Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras

Lifting KK-elemen ts, asy mptotical unitary equiv alence and classiﬁcat ion of simple C*-algebras Huaxin Lin and Zhuang Niu No vem b er 20 , 2018 Abstract Let A and C b e t wo unital simple C*-algebras with tracial rank zero. Supp ose that C is amenable and satisﬁes the Universa l Coeﬃcient Theorem. Denote by K K e ( C, A ) ++ the set of those κ for whic h κ ( K 0 ( C ) + \ { 0 } ) ⊂ K 0 ( A ) + \ { 0 } and κ ([1 C ]) = [1 A ]. Supp ose that κ ∈ K K e ( C, A ) ++ . W e show th at there is a unital monomorphism φ : C → A such that [ φ ] = κ. Supp ose that C is a unital AH-algebra and λ : T( A ) → T f ( C ) is a con tinuous aﬃne map for whic h τ ( κ ([ p ])) = λ ( τ )( p ) for all pro jections p in all matrix alg ebras of C and any τ ∈ T( A ) , where T( A ) is th e simplex of tracial states of A and T f ( C ) is the con vex set of faithful tracial states of C . W e prov e that there is a unital monomorphism φ : C → A such th at φ induces both κ and λ. Supp ose that h : C → A is a unital monomorphism and γ ∈ Hom( K 1 ( C ) , A ﬀ ( A )) . W e show t hat there exists a unital monomorphism φ : C → A suc h that [ φ ] = [ h ] in K K ( C, A ) , τ ◦ φ = τ ◦ h for all tracial states τ and the asso ciated rotation map can b e giv en b y γ . Denote by K K T ( C, A ) ++ the set of compatible pairs ( κ, λ ) , where κ ∈ K L e ( C, A ) ++ and λ is a continuous aﬃne map from T( A ) to T f ( C ) . T ogether with a result of asymptotic unitary equiv alence in [15], th is provides a b ijection from the asymptotic unitary equiv alence classes of unital monomorphisms from C to A to ( K K T ( C , A ) ++ , H om( K 1 ( C ) , A ﬀ (T( A ))) / R 0 ) , where R 0 is a subgroup related to v anishing rotation maps. As an app lication, com bin ing with a result of W. Winter ([24]), we show that tw o un ital amenable simple Z -stable C*-algebras are isomo rphic if they ha ve the same Elliott inv ariant and the tensor pro du ct s of these C*-algebras with an y UH F-algebras hav e tracial rank zero. In particular, if A and B are tw o u nital separable simple Z - stable C*-algebras whic h are inductive limits of C*-algebras of type I with u nique tracial states, then they are isomorphic if and only if th ey ha ve iso morphic Elliott inv arian t . 1 In tro duction Let A a nd B b e t wo unital separa ble amenable simple C* -algebra s satis fying the Univ ersal Co eﬃcient Theorem. It has been sho wn (see [12] and [17], a lso [2] and [3]) that, if in addition, A and B hav e tracial r ank one or zero, then A a nd B are iso morphic if their Ellio tt inv ariant are isomo rphic. There are interesting s imple a menable C*-algebr as with stable rank one whic h do not have tracial ra nk zero or one and some classiﬁcation theo rem have bee n established to o (see for example [7], [6], [20] and [5]). One of the in teresting classes of simple amena ble C*-algebr as which satisfy the UCT are those simple ASH-alge bras (approximate sub-ho mo geneous C*-algebr as), or even more general, tho s e simple C*-a lgebras which are inductive limits of type I C*-alg ebras. In the case of 1 unital Z -stable simple ASH-algebr as, if in addition, their pro jections se parate traces, a class iﬁcation theorem can b e given (se e [2 4] and [16]). More precisely , let A and B b e tw o unital simple Z -stable ASH-algebra s whose pro jectio ns se pa rate the traces . Then A ∼ = B if and only if ( K 0 ( A ) , K 0 ( A ) + , [1 A ] , K 1 ( A )) is isomo rphic to ( K 0 ( B ) , K 0 ( B ) + , [1 B ] , K 1 ( B )) provided that K i ( A ) are ﬁnitely generated, or K i ( A ) contains its torsion part as a direct summand. It should b e noted tha t there are known exa mples that ASH-alg ebras whose pr o jections separate the trace s but hav e r eal rank other than zero. In par ticular, A a nd B may not hav e any no n-trivial pro jections as lo ng as ea ch one ha s a unique tra cial state (for instance, the Jiang-Su a lgebra Z ). It is clea rly impo rtant to remov e the ab ov e mentioned restriction on K - theo ry . The or iginal purp ose o f this pap er was to remov e these restr ictions. One of the technical to ols used in the pro of of [24] and [16] is a theor em which determines when tw o uni- tal mono morphisms from a unital AH-algebra C to a unita l simple C*-alge bra A are asymptotically unitarily equiv alent. In the c a se tha t A has tracial rank zero , it was s hown in [15] that tw o such monomorphisms are asymptotically unitarily equiv alent if they induce the same ele men t in K K ( C, A ) and the same a ﬃne map on the tracial sta te spaces, and the rotatio n map asso cia ted with these tw o monomorphisms v a nishes. It is equally impo rtant to determine the r ange o f a symptotical unitary equiv a lence cla sses of thos e monomorphisms. In fact, the ab ove mentioned restr iction can b e remov ed if the range ca n be determined. The ﬁrst ques tion we need to answer is the following: Let A and C be tw o unital simple amena ble C*-algebr as with tra cial r ank zer o and let C s a tisfy the UCT. Suppose that κ ∈ K K e ( C, A ) ++ ( i.e., those κ ∈ K K ( C, A ) such that κ ( K 0 ( C ) + \ { 0 } ) ⊆ K 0 ( A ) + \ { 0 } with κ ([1 C ]) = [1 A ]) . Is there a unital monomorphism φ : C → A such that [ φ ] = κ ? It is known (see [9]) that if K i ( C ) is ﬁnitely g enerated then the a ns w er is aﬃrmative. It was prov ed in [10] that there exists a unital monomorphism φ : C → A such tha t [ φ ] − κ = 0 in K L ( C, A ) . The problem remained op en whether one can choo se φ so that [ φ ] = κ in K K ( C, A ) . It also has b e en known that ther e a re several signiﬁcant consequences if such φ ca n b e found. The se c ond ques tion is the following: Let φ : C → A be a unital monomorphism and γ ∈ Hom( K 1 ( C ) , Aﬀ (T( A ))) . Can we ﬁnd a unital monomor phism ψ : C → B with [ ψ ] = [ φ ] in K K ( C, A ) and the asso ciated ro tation map is γ ? In this pap er, we will give aﬃrmative answers to b o th questions. Among other consequence s , we g ive the following: Let A and B b e tw o unital simple C*-algebr as whic h are inductive limits of t ype I C* - algebra s with unique tra cial states. Suppo se that ( K 0 ( A ) , K 0 ( A ) + , [1 A ] , K 1 ( A )) ∼ = ( K 0 ( B ) , K 0 ( B ) + , [1 B ] , K 1 ( B )) . Then A and B are Z -stably isomo rphic (see 5 .6 b elow). F or the ﬁrst question, as men tioned ab ov e, an aﬃrmative a ns w er w as kno wn fo r C with ﬁnitely generated K - theory . Passing to inductiv e limits, the ﬁrst author show ed in [1 0] that any strictly p ositive element in K L ( C, A ) can b e lift for any unital AH-alg ebra C . How ever, since the K K -functor is not co n tin uous with resp ect to inductive limits, it r e mained somewha t mysterious how to mov e from K L ( C, A ) to K K ( C, A ) until a clue was given by a pap er of Kishimoto and Kumjian ([8]), where they s tudied simple A T -algebr a s with real ra nk zero . The imp orta n t adv antage o f simple A T -a lgebras is that their K -theor y are torsion free. In this case a hidden Bott-like map was revealed. K ishimoto and Kumjian navigated this h urdle using the so-called Basic Homotopy Lemma in simple C*-algebr a s of re al ra nk zero . W e will take the idea of Kishimoto and Kumjian to the case that 2 the domain a lgebras are no lo nger a ssumed the same as the targets. Mo re generally , w e will not assume that C has real rank zer o, nor will we as sume it is simple. F urthermore , we will allow C to have torsio n in its K - theo ry . Therefore, for the g e ne r al case , the Bott-like maps inv olve the K -theory with co eﬃcient and demand a m uch more general Basic Homotopy Lemma. F or this we apply the r ecent established results in [14] and [15]. It is also int eresting to note that the second pro blem is clo sely r elated to the ﬁrst one a nd its pro of are als o closely related. W e will c onsider the ca se that C is a g eneral unital AH-a lgebra (which ma y hav e a r bitrary stable ra nk and other proper ties even in the case it is simple (see, fo r example, [2 2], [23] and [4])). Denote by T f ( C ) the con vex set of all faithful tracial states of C . Let κ ∈ K K e ( C, A ) ++ and let λ : T( A ) → T f ( C ) b e a contin uous aﬃne map. W e s ay that λ is compatible with κ if λ ( τ )( p ) = τ ( κ ([ p ])) for all pr o jections p in matrix algebras of C and for all τ ∈ T( A ) . W e a ctually show that for an y suc h compatible pair ( κ, λ ) , there exists a unital monomorphism φ : C → A such that [ φ ] = κ and φ T = λ, where φ T : T( A ) → T f ( C ) is the induced co nt inu ous aﬃne map induced by φ. It w o rth to p oint out that the information λ is essen tial s ince there a re examples of compact metric s pa ces X , unital simple AF-alge br as and κ ∈ K e ( C, A ) ++ for whic h there is no unital monomor phis m h such that [ h ] = κ (see [18]). F urthermor e, given a pair ([ φ ] , φ T ) and γ ∈ Hom( K 1 ( C ) , Aﬀ (T( A )), there is a unital monomorphism ψ : C → A suc h that ([ ψ ] , ψ T ) = ([ φ ] , φ T ) and a ro tation map from K 1 ( C ) to Aﬀ (T( A )) asso ciated with φ a nd ψ is exactly γ . The pa pe r is org anized as follows: Preliminaries and notation are giv en in Section 2. In Sectio n 3, it is sho wn that any element κ ∈ K K e ( C, A ) ++ for which is compatible with a contin uous aﬃne map λ : T( A ) → T f ( C ) can be represented b y a monomorphism α if C is a unital AH-algebr a and A is a unital simple C* -algebra with tracial rank zero. It is also shown that if A = C , then α can b e chosen as a n automorphism. Then, in Section 4, w e pr ov e that, for any monomo rphism ι : C → A , o ne can realize any homomo r phism ψ from K 1 ( C ) to Aﬀ (T( A )) as a rotation map without changing the K K -class of ι , that is, there is a mo no morphism α : C → A such that [ ι ] = [ α ] in K K ( C, A ) and ˜ η ι,α = ψ . Moreov er , we also give a descriptio n of the asymptotical unita r y equiv alence class of the maps inducing the same K K -element. In Section 5, we give an applica tion of the results in the previous sections to the classiﬁca tion progra m. Combined with the work [24] o f W. Winter and that of [15], it is shown that cer tain Z -stable C*- algebras can b e c lassiﬁed by their K -theory information. Ac knowledgmen ts . Mos t of this rese a rch were conducted when b oth a uthors were visiting the Fields Institute in F all 2007. W e wish to thank the Fields Institute for the excellent w o rking environmen t. The research of s econd named a uthor is s uppo rted by an NSERC Postdocto ral F ellowship. 2 Preliminaries and Notation 2.1. Let A be a unital stably ﬁnite C*-alge br a. Deno te b y T( A ) the simplex of tr acial states of A and denote by Aﬀ(T( A )) the space of all real aﬃne contin uous functions on T( A ) . Supp ose that τ ∈ T( A ) is a tra c ial state. W e will also use τ for the trace τ ⊗ T r on M k ( A ) = A ⊗ M k ( C ) (for every integer k ≥ 1), where T r is the standa rd trace o n M k ( C ) . A tra ce τ is faithful if τ ( a ) > 0 for any a ∈ A + \ { 0 } . Denote by T f ( A ) the co nvex subset o f T( A ) which consists of all faithful tracial states. Denote by M ∞ ( A ) the set ∞ [ k =1 M k ( A ) , where M k ( A ) is regarded as a C*-subalgebra of M k +1 ( A ) by the 3 embedding M k ( A ) ∋ a 7→ a 0 0 0 ! ∈ M k +1 ( A ) Deﬁne the p ositive homo morphism ρ A : K 0 ( A ) → Aﬀ(T( A )) b y ρ A ([ p ])( τ ) = τ ( p ) for any pro jection p in M k ( A ) . W e also denote b y S( A ) the susp ension of A , U( A ) the unitary gro up of A , and U 0 ( A ) the c onnected comp onent of U( A ) con ta ining the identit y . Suppo se that C is another unita l C*-algebra and φ : C → A is a unital *-homomorphism. Denote by φ T : T( A ) → T( C ) the contin uous a ﬃne map induced by φ, i.e., φ T ( τ )( c ) = τ ◦ φ ( c ) for all c ∈ C and τ ∈ T( A ) . Deﬁnition 2.2. Let A b e a unital C* -algebra and let B ⊆ A be a unital C*-subalgebra . F or a n y u ∈ U( A ) , the *-homomor phism Ad( u ) is deﬁned b y Ad( u ) : B ∋ b 7→ u ∗ bu ∈ A. Denote by Inn( B , A ) the clos ure of { Ad ( u ); u ∈ U( A ) } in Hom( B , A ) with the p o int wise conv er gence top olo gy . Note that Inn( A ) ⊆ Inn( B , A ). Deﬁnition 2. 3 . Let A a nd B b e t wo unital C*- algebra s , and let ψ and φ b e t wo unital *-mono morphisms from B to A . Then the mapping torus M ψ ,φ is the C*-algebr a deﬁned b y M ψ ,φ := { f ∈ C([0 , 1] , A ); f (0) = ψ ( b ) and f (1 ) = φ ( b ) for some b ∈ B } . If B ⊆ A is a unital C*-subalgebr a with ι the inclusion map, then, for any unital *-monomor phism α from B to A , w e also denote M ι,α by M α . F or a n y ψ , φ ∈ Hom( B , A ), denoting by π 0 the ev a lua tion of M ψ ,φ at 0 , we ha ve the short exact sequence 0 / / S( A ) ı / / M ψ ,φ π 0 / / B / / 0 , and hence the six-term exact sequence K 0 (S( A )) ı 0 / / K 0 ( M ψ ,φ ) [ π 0 ] 0 / / K 0 ( B ) [ ψ ] 0 − [ φ ] 0   K 1 ( B ) [ ψ ] 1 − [ φ ] 1 O O K 1 ( M ψ ,φ ) [ π 0 ] 1 o o K 1 (S( A )) . ı 1 o o If [ ψ ] ∗ = [ φ ] ∗ , (in par ticula r, if B ⊆ A a nd α ∈ Inn( B , A )), then the six-term ex a ct se q uence a bove breaks down to the follo wing tw o extensions: η 0 ( M ψ ,φ ) : 0 / / K 1 ( A ) / / K 0 ( M ψ ,φ ) / / K 0 ( B ) / / 0 , and η 1 ( M ψ ,φ ) : 0 / / K 0 ( A ) / / K 1 ( M ψ ,φ ) / / K 1 ( B ) / / 0 . Moreov er, if [ ψ ] = [ φ ] in K L ( B , A ), the tw o extensions above ar e pure. 2.4. Supp ose that, in addition, τ ◦ φ = τ ◦ ψ for all τ ∈ T( A ) . (2.1) 4 F or a n y piecewise smooth pa th u ( t ) ∈ M ψ ,φ , the in tegral R φ,ψ ( u ( t ))( τ ) = 1 2 π i Z 1 0 τ ( ˙ u ( t ) u ∗ ( t ))d t deﬁnes an aﬃne function on T( A ) , and it dep ends only on the homotopy class of u ( t ). Therefore , it induces a map, denoted b y R ψ ,φ , from K 1 ( M α ) to Aﬀ (T( A )), and w e call it the rotation map. The ma p R ψ ,φ is in fact a homomorphism. 2.5. If p and q b e tw o m utually orthogonal pro jections in M l ( A ) for some integer l ≥ 1, deﬁne a unitary u ∈ U( ^ M l (S( A ))) b y u ( t ) = ( e 2 π it p + (1 − p ))( e − 2 π it q + (1 − q )) for t ∈ [0 , 1] . One computes that Z 1 0 τ ( du ( t ) dt u ( t ) ∗ ) dt = τ ( p ) − τ ( q ) for all t ∈ [0 , 1] . Note that if v ( t ) ∈ ^ M l (S A ) is a nother pie c e wise smoo th unitary which is homotopic to u ( t ) , then, a s mentioned ab ov e, Z 1 0 τ ( dv ( t ) dt ) dt = τ ( p ) − τ ( q ) . It follows that, for any tw o pro jections p and q in M l ( A ) , R φ,ψ ( ı 1 ([ p ] − [ q ]))( τ ) = τ ( p ) − τ ( q ) for all τ ∈ T( A ) . In other w ords, R φ,ψ ( ı 1 ([ p ] − [ q ])) = ρ A ([ p ] − [ q ]) . Thu s one has, exactly as in 2.2 of [8], the follo wing: Lemma 2.6. When ( 2.1 ) holds, the fol lowing diagr am c ommutes: K 0 ( A ) ı 1 − → K 1 ( M φ,ψ ) ρ A ց ւ R φ,ψ Aﬀ(T( A )) Deﬁnition 2. 7 . Let A and C b e t w o unital C*-alge br as and let φ, ψ : C → A b e t w o unital homomorphisms. W e say that φ and ψ ar e asymptotically unitar ily equiv alent if there exists a contin uous path o f unitaries { u ( t ) : t ∈ [0 , ∞ ) } such that lim t →∞ Ad( u ( t )) ◦ φ ( c ) = ψ ( c ) for all c ∈ C . Deﬁnition 2.8. Le t A b e a unital C*-algebr a and let C b e a separable C*-algebra which satisﬁes the Universal Co eﬃcient Theorem. By [1] of D˘ ad˘ arlat and Loring, K L ( C, A ) = Ho m Λ ( K ( C ) , K ( A )) , (2.2) where, for an y C*-alg ebra B , K ( B ) = ( K 0 ( B ) ⊕ K 1 ( B )) ⊕ ( ∞ M n =2 ( K 0 ( B , Z /n Z ) ⊕ K 1 ( B , Z /n Z ))) . 5 W e will iden tify the t wo ob jects in (2.2). Note that one may view K L ( C, A ) as a quotient of K K ( C, A ) . Denote by K L ( C, A ) ++ the set of those ¯ κ ∈ Hom Λ ( K ( C ) , K ( A )) suc h that ¯ κ ( K 0 ( C ) + \ { 0 } ) ⊆ K + 0 ( A ) \ { 0 } . Denote by K L e ( C, A ) ++ the set of those elements ¯ κ ∈ K L ( C, A ) ++ such that ¯ κ ([1 C ]) = [1 A ] . Suppo se that b oth A a nd C are unital and T( C ) 6 = ∅ and T( A ) 6 = ∅ . Let λ : T ( A ) → T( C ) be a con tin uous aﬃne ma p. W e say λ is compatible with ¯ κ if for any pro jection p ∈ M ∞ ( C ) , one has that λ ( τ )( p ) = τ ( ¯ κ ([ p ]) for all τ ∈ T( A ) . Denote by K LT ( C, A ) ++ the set o f those pair s ( ¯ κ , λ ) , where ¯ κ ∈ K L e ( C, A ) ++ and λ : T( A ) → T f ( C ) is a contin uous aﬃne map which is compatible with ¯ κ . Deﬁnition 2.9. Denote by K K ( C, A ) ++ the set of those elements κ ∈ K K ( C , A ) such that its ima ge ¯ κ is in K L ( C, A ) ++ . Denote b y K K e ( C, A ) ++ the set of those κ ∈ K K ( C, A ) ++ for which ¯ κ ∈ K L e ( C, A ) ++ . Denote by K K T ( C, A ) ++ the set of pairs ( κ, λ ) suc h that ( ¯ κ, λ ) ∈ K LT ( C, A ) ++ . 2.10. Let A and B b e t wo unital C*- a lgebras. Let h : A → B b e a homomorphism and v ∈ U( B ) suc h that [ h ( g ) , v ] = 0 f or an y g ∈ A. W e then hav e a homomorphism h : A ⊗ C( T ) → B by f ⊗ g 7→ h ( f ) g ( v ) for any f ∈ A and g ∈ C( T ). The tensor pro duct induces t wo injectve homomor phis ms: β (0) : K 0 ( A ) → K 1 ( A ⊗ C( T )) , and β (1) : K 1 ( A ) → K 0 ( A ⊗ C( T )) . The se c ond one is the usual Bott map. No te, in this w ay , one writes K i ( A ⊗ C( T )) = K i ( A ) ⊕ β ( i − 1) ( K i − 1 ( A )) . W e use d β ( i ) : K i ( A ⊗ C( T )) → β ( i − 1) ( K i ( A )) for the pro jection. F or ea ch integer k ≥ 2, one also hav e the following injective homo morphisms: β ( i ) k : K i ( A, Z /k Z ) → K i − 1 ( A ⊗ C( T ) , Z /k Z ) , i = 0 , 1 . Thu s, we write K i ( A ⊗ C( T ) , Z /k Z ) = K i ( A, Z /k Z ) ⊕ β ( i − 1) ( K i − 1 ( A ) , Z /k Z ) . Denote by d β ( i ) k : K i ( A ⊗ C( T ) , Z /k Z ) → β ( i − 1) ( K i ( A ) , Z /k Z ) similar to that o f d β ( i ) . If x ∈ K ( A ), we use β ( x ) for β ( i ) ( x ) if x ∈ K i ( A ) and for β ( i ) k ( x ) if x ∈ K i ( A, Z /k Z ). Thus w e have a map β : K ( A ) → K ( A ⊗ C( T )) as well as b β : K ( A ⊗ C( T )) → β ( K ). Therefore , we ma y write K ( A ⊗ C( T )) = K ( A ) ⊕ β ( K ( A )). On the other hand, h induces homomorphisms h ∗ i,k : K i ( A ⊗ C( T ) , Z /k Z ) → K i ( B , Z /k Z ) , k = 0 , 2 , ..., and i = 0 , 1. 6 W e use Bott( h, v ) for all homomor phisms h ∗ i,k ◦ β ( i ) k , and we use bott 1 ( h, v ) for the homomor phis m h 1 , 0 ◦ β (1) : K 1 ( A ) → K 0 ( B ), and bott 0 ( h, v ) for the homomorphism h 0 , 0 ◦ β (0) : K 0 ( A ) → K 1 ( B ). W e also use b ott( u, v ) for the Bo tt elemen t when [ u , v ] = 0. 2.11. Given a ﬁnite subset P ⊂ K ( A ), there exists a ﬁnite subset F ⊂ A a nd δ 0 > 0 such that Bott( h, v ) | P is well deﬁned if k [ h ( a ) , v ] k < δ 0 for all a ∈ F . See 2.11 of [1 5] and 2 .10 of [14] for mor e details. Deﬁnition 2. 1 2. A unital simple C*-a lgebra A has tracial r ank zer o, denoted by TR( A ) = 0, if for any ﬁnite subset F ⊂ A , a ny ε > 0 , and nonzero a ∈ A + , there are nonzero pro jectio n p ∈ A and ﬁnite dimensio nal C*-subalge br a F with 1 F = p , suc h that 1. k [ x , p ] k ≤ ε for any x ∈ F , 2. for any x ∈ F , there is x ′ ∈ F such that k pxp − x ′ k ≤ ε , and 3. 1 − p is Murray-von Neumann equiv alent to a pr o jection in aAa . 2.13. Finally , we will write a ≈ ε b if k a − b k < ε. 3 KK-lifting Let A be a unital C* -algebr a . Fix 0 < δ p ≤ min { δ 1 , δ 2 } where δ 1 and δ 2 are the constant of δ of Lemma 9.6 and Lemma 9 .7 of [15] respe ctively . Note that δ p is universal a nd δ p < 1 / 4 (therefor e , the Bott e le men t bott( u, v ) is well-deﬁned for any unitaries u and v with k [ u, v ] k < δ p ). F or some integer l ≥ 1, let z b e a unitary in M l ( A ), and let U ( t ) b e a path o f uniarie s U ( t ) ∈ C([0 , 1] , U(M l ( A ))) with U (0) = 1 and k [ U (1) , z ] k ≤ δ p , we hav e that U (1) z U ∗ (1) z ∗ = exp( iω ) where ω = (1 /i )(log ( U (1) z U ∗ (1) z ∗ )) ∈ A s.a . Deﬁne the e lemen t p ( U, z ) as follows: p ( U, z )( t ) =    U ((8 / 7 ) t ) z U ∗ ((8 / 7) t ) z ∗ for all t ∈ [0 , 7 / 8]; exp( i 8(1 − t ) ω ) for all t ∈ [7 / 8 , 1] . Then, p ( U , z )( t ) deﬁnes a lo op o f unitaries in A , and this gives a well-deﬁned element in K 1 (S( A )) ∼ = K 0 ( A ). Remark 3.1. F or a path o f uniaries U ( t ) ∈ C([0 , 1] , U(M l ( A ))) for some integer l ≥ 1 . Denote b y S ( t ) := U ( t ) U ∗ ( t ) ! and Z := z z ! . If U (0) = 1 and k [ U (1) , z ] k ≤ δ p , then S (0) = 1 and k [ S (1) , Z ] k ≤ δ p , and w e can consider the K -element [ p ( S, Z )] and we hav e [ p ( S, Z )] = [ p ( U, z )] + [ p ( U ∗ , z )] . 7 Denote by R ( t, s ) := U ( t ) 1 ! R ( s ) 1 U ∗ ( t ) ! R ∗ ( s ) where R ( s ) = cos( π s 2 ) sin( π s 2 ) − sin( π s 2 ) cos( π s 2 ) ! . Then o ne has that R ( t, 0) = S ( t ), R ( t, 1) = 1, and for any s ∈ [0 , 1], R (0 , s ) = 1 and k [ R (1 , s ) , Z ] k ≤ 2 δ p . Therefore, the path p ( S, Z )( t ) is homotopic to the identit y in M 2 ( ] S( A )) by a small per turbation of W ( t, s ) := R ( t, s ) Z R ∗ ( t, s ) Z ∗ , and hence [ p ( U, z )] + [ p ( U ∗ , z )] = [ p ( S, Z )] = 0 in K 1 (S( A )) . Lemma 3. 2. L et U ( t ) and V ( t ) b e two c ontinuous and pie c ewise smo oth p aths of u nitaries. L et z 1 and z 2 b e unitaries in A with k [ U (1) , z 1 ] k < δ p and k [ V (1) , z 2 ] k < δ p . If [ p ( U, z 1 )] = [ p ( V , z 2 )] , t hen one has that bo tt( U (1 ) , z 1 ) = b o tt( V (1) , z 2 ) . Pr o of. Consider the path of unitaries S ( t ) = diag( U ( t ) , V ∗ ( t )) and the unitary Z := diag( z 1 , z 2 ) in M 2 ( A ). By Remark 3.1, one has [ p ( V ∗ , z 2 )] + [ p ( V , z 2 )] = 0 , and hence [ p ( S ∗ , Z )] = [ p ( U ∗ , z 1 )] + [ p ( V , z 2 )] = 0 . Therefore, p ( S ∗ , Z ) is homotopic to the identit y in a matr ix algebra of ^ (S( A )). Hence, pa ssing to the matrix algebra if necessary , ther e is a contin uous path of unitar ies W ( t, s ) such that W ( t, 0) = 1 = W ( t, 1) for any t ∈ [0 , 1] and W (1 , t ) = p ( S ∗ , Z )( t ) for an y t ∈ [0 , 1], a nd W ( t, 1 ) = 1. Note that p ( S ∗ , Z )( t ) = S ∗ ( 8 7 t ) Z S ( 8 7 t ) Z ∗ for t ∈ [0 , 7 / 8] and k p ( S ∗ , Z )(7 / 8) − 1 k < δ p . Thu s, by Lemma 9 .6 of [1 5], 0 = bo tt( S (1) , Z ) = bo tt( U (1 ) , z 1 ) − b o tt( V (1) , z 2 ) , as desir ed. Lemma 3. 3 . Le t U ( t ) and V ( t ) b e two c ontinuous and pie c ewise smo oth p athes of u nitaries. L et z 1 and z 2 b e unitaries in A with k [ U ( t ) , z 1 ] k < δ p and k [ V ( t ) , z 2 ] k < δ p . If b ott( U (1) , z 1 ) = b ott( V (1) , z 2 ) , then one has that [ p ( U, z 1 )] = [ p ( V , z 2 )] . 8 Pr o of. Consider the path of unitaries W ( t ) := diag( U ( t ) , V ∗ ( t )) and unitary Z := diag( z 1 , z 2 ) in M 2 ( A ). Then, one has that b ott( W (1) , Z ) = b ott( U (1 ) , z 1 ) − b ott( V (1) , z 2 ) = 0. By Lemma 9.7 of [15], one has [ p ( W ∗ , Z )] = 0 , and hence [ p ( U ∗ , z 1 )] + [ p ( V , z 2 )] = 0 . (3.1) On the other hand, by Remark 3.1, [ p ( V ∗ , z 2 )] + [ p ( V , z 2 )] = 0 . T ogether with (3.1), one has [ p ( U, z 1 )] = [ p ( V , z 2 )] , as desir ed. 3.4. Deno te b y C ( A ) the subset of K 1 (S( A )) consisting of [ p ( U, z )( t )] for a path of unitaries U ( t ) ∈ C([0 , 1] , M l ( A )) and a unitar y z ∈ M l ( A ) for s o me l ≥ 1 with k [ U (1 ) , z ] k < δ p . It is easy to verify that C ( A ) is a subgroup. It follows from Lemma 3 .2 and Lemma 3.3 that there is an injectiv e map Λ : C ( A ) → K 0 ( A ) deﬁned b y Λ : [ p ( U, z )] 7→ b ott ( U (1) , z ) . Moreov er, the map Λ is a ho momorphism. Lemma 3. 5 . Consider a p ath of un itaries U ( t ) ∈ C([0 , 1] , M ∞ ( A )) with U (0) = 1 and k [ z , U (1)] k < δ p . Then we have τ ([ p ( U, z )]) = τ (b ott 1 ( U (1) , z )) for any τ ∈ T( A ) if one c onsiders [ p ( U, z )] as an element in K 0 ( A ) ∼ = K 1 (S( A )) . In other wor ds, τ (Λ( h )) = τ ( h ) for any h ∈ C ( A ) . Pr o of. Denote by ω = U (1) z U ∗ (1) z ∗ , and note that [ p ( U, z )] ∈ K 1 (S( A )) = K 0 ( A ) . Denote b y r ( t ) = exp(log( ω )(1 − t )) . 9 W e then ha ve that (see 2.5), for any τ ∈ T( A ), τ ( p ( U, z )) = 1 2 π i Z 7 / 8 0 τ (( U ((8 / 7) t ) z U ∗ ((8 / 7) t ) z ∗ ) ′ ( z U ((8 / 7) t ) z ∗ U ∗ ((8 / 7) t )) dt + 1 2 π i Z 1 7 / 8 τ ( ˙ r (8( t − 7 / 8)) r ∗ (8( t − 7 / 8))) dt = 1 2 π i Z 7 / 8 0 τ (( U ((8 / 7) t ) z U ∗ ((8 / 7) t )) ′ ( U ((8 / 7 ) t ) z ∗ U ∗ ((8 / 7) t )) dt + 1 2 π i Z 1 7 / 8 τ ( ˙ r (8( t − 7 / 8)) r ∗ (8( t − 7 / 8))) dt = 1 2 π i Z 1 7 / 8 τ ( ˙ r (8( t − 7 / 8)) r ∗ (8( t − 7 / 8))) dt (b y Lemma 4.2 of [15 ]) = − 1 2 π i τ (log ( w )) = τ (b ott( U (1) , z )) (by Theorem 3.6 of [15]) , as desir ed. Deﬁnition 3.6. Let A b e a unital C* -algebra with T( A ) 6 = ∅ . W e say that A has Prop erty (B1) if the following holds: F o r an y unitary z ∈ U(M k ( A )) (for some integer k ≥ 1) with sp( z ) = T , there is a non-decrea s ing function 1 / 4 > δ z ( t ) > 0 on [0 , 1] with δ z (0) = 0 such that for any x ∈ K 0 ( A ) with | τ ( x ) | ≤ δ z ( ε ) for all τ ∈ T( A ) , there exists a unitary u ∈ M k ( A ) s uch that k [ u, z ] k < min { ε, 1 4 } and bott 1 ( u, z ) = x. (3.2) Let C be a unital sepa rable C* - algebra. L et 1 / 4 > ∆ c ( t, F , P 0 , P 1 , h ) > 0 be a function deﬁned on t ∈ [0 , 1] , the family of a ll ﬁnite subsets F ⊂ C, the family of all ﬁnite subsets P 0 ⊂ K 0 ( C ) , and family of all ﬁnite subsets P 1 ⊂ K 1 ( C ) , and the set of a ll unital mono morphisms h : C → A. W e sa y that A has Pr op erty (B2) a sso ciated with C and ∆ c if the follo wing holds: F or any unital mono morphism h : C → A, any ε > 0 , any ﬁnite subset F ⊂ C, any ﬁnite subset P 0 ⊂ K 0 ( C ) , and any ﬁnite subse t P 1 ⊂ K 1 ( C ) , there ar e ﬁnitely generated subgr o ups G 0 ⊂ K 0 ( C ) and G 1 ⊂ K 1 ( C ) with G 0 and G 1 the s e ts of genera tors resp ectively and P 0 ⊂ G 0 and P 1 ⊂ G 1 satisfying the following: fo r any homomorphisms b 0 : G 0 → K 1 ( A ) and b 1 : G 1 → K 0 ( A ) s uch that | τ ◦ b 1 ( g ) | < ∆ c ( ε, F , P 0 , P 1 , h ) (3.3) for any g ∈ G 1 and a ny τ ∈ T( A ), there exists a unitary u ∈ U( A ) such tha t bo tt 0 ( h, u ) | P 0 = b 0 | P 0 , b ott 1 ( h, u ) | P 1 = b 1 | P 1 and (3.4) k [ h ( c ) , u ] k < ε for all c ∈ F . (3.5) Remark 3. 7. Note tha t in the deﬁnition of Prop erty (B2), b 0 and b 1 are deﬁned on G 0 and G 1 , r esp ectively , not on the subgroups generated by P 0 and P 1 . One should als o note that if A has Prop erty (B2) a sso ciated with any C*-subalge br a C( T ), then A also has Prop e r ty (B1). 10 Remark 3.8. Let A be a C*-alg ebra with Prop erty (B2) ass o ciated with C and ∆ c ( t, F , P 0 , P 1 , h ). Then the function ∆ ′ c and the subgroups G 0 and G 1 can be c hosen so that they only dep end on the unitary conjugate class of the unital embedding h . Indeed, pick o ne representativ e h λ for each co njugate class of the unital embedding of C to A , and deﬁne the function ∆ ′ c ( t, F , P 0 , P 1 , h ) = ∆ c ( t, F , P 0 , P 1 , h λ ) where h λ is in the unitary conjugate clas s of h . It is clear that ∆ ′ c only dep ends o n the conjugate c la ss of h . Let us show that A has P r op erty (B2) asso ciated with C and ∆ ′ c and G 0 and G 1 can b e chosen so that they only depe nd on h λ . Fix a unital monomorphism h : C → A, an y ε > 0 , a ﬁnite subset F ⊂ C, a ﬁnite subset P 0 ⊂ K 0 ( C ) , and a ﬁnite subset P 1 ⊂ K 1 ( C ). There is a unitary w ∈ A suc h that Ad( w ) ◦ h = h λ for some λ . Since A ha s Pr op erty (B2) as so ciated with C and ∆ c , there a re ﬁnitely generated subgroups G 0 ⊂ K 0 ( C ) and G 1 ⊂ K 1 ( C ) with G 0 and G 1 the sets of genera tors res pectively and P 0 ⊂ G 0 and P 1 ⊂ G 1 satisfying the following: for any homomor phisms b 0 : G 0 → K 1 ( A ) a nd b 1 : G 1 → K 0 ( A ) such that | τ ◦ b 1 ( g ) | < ∆ c ( ε, F , P 0 , P 1 , h λ ) (3.6) for any g ∈ G 1 and a ny τ ∈ T( A ), there exists a unitary u ∈ U( A ) such tha t bo tt 0 ( h λ , u ) | P 0 = b 0 | P 0 , b ott 1 ( h λ , u ) | P 1 = b 1 | P 1 and (3.7) k [ h λ ( c ) , u ] k < ε for all c ∈ F . (3.8) Then, bo tt 0 ( h, wuw ∗ ) = b o tt 0 ( h λ , u ) | P 0 = b 0 | P 0 , b ott 1 ( h, wuw ∗ ) = b o tt 1 ( h λ , u ) | P 1 = b 1 | P 1 (3.9) and k [ h ( c ) , wuw ∗ ] k < ε for all c ∈ F . (3.10) In other w ords, A has Pr op erty (B2 ) ass o ciated with ∆ ′ c . Ther efore, we can alw ays assume that the function ∆ c and the subgroups G 0 and G 1 only dep end on the conjugate classes of em beddings of C in to A . Lemma 3. 9 . Le t A b e a u n ital C*-algebr a which c ontains a p ositive element b with sp( b ) = [0 , 1] , and assume that A has Pr op erty (B1) . The r e ex ist s δ > 0 such t hat for any a ∈ K 0 ( A ) , if | τ ( a ) | ≤ δ for any τ ∈ T( A ) , then one has a = [ p ( U, z )] ∈ K 1 (S( A )) for some unitary z ∈ M k ( A ) (for some k ≥ 1 ) and some U ( t ) ∈ C([0 , 1] , U(M k ( A ))) with U (0) = 1 and k [ U (1 ) , z ] k ≤ δ p . In other wor ds, a ∈ C ( A ) . Pr o of. Since A has Prop erty (B1 ), for any 1 / 4 > ε > 0 and any unita ry z in a matrix of A with sp( z ) = T , there is a non-decrea sing p o sitive function δ z ( t ) suc h that if a ∈ K 0 ( A ) with | τ ( a ) | ≤ δ z ( ε ) for any trace τ , then there is a unitary u in the matrix of A such that k [ u, z ] k < ε and bott 1 ( u, z ) = a . F or a n y ε , there is δ e ( ε ) such that if | x − y | ≤ δ e ( ε ) with x, y ∈ { c ∈ C ; | c − 1 | ≤ 1 2 } , one has that | log( x ) − log( y ) | ≤ ε . W e also regard δ e ( ε ) as a p ositive function o f ε with δ e (0) = 0 and δ e ( ε ) > 0 if ε > 0. 11 F or a n y natural num ber k and any ε > 0, deﬁne ∆ z ( ε, k ) := ( min { δ z ( 1 2 ∆ z ( ε, k − 1)) , 1 2 δ e ( 1 2 ∆ z ( ε, k − 1)) , δ p , ε } if k ≥ 2 , min { δ z ( ε ) , δ p } if k = 1 . It is a p ositive function of ε and k with ∆ z ( ε, k ) > 0 if ε > 0. Fix 0 < ε < 1 / 12. Then, there exists m ∈ N such that there there is a par tition 0 = t 0 < t 1 < · · · < t m − 1 < t m = 1 such that for an y W ( t ) = ( e 2 π it p + (1 − p ))( e − 2 π it q + (1 − q )) ∈ U ∞ ( ] S( A )) wher e p and q are any pro jections, one has that k W ( t i − 1 ) − W ( t i ) k ≤ ε for each 1 ≤ i ≤ m . Fix this partition. Let z 0 = exp( i 2 π b ) . Since s p( b ) = [0 , 1] , z 0 is a unitary in A with sp( z 0 ) = T . Denote b y δ = 1 2 min { ∆ z 0 ( ε, 2 m ) , ∆ z 0 ( ε, 2 m − 1) , ..., ∆ z 0 ( ε, 1) } . Let a be a n elemen t in K 0 ( A ) with | τ ( a ) | < δ for any τ ∈ T( A ) . Without loss of gener a lity , we ma y assume that a = [ p ] − [ q ] for some pro jections p, q ∈ M k ( A ) for some integer k ≥ 1 . Note tha t | τ ( p ) − τ ( q ) | < δ . Put W 0 ( t ) = ( e 2 π it p + (1 − p ))( e − 2 π it q + (1 − q )) . T o simplify nota tion, b y replacing A b y M k ( A ) , without loss of generality , w e may assume that p, q ∈ A and W 0 ( t ) ∈ A for each t ∈ [0 , 1] . It follows from Lemma 8 of [19] that, for some large n ≥ 1 , there is a unitary V t ∈ M n +1 ( A ) , for ea ch t ∈ [0 , 1 ] , k V ∗ t z V t − W ( t ) z k < δ / 2 , (3.11) where W ( t ) = diag ( W 0 ( t ) , n z }| { 1 , 1 , ..., 1) , and z = diag( z 0 , ω , ω 2 , ..., ω n ) . and where ω = e 2 π i/n +1 . W e a ssert that o ne can ﬁnd unitaries V 0 , V 1 , ..., V m such that k V i z V ∗ i − W ( t i ) z k < ∆ z 0 ( ε, 2 m + 1 − 2 i ) for any 0 ≤ i ≤ m, and bo tt 1 ( V ∗ i +1 V i , z ) = 0 for a n y 0 ≤ i ≤ m − 1 . Assume that the unitaries V 0 , V 1 , ..., V i satisfy the condition ab ov e. As indicated above, one can ﬁnd a unitary V i +1 such that   V i +1 z V ∗ i +1 − W ( t i +1 ) z   < min { 1 2 δ e ( 1 2 ∆ z 0 ( ε, 2 m − 2 i )) , 1 2 ∆ z 0 ( ε, 2 m − 2 i − 1) } , and denote b i := b ott 1 ( V ∗ i +1 V i , z ) . 12 By Theor em 3.6 o f [15], o ne has that for any τ ∈ T( A ) | τ ( b i ) | = 1 2 π   τ (log ( z ∗ V ∗ i +1 V i z V ∗ i V i +1 ))   = 1 2 π   τ (log ( V i +1 z ∗ V ∗ i +1 V i z V ∗ i ))   . Note that   V i +1 z ∗ V ∗ i +1 V i z V ∗ i − z ∗ W ∗ ( t i +1 ) W ( t i ) z   < 1 2 δ e ( 1 2 ∆ z 0 ( ε, 2 m − 2 i )) + ∆ z 0 ( ε, 2 m − 2 i + 1) < δ e ( 1 2 ∆ z 0 ( ε, 2 m − 2 i )) . Therefore, we hav e   log( V i +1 z ∗ V ∗ i +1 V i z V ∗ i ) − log ( z ∗ W ∗ ( t i +1 ) W ( t i ) z )   < 1 2 ∆ z 0 ( ε, 2 m − 2 i ) , and hence | τ ( b i ) | < 1 2 π | τ (log ( W ∗ ( t i +1 ) W ( t i ))) | + 1 2 ∆ z 0 ( ε, 2 m − 2 i ) = 1 2 π ( t i +1 − t i ) | ( τ ( p ) − τ ( q )) | + 1 2 ∆ z 0 ( ε, 2 m − 2 i ) < ∆ z 0 ( ε, 2 m − 2 i ) < δ z 0 ( 1 2 ∆ z 0 ( ε, 2 m − 2 i − 1)) . Since A has Pro per t y (B1), there is a unitary v ′ i +1 ∈ A s uch that   v ′ i +1 z 0 ( v ′ i +1 ) ∗ − z 0   < 1 2 ∆ z 0 ( ε, 2 m − 2 i − 1) and bott 1 ( v ′ i +1 , z 0 ) = b i . Set v i +1 = diag( v ′ i +1 , 1 , ..., 1) . W e then ha ve   v i +1 z v ∗ i +1 − z   =   v ′ i +1 z 0 ( v ′ i +1 ) ∗ − z 0   < 1 2 ∆ z 0 ( ε, 2 m − 2 i − 1) and bo tt 1 ( v i +1 , z ) = b ott 1 ( v ′ i +1 , z 0 ) = b i . Therefore   ( V i +1 v i +1 ) z ( v ∗ i +1 V ∗ i +1 ) − W ( t i +1 ) z   < 1 2 ∆ z 0 ( ε, 2 m − 2 i − 1) +   V i +1 z V ∗ i +1 − W ( t i +1 ) z   < ∆ z 0 ( ε, 2 m − 2 i − 1) and bo tt 1 (( v ∗ i +1 V ∗ i +1 ) V i , z ) = bott 1 ( v ∗ i +1 , z ) + b ott 1 ( V ∗ i +1 V i , z ) = 0 . By repla c ing V i +1 by V i +1 v i +1 , we prov ed the asser tion. Then, for ea ch V i and V i +1 , there is a path of unitary V ( i ) ( t ) such that k [ V ( i ) ( t ) , z ] k < ε, V ( i ) (0) = 1 , V ( i ) (1) = V ∗ i +1 V i . 13 By setting U ( i ) ( t ) = V i +1 V ( i ) (1 − t ), w e have that U ( i ) (0) = V i , U ( i ) (1) = V i +1 , and for eac h t ,    U ( i ) ( t ) z ( U ( i ) ( t )) ∗ − W ( t ) z    =    V i +1 V ( i ) (1 − t ) z ( V ( i ) (1 − t )) ∗ V ∗ i +1 − W ( t ) z    < ε +   V i +1 z V ∗ i +1 − W ( t ) z   ≤ 2 ε ≤ 1 4 . By connecting all U ( i ) ( t ), we get a path of unitar y U ( t ) such that for an y t , k U ( t ) z U ∗ ( t ) − W ( t ) z k < 1 4 . In par ticular, a = [ W ( t )] 1 = [ p ( U, z )] . Moreov er, k [ U (1 ) , z ] k = k [ V m , z ] k < δ p , as desir ed. Corollary 3.10. L et A b e a unital sep ar able simple C*-algebr a with TR( A ) = 0 . Then ther e exists δ > 0 such that for any element a ∈ K 0 ( A ) , if | τ ( a ) | ≤ δ for any τ ∈ T( A ) , then one has a = [ p ( U, z )] ∈ K 1 (S( A )) for some unitary z ∈ M k ( A ) and some U ( t ) ∈ C([0 , 1 ] , U(M k ( A ))) (for some k ≥ 1 ) with U (0 ) = 1 and k [ U (1 ) , z ] k < δ p . Pr o of. By Lemma 5 .2 of [1 5], A has Prop erty (B1). Then, the statement follows from Lemma 3.9. Corollary 3.11 . L et A b e a unital sep ar able simple C*-algebr a with TR( A ) = 0 . Then C ( A ) = K 0 ( A ) . Pr o of. Denote by δ the constant of Coro llary 3.10. F or any a ∈ K 0 ( A ), since K 0 ( A ) is tracially appr oximately divisible (see Deﬁnition 3.15), one has a = a 1 + a 2 + · · · + a n with | τ ( a i ) | < δ for ea ch 1 ≤ i ≤ n . Therefore, a i ∈ C ( A ). Since C ( A ) is a gr oup, one has that a ∈ C ( A ), a s desired. Deﬁnition 3.12. F or an y unitary u in a C*-algebra A , denote b y R ( u, t ) the unitary path in M 2 ( A ) deﬁned b y R ( u, t ) := u 0 0 1 ! cos( π t 2 ) sin( π t 2 ) − sin( π t 2 ) cos( π t 2 ) ! u ∗ 0 0 1 ! cos( π t 2 ) − sin( π t 2 ) sin( π t 2 ) cos( π t 2 ) ! , t ∈ [0 , 1] . Note that R ( u, 0) = 1 0 0 1 ! and R ( u, 1) = u 0 0 u ∗ ! 14 Lemma 3.13. L et A b e a unital C*-algebr a. L et B ⊆ A b e a unital sep ar able C*-sub algebr a. Write K 0 ( B ) + = { k 1 , k 2 , ..., k n , ... } and K 1 ( B ) = { h 1 , h 2 , ..., h n , ... } , and denote by K n = < k 1 , ..., k n > the sub gr oup gener ate d by { k 1 , ..., k n } , and by H n = < h 1 , ..., h n > the sub gr oup gener ate d by { h 1 , ..., h n } . L et {F i } b e an incr e asing family of ﬁ nite subsets whose union is dense in B . Assume that for e ach i , ther e exist a pr oje ction p i ∈ M r i ( F i ) and a unitary z i ∈ M r i ( F i ) with [ p i ] 0 = k i and [ z i ] 1 = h i . L et { u n } b e a se quenc e of unitaries such that    [ u ( r n ) n +1 , a ]    ≤ δ p 2 n +1 for any a ∈ M r n ( w ∗ n F n w n ) , wher e w n = u 1 · · · u n and u ( r j ) k = diag( r j z }| { u k , u k , ..., u k ) . Then α = lim n →∞ Ad( w n ) deﬁnes a monomorphism fr om B to A , and the ex tensions η 0 ( M α ) and η 1 ( M α ) ar e determine d by the inductive limits 0 K 1 ( A ) K 1 ( A ) ⊕ K n +1 K n +1 0 0 K 1 ( A ) K 1 ( A ) ⊕ K n K n 0 / / / / / / / / / / / / / / / / ι n,n +1 O O ι n,n +1 O O γ 0 n ` ` A A A A A A A A A A and 0 K 0 ( A ) K 0 ( A ) ⊕ H n +1 H n +1 0 0 K 0 ( A ) K 0 ( A ) ⊕ H n H n 0 / / / / / / / / / / / / / / / / ι n,n +1 O O ι n,n +1 O O γ 1 n ` ` A A A A A A A A A A r esp e ct ively, wher e γ 0 n : K n → K 1 ( A ) is deﬁne d by k i 7→ [( w ∗ n p i w ∗ n ) u n +1 ( w ∗ n p i w n ) + (1 − w ∗ n p i w n )] and γ 1 n : H n → K 0 ( A ) is deﬁne d by h i 7→ [ p ( R ∗ ( u n +1 , t ) , w ∗ n z i w n )] . Pr o of. It is clear that we may assume that r n ≤ r n +1 , n = 1 , 2 , ... Note that for any a ∈ F n , k Ad( w n + k )( a ) − Ad( w n )( a ) k ≤ k X i =1 k Ad( u 1 u 2 · · · u n + i )( a ) − Ad( u 1 u 2 · · · u n + i − 1 )( a ) k ≤ k X i =1 1 2 n + i ≤ 1 2 n . Therefore lim Ad( w n ) exists if n → ∞ . D enote by α n = Ad( w n ), and deno te its limit by α . N ote that α is a monomorphism. Moreov er, [ α ] = [ ı ] in K L ( B , A ) , (3.12) 15 where ı : B → A is the em b edding. It follo ws that the six-term exact s equence in 2.3 splits. T o simplify notation, without loss o f generality , in what follows, we ma y replace α ⊗ id M r i by α, α n ⊗ id M r i by α n , u ( r n ) n by u n , and w ( r n ) n by w n resp ectively (and write p i ∈ A ). Consider K 0 ( M α ( A )) ﬁr s t. Fix n , and note that k α n ( p i ) − α ( p i ) k ≤ 1 4 , for any 1 ≤ i ≤ n. Therefore, for each i , there is a unitary v i with k v i − 1 k ≤ 1 2 such that α ( p i ) = v ∗ i α n ( p i ) v i . In particular, there is a path r ( n ) i ( t ) of pro jection with r ( n ) i (0) = α n ( p i ), r ( n ) i (1) = α ( p i ), and    r ( n ) i ( t ) − α ( p i )    ≤ 1 2 . Then there is a homomorphism ψ (0) n : K 1 ( A ) ⊕ K n → K 0 ( M α ) deﬁned b y ψ (0) n : ([ Q ( t )] , [ p i ]) 7→ [ Q ( t )] + [ P ( n ) i ( t )] , where P ( n ) i ( t ) is the path P ( n ) i ( t ) = ( R ∗ ( w n , 2 t ) p i R ( w n , 2 t ) 0 ≤ t ≤ 1 2 r ( n ) i (2 t − 1) 1 2 ≤ t ≤ 1 , and [ Q ( t )] ∈ K 0 (S( A )). Since [ π ] 0 ◦ ψ (0) n = id K n , the map ψ (0) n is injective. W e then ha ve ψ (0) n +1 ([ Q ( t )] , [ p i ]) − ψ (0) n ([ Q ( t )] , [ p i ]) = [ P ( n +1) i ( t )] − [ P ( n ) i ( t )] ∈ K 0 (S( A )) . (3.13) Moreov er, it is ea sy to see that [ P ( n +1) i ( t )] − [ P ( n ) i ( t )] = [ R ∗ ( u n +1 , t ) w ∗ n p i w n R ( u n +1 , t )] − [ w ∗ n p i w n ] ∈ K 0 (S( A )) . Deﬁne homo mo rphism ψ (0) n,n +1 : K 1 ( A ) ⊕ K n → K 1 ( A ) ⊕ K n +1 by ψ (0) n,n +1 : ( Q ( t ) , [ p i ( t )]) 7→ ([ Q ( t )] + ([ P ( n ) i ( t )] − [ P ( n +1) i ( t )]) , [ p i ( t )]) . By the construction we hav e that ψ (0) n +1 ◦ ψ (0) n,n +1 = ψ (0) n . Thus, we obtain a homomorphism ψ (0) : lim − → ( K 0 ( M α n ) , ψ n,n +1 ) → K 0 ( M α ( A )) . Let us show that ψ (0) is s urjective. F or each pro jection p ( t ) ∈ M α , we can assume that [ p (0)] 0 = k i = [ p i ] 0 ∈ K n for some i ≤ n , and if denote by v the par tial isometr y with v ∗ p i v = p (0), then k w ∗ n v w n − v k ≤ 1 2 . Then there is a path of partial iso metries v ( t ) ∈ M α such that v (0) = v a nd v (1) = α ( v ). Then h := [ p ( t )] − [ v ∗ ( t ) P ( n ) i ( t ) v ( t )] is an element in K 0 (S( A )) a nd [ p ( t )] 0 = ψ (0) ( h, k i ). Therefore ψ (0) is surjective. The injectivity of ψ (0) follows from the injectivit y of each ψ (0) n . Thus, ψ (0) is a n isomor phis m. Let us sho w that ψ (0) n,n +1 has the desired form. Consider the inv er tible e le men t c = ( w ∗ n p i w n ) u n +1 ( w ∗ n p i w n ) + (1 − ( w ∗ n p i w n )) ∈ A (whic h is close to a unitary). Let us calculate the co rresp onding element in S( A ). Consider the path Z ( t ) = w ∗ n p i w n 0 0 w ∗ n p i w n ! R ( u n +1 , t ) w ∗ n p i w n 0 0 w ∗ n p i w n ! + 1 − w ∗ n p i w n 0 0 1 − w ∗ n p i w n ! . 16 Note that Z (0) = diag(1 , 1 ) and Z (1) = diag( c, c ∗ ) and Z ( t ) is inv ertible for any t with k Z ∗ ( t ) Z ( t ) − 1 k ≤ 1 2 n . Let e ( t ) = ( Z ∗ ( t ) Z ( t )) − 1 2 Z ∗ ( t ) 1 0 0 0 ! Z ( t )( Z ∗ ( t ) Z ( t )) − 1 2 . Then the elemen t in K 0 (S( A )) which cor r esp onds to c is [ e ] − [ 1 0 0 0 ! ]. Howev er , since      " w ∗ n p i w n 0 0 w ∗ n p i w n ! , R ( u n +1 , t ) #      ≤ 2 k [ w ∗ n p i w n , u n +1 ] k ≤ 1 2 n , a direct calculation shows e ( t ) ≈ 2 / (2 n − 1) w ∗ n p i w n 0 0 w ∗ n p i w n ! R ∗ ( u n +1 , t ) w ∗ n p i w n 0 0 0 ! R ( u n +1 , t ) w ∗ n p i w n 0 0 w ∗ n p i w n ! + 1 − w ∗ n p i w n 0 0 0 ! ≈ 2 / 2 n R ∗ ( u n +1 , t ) w ∗ n p i w n 0 0 w ∗ n p i w n ! w ∗ n p i w n 0 0 0 ! w ∗ n p i w n 0 0 w ∗ n p i w n ! R ( u n +1 , t ) + 1 − w ∗ n p i w n 0 0 0 ! = R ∗ ( u n +1 , t ) w ∗ n p i w n 0 0 0 ! R ( u n +1 , t ) + 1 − w ∗ n p i w n 0 0 0 ! , and therefor e , for n > 2, [ e ] − [ 1 0 0 0 ! ] = [ R ∗ ( u n +1 , t ) w ∗ n p i w n 0 0 0 ! R ( u n +1 , t )] − [ w ∗ n p i w n 0 0 0 ! ] = [ R ∗ ( u n +1 , t ) w ∗ n p i w n R ( u n +1 , t )] − [ w ∗ n p i w n ] = [ P ( n +1) i ( t )] − [ P ( n ) i ( t )] . Therefore, the corresp onding element of [ P ( n +1) i ( t )] − [ P ( n ) i ( t )] in K 1 ( A ) is ( w ∗ n p i w n ) u n +1 ( w ∗ n p i w n ) + (1 − ( w ∗ n p i w n )) , and hence the map ψ (0) n,n +1 has the desired form. Now, let us consider K 1 ( M α ). F o r each n , consider the unitaries { z i ; 1 ≤ i ≤ n } . Note that k α ( z i ) − α n ( z i ) k ≤ 1 2 n for each i . Then, there a re paths of unitaries s ( n ) i such that s ( n ) i (0) = α n ( v i ), s ( n ) i (1) = α ( v i ), and    s ( n ) i ( t ) − α ( v i )    ≤ 1 2 n − 1 for each t ∈ [0 , 1]. Deﬁne a ho momorphism ψ (1) n : K 0 ( A ) L H n → K 1 ( M α ( A )) by ψ (1) n : ([ S ( t )] , z i ) 7→ [ S ( t )] + [ V ( n ) i ( t )] 17 where V ( n ) i is the path V ( n ) i ( t ) = ( R ∗ ( w n , 2 t ) z i R ( w n , 2 t ) 0 ≤ t ≤ 1 2 s ( n ) i (2 t − 1) 1 2 ≤ t ≤ 1 , and S ( t ) is a unitary in U ∞ ( ] S( A )). W e then ha ve ψ n +1 ([ S ( t )] , [ v i ]) − ψ n ([ S ( t )] , [ v i ]) = [ V ( n +1) i ( t )] − [ V ( n ) i ( t )] = [ p ( R ∗ ( u n +1 , t ) , w ∗ n v i w n )] ∈ K 1 (S( A )) . Deﬁne homo mo rphism ψ (1) n,n +1 : K 0 ( A ) ⊕ H n → K 0 ( A ) ⊕ H n +1 by ψ (0) n,n +1 : ( S ( t ) , [ v i ]) 7→ ([ S ( t )] + ([ V ( n ) i ( t )] − [ V ( n +1) i ( t )]) , [ v i ]) . By the construction we hav e that ψ (1) n +1 ◦ ψ (1) n,n +1 = ψ (1) n . Thus, there is a homomor phism ψ (1) : lim − → ( K 0 ( M α n ) , ψ n,n +1 ) → K 0 ( M α ) . By the same argument as that for K 0 ( M ) α , the map ψ (1) is an isomo rphism, and moreover, it has the des ired form. 3.14. Let 0 / / G / / E π / / H / / 0 be a sho r t ex act sequence o f ab elian gr oups with H co un table. W e assume that the extens io n is pur e , i.e., for any ﬁnitely genera ted subgroup H ′ ⊆ H , there is a homomorphism θ ′ : H ′ → E such that π ◦ θ ′ = id H ′ . W rite H = { h 1 , h 2 , ... } . Co ns ider H n = < h 1 , ..., h n > , the subgro up generated b y { h 1 , ..., h n } . Since E is a pure extension, ther e is a map θ n : H n → E such that π ◦ θ n = id H n . L e t us call this map θ n a partial splitting map. Denote by ι n the inclusio n map from H n to H n +1 , and set the ma p γ n : H n → E by γ n = θ n +1 ◦ ι n − θ n . It is clear that γ n ( H n ) ⊆ G ⊆ E . Ther efore, let us regard γ n as a map from H n to G . Deﬁnition 3 . 15. A par tially ordered group G with an order unit is tracially approximately divisible if for any a ∈ G , any ε > 0, and an y natural n umber n , there ex ist b ∈ G suc h that | τ ( a − nb ) | ≤ ε for any state τ of G . Lemma 3.16 . With the setting as 3.14, if, in additio n, G is a p artial ly or der e d gr oup with an or der u nit which is tr acial ly appr oximately divisible, then, for any ε > 0 and any given p artial splitting map θ n with π ◦ θ n = id H n , one c an cho ose a p artial splitting map θ n +1 such that | τ ( γ n ( h i )) | < ε for any 1 ≤ i ≤ n and any state τ of G . Pr o of. Pic k any partial splitting map θ ′ n +1 : H n +1 → E 1 , and consider the map γ ′ n : H n → G deﬁned b y γ ′ n = θ ′ n +1 ◦ ι n − θ n . Consider the Q -linear map γ ′ n ⊗ id : H n O Q → G O Q . 18 Then it has an extens tio n to H n +1 N Q . That is, there is an linear map ψ : H n +1 N Q → G N Q such that G N Q H n +1 N Q ψ o o H n N Q γ ′ n ⊗ id f f L L L L L L L L L L ι n ⊗ id O O commutes. Since H n +1 is ﬁnitely g enerated, H n +1 ∼ = Z k n +1 L T n +1 for so me ﬁnite ab elia n gro up T n +1 . Denote by { e 1 , e 2 , ..., e k n +1 } the standar d genera tors for the torsion free part of H n +1 . Then, for each 1 ≤ i ≤ k n +1 , w e hav e ψ ( e i ) = l i X j =1 r ( i ) j g ( i ) j for some r ( i ) j ∈ Q a nd g ( i ) j ∈ G . W rite ( ι n ( h s )) free = P i m ( s ) i e i , and deno te b y m = ma x { m ( s ) i ; 1 ≤ s ≤ n, 1 ≤ i ≤ k n +1 } . Since G is tra cially approximately divisible, for each e i , one can ﬁnd p i ∈ G s uch that τ ( ψ ( e i ) − p i ) < ε k n +1 m for any state τ of G . Deﬁne the map φ : H n +1 → K 0 ( A ) ⊂ E 1 by sending e i to p i and T n +1 to { 0 } , and let us consider the map θ n +1 := θ ′ n +1 − φ. Then, the map θ n +1 satisﬁes the lemma. Indeed, it is clear that π ◦ θ n +1 = id H n +1 . Mo reov er, for a ny h s and an y τ , w e hav e | τ ( θ n +1 ◦ ι n ( h s ) − θ n ( h s )) | =   τ ( θ ′ n +1 ◦ ι n ( h s ) − φ ◦ ι n ( h s ) − θ n ( h s ))   = | τ ( γ ′ n ( h s ) − φ ◦ ι n ( h s )) | = | τ ( ψ ◦ ι n ( h s ) − φ ◦ ι n ( h s )) | =      X i m ( s ) i τ ( ψ ( e i ) − p i )      < ε, as desir ed. Theorem 3.1 7. L et A b e a unital simple C*-algebr a and let B ⊆ A b e a unital sep ar able C*-sub algebr a. Supp ose that A c ontains a p ositive element b with sp( b ) = [0 , 1] , and A has Pr op erty (B1) and Pr op erty (B2) asso ciate d with B and ∆ B and K 0 ( A ) is tr acial ly appr oximately divisible. F or any E 0 ∈ Pext(K 1 (A) , K 0 (B)) and E 1 ∈ Pext(K 0 (A) , K 1 (B)) , ther e exists α ∈ Inn( B , A ) su ch t hat η 0 ( M α ) = E 0 and η 1 ( M α ) = E 1 . Pr o of. W rite K 0 ( B ) + = { k 1 , k 2 , ..., k n , ... } and K 1 ( B ) = { h 1 , h 2 , ..., h n , ... } , 19 and cons ide r the subg roups K n := < k 1 , ..., k n > and H n := < h 1 , ..., h n > . Let {F i } b e a n increa sing family o f ﬁnite subsets in the unit ball of B with the union dense in the unit ball of B . Assume that for eac h i , there is a pro jection p i ∈ M r i ( F i ) and unitary z i ∈ M r i ( F i ) with [ p i ] 0 = k i and [ z i ] 1 = h i . W e ma y assume tha t r i ≤ r i +1 , i ∈ N . W e a ssert that ther e are unitaries { u n } and diagrams 0 K 1 ( A ) E 0 K 0 ( B ) 0 0 K 1 ( A ) K 1 ( A ) ⊕ K n +1 K n +1 0 0 K 1 ( A ) K 1 ( A ) ⊕ K n K n 0 . . . . . . . . . . . . . . . . . . . . . . . . / / / / π n / / θ 0 n +1 o o / / / / / / π / / / / / / / / π / / θ 0 n o o / / ι n,n +1 O O ι n,n +1 O O γ 0 n ` ` A A A A A A A A A A O O O O O O O O ` ` A A A A A A A A A A A O O O O ` ` A A A A A A A A A A A and 0 K 0 ( A ) E 1 K 0 ( B ) 0 0 K 0 ( A ) K 0 ( A ) ⊕ H n +1 H n +1 0 0 K 0 ( A ) K 0 ( A ) ⊕ H n H n 0 . . . . . . . . . . . . . . . . . . . . . . . . / / / / π n / / θ 1 n +1 o o / / / / / / π / / / / / / / / π / / θ 1 n o o / / ι n,n +1 O O ι n,n +1 O O γ 1 n ` ` A A A A A A A A A A O O O O O O O O ` ` A A A A A A A A A A A O O O O ` ` A A A A A A A A A A A such that k [ u n +1 , a ] k ≤ δ p r 2 n · 2 n +1 for any a ∈ M r n ( w ∗ n F n w n ), where w n = u 1 · · · u n and u 1 = 1. The image of each γ 1 n lies inside C ( A ) so that Λ ◦ γ 1 n is well-deﬁned, and bo tt 1 ( w ∗ n z i w n , u n +1 ) = Λ ◦ γ 1 n ( h i ) and bo tt 0 ( w ∗ n p i w n , u n +1 ) = γ 0 n ( k i ) . Moreov er, each partia l splitting map θ i n ( i = 0 , 1) ca n b e extended to a partial splitting ma p ˜ θ i n ( i = 0 , 1) deﬁned on the s ubgroup g e ne r ated b y K n ∪ G n 0 or H n ∪ G n 1 , where G i ( i = 0 , 1 ) is the set of ge ne r ators of G i 20 ( i = 0 , 1) of Deﬁnition 3.6 w ith r esp ect to δ p r 2 n +1 · 2 n +1 , F n , P ( n ) 0 , P ( n ) 1 , and ı , wher e P 0 = { [ p 1 ] , [ p 2 ] , ..., [ p n ] } a nd P 1 = { [ z 1 ] , [ z 2 ] , ..., [ z n ] } . Denote b y the subgroups g enerated by K n ∪ G 0 and H n ∪ G 1 by e K n and e H n resp ectively . Assume that w e ha ve construc ted the unitaries { u 1 = 1 , u 2 , ..., u n } and the diagrams 0 K 1 ( A ) E 0 K 1 ( B ) 0 0 K 1 ( A ) K 1 ( A ) ⊕ K n K n 0 0 K 1 ( A ) K 1 ( A ) ⊕ K n − 1 K n − 1 0 . . . . . . . . . . . . / / / / π n / / θ 0 n o o / / / / / / π / / / / / / / / π / / θ 0 n − 1 o o / / ι n − 1 ,n O O ι n − 1 ,n O O γ 0 n − 1 ` ` A A A A A A A A A A O O O O O O O O ` ` A A A A A A A A A A A and 0 K 0 ( A ) E 1 K 0 ( B ) 0 0 K 0 ( A ) K 0 ( A ) ⊕ H n H n 0 0 K 0 ( A ) K 0 ( A ) ⊕ H n − 1 H n − 1 0 . . . . . . . . . . . . / / / / π n − 1 / / θ 1 n o o / / / / / / π / / / / / / / / π / / θ 1 n − 1 o o / / ι n − 1 ,n O O ι n − 1 ,n O O γ 1 n − 1 ` ` A A A A A A A A A A O O O O O O O O ` ` A A A A A A A A A A A satisfying the above a ssertion. Denote by δ n = ∆ B ( δ p r 2 n +1 · 2 n +1 , F n , P ( n ) 0 , P ( n ) 1 , ad( w n ) ◦ ı ) , where P 0 = { [ p 1 ] , [ p 2 ] , ..., [ p n ] } a nd P 1 = { [ z 1 ] , [ z 2 ] , ..., [ z n ] } . Set w ( r n ) n = diag( r n z }| { w n , w n , ..., w n ) , n = 1 , 2 , ... W e no te that [ p i ] = [( w ( r n ) n ) ∗ p i w ( r n ) n ] and [ z i ] = [( w ( r n ) n ) ∗ z i w ( r n ) n ] , i = 1 , 2 , ... Since E 0 and E 1 are pure extensions, there ar e partial splitting maps ˜ θ 0 n +1 : e K n +1 → E 0 and ˜ θ 1 n +1 : e H n +1 → E 1 . Since K 0 ( A ) is tracia lly appr oximately divisible, by Lemma 3.1 6, the par tial splitting map ˜ θ 1 n +1 can b e chosen so that for any g ∈ G n i ∪ { g 1 , ..., g n } ,   τ ( γ 1 n ( g ))   ≤ min { δ, δ n } , for all τ ∈ T( A ) , where δ is the cons tant of Le mma 3 .9 (since A has Pro per ty (B1)). Note that the maps γ 0 n and γ 1 n are deﬁned o n e K n and e H n resp ectively , in particular , on G 0 and G 1 resp ectively . By Lemma 3 .9, one has γ 1 n ( h i ) = [ U ∗ ( t ) z U ( t ) z ∗ ] 1 ∈ K 1 (S( A )) 21 for a unitary z ∈ A and a path U ( t ) ∈ C([0 , 1] , U ∞ ( A )) with U 0 = 1 and k [ U (1) , z ] k ≤ δ p . Ther efore, the map Λ ◦ γ 1 n | H n : H n → K 0 ( A ) is well-deﬁned, and by Lemma 3.5,   τ (Λ ◦ γ 1 n ( h i ))   =   τ ( γ 1 n ( h i ))   ≤ δ n for any τ ∈ T( A ) and 1 ≤ i ≤ n . Put b 0 = γ 0 n and b 1 = Λ ◦ γ 1 n . By the assumption tha t A has Prop erty (B2) asso ciated with B and ∆ B , there is a unitary u n +1 ∈ A such that k [ u n +1 , a ] k ≤ 1 r 2 n · 2 n +1 for any a ∈ M r n ( w ∗ n F n w n ) and bo tt 1 ( w ∗ n z i w n , u n +1 ) = Λ ◦ γ 1 n ( h i ) and bo tt 0 ( w ∗ n p i w n , u n +1 ) = γ 0 n ( k i ) . Denote by θ i n +1 ( i = 0 , 1) the restr ic tion of ˜ θ i n +1 ( i = 0 , 1) to K n +1 and H n +1 resp ectively . Repea ting this pro cedure, we get a sequence o f unitaries { u n } and diagrams satisfying the asser tion. By Lemma 3.13, the inner automorphisms { Ad( u 1 · · · u n ) } converge to a monomorphism α , a nd the extension η 0 ( M α ) and η 1 ( M α ) ar e determined by the inductive limits of 0 K 1 ( A ) K 1 ( A ) ⊕ K n +1 K n +1 0 0 K 1 ( A ) K 1 ( A ) ⊕ K n K n 0 / / / / / / / / / / / / / / / / ι n,n +1 O O ι n,n +1 O O ˜ γ 0 n ` ` A A A A A A A A A A and 0 K 0 ( A ) K 0 ( A ) ⊕ H n +1 H n +1 0 0 K 0 ( A ) K 0 ( A ) ⊕ H n H n 0 / / / / / / / / / / / / / / / / ι n,n +1 O O ι n,n +1 O O ˜ γ 1 n ` ` A A A A A A A A A A resp ectively , where ˜ γ 0 n ( k i ) = ˜ γ 0 n ([ p i ]) = [( w ∗ n p i w n ) u n +1 ( w ∗ n p i w n ) + (1 − w ∗ n p i w n )] and ˜ γ 1 n ( h i ) = ˜ γ 1 n ([ z i ]) = [ p ( R ∗ ( u n +1 , t ) , w ∗ n z i w n )] , and therefor e , ˜ γ 0 n ( k i ) = b o tt 0 ( w ∗ n p i w n , u n +1 ) = γ 0 n ( k i ) , 1 ≤ i ≤ n, that is, ˜ γ 0 n = γ 0 n . F or each cross ma p ˜ γ 1 n , one has Λ ◦ ˜ γ 1 n ( h i ) = b o tt( w ∗ n z i w n , R ( u n +1 , 1)) = b ott( w ∗ n z i w n , u n +1 ) = Λ ◦ γ 1 n ( h i ) , 1 ≤ i ≤ n. Since Λ is injective, w e have that ˜ γ 1 n = γ 1 n . Hence, one has that η 0 ( M α ) = E 0 and η 1 ( M α ) = E 1 , as desired. 22 Prop ositio n 3.18 . L et C b e a unital AH-algebr a and let A b e a u nital sep ar able s imple C*-algebr a with TR( A ) = 0 . S upp ose that ther e is a unital monomorphism h : C → A. Then A has Pr op erty (B2) asso ciate d with C for some ∆ C as describ e d in 3.6. Pr o of. Fix ε > 0 , ﬁnite subset F ⊂ C, P 0 ⊂ K 0 ( C ) and P 1 ⊂ K 1 ( C ) . W rite C = lim − → ( C n , ψ n ) so that C n and ψ n satisfy the conditions in 7.2 of [15]. Let ∆ B ( ε, F , P 0 , P 1 , h ) be δ a s r equired b y Lemma 7.5 o f [15] for the above ε, F , P = P 0 ∪ P 1 . Let n ≥ 1 b e an integer in 7.5 of [15] so we ma y assume that P ⊂ ∪ i =0 , 1 ( ψ n, ∞ ) ∗ i ( K i ( C n )) and let k ( n ) ≥ n be as in 7.5 of [15]. Put G i = ( ψ k ( n ) , ∞ ) ∗ i ( K i ( C k ( n ) ) , i = 0 , 1 . In particular, G i is ﬁnitely generated and P i ⊂ G i , i = 0 , 1 . Let b i : G i → K i − 1 ( A ) b e giv e n, i = 0 , 1 . W r ite K i ( C k ( n ) ⊗ C( T )) = K i ( C k ( n ) ) ⊕ β ( K i − 1 ( C k ( n ) )) , i = 0 , 1 a nd (3.14) K ( C k ( n ) ⊗ C( T )) = K ( C k ( n ) ) ⊕ β ( K ( C k ( n ) )) (3.15) (see 2.10 of [14]). Deﬁne κ (0) : K ( C k ( n ) ) → K ( A ) by κ (0) = [ h ◦ ψ k ( n ) , ∞ ] . Deﬁne κ (1) i : β ( K i ( C k ( n ) )) → K i − 1 ( A ) by κ (1) i ◦ β ( x ) = b i − 1 ( x ) for x ∈ K i − 1 ( C k ( n ) ) , i = 0 , 1 . Since C s a tisﬁes the UCT, there is κ (1) ∈ K K (S( C k ( n ) ) , A ) suc h that Γ( κ (1) ) = κ (1) i . Deﬁne κ : K ( C k ( n ) ⊗ C( T )) → K ( A ) by κ | K ( C k ( n ) ) = κ (0) and κ | β ( K ( C k ( n ) ) = κ (1) . The lemma then follows from Lemma 7 .5 of [1 5]. Lemma 3 .19. L et B b e a unital sep ar able simple amenable C*-algebr a with TR( B ) = 0 which satisﬁes the UCT, and let A b e a un ital simple C*-algeb r a with r e al r ank zer o, stable r ank one and we akly unp erfor ate d K 0 ( A ) . Supp ose t hat ¯ κ ∈ K L ( B , A ) ++ with ¯ κ ([1 B ]) ≤ [1 A ] in K 0 ( A ) . Then ther e is a monomorphism α : B → A such that [ α ] = ¯ κ in K L ( B , A ) . (3.16) Pr o of. It follo ws from the classiﬁca tion theorem ([12]) that B is a unital simple AH-algebra with slow dimension growth and with real rank zero . Then the lemma follows from Theo rem 4.6 of [10] immediately . Theorem 3.20. L et A and B b e un ital simple C*-algebr as with TR( A ) = 0 and TR( B ) = 0 . Assume that B is sep ar able amenable and satisﬁes the UCT. Then, for any κ ∈ K K ( B , A ) ++ with κ ([1 B ]) ≤ [1 A ] in K 0 ( A ) , t her e is a monomorph ism α : B → A such that [ α ] = κ in K K ( B , A ) . Pr o of. Denote by ¯ κ the ima ge of κ in K L ( B , A ) + . It follows from Lemma 3.19 that there exis ts α 1 ∈ Hom( B , A ) such that [ α 1 ] K L = ¯ κ . B y considering the cut-down of A b y α 1 (1 B ), we can regar d B as a unital C*-suba lgebra of A with e mbedding α 1 . Since κ − [ α 1 ] K K ∈ P ext( K ∗ ( B ) , K ∗ +1 ( A )), by Theorem 3.17, there is an approximately inner monomorphism α 2 of from B to A such that [ α 2 ◦ α 1 ] K K − [ α 1 ] K K = κ − [ α 1 ] K K . Then, α := α 2 ◦ α 1 is the desired homo morphism. 23 Let us recall the following theorem from [18]: Theorem 3.2 1 (Theorem 5.2 of [18]) . L et C b e a un ital AH-algebr a and let A b e a unital simple C*-algebr a with TR( A ) = 0 . Su pp ose that t her e is a p air κ ∈ K L e ( C, A ) ++ and a c ontinu ous aﬃne map λ : T( A ) → T f ( C ) which is c omp atible with κ. Then ther e exists a unital monomorphism h : C → A such that [ h ] = κ and h T = λ. Theorem 3.22. L et C b e a un it al A H-algebr a and let A b e a unital simple C*-algebr as with TR( A ) = 0 . Supp ose that ( κ, λ ) ∈ K K T ( C, A ) ++ . Then ther e is a monomorphism φ : C → A su ch that [ φ ] = κ and φ T = λ. Pr o of. It follows from Theor e m 3.21 that there exists a unital monomorphism ψ : C → A s uc h that [ ψ ] K L = ¯ κ in K L ( C, A ) and φ T = λ. Then κ − [ ψ ] ∈ Pext( K ∗ ( C ) , K ∗ ( A )) . As in the pro of of 3.20, we obta in a unital monomo rphism α : ψ ( C ) → A for which [ α ◦ ψ ] = [ κ ] i n K K ( C , A ) and there exists a se quence of unitaries { u n } ⊂ U( A ) such that lim n →∞ ad u n ◦ φ ( c ) = α ◦ φ ( c ) for all c ∈ C . Put φ = α ◦ ψ . Then λ ( τ )( c ) = τ ◦ ψ ( c ) = lim n →∞ τ (ad u n ◦ ψ ( c )) (3.17) = τ ( α ◦ ψ ( c )) = τ ◦ φ ( c ) for a ll c ∈ C . (3.18) It follows that φ T = λ, as desir ed. Remark 3.2 3 . It was shown in [18] that ther e are compact metric space X , unital simple AF-algebr a s A and ¯ κ ∈ K L e (C( X ) , A ) ++ for which there is no unital homomor phism h : C( X ) → A s o that [ h ] = ¯ κ. Thus the information on λ is essential in g e neral. In the case that C is also simple and has real rank zero, then the map λ is c o mpletely determined by ¯ κ since ρ C ( K 0 ( C )) is dense in Aﬀ (T( C )) and T( C ) = T f ( C ) . If the C*-algebra C is real ra nk zero and exact, without assuming the simplicity , then T( C ) = S( K 0 ( C )), and hence one c an deﬁne the map r : T( A ) → T( C ) by factoring through S( K 0 ( A )). It is obviously that r is compatible with ¯ κ . Mo r eov er , r ( τ ) is faithful on C for any τ ∈ T( A ). Indeed, if r ( τ )( c ) = 0 for so me no nzero p ositive ele ment c ∈ C , then r ( τ )( cC c ) = { 0 } . In particula r , r ( τ )( p ) = 0 for s ome nonzero pro jection p ∈ cC c , which contradicts the s trict po sitivity of ¯ κ . Lemma 3. 2 4. L et X b e a Banac h sp ac e, and let { α n } b e a se quenc e of isometries. If e ach α n is invertible, and lim n →∞ α n ( x ) and lim n →∞ α − 1 n ( x ) exist for any x ∈ X , then α := lim n →∞ α n is invertible. 24 Pr o of. Denote b y β = lim n →∞ α − 1 n . It is clear that α and β are isometries. Fix a n elemen t x ∈ X . F or any ε > 0, there exists N such that   β ( x ) − α − 1 n ( x )   ≤ ε and k α ◦ β ( x ) − α n ◦ β ( x ) k ≤ ε for any n ≥ N . Then k α ◦ β ( x ) − x k ≤   α ◦ α − 1 n ( x ) − x   +   α ◦ β ( x ) − α ◦ α − 1 n ( x )   ≤   α ◦ α − 1 n ( x ) − α n ◦ α − 1 n ( x )   + ε ≤   α ◦ α − 1 n ( x ) − α ◦ β ( x )   + k α ◦ β ( x ) − α n ◦ β ( x ) k +   α n ◦ β ( x ) − α n ◦ α − 1 n ( x )   + ε ≤ 4 ε. Since ε is arbitra ry , one has that α ◦ β ( x ) = x , and hence α ◦ β = id. The arg ument same a s ab ov e shows that β ◦ α = id. Therefor e, α is in vertible, as desired. Deﬁnition 3.25. Let A b e a unital C*-alge bra. Denote by K K − 1 e ( A, A ) ++ the s e t o f those elements κ ∈ K K e ( A, A ) ++ such that κ induces an or dered isomorphism betw een K 0 ( A ) and isomorphisms b etw een K ( A ). Corollary 3 .26. L et A b e a u nital sep ar able amenable simple C*-algebr a with TR( A ) = 0 and satisﬁes the UCT. Then, for any κ ∈ K K − 1 e ( A, A ) ++ , ther e exists an automorphism α ∈ Aut( A ) such that [ α ] = κ. Pr o of. By Theorem 3.2 0 and its remark , there is a mono morphism α = α 2 ◦ α 1 such that [ α ] = κ in K K ( A, A ). Let us show that α ca n b e chosen to be an automorphism. Let us ﬁrst show that α 1 can b e c hosen to b e an automorphism. W e ﬁrst c ho o se a unital monomorphism α ′ 1 : A → A s o that [ α ′ ] = κ b y 3.20 and its remark. By the UCT, there is κ 1 ∈ K K − 1 e ( A, A ) ++ such that κ 1 × κ = κ × κ 1 = [id A ] . So there is a unital monomorphism β : A → A suc h that [ β ] = κ 1 . The n [ β ◦ α ′ ] = [ α ′ ◦ β ] = [id A ] . B y the uniquenes s theore m (2.3 of [11]), b oth β ◦ α ′ and α ′ ◦ β a re approximately unitar ily e q uiv alent to the identit y . Then, b y a s tandard intert wining a rgument of Elliott, one obtains an isomor phism α 1 : A → A whic h is approximately unitarily equiv alent to α ′ (see for exa mple Theorem 3.6 of [11]). So in particular, [ α 1 ] = κ in K L ( A, A ) . (This als o follows from the pro of of Theorem 3.7 of [11] tha t there is an isomorphism α 1 : A → A suc h that [ α 1 ] = ¯ κ in K L ( A, A ) , using that fact that A is pre-class iﬁa ble in the sens e o f [11], see Theor em 4.2 of [12]). . Consider the map α 2 . No te that if A = B , then, in the pro of of Theo rem 3.1 7, the union of ﬁnite subsets F n is dense in A , and the inner automorphis ms { Ad( u 1 · · · u n ) } satisfy Lemma 3.2 4. Therefore, by Lemma 3.24, the monomorphism α 2 = lim n →∞ Ad( u 1 · · · u n ) is an automorphism o f A . Therefore, the map α = α 2 ◦ α 1 is a n automorphis m of A . 4 Rotation maps Lemma 4.1. L et H b e a ﬁnitely gener ate d ab elian gr oup, and let A b e a C*-algebr a with ρ A ( K 0 ( A )) dense in Aﬀ (T( A )) . L et ψ ∈ Hom( H , Aﬀ (T( A ))) . Fix { g 1 , ..., g n } ⊆ H . Then, for any ε > 0 , t her e is a homomorphism h : H → K 0 ( A ) such that | ψ ( g i ) − ρ A ( h ( g i )) | < ε 25 for any 1 ≤ i ≤ n . Pr o of. Since the map ψ factors through H/H T or , we may as sume that H = L k Z for so me k a nd pro ve the lemma for an { g i } replaced b y the standard base { e i ; i = 1 , ..., k } . Since ρ A ( K 0 ( A )) is dense in Aﬀ (T( A )), there exist a 1 , ..., a n ∈ K 0 ( A ) s uch that | ψ ( e i ) − ρ A ( a i ) | < ε, i = 1 , ..., k . The the map h : H → K 0 ( A ) deﬁned b y e i 7→ a i satisﬁes the lemma. Theorem 4.2 . L et A b e a u nital simple C*-algebr a with ρ A ( K 0 ( A )) dense in Aﬀ (T( A )) . Assume t hat ther e is a p ositive element b ∈ A with sp( b ) = [0 , 1] . L et B ⊆ A b e a un ital C*-sub algebr a and denote by ι the inclusion map. Supp ose that A has Pr op erty (B1) and Pr op erty (B2) asso ciate d with B and ∆ B . F or any ψ ∈ Hom( K 1 ( B ) , Aﬀ (T( A ))) , ther e exists α ∈ Inn( B , A ) such that ther e ar e maps θ i : K i ( B ) → K i ( M α ) with π 0 ◦ θ i = id K i ( B ) , i = 0 , 1 , and t he r otation map R ι,α : K 1 ( M α ) → Aﬀ (T( A )) is given by R ι,α ( c ) = ρ A ( c − θ 1 ([ π 0 ] 1 ( c ))) + ψ ([ π 0 ] 1 ( c )) , ∀ c ∈ K 1 ( M α ( A )) . In other wor ds, [ α ] = [ ι ] in K K ( B , A ) , and the r otation map R ι,α : K 1 ( M α ) → Aﬀ (T( A )) is given by R ι,α ( a, b ) = ρ A ( a ) + ψ ( b ) for some identiﬁc ation of K 1 ( M α ) with K 0 ( A ) ⊕ K 1 ( B ) . Pr o of. W rite K 1 ( B ) = { g 1 , g 2 , ..., g n , ... } , K 0 ( B ) + = { k 1 , k 2 , ..., k n , ... } , and denote b y H n = < g 1 , ..., g n >, the subgroup g enerated b y { g 1 , ..., g n } , and K n = < k 1 , ..., k n >, the subgr oup generated b y { k 1 , ..., k n } . Let {F i } b e an increasing family o f ﬁnite subsets with the union dense in B . Assume that for each i , there is a unitary z i and a pro jection p i in M r i ( F i ) for some natura l n umber r i such that [ z i ] 1 = g i and [ p i ] = k i . Without loss of generality , we may assume that r i ≤ r i +1 , i ∈ N . In what follo ws, if v ∈ A, by v ( m ) , we mean v ( m ) = diag( m z }| { v , v , ..., v ) . W e ass e rt that ther e are maps { h i : H i → K 0 ( A ); i ∈ N } and unitaries { u i ; i ∈ N } such that for any n , if denoted by w n = u 1 · · · u n − 1 (assume u − 1 = u 0 = 1), one has 1. F or any x ∈ { g 1 , ..., g i } ∪ G n 1 , | ρ A ◦ h n ( x ) − ψ ( x ) | < δ n δ 2 n where δ n = ∆ B ( δ p r 2 n · 2 n +1 , F n , P (0) n , P (1) n , Ad( w n − 1 ) ◦ ι ) for P (0) n = { k 1 , k 2 , ..., k n } and P (1) n = { g 1 , g 2 , ..., g n } , δ the constant of Lemma 3.9 (since A has Pro p er t y (B1)), and G n 1 is the set of generato r s of G 1 of Deﬁnition 3.6 with resp ect to δ p r 2 n · 2 n +1 , F n , P (0) n , P (1) n , and ι ). 26 2. F or any a ∈ w ∗ n − 1 F n w n − 1 , k [ u n , a ] k ≤ δ p r 2 n · 2 n +1 where w n = u 1 ...u n , and moreover, the image of φ n := h n +1 | H n − h n lies ins ide C ( A ) so that Λ ◦ φ n is well-deﬁned, and bo tt 1 (( w ( r n ) n − 1 ) ∗ z i w ( r n ) n − 1 , u n ) = Λ ◦ φ n ( g i ) and bo tt 0 (( w ( r n ) n − 1 ) ∗ p i w ( r n ) n − 1 , u n ) = 0 for any i = 1 , ..., n . If n = 1, since ρ A ( K 0 ( A )) is dense in Aﬀ (T( A )), b y Lemma 4.1, there is a map h 1 : < { g 1 } ∪ G 1 > → K 0 ( A ) such that for a ny x ∈ { g 1 } ∪ G 1 | ψ ( x ) − ρ A ( h 1 ( x )) | < δ 1 δ 2 , and a map h 2 : < { g 1 , g 2 } ∪ G 2 > → K 0 ( A ) such that | ψ ( x ) − ρ A ( h 2 ( x )) | ≤ δ 1 δ 2 2 for any x ∈ { g 1 , g 2 } ∪ G 2 . F or the function φ 1 = h 2 | H 1 − h 1 , we hav e that | τ ( φ 1 ( g 1 )) | < min { δ, δ 1 } for an y τ ∈ T( A ) and hence φ 1 ( g 1 ) ∈ C ( A ) by Lemma 3.9. By Lemma 3.5, | τ (Λ ◦ φ 1 ( x )) | = | τ ( φ 1 ( x )) | < δ 1 for any τ ∈ T( A ) a nd any x ∈ G 1 . Since A has Prop erty (B2 ) asso c ia ted with B a nd ∆ B , there is a unitary u 1 ∈ U( A ) such that k [ u 1 , a ] k ≤ δ p r 2 1 · 2 2 , ∀ a ∈ F 1 , and bo tt 1 ( z 1 , u 1 ) = Λ ◦ φ 1 ( g 1 ) and bo tt 0 ( p 1 , u 1 ) = 0 . Assume that w e hav e constructed the maps { h i : H i → K 0 ( A ); i = 1 , ..., n } and unitaries { u i ; i = 1 , ..., n − 1 } satisfying the as s ertion ab ov e. By by Lemma 4.1, there is a function h n +1 : < { g 1 , ..., g n +1 } ∪ G n 1 ∪ G n +1 1 > → K 0 ( A ) such that for a ny x ∈ { g 1 , ..., g n +1 } ∪ G n 1 ∪ G n +1 1 , | ρ A ◦ h n +1 ( x ) − ψ ( x ) | < δ n +1 δ 2 n +1 where δ n +1 = ∆ B ( δ p r 2 n · 2 n +2 , F n +1 , P (0) n +1 , P (1) n +1 , Ad( w n ) ◦ ι ) for P (0) n +1 = { k 1 , k 2 , ..., k n +1 } and P (1) n +1 = { g 1 , g 2 , ..., g n +1 } . Recall that φ n = h n +1 | H n − h n . Then, a direct c alculation shows that for an y τ ∈ T( A ), | τ (Λ ◦ φ n )( x ) | = | τ ( φ n ( x )) | < min { δ, δ n +1 } for any x ∈ G n 1 . Since A Pr op erty (B2 ) asso ciated with B a nd ∆ B , there is a unitar y u n ∈ U( A ) such that   [ u n , w ∗ n − 1 aw n − 1 ]   ≤ δ p r 2 n · 2 n +1 , ∀ a ∈ F n , 27 and bo tt 1 (Ad( w n − 1 ) ◦ ι, u n ) | P (1) n = Λ ◦ φ n and b o tt 0 (Ad( w n − 1 ) ◦ ι, u n ) | P (0) n = 0 . This pr ov es the a ssertion. By Lemma 3.13, Ad( w n ) con verge to a monomorphism α : B → A . Moreover, the extension η 0 ( M α ) is trivial, and η 1 ( M α ) is determined by the inductive limit of 0 K 0 ( A ) K 0 ( A ) ⊕ H n +1 H n +1 0 0 K 0 ( A ) K 0 ( A ) ⊕ H n H n 0 / / / / / / / / / / / / / / / / ι n,n +1 O O ι n,n +1 O O γ n ` ` A A A A A A A A A A where γ n ( g i ) = [ p ( R ∗ ( u n +1 , t ) , w ∗ n z i w n )]. Howev er, since Λ ◦ γ n ( g i ) = b o tt 1 (Ad( w n − 1 ) ◦ ι, u n )( g i ) = Λ ◦ φ n ( g i ) , by the injectivit y o f Γ, w e hav e that γ n = φ n . W e a ssert that η 1 ( M α ) is also trivial. F or an y n , deﬁne a map θ ′ n : H n → K 1 ( M α ) by θ ′ n ( g ) = ( h n ( g ) , g )) . W e then ha ve θ ′ n +1 ◦ ι n,n +1 ( g ) − θ ′ n ( g ) = ( h n +1 ◦ ι n,n +1 ( g ) , ι n,n +1 ( g )) − ( h n ( g ) + φ n ( g ) , ι n,n +1 ( g )) = ( h n +1 ◦ ι n,n +1 ( g ) , ι n,n +1 ( g )) − ( h n ( g ) + ( h n +1 ( ι n,n +1 ( g )) − h n ( g )) , ι n,n +1 ( g )) = 0 , and hence ( θ ′ n ) deﬁne a ho mo morphism θ 1 : K 1 ( B ) → K 1 ( M α ). Moreover, since π ◦ θ 1 = id K 1 ( B ) , the extension η 1 ( M α ) splits. Therefor e, [ α ] = [ ι ] in K K ( B , A ). Let us ca lculate the corres po nding rotatio n map. P ick [ z i ] = g i for some g i ∈ H m . T o simplify notation, without los s of ge ne r ality , we will use α for α ⊗ id M r ( C ) on B ⊗ M r ( C ) for an y integer r ≥ 1 . T ake a path v ( t ) in M 2 r i ( A ) fr o m z i to α ( z i ) as follows v ( t ) = R ∗ (2 t ) 1 0 0 ( w ( r ( i )) m ) ∗ ! R (2 t ) z i 0 0 1 ! R ∗ (2 t ) 1 0 0 w ( r ( i )) m ! R (2 t ) for t ∈ [0 , 1 / 2] and v ( t ) = ( w ( r ( i )) m ) ∗ z i w ( r ( i )) m exp((2 t − 1) c m ) for t ∈ [1 / 2 , 1] , where R ( t ) = cos( π t 2 ) sin( π t 2 ) − sin( π t 2 ) cos( π t 2 ) ! and c m = log(( w ( r ( i )) m ) ∗ z ∗ i w ( r ( i )) m α ( z i )). 28 Then, for a n y τ ∈ T( A ), w e ha ve 1 2 π i Z 1 0 τ ( ˙ v ( t ) v ∗ ( t ))d t = τ (log (( w ( r ( i )) m ) ∗ z ∗ i w ( r ( i )) m α ( z i ))) = lim n →∞ τ (log (( w ( r ( i )) m ) ∗ z ∗ w ( r ( i )) m Ad w ( r ( i )) n ( z i ))) = lim n →∞ n X k =0 τ (log ((( w ( r ( i )) m + k − 1 ) ∗ z ∗ i w ( r ( i )) m + k − 1 )(( w ( r ( i )) m + k ) ∗ z i w ( r ( i )) m + k ))) = ∞ X k =0 τ (log ((( w ( r ( i )) m + k − 1 ) ∗ z ∗ i w ( r ( i )) m + k − 1 )(( w ( r ( i )) m + k ) ∗ z i w ( r ( i )) m + k ))) = ∞ X k =0 τ (log ((( w ( r ( i )) m + k − 1 ) ∗ z ∗ i w ( r ( i )) m + k − 1 )( u ( r ( i )) m + k (( w ( r ( i )) m + k − 1 ) ∗ z i w ( r ( i )) m + k − 1 ) u ( r ( i )) m + k )) = ∞ X k =0 τ (b ott 1 (( w ( r ( i )) m + k − 1 ) ∗ z ∗ i w ( r ( i )) m + k − 1 , u m + k )) . Then, by the choice of { u n } , we hav e 1 2 π i Z 1 0 τ ( ˙ v ( t ) v ∗ ( t ))d t = ∞ X k =0 τ (Λ ◦ φ m + k ( g i )) = ∞ X k =0 τ ( φ m + k ( g i )) = ∞ X k =0 τ ( h m + k +1 ( g i ) − h m + k ( g i )) = ψ ( g i ) − τ ( h m ( g i )) . Thu s, R α 1 ([ v ]) = ψ ( g i ) − ρ A ◦ h m ( g i ) = ψ ([ π 0 ( v )]) + ρ A ( − h m ( g i )) = ψ ([ π 0 ( v )]) + ρ A ((0 , g i ) − θ ′ m ( g i )) = ψ ([ π 0 ( v )]) + ρ A ([ v ] − θ 1 ([ π 0 ( v )])) , as desir ed. Corollary 4.3. L et A b e a unital simple C*-algebr a with TR( A ) = 0 and let B ⊆ A b e an AH - algebr a and denote by ι the inclusion map. F or any ψ ∈ Hom( K 1 ( B ) , Aﬀ (T( A ))) , ther e exists α ∈ Inn( B , A ) such that t her e ar e maps θ i : K i ( B ) → K i ( M α ) with π 0 ◦ θ i = id K i ( B ) , i = 0 , 1 , and the r otation map R ι,α : K 1 ( M α ) → Aﬀ (T( A )) is given by R ι,α ( c ) = ρ A ( c − θ 1 ([ π 0 ] 1 ( c ))) + ψ ([ π 0 ] 1 ( c )) , ∀ c ∈ K 1 ( M α ( A )) . In other wor ds, [ α ] = [ ι ] in K K ( B , A ) and the r otation map R ι,α : K 1 ( M α ) → Aﬀ (T( A )) is given by R ι,α ( a, b ) = ρ A ( a ) + ψ ( b ) 29 for some identiﬁc ation of K 1 ( M α ) with K 0 ( A ) ⊕ K 1 ( B ) . Pr o of. It follows directly from Lemma 5.2 of [15], Theore m 4.2, and Pr op osition 3.18. Deﬁnition 4. 4. Fix tw o unital C* -algebra s A and B with T( A ) 6 = ∅ . Let R 0 be the subset of Hom( K 1 ( B ) , Aﬀ(T( A ))) consisting of those homomorphisms h ∈ Hom( K 1 ( B ) , Aﬀ(T( A ))) such that there e x ists a ho mo morphism d : K 1 ( B ) → K 0 ( A ) s uch that h = ρ A ◦ d. Then R 0 is cle a rly a subg roup of Ho m( K 1 ( B ) , Aﬀ(T( A ))) . The following is a v ar iation of Theo rem 10 .7 o f [15]. Theorem 4.5. L et C b e a unital AH-algebr a and let A b e a un ital s ep ar able simple C*-algebr a with TR( A ) = 0 . Supp ose that φ, ψ : C → A ar e two unital monomorphi sms such that [ φ ] = [ ψ ] in K K ( C, A ) and, for al l τ ∈ T( A ) , τ ◦ φ = τ ◦ ψ . Supp ose also that ther e exists a homomorph ism θ ∈ Hom( K 1 ( C ) , K 1 ( M φ,ψ )) with ( π 0 ) ∗ 1 ◦ θ = id K 1 ( C ) such that R φ,ψ ◦ θ ∈ R 0 . (4.1) Then φ and ψ ar e asymptotic al ly unitarily e quivalent. Pr o of. W e may write K 1 ( M φ,ψ ) = K 0 ( A ) ⊕ θ ( K 1 ( C )) . Let h = R φ,ψ ◦ θ. If there is a homomorphis m d : K 1 ( C ) → K 0 ( A ) s uch that h = ρ A ◦ d, deﬁne θ ′ = θ − d. Note that θ is a homo morphism, and ( π 0 ) ∗ 1 ◦ θ = id K 1 ( C ) . Then R φ,ψ ◦ θ ′ = R φ,ψ ◦ θ − ρ A ◦ d = 0 . Since [ φ ] = [ ψ ] in K K ( C, A ) , there exists a n elemen t θ 0 ∈ Hom Λ ( K ( C ) , K ( M φ,ψ )) such that [ π 0 ] ◦ [ θ 0 ] = [id K ( C ) ] . Deﬁne θ ′ 0 : K i ( C ) → K i ( M φ,ψ ) by θ ′ 0 | K 1 ( C ) = θ ′ and θ ′ 0 | K 0 ( C ) = θ 0 | K 0 ( C ) . By the UCT, there exis ts θ ′′ 0 ∈ K L ( C, M φ,ψ ) such that Γ( θ ′′ 0 ) = θ ′ 0 , where Γ is the map fr o m K L ( C, M φ,ψ ) onto Ho m( K ∗ ( C ) , K ∗ ( M φ,ψ )) . W e will use the ide ntiﬁcation K L (S C, M φ,ψ ) = Hom Λ ( K (S C ) , K ( M φ,ψ )) . (see [1]). Put x 0 = [ π 0 ] ◦ θ ′′ 0 − [id K ( C ) ] . 30 Then Γ( x 0 ) = 0 . Deﬁne θ 1 = θ ′′ 0 − θ 0 ◦ x 0 ∈ Hom Λ ( K ( C ) , K ( M φ,ψ )) . Then one computes that [ π 0 ] ◦ θ 1 = [ π 0 ] ◦ θ ′′ 0 − [ π 0 ] ◦ θ 0 ◦ x 0 (4.2) = ([id K ( C ) ] + x 0 ) − [id K ( C ) ] ◦ x 0 (4.3) = [id K ( C ) ] + x 0 − x 0 = [id K ( C ) ] . (4.4) Moreov er, Γ( θ 1 ) | K 1 ( C ) = θ ′ . In par ticular, R φ,ψ ◦ ( θ 1 ) K 1 ( C ) = 0 . It follows from T he o rem 10 .7 of [15] that φ and ψ are asy mpto tica lly unitarily equiv ale n t. Remark 4.6. In 4.5, the condition that such θ exists is also necessary . This follows fro m Theorem 9 .1 of [1 5]. Deﬁnition 4. 7. Let A b e a unital C* - algebra, and let C be a unital separ able C*-a lgebra. Denote b y Mon e asu ( C, A ) the set of asymptotically unitary equiv alence classes of unital monomorphisms. Denote by K the map from Mon e asu ( C, A ) in to K K e ( C, A ) ++ deﬁned by φ 7→ [ φ ] for all φ ∈ Mon e asu ( C, A ) . Let κ ∈ K K e ( C, A ) ++ . Denote b y h κ i the classes of φ ∈ Mon e asu ( C, A ) suc h that K ( φ ) = κ . Denote by f K the map from Mon e asu ( C, A ) in to K K T ( C, A ) ++ deﬁned by φ 7→ ([ φ ] , φ T ) for all φ ∈ Mon e asu ( C, A ) . Denote by h κ, λ i the classes of φ ∈ Mon e asu ( C, A ) suc h that f K ( φ ) = ( κ, λ ) . Theorem 4.8. L et C b e a unital AH-algebr a and let A b e a un ital s ep ar able simple C*-algebr a with TR( A ) = 0 . Then the map f K : Mon e asu ( C, A ) → K K T ( C, A ) ++ is surje ctive. Mor e over, for e ach ( κ, λ ) ∈ K K T ( C, A ) ++ , ther e exists a bije ction η : h κ, λ i → Hom( K 1 ( C ) , Aﬀ (T( A ))) / R 0 . Pr o of. It follows from 3.22 tha t f K is surjectiv e. Fix a pair ( κ, λ ) ∈ K K T ( C, A ) ++ and ﬁx a unita l monomorphism φ : C → A such that [ φ ] = κ a nd φ T = λ. F or any homomorphism γ ∈ Hom( K 1 ( C ) , Aﬀ (T( A ))), it follo ws from 4 .2 that ther e exists a unital monomorphism α ∈ Inn( φ ( C ) , A ) with [ α ◦ φ ] = [ φ ] in K K ( C, A ) such tha t there ex ists a homomorphism θ : K 1 ( C ) → M φ,α ◦ φ with ( π 0 ) ∗ 1 ◦ θ = id K 1 ( C ) such that R φ,α ◦ φ ◦ θ = γ . Let ψ = α ◦ φ. Then R φ,ψ ◦ θ = γ . Note also since α ∈ Inn( φ ( C ) , A ) , ψ T = φ T . In particular, f K ( ψ ) = f K ( φ ) . Suppo se that θ ′ : K 1 ( A ) → K 1 ( M φ,ψ ) such that ( π 0 ) ∗ 1 ◦ θ ′ = id K 1 ( C ) . Then ( θ ′ − θ )( K 1 ( C )) ⊂ K 0 ( A ) . It follows that R φ,ψ ◦ θ ′ − R φ,ψ ◦ θ ∈ R 0 . 31 Thu s we obtain a well-deﬁned map η : h [ φ ] , φ T i → Hom( K 1 ( A ) , Aﬀ(T( A ))) / R 0 . F ro m what we hav e prov ed, the map η is surjective. Suppo se that φ 1 , φ 2 : C → A are t wo unital mono mo rphisms such that φ 1 , φ 2 ∈ h [ φ ] , φ T i and R φ,φ 1 ◦ θ 1 − R φ,φ 2 ◦ θ 2 ∈ R 0 , where θ 1 : K 1 ( C ) → K 1 ( M φ,φ 1 ) and θ 2 : K 1 ( C ) → K 1 ( M φ,φ 2 ) a re homo morphisms s uc h tha t ( π 0 ) ∗ 1 ◦ θ 1 = ( π 0 ) ∗ 1 ◦ θ 2 = id K 1 ( C ) , r e s pec tiv ely . Thus there is δ : K 1 ( C ) → K 0 ( A ) such that R φ,φ 2 ◦ θ 2 − R φ,φ 1 ◦ θ 1 = ρ A ◦ δ. Fix z ∈ K 1 ( C ) a nd let u ∈ M k ( C ) s uc h that [ u ] = z in K 1 ( C ) . Le t U ( θ 1 , z )( t ) ∈ U(M N ( M φ,φ 1 )) b e a unitary represented by θ 1 ( z ) . W e may assume that N ≥ k . It is easy to see that U ( θ 1 , z )( t ) is homo topic to a unitary V ( θ 1 , z )( t ) ∈ U(M N ( M φ,φ 1 )) with V ( θ 1 , z )(0) = diag( φ ( u ) , 1 N − k ) and V ( θ 2 , z )(1) = diag ( φ 1 ( u ) , 1 N − k ) (b y enlarg ing the size of matrice s if necessa ry). In particula r, [ V ( θ 1 , z )] = θ 1 ( z ) in K 1 ( M φ,φ 1 ) . Similarly , w e ma y assume that there is a unitary V ( θ 2 , z ) ∈ U(M N ( M φ,φ 2 )) with V ( θ 2 , z )(0) = dia g( φ ( u ) , 1 N − k ) and V ( θ 2 , z )(1) = diag( φ 2 ( u ) , 1 N − k ) which is r epresented b y θ 2 ( z ) in K 1 ( M φ,φ 2 ) . Now deﬁne V ( θ 3 , z )( t ) =    V ( θ 1 , z )(1 − 2 t ) for t ∈ [0 , 1 / 2] V ( θ 2 , z )(2 t − 1) for t ∈ (1 / 2 , 1] , (4.5) Note that V ( θ 3 , z ) ∈ U(M N ( M φ 1 ,φ 2 )) . Deﬁne the homomorphism θ 3 : K 1 ( C ) → K 1 ( M φ 1 ,φ 2 ) by θ 3 ( z ) = [ V ( θ 3 , z )] for all z ∈ K 1 ( C ) . It follows that ( π 0 ) ∗ 1 ◦ θ 3 = ( π 0 ) ∗ 1 ◦ θ 1 = id K 1 ( C ) . Moreov er, a dirct calculation shows that R φ 1 ,φ 2 ◦ θ 3 = R φ,φ 2 ◦ θ 2 − R φ,φ 1 ◦ θ 1 = ρ A ◦ δ. It follo ws from Theorem 4.5 that ψ 1 and ψ 2 are as ymptotically unitarily equiv alent. Therefo re, η is o ne to one. Corollary 4 .9. Le t C b e a unital sep ar able amenable s imple C*-algebr a with TR( C ) = 0 which satisﬁes the U CT and let A b e a un ital simple C*-algebr a with TR( A ) = 0 . Then t he map K : Mon e asu ( C, A ) → K K e ( C, A ) ++ is surje ctive. Mor e over, the map η : h [ φ ] i → Hom( K 1 ( C ) , Aﬀ (T( A ))) / R 0 is bije ctive. Pr o of. It follows fro m 4.8 that it suﬃces to p oint out that in this case ρ C ( K 0 ( C )) is dense in Aﬀ (T( C )) and T f ( C ) = T( C ) . 32 Corollary 4.10. L et A b e a unital sep ar able amenable simple C*-algebr a with TR( A ) = 0 which satisﬁes the UCT. Then the map K : Aut( A ) → K K − 1 e ( A, A ) ++ is surje ctive. Mor e over, the map η : h id A i → Hom( K 1 ( A ) , Aﬀ (T( A ))) / R 0 is bije ctive. Pr o of. This follows from 3.2 6 and 4.2 as in the pro of of 4.8. 5 Classiﬁcation of simple C*-algebras Denote by Q the UHF alg e br a with K 0 ( Q ) = Q with [1 Q ] = 1. If p is a superna tur al n umber, let M p denote the UHF-algebra as s o ciated with p . Lemma 5. 1 . L et B b e a un it al sep ar able amenable simple C*-alge br a such that B ⊗ Q has tr acial r ank zer o. L et A b e unital sep ar able amenable simple C*-algebr as with t r acial r ank zer o satisfying the UCT, and let ϕ 1 , ϕ 2 : A → B b e two homomorphi sms. Supp ose that φ, ψ : A → B ⊗ Q ar e two unital monomorp hisms su ch that [ φ ] = [ ψ ] in K K ( A, B ⊗ Q ) . Supp ose that φ induc es an aﬃne home omorphism φ T : T( B ⊗ Q ) → T( A ) by φ T ( τ )( a ) = τ ◦ φ ( a ) for al l a ∈ A s.a and for al l τ ∈ T( B ⊗ Q ) . Then ther e exists an automorph ism α ∈ Aut( φ ( A ) , φ ( A )) with [ α ] = [id φ ( A ) ] in K K ( φ ( A ) , φ ( A )) such that α ◦ φ and ψ ar e str ongly asymptotic al ly unitarily e quivalent. Pr o of. The pro of is exactly the sa me as that of the pr o of of Lemma 3.2 o f [16] but we apply 4.2 instead of 4.1 o f [8] so that the r estriction on K i ( A ) ( i = 0 , 1 ) can b e remov ed. Deﬁnition 5.2. Le t C and A be tw o unita l C* - algebra s . Let φ, ψ : C → A be tw o homomorphisms. Recall that φ and ψ are sa id to be strong ly asymptotically unitar ily equiv a lent if ther e exists a con tinuous pa th of unitaries { u ( t ) : t ∈ [0 , ∞ ) } such that u (0) = 1 and lim t →∞ ad u ( t ) ◦ φ ( c ) = ψ ( c ) for a ll c ∈ C. Lemma 5.3 (Theo r em 3.4 of [16]) . L et A and B b e two unital sep ar able amenable simple C*-algebr as satisfying t he UCT. L et p and q b e sup ernatur al numb ers of inﬁnite t yp e such that M p ⊗ M q ∼ = Q. Supp ose that A ⊗ M p , A ⊗ M q , B ⊗ M p and B ⊗ M q have tr acial r ank zer o. L et σ p : A ⊗ M p → B ⊗ M p and ρ q : A ⊗ M q → B ⊗ M q b e two unital isomorphisms. Su pp ose [ σ ] = [ ρ ] in K K ( A ⊗ Q , B ⊗ Q ) , wher e σ = σ p ⊗ id M q and ρ = ρ q ⊗ id M p . Then ther e is an automorphi sm α ∈ Aut( σ p ( A ⊗ M p )) such that [ α ◦ σ p ] = [ σ p ] in K K ( A ⊗ M p , B ⊗ M p ) , and α ◦ σ p ⊗ id M q is str ongly asymptotic al ly unitarily e quivalent to ρ. 33 Pr o of. It follo ws fro m 5.1 that ther e exists β ∈ Aut( B ⊗ Q ) suc h that β ◦ σ is strongly asymptotically unitar ily equiv alent to ρ. Mo reov er, [ β ] = [id B ⊗ Q ] in K K ( B ⊗ Q, B ⊗ Q ) . Now consider tw o homomorphisms σ p and β ◦ σ p . One has [ β ◦ σ p ] = [ σ p ] in K K ( A ⊗ M p , B ⊗ Q ) . Since σ p is an isomorphism, it is easy to see that σ T : T( B ⊗ Q ) → T( A ⊗ M p ) is an aﬃne homeo morphism. By applying 5.1 again, one obtains α ∈ Aut( σ p ( A ⊗ M p )) such that [ α ] = [id σ p ] in K K ( σ p ( A ⊗ M p ) , σ p ( A ⊗ M p )) and α ◦ σ p is s tr ongly asymptotica lly unitarily equiv alent to β ◦ σ p . Deﬁne β ◦ σ p ⊗ id M q : A ⊗ M p ⊗ M q → ( B ⊗ Q ) ⊗ M q . It is easy to see that β ◦ σ p ⊗ id M q is str ongly appr oximately unitarily equiv a lent to β ◦ σ . Note that σ ( A ⊗ M p ) = B ⊗ M p . Le t σ ′ = α ◦ σ p ⊗ id M q . It follows that σ ′ is strongly as ymptotically unitar ily equiv alent to β ◦ σ . Cons equently σ ′ is s tr ongly asymptotica lly unitarily equiv alent to ρ . Theorem 5.4. L et A and B b e two u nital sep ar able amenable simple C*-algebr as satisfying the UCT. Supp ose that ther e is an isomorphi sm κ : ( K 0 ( A ) , K 0 ( A ) + , [1 A ] , K 1 ( A )) → ( K 0 ( B ) , K 0 ( B ) + , [1 B ] , K 1 ( B )) . Supp ose also t hat ther e is a p air of supp er-natur al nu mb ers p and q of inﬁnite t yp e whi ch ar e r elative prime such that M p ⊗ M q ∼ = Q and TR( A ⊗ M p ) = TR( A ⊗ M q ) = TR( B ⊗ M p ) = TR( B ⊗ M q ) = 0 . Then A ⊗ Z ∼ = B ⊗ Z . Pr o of. The pro of is exactly the same as that of Theore m 3.5 of [16] but we no w apply 5.3 instead. Corollary 5.5 (Coro llary 8.3 of [24]) . L et A and B b e two unital sep ar able simple ASH -algebr as whose pr oje ctions sep ar ate tr ac es which ar e Z -stable. Supp ose that ( K 0 ( A ) , K 0 ( A ) + , [1 A ] , K 1 ( A )) → ( K 0 ( B ) , K 0 ( B ) + , [1 B ] , K 1 ( B )) . Then A ∼ = B . Pr o of. As in the pro of o f 6.3 of [24], A ⊗ C a nd B ⊗ C have tra cial r a nk zero for any unital UHF-algebra C . Thus Theorem 5 .4 applies. Corollary 5.6. L et A and B b e two un ital sep ar able simple Z -s t able C*-algebr as which ar e inductive limits of typ e I C*-algebr as with unique tr acial states. Supp ose that ( K 0 ( A ) , K 0 ( A ) + , [1 A ] , K 1 ( A )) → ( K 0 ( B ) , K 0 ( B ) + , [1 B ] , K 1 ( B )) . Then A ∼ = B . Pr o of. F or any UHF-a lgebra C , A ⊗ C is a pproximately divisible. Since A has a unique tr acial state, so do e s A ⊗ C . There fo re pro jections of A ⊗ C separa te tr aces. It follo ws from [21] that A ⊗ C has rea l rank zero, stable rank o ne and weakly unp erfora ted K 0 ( A ⊗ C ) . Moreov er , A ⊗ C is a lso an inductiv e limit of t yp e I C*-algebr as. It follows from Theorem 4.15 and Prop osition 5.4 o f [13] tha t TR( A ⊗ C ) = 0 . Ex actly the s ame argument shows that TR( B ⊗ C ) = 0 . 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Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras

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