moreover (by the proof of Theorem 21.10 of [21]), ϕ p carries Γ p . In thesame way, Γ induces an isomorphismΓ q : Ell( A ⊗ M q ) → Ell( B ⊗ M q )and there is an isomorphism ψ q : A ⊗ M q → B ⊗ M q which induces Γ q . Put ϕ = ϕ p ⊗ id M q : A ⊗ Q → B ⊗ Q and ψ = ψ q ⊗ id M p : A ⊗ Q → B ⊗ Q. Note that( ϕ ) ∗ i = ( ψ ) ∗ i ( i = 0 ,
1) and ϕ T = ψ T (all four of these maps are induced by Γ). Note that ϕ T and ψ T are affine homeomorphisms.Since K ∗ i ( B ⊗ Q ) is divisible, we in fact have [ ϕ ] = [ ψ ] (in KK ( A ⊗ Q, B ⊗ Q )). It follows fromLemma 29.3 that there is an automorphism β : B ⊗ Q → B ⊗ Q such that[ β ] = [id B ⊗ Q ] in KK ( B ⊗ Q, B ⊗ Q )and such that ϕ and β ◦ ψ are asymptotically unitarily equivalent. Since K ( B ⊗ Q ) is divisible, H ( K ( A ⊗ Q ) , K ( B ⊗ Q )) = K ( B ⊗ Q ) . It follows that ϕ and β ◦ ψ are strongly asymptoticallyunitarily equivalent. Note also in this case β T = (id B ⊗ Q ) T . Let ı : B ⊗ M q → B ⊗ Q be defined by ı ( b ) = b ⊗ b ∈ B. We consider the pair β ◦ ı ◦ ψ q and ı ◦ ψ q . Applying Lemma 29.4, we obtain an automorphism α : B ⊗ M q → B ⊗ M q such that ı ◦ α ◦ ψ q and β ◦ ı ◦ ψ q are asymptotically unitarily equivalent (in B ⊗ Q ). So, by Lemma 29.3,they are strongly asymptotically unitarily equivalent in B ⊗ Q. Moreover,[ α ] = [id B ⊗ M q ] in KK ( B ⊗ M q , B ⊗ M q ) . We will show that β ◦ ψ and ( α ◦ ψ q ) ⊗ id M p are strongly asymptotically unitarily equivalent.Define β = ( β ◦ ı ◦ ψ q ) ⊗ id M p : B ⊗ Q ⊗ M p → B ⊗ Q ⊗ M p . Let j : Q → Q ⊗ M p be definedby j ( b ) = b ⊗ . There is an isomorphism s : M p → M p ⊗ M p such that the homomorphismid M q ⊗ s : M q ⊗ M p (= Q ) → M q ⊗ M p ⊗ M p (= Q ⊗ M p ) induces (id M q ⊗ s ) ∗ = j ∗ . In thiscase, [id M q ⊗ s ] = [ j ] . Since K ( M p ) = 0 , by Theorem 27.5, id M q ⊗ s is strongly asymptoticallyunitarily equivalent to j. It follows that ( α ◦ ψ q ) ⊗ id M p and ( β ◦ ı ◦ ψ q ) ⊗ id M p are stronglyasymptotically unitarily equivalent (note that ı ◦ α ◦ ψ q and β ◦ ı ◦ ψ q are strongly asymptoticallyunitarily equivalent). Consider the C ∗ -subalgebra C = β ◦ ψ (1 ⊗ M p ) ⊗ M p ⊂ B ⊗ Q ⊗ M p . In C, β ◦ ϕ | ⊗ M p and j are strongly asymptotically unitarily equivalent, where j : M p → C is67efined by j ( a ) = 1 ⊗ a for all a ∈ M p . In particular, there exists a continuous path of unitaries { v ( t ) : t ∈ [0 , ∞ ) } ⊂ C such thatlim t →∞ Ad v ( t ) ◦ β ◦ ϕ (1 ⊗ a ) = 1 ⊗ a for all a ∈ M p . (e 29.18)It follows that β ◦ ψ and β are strongly asymptotically unitarily equivalent. Therefore β ◦ ψ and ( α ◦ ψ q ) ⊗ id M p are strongly asymptotically unitarily equivalent. Finally, we conclude that( α ◦ ψ q ) ⊗ id M p and ϕ are strongly asymptotically unitarily equivalent. Note that α ◦ ψ q is anisomorphism which induces Γ q . Let { u ( t ) : t ∈ [0 , } be a continuous path of unitaries in B ⊗ Q with u (0) = 1 B ⊗ Q such thatlim t →∞ Ad u ( t ) ◦ ϕ ( a ) = α ◦ ψ q ⊗ id M p ( a ) for all a ∈ A ⊗ Q. One then obtains a unitary suspended isomorphism which lifts Γ along Z p,q (see [56]). It followsfrom Theorem 7.1 of [56] that A ⊗ Z and B ⊗ Z are isomorphic. Definition 29.6.
Denote by N the class of those unital simple C ∗ -algebras A in N for which A ⊗ M p ∈ N ∩ B for any supernatural number p of infinite type.Of course N contains all unital simple amenable C ∗ -algebras in B which satisfy the UCT.It contains all unital simple inductive limits of C ∗ -algebras in C . It should be noted that, byTheorem 19.3 of [21], N = N . Corollary 29.7.
Let A and B be two C ∗ -algebras in N . Then A ⊗ Z ∼ = B ⊗ Z if and only if Ell( A ⊗ Z ) ∼ = Ell( B ⊗ Z ) . Proof.
This follows from Theorem 29.5 immediately.
Theorem 29.8.
Let A and B be two unital separable simple amenable Z -stable C ∗ -algebraswhich satisfy the UCT. Suppose that gT R ( A ⊗ Q ) ≤ and gT R ( B ⊗ Q ) ≤ . Then A ∼ = B ifand only if Ell( A ) ∼ = Ell( B ) . Proof.
It follows from Corollary 19.3 of [21] that A ⊗ U, B ⊗ U ∈ B for any UHF-algebra U ofinfinite type. The theorem follows immediately by Corollary 29.7. Corollary 29.9.
Let A and B be two unital separable simple C ∗ -algebras which satisfy the UCT.Suppose that gT R ( A ) ≤ and gT R ( B ) ≤ . Then A ∼ = B if and only if Ell( A ) ∼ = Ell( B ) . Corollary 29.10.
Let A and B be two unital simple C ∗ -algebras in B ∩ N . Then A ∼ = B if andonly if Ell( A ) ∼ = Ell( B ) . Proof.
It follows from Theorem 10.7 of [21] that A ⊗ Z ∼ = A and B ⊗ Z ∼ = B. The corollary thenfollows from Theorem 29.8.
Remark 29.11.
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