The Automorphism group of a simple Z -stable C ∗ -algebra
aa r X i v : . [ m a t h . OA ] N ov THE AUTOMORPHISM GROUP OF A SIMPLE Z -STABLE C ∗ -ALGEBRA PING WONG NG AND EFREN RUIZ
Abstract.
We study the automorphism group of a simple, unital, Z -stable C ∗ -algebra.We show that Inn ( A ) is a simple topological group and Inn( A )Inn ( A ) is isomorphic (as topo-logical groups) to the inverse limit of quotient groups of K ( A ), where A is a Z -stable C ∗ -algebra satisfying the following property: for every UHF algebra B , A ⊗ B is a nu-clear, separable, simple, tracially AI algebra satisfying the Universal Coefficient Theorem(UCT) of Rosenberg and Schochet. By the recent results of Lin and Winter, ordered K -theory, traces, and the class of the unit is a complete isomorphism invariant for this classof C ∗ -algebras. Introduction
Denote the group of automorphisms of A equipped with the topology of pointwise conver-gence by Aut( A ), denote the closure of the group of inner automorphisms of A by Inn( A ),and denote the closure of the group of inner automorphisms of A whose implementing uni-taries are connected to 1 A via a norm continuous path of unitaries by Inn ( A ). We thenhave that Aut( A ) decomposes into the following series of closed normal subgroupsInn ( A ) ✁ Inn( A ) ✁ Aut( A ) . In [7], Elliott and Rørdam showed that for a simple, unital C ∗ -algebra A that either isreal rank zero, stable rank one and weakly unperforated, or is purely infinite, Inn ( A ) is asimple topological group (no non-trivial closed normal subgroup). They also showed that Inn( A )Inn ( A ) is totally disconnected when A is a simple AT algebra with real rank zero. Hence,when A is a simple AT algebra with real rank zero, the results of Elliott and Rørdam givea structure theorem for Aut( A ) since Aut( A ) fits into the following exact sequence { } → Inn( A ) → Aut( A ) → Aut( K ∗ ( A )) + , → { } . In their paper, they asked if Inn ( A ) is a simple topological group and if Inn( A )Inn ( A ) is totallydisconnected for every simple, unital C ∗ -algebra A .Recently, the authors in [21] and [22] proved Inn ( A ) is a simple topological group and Inn( A )Inn ( A ) is totally disconnected for all nuclear, separable, simple, tracially AI algebras satis-fying the UCT and for all nuclear, purely infinite, separable, simple C ∗ -algebras satisfyingthe UCT. In this paper, we generalize the results of [21] and [22]. We show that Inn( A )Inn ( A ) is Date : November 21, 2018.2000
Mathematics Subject Classification.
Primary: 46L35.
Key words and phrases. automorphism, topological groups. isomorphic to an inverse limit of discrete abelian groups (similar to the one used by Elliottand Rørdam in [7]) for a nuclear, separable, simple Z -stable C ∗ -algebra A such that foreach supernatural number p of infinite type, A ⊗ M p is a tracially AI algebra that satisfiesthe UCT. (Here, M p is the UHF algebra with supernatural number p .) In fact, we showthat Inn( A )Inn ( A ) is isomorphic (as topological groups) to the inverse limit of quotient groups of K ( A ), where the quotient groups are given the discrete topology and the inverse limit isgiven the inverse limit topology. Consequently, Inn( A )Inn ( A ) is totally disconnected. Moreover, weshow that Inn ( A ) is a simple topological group for any separable, simple, unital, Z -stable C ∗ -algebra that is either nuclear and quasidiagonal or is exact and has a unique tracialstate.It turns out that a large class of simple C ∗ -algebras satisfies the above condition. Wegive examples of C ∗ -algebras in this class.(1) Nuclear, separable, simple, unital, tracially AI algebras which satisfy the UCT.(2) Simple, unital, Z -stable AH algebras (see Corollary 11.12 of [16]).(3) The Jiang-Su algebra Z ([9]).(4) Simple, unital Z -stable C ∗ -algebras which are locally type I with unique tracialstate (see Corollary 5.6 of [17]; see also Corollary 8.2 of [33]).(5) Simple, unital Z -stable, ASH-algebras A such that T ( A ) = S [1] ( K ( A )), where S [1] ( K ( A )) is the state space of K ( A ) (see Corollary 5.5 of [17]; also see Corollaries6.3 and 8.3 of [33]).(6) Simple, unital A T D algebras (an A T D algebra is an inductive limit of dimensiondrop circle algebras; see Definitions 2.3 and 2.4 of [18] ; and see Theorem 4.2 of[18]).The paper is organized as follows: In Section 2, we show that for a separable, simple,unital, Z -stable C ∗ -algebra A that is either nuclear and quasidiagonal or is exact and has aunique tracial state, Inn ( A ) is a simple topological group. In Section 3, we introduce theBott Maps which are defined by Lin in [13] and we present some technical results Theorem3.16, Corollary 3.17, and Theorem 3.22. In Section 4, we show that the topological group Inn( A )Inn ( A ) is isomorphic to the inverse limit of discrete abelian groups, where the inverse limitis given the inverse limit topology. We also provide a partial structure theorem for Aut( A ).2. Simplicity of
Inn ( A )We first start with some notation that will be used throughout the paper. Let A be a C ∗ -algebra and let p and q be projections in A .(1) If p and q are Murray-von Neumann equivalent , i.e., there exists v ∈ A such that v ∗ v = p and vv ∗ = q , we write p ∼ q .(2) If there exists v ∈ A such that v ∗ v = p and vv ∗ ≤ q , we write p - q .(3) If there exists v ∈ A such that v ∗ v = p , vv ∗ ≤ q , and vv ∗ = q , we write p (cid:16) q .(4) For a, b ∈ A , ( a, b ) = aba ∗ b ∗ .(5) If A is unital, then denote the norm closure of the commutator subgroup of U ( A )by CU ( A ) and denote the norm closure of the commutator subgroup of U ( A ) by CU ( A ) . HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 3 Definition 2.1.
Let p and q be supernatural numbers. Set Z p , q = (cid:8) f ∈ C ([0 , , M p ⊗ M q ) : f (0) ∈ M p ⊗ M q and f (1) ∈ M p ⊗ M q (cid:9) . (Here, M p is the UHF algebra with supernatural number p . Similar for M q .)We shall regard Z p , q (and any tensor product with it) as C ([0 , C ([0 , Lemma 2.2.
Let A be a separable, simple, unital C ∗ -algebra and let C be a UHF algebra.Then A ⊗ C is Z -stable and hence either purely infinite or stably finite. Moreover, if A ⊗ C is (stably) finite then it has the following properties:(1) stable rank one(2) cancellation of projections(3) strict comparison of positive elements when A is, additionally, exact(4) weak unperforation(5) K -injectivity(6) the (SP) property(7) For every nonzero projection p ∈ A ⊗ C , for every n ≥ , p ( A ⊗ C ) p contains a unitalsub- C ∗ -algebra which is isomorphic to M n ⊕ M n +1 .(8) If p, q are nonzero projections in A ⊗ C , then there exist nonzero projections p ′ , q ′ in p ( A ⊗ C ) p and q ( A ⊗ C ) q respectively such that p ′ ∼ q ′ .Proof. These results are contained in [2], [8], [9], [25], [26], [27] and [24]. (cid:3)
Lemma 2.3.
Let A be a unital Z -stable C ∗ -algebra. Then T ⊆ CU ( A ) , i.e., CU ( A ) contains all scalar unitaries.Proof. Since A ∼ = A ⊗ Z , it is enough to show that T ⊆ CU ( A ⊗ Z ) . Clearly, CU (1 A ⊗ Z ) = CU (1 A ⊗ Z ) ⊆ CU ( A ⊗ Z ) . By [20], CU (1 A ⊗ Z ) contains all scalar unitaries. Therefore, T ⊆ CU ( A ⊗ Z ) . (cid:3) The next lemma can be proven using a spectral theory argument.
Lemma 2.4.
For every ǫ > , there exists δ > such that for any unital C ∗ -algebra A , if(1) p , p , ..., p n are pairwise orthogonal projections in A ,(2) q , q , ..., q n are pairwise orthogonal projections in A ,(3) α , α , ..., α n are scalars (complex numbers) with norm one,(4) | α i − α j | ≥ ǫ for i = j , and(5) k ( α p + α p + · · · + α n p n ) − ( α q + α q + · · · + α n q n ) k < δ then k p i − q i k < ǫ and p i ∼ q i in A for ≤ i ≤ n . Lemma 2.5.
There exists a ∗ -isomorphism Φ :
Z → Z ⊗ Z and there exists a sequence ofunitaries { u n } ∞ n =1 in U ( Z ⊗ Z ) such that for all a ∈ Z , lim n →∞ k Φ( a ) − u n ( a ⊗ Z ) u ∗ n k = 0 . Proof.
The result follows from Theorems 7.6 and 8.7 of [9]. (cid:3)
Lemma 2.6.
Let B be a simple, separable, unital C ∗ -algebra and let G be a closed normalsubgroup of U ( B ⊗ Z ⊗ Z ) that properly contains T . Then G contains CU (1 B ⊗Z ⊗ Z ) . PING WONG NG AND EFREN RUIZ
Proof.
By Theorem 3 of [8], B ⊗ Z ⊗ Z ( ∼ = B ⊗ Z ) is either purely infinite or stably finite.If B ⊗ Z ⊗ Z is purely infinite, then by Theorem 2.4 of [7], G = U ( B ⊗ Z ⊗ Z ) ; and thus, G contains CU (1 B ⊗Z ⊗ Z ) . Hence, we may assume that B ⊗ Z ⊗ Z (and hence B ; seeLemma 3.3 of [8]) is stably finite.Set A = B ⊗ Z . Let u be an element of G \ T and Φ : Z → Z ⊗ Z be the ∗ -isomorphismgiven in Lemma 2.5. Then by Lemma 2.5, there exist w ∈ A and a sequence of unitaries { u n } ∞ n =1 ⊆ U ( A ⊗ Z ) such that (id B ⊗ Φ)( w ) = u andlim n →∞ k u ∗ n (id B ⊗ Φ)( w ) u n − w ⊗ Z k = 0 . Since id B ⊗ Φ is an isomorphism and since u ∈ U ( A ⊗ Z ) \ T , w ∈ U ( A ) \ T . Since G is aclosed normal subgroup of U ( A ⊗ Z ) , u ∗ n uu n = u ∗ n (id B ⊗ Φ)( w ) u n ∈ G which implies that w ⊗ Z ∈ G . Hence, G contains a unitary of the form x = w ⊗ Z where w ∈ U ( A ) \ T .Let p , q be relatively prime supernatural numbers of infinite type. By Theorem 3.4 of[28], Z is a C*-inductive limit Z = S ∞ n =1 Z n where Z n ∼ = Z p , q for all n ≥ Z N . Wewill prove that G contains a nonscalar unitary in CU (1 A ⊗ Z N ) = CU (1 A ⊗ Z N ) .Now x ∈ U ( A ⊗ Z N ) ; in particular, x = w ⊗ Z N ∈ A ⊗ Z N ∼ = A ⊗ Z p , q . Since T ⊆ G ,multiplying x by a scalar if necessary, we may assume that 1 is in the spectrum of w . Hence,1 will be in the spectrum of x . Since w is not in T , the spectrum w and x contains a pointother than 1.Case 1: Suppose that the spectrum of w contains a point α = − , , i, − i . To proceed,recall the following matrix computation: (cid:20) (cid:21) = " √ √ √ − √ − (cid:21) " √ √ √ − √ . Let { u ( t ) } t ∈ [0 , be the continuous path of unitaries in M given by(2.1) u ( t ) = " √ √ √ − √ iπt ) (cid:21) " √ √ √ − √ for all t ∈ [0 , u (0) = 1 M and u (1) = (cid:20) (cid:21) . Also, for β, γ ∈ T , if { v ( t ) } t ∈ [0 , is the continuous path of unitaries in M that is given by(2.2) v ( t ) = u ( t ) (cid:20) β γ (cid:21) u ( t ) ∗ (cid:20) β γ (cid:21) for all t ∈ [0 , v (0) = 1 M and v (1) = (cid:20) βγ βγ (cid:21) . Moreover, by a direct computa-tion, we get that if β = ± γ then for all t ∈ (0 , v ( t ) are distinct andare complex conjugates of each other; also, for 0 < s, t < s = t , the set of eigenvaluesof v ( s ) is different from the set of eigenvalues of v ( t ).Let h , h : [0 , → T be the unique continuous functions such that h (0) = h (0) =1 , h (1) = α , h (1) = α , and { h , h } are the eigenfunctions of the continuous path ofunitaries { u ( t )diag( α, α ) u ( t ) ∗ diag( α, α ) } t ∈ [0 , , i.e., for all t ∈ [0 , { h ( t ) , h ( t ) } is the HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 5 set of (distinct when 0 < t <
1) eigenvalues of u ( t )diag( α, α ) u ( t ) ∗ diag( α, α ). Define twocontinuous functions g , g : [0 , → R by g ( t ) = α if t ∈ (cid:2) , (cid:3) t ∈ (cid:2) , (cid:3) ∪ (cid:2) , (cid:3) h (5 t −
1) if t ∈ (cid:2) , (cid:3) h ( − t + 4) if t ∈ (cid:2) , (cid:3) g ( t ) = α if t ∈ (cid:2) , (cid:3) t ∈ (cid:2) , (cid:3) ∪ (cid:2) , (cid:3) h (5 t −
1) if t ∈ (cid:2) , (cid:3) h ( − t + 4) if t ∈ (cid:2) , (cid:3) . Claim 1: For every ǫ >
0, there exist pairwise orthogonal nonzero projections r , r ∈ C [0 , ⊗ A ⊗ M p ⊗ M q such that if p , p are nonzero projections in r ( C [0 , ⊗ A ⊗ M p ⊗ M q ) r , r ( C [0 , ⊗ A ⊗ M p ⊗ M q ) r respectively with p ∼ p , then there exists a unitary w ′ ∈ G such that the following hold:(a) w ′ (1 − ( p + p )) = (1 − ( p + p )) w ′ = 1 − ( p + p ) in C [0 , ⊗ A ⊗ M p ⊗ M q and(b) k w ′ ( t ) − ( g ( t ) p ( t ) + g ( t ) p ( t ) + 1 − ( p ( t ) + p ( t ))) k < ǫ for all t ∈ [0 , ǫ > ǫ < /
2. Plug min n | − α | , | − α | , | α − α | , ǫ o into Lemma 2.4 to get a positive real number δ ′ . Let δ = min n ǫ , δ ′ , | − α | , | − α | , | α − α | o . Contracting δ > γ , γ ∈ T , if | γ − γ | < δ , then | γ − γ | < ǫ . By Lemma 2.2,there exist nonzero projections r ′ , s ′ in A ⊗ M p ⊗ M q such that the following hold:(1) r ′ is contained in the hereditary sub- C ∗ -algebra of A ⊗ M p ⊗ M q generated by f ( w ⊗ M p ⊗ M q ), where f is a continuous real-valued function on C with 0 ≤ f ≤ f (1) = 1 and f vanishes outside of a small precompact open neighbourhood O of1.(2) s ′ is contained in the hereditary sub- C ∗ -algebra of A ⊗ M p ⊗ M q that is generated by g ( w ⊗ M p ⊗ M q ), where g is a continuous real-valued function on C with 0 ≤ g ≤ g ( α ) = 1 and g vanishes outside of a small precompact neighbourhood O of α .(3) O ∩ O = ∅ .(4) r ′ ∼ s ′ in A ⊗ M p ⊗ M q .(5) For all s, t ∈ { r ′ ( w ⊗ M p ⊗ M q ) , ( w ⊗ M p ⊗ M q ) r ′ , r ′ ( w ⊗ M p ⊗ M q ) r ′ , r ′ } , k s − t k < δ. (6) For all s, t ∈ { s ′ ( w ⊗ M p ⊗ M q ) , ( w ⊗ M p ⊗ M q ) s ′ , s ′ ( w ⊗ M p ⊗ M q ) s ′ , αs ′ } , k s − t k < δ. Define r , s ∈ C [0 , ⊗ A ⊗ M p ⊗ M q by r = r ′ ⊗ C [0 , and s = s ′ ⊗ C [0 , . Let p , q be projections in A ⊗ M p ⊗ M q that satisfy the hypotheses of the statement of Claim 1.From the definition of r , s and the properties of r ′ , s ′ , we know that the following musthold for p , q : PING WONG NG AND EFREN RUIZ (7) p is contained in the hereditary sub- C ∗ -algebra of C [0 , ⊗ A ⊗ M p ⊗ M q generatedby f ( x ), and q is contained in the hereditary sub- C ∗ -algebra of C [0 , ⊗ A ⊗ M p ⊗ M q generated by g ( x ), where f, g are the functions in the definition of r ′ , s ′ .(8) For all s, t ∈ { p x, xp , p xp , p } , k s − t k < δ. (9) For all s, t ∈ { q x, xq , q xq , αq } , k s − t k < δ. Since p and q are orthogonal projections and since p ∼ q , there exists v ∈ U ( C [0 , ⊗ A ⊗ M p ⊗ M q ) such that v p v ∗ = v ∗ p v = q and v (1 − ( p + q )) = (1 − ( p + q )) v =1 − ( p + q ). We may assume that p v = v ∗ q .By (7)–(9) and the definition of v , we get k x ∗ v xv ∗ p − αp k < δ , k x ∗ v xv ∗ q − αq k < δ , and k x ∗ v xv ∗ (1 − ( p + q )) − (1 − ( p + q )) k < δ. The above inequalities implies that(2.3) k x ∗ v xv ∗ − ( αp + αq + 1 − ( p + q )) k < δ. Recall that if a, b are elements in a unital C ∗ -algebra such that a is invertible and k a − b k < k a − k then b is also invertible in the C ∗ -algebra. From this, (2.3) and the definition of δ , it follows that the spectrum of x ∗ v xv ∗ is contained in three pairwise disjoint openballs with centres 1 , α and α . Since the spectrum is a compact set, we may assume thatthe closures of the three open balls are also pairwise disjoint. In particular, we can takeeach open ball to have radius 10 δ . Hence, there exist pairwise disjoint self-adjoint partialisometries x , y , z ∈ C [0 , ⊗ A ⊗ M p ⊗ M q and there exist pairwise disjoint projections c , d , e ∈ C [0 , ⊗ A ⊗ M p ⊗ M q such that the following hold:(10) x ∗ v xv ∗ = x + y + z .(11) x , y , z are elements of the open balls about α, α, δ . (Of course, we are really applying the continuous functional calculus to x ∗ v xv ∗ .)(12) x ∗ x = x x ∗ = c , y ∗ y = y y ∗ = d and z ∗ z = z z ∗ = e .(13) c + d + e = 1 C [0 , ⊗ A ⊗ M p ⊗ M q .By (11) and (12), we have that(2.4) k x ∗ v xv ∗ − ( αc + αd + e ) k < δ. Together with (2.3), we get(2.5) k αp + αq + 1 − ( p + q ) − ( αc + αd + e ) k < δ. Hence, by (2.5), the definition of δ and Lemma 2.4, we have that c , d , e is Murray-vonNeumann equivalent and close to p , q , − ( p + q ) respectively.Let w ∈ U ( C [0 , ⊗ A ⊗ M p ⊗ M q ) be a unitary such that w c w ∗ = p , w d w ∗ = q and w e w ∗ = 1 − ( p + q ). We can choose w to be close to 1. Then, by (2.4),(2.6) k w x ∗ v xv ∗ w ∗ − ( αp + αq + 1 − ( p + q )) k < δ. Let x be the unitary in U ( C [0 , ⊗ A ⊗ M p ⊗ M q ) that is given by x = ( w x ∗ v xv ∗ w ∗ ) v ( w x ∗ v xv ∗ w ∗ ) ∗ v ∗ . HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 7 By (10)–(13), (2.6), and the definitions of v , w , x and δ , the following hold:(14) x p = p x = p x p and k x p − α p k < δ .(15) x q = q x = q x q and k x q − α q k < δ .(16) x (1 − ( p + q )) = (1 − ( p + q )) x = 1 − ( p + q ).We now define elements v , w , v ∈ U ( A ⊗ Z p , q ) . To simplify notation, assume that p , q , v , w ∈ C [0 , ⊗ A ⊗ M p ⊗ M q . The general case is similar. To define v , we firstconsider the norm-continuous path of unitaries { u ( t ) } t ∈ [0 , in ( p + q )( A ⊗ M p ⊗ M q )( p + q )given by (2.1) with the standard system of matrix units for M as e , = p , e , = q , e , = p v = v ∗ q . The last equality follows from the definition of v . Define v by v ( t ) = v if t ∈ (cid:2) , (cid:3) t ∈ (cid:2) , (cid:3) ∪ (cid:2) , (cid:3) u (5 t −
1) + 1 − ( p + q ) if t ∈ (cid:2) , (cid:3) u ( − t + 4) + 1 − ( p + q ) if t ∈ (cid:2) , (cid:3) . Note that v ∈ U ( A ⊗ Z ) . Let w , v ∈ U ( A ⊗ Z p , q ) be unitaries chosen so that w ( t ) = w for t ∈ (cid:2) , (cid:3) , w (0) = w (1) = 1, v ( t ) = v for t ∈ (cid:2) , (cid:3) , and v (0) = v (1) = 1.Since x ∈ G and G is a normal subgroup of U ( A ⊗ Z p , q ) , w = ( w x ∗ v xv ∗ w ∗ ) v ( w x ∗ v xv ∗ w ∗ ) ∗ v ∗ is an element of G . Hence, it follows, by the definitions above, that the following hold:(17) w (1 − ( p + q )) = (1 − ( p + q )) w = 1 − ( p + q ).(18) k w − ( αp + αq + 1 − ( p + q )) v ( αp + αq + 1 − ( p + q )) ∗ v ∗ k < δ .(19) The eigenvalues of ( αp + αq + 1 − ( p + q )) v ( t )( αp + αq + 1 − ( p + q )) ∗ v ∗ ( t )are given by { g ( t ) , g ( t ) } for all t ∈ [0 , g ( t ) = g ( t ) for t ∈ (cid:2) , (cid:3) and g ( t ) = g ( t ) = 1 for t ∈ (cid:2) , (cid:3) ∪ (cid:2) , (cid:3) , there exist twoprojections p , q ∈ C [0 , ⊗ A ⊗ M p ⊗ M q such that( αp + αq + 1 − ( p + q )) v ( αp + αq + 1 − ( p + q )) ∗ v ∗ = g p + g q + 1 − ( p + q ) . Indeed, p , q can be chosen to be projections inside the copy of M that is generated by { p , q , v p } ; and they can be chosen to satisfy p ∼ p , q ∼ q . Hence, there exists aunitary v ∈ U ( A ⊗ Z p , q ) such that v ( αp + αq + 1 − ( p + q )) v ( αp + αq + 1 − ( p + q )) ∗ v ∗ v ∗ = g p + g q + 1 − ( p + q )and v (1 − ( p + q )) = (1 − ( p + q )) v = 1 − ( p + q ) . Set w ′ = v w v ∗ . Then w ′ ∈ G is a unitary that satisfies the statement of Claim 1. Thiscompletes the proof of Claim 1.Claim 2: Suppose that r, s are pairwise orthogonal projections in C [0 , ⊗ A ⊗ M p ⊗ M q which are Murray-von Neumann equivalent in C [0 , ⊗ A ⊗ M p ⊗ M q . Then g r + g s + 1 − ( r + s ) ∈ G. We will show that g r + g s + 1 − ( r + s ) can be approximated arbitrarily close by elementsof G . Let ǫ > ǫ into Claim 1 to get nonzero orthogonal projections r , r ∈ C [0 , ⊗ A ⊗ M p ⊗ M q . Decompose r, s into pairwise orthogonal projections r = r , + r , + .... + r ,n PING WONG NG AND EFREN RUIZ and s = s , + s , + .... + s ,n so that the following hold:(i) r ,j and s ,j are projections in C [0 , ⊗ A ⊗ M p ⊗ M q for all j ≥ r ,j ∼ s ,j in C [0 , ⊗ A ⊗ M p ⊗ M q for all j ≥ r ,j - r ∼ r in C [0 , ⊗ A ⊗ M p ⊗ M q for all j ≥ C [0 , ⊗ A ⊗ M p ⊗ M q has cancellation of projections and since G is a normal subgroupof U ( A ⊗ Z ) , it follows, by Claim 1, that for j ≥
1, there exists a unitary w ,j ∈ G suchthat the following conditions hold:(iv) w ,j (1 − ( r ,j + s ,j )) = (1 − ( r ,j + s ,j )) w ,j = 1 − ( r ,j + s ,j ) for all j ≥ k w ,j − ( g r ,j + g s ,j + 1 − ( r ,j + s ,j )) k < ǫ for all j ≥ w = w , w , ...w ,n . Then k w − ( g r + g s + 1 − ( r + s )) k < ǫ. Since ǫ > G is closed, g r + g s + 1 − ( r + s ) ∈ G as required. This completes the proof of Claim 2.We now complete the proof that CU (1 A ⊗ Z ) ⊆ G for Case 1. Choose two nonzeroorthogonal projections r, s ∈ C [0 , ⊗ A ⊗ M p ⊗ M q such that r ∼ s in C [0 , ⊗ A ⊗ M p ⊗ M q .By Claim 2, w = g r + g s + 1 − ( r + s ) ∈ G . Note that w is also a unitary in CU (1 A ⊗ Z )which is not a scalar multiple of the identity. By [20], CU ( Z ) / T is a simple topologicalgroup. Hence, CU (1 A ⊗ Z ) ⊆ G as required.Case 2: Suppose that the spectrum of w (and hence the spectrum of x ) is contained in {− , , i, − i } . Recall that w is not in T and its spectrum contains 1. For simplicity, let usassume that the spectrum of w is { , α } where α ∈ {− , i, − i } . (The proofs for the othercases are similar.) Hence, there exist nonzero orthogonal projections p, q ∈ A such that w = p + αq . Therefore, x = p ⊗ Z p , q + αq ⊗ Z p , q .Let r, s ∈ A ⊗ M p ⊗ M q be nonzero orthogonal projections such that r is orthogonal to p , s is orthogonal to q , r ∼ s (cid:16) p ⊗ M p ⊗ M q and s (cid:16) q ⊗ M p ⊗ M q in A ⊗ M p ⊗ M q . Then thereexist projections r ′ , s ′ ∈ A ⊗ M p ⊗ M q and v ∈ A ⊗ M p ⊗ M q such that r ∼ r ′ (cid:12) p ⊗ M p ⊗ M q , s ∼ s ′ (cid:12) q ⊗ M p ⊗ M q , and v ∗ v = r ′ and vv ∗ = s ′ .Let { u ( t ) } t ∈ [0 , be the norm-continuous path of unitaries in M as in (2.1) except thatthe canonical system of matrix units is taken to be e , = r ′ , e , = s ′ and e , = v . Definethe norm-continuous path of unitaries { w ( t ) } t ∈ [0 , in U ( A ⊗ M p ⊗ M q ) by w ( t ) = ( u (2 t ) + 1 − ( r ′ + s ′ ) ⊗ C [0 , if t ∈ (cid:2) , (cid:3) u ( − t + 2) + 1 − ( r ′ + s ′ ) ⊗ C [0 , if t ∈ (cid:2) , (cid:3) . Note that w ∈ U ( A ⊗Z p , q ) . Since G is a closed normal subgroup of U ( A ⊗Z ) , w ′ = wxw ∗ x ∗ is an element of G ∩ U ( A ⊗ Z p , q ) . Moreover, we have that w ′ is not a scalar multiple ofthe identity, w ′ (1 − ( r ′ + s ′ ) ⊗ C [0 , ) = (1 − ( r ′ + s ′ ) ⊗ C [0 , ) w ′ = 1 − ( r ′ + s ′ ) ⊗ C [0 , in C [0 , ⊗ A ⊗ M p ⊗ M q and w ′ (0) = w ′ (1) = 1.Let f , g : [0 , → T be two continuous functions such that for all t ∈ [0 , { f ( t ) , g ( t ) } are the eigenvalues of ( r ′ + s ′ ) w ′ ( t )( r ′ + s ′ ). Note that f (0) = f (1) = g (0) = g (1) = 1.Set w ′′ = f r ′ + g s ′ + 1 − ( r ′ + s ′ ) ⊗ C [0 , . Then w ′′ ∈ U ( A ⊗ Z p , q ). Note that f r ′ + g s ′ ∈ HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 9 C [0 , ⊗ C ∗ ( r ′ , s ′ , v ) ∼ = C [0 , ⊗ M . Hence, by [30], w ′ and w ′′ are approximately unitarilyequivalent in C [0 , ⊗ A ⊗ M p ⊗ M q . Moreover, since w ′ (0) = w ′ (1) = w ′′ (0) = w ′′ (1) = 1,we can choose the implementing unitaries to also have unit value at the endpoints and beelements of U ( A ⊗ Z p , q ) . Hence, since G is a closed normal subgroup of U ( A ⊗ Z ) , w ′′ ∈ G .Since f ( t ) = f (1 − t ), g ( t ) = g (1 − t ) for all t ∈ [0 , w ′′′ = f r + g s + 1 − ( r + s ) ⊗ C [0 , ∈ U (1 A ⊗ Z p , q ) . Note also that w ′′′ is not a scalar multiple of the identity. Since r ∼ r ′ , s ∼ s ′ , r is orthogonal to r ′ , s is orthogonal to s ′ , and w ′′′ (0) = w ′′ (0) = w ′′′ (1) = w ′′ (1) = 1, we have that w ′′ and w ′′′ are approximately unitarily where we can choose theunitaries to have unit value at endpoints and be elements in U ( A ⊗ Z ) . Hence, since G is a closed normal subgroup of U ( A ⊗ Z ) , we must have that w ′′′ ∈ G , i.e., G contains anelement of U (1 A ⊗ Z ) which is not a scalar multiple of the identity. By [20], U (1 A ⊗ Z ) / T is a simple topological group. Hence, U (1 A ⊗ Z ) ⊆ G as required. This completes theproof for Case 2 and hence, all the cases. (cid:3) Recall that for a C ∗ -algebra C and for elements a, b ∈ C , ( a, b ) is defined to be ( a, b ) = aba ∗ b ∗ . Lemma 2.7.
Consider the supernatural numbers p = 2 ∞ and q = 3 ∞ . Let A be a simpleunital C ∗ -algebra. Let G be a closed normal subgroup of U ( A ⊗Z p , q ) that contains CU (1 A ⊗Z p , q ) and let u i , v i : [0 , → U ( A ⊗ Z p , q ) ( ≤ i ≤ n ) be norm-continuous paths. Define w by w = n Y i =1 ( u i , v i ) = ( u , v )( u , v ) ... ( u n , v n ) . Note that w ∈ CU ( C [0 , ⊗ A ⊗ M p ⊗ M q ) ⊆ CU ( A ⊗ Z p , q ) . If w (0) = 1 , then w ∈ G .Proof. Since G is a closed subset of U ( A ⊗ Z p , q ) , we may assume that there exists δ > δ <
1) such that for all t ∈ [0 , δ ), w ( t ) = 1. By definition of q , M q = ∞ [ j =1 M j with connecting maps defined by a diag( a, a, a ).As an intermediate step, we will work inside C [0 , ⊗ A ⊗ M . Let x , x ,i , x ,i , x ,i ∈ C [0 , ⊗ A ⊗ M be given as follows: x = − x ,i = u i v ∗ i
00 0 1 x ,i = v i
00 0 1 x ,i = u i
00 0 1 . Hence, in C [0 , ⊗ A ⊗ M , we have that u i v i u ∗ i v ∗ i
00 0 1 = x ,i x ,i x x ∗ ,i x ∗ x ∗ ,i = x ,i ( x ,i , x ) x ∗ ,i and u ∗ i v ∗ i u i v i
00 0 1 = x ∗ ,i x ∗ ,i x x ,i x ∗ x ,i = x ∗ ,i ( x ∗ ,i , x ) x ,i . Hence,(2.7) ( u i , v i ) 0 00 ( u i , v i ) 00 0 1 = x ,i x ,i [ x ,i ( x ,i , x ) x ∗ ,i ][ x ∗ ,i ( x ∗ ,i , x ) x ,i ] x ∗ ,i x ∗ ,i = x ,i x ,i x ,i ( x ,i , x ) x ∗ ,i ( x ∗ ,i , x ) x ,i x ∗ ,i x ∗ ,i . Let y , y ,i , y ,i , y ,i ∈ C [0 , ⊗ A ⊗ M be given as follows: y = − y ,i = u i
00 0 v ∗ i y ,i = v i y ,i = u i . Hence, in C [0 , ⊗ A ⊗ M , we have that v i u i
00 0 u ∗ i v ∗ i = y y ∗ ,i y ∗ y ,i = ( y , y ∗ ,i )and u ∗ i v ∗ i
00 0 u i v i = y ∗ ,i y ∗ ,i y y ,i y ∗ y ,i = y ∗ ,i ( y ∗ ,i , y ) y ,i . Hence,(2.8) ( u i , v i ) 0 00 ( u i , v i ) 00 0 ( u i , v i ) = y ,i y ,i ( u i , v i ) 0 00 u i v i u ∗ i v ∗ i
00 0 1 ( y , y ∗ ,i )[ y ∗ ,i ( y ∗ ,i , y ) y ,i ] y ∗ ,i y ∗ ,i . From this and (2.7), we have in C [0 , ⊗ A ⊗ M that(2.9) w = n Y i =1 ( u i , v i ) 0 00 ( u i , v i ) 00 0 ( u i , v i ) = n Y i =1 y ,i y ,i ( u i , v i ) 0 00 ( u i , v i ) 00 0 1 ( y , y ∗ ,i )[ y ∗ ,i ( y ∗ ,i , y ) y ,i ] y ∗ ,i y ∗ ,i . We now use the above to manufacture elements in CU ( A ⊗ Z p , q ) . Let h : [0 , → [0 , δ ]be the continuous function that is given by h ( t ) = ( t for t ∈ [0 , δ ] δ for t ∈ [ δ, . For 1 ≤ i ≤ n , 2 ≤ j ≤
4, let x ′ , y ′ , x ′ j,i , y ′ j,i : [0 , → U ( A ⊗ M ) be continuous functionssuch that the following hold:(i) x ′ ( t ) = x ( t ), y ′ ( t ) = y ( t ), x ′ j,i ( t ) = x j,i ( t ) and y ′ j,i ( t ) = y j,i ( t ) for t ∈ [ δ, HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 11 (ii) x ′ (0) = y ′ (0) = x ′ j,i (0) = y ′ j,i (0) = 1.(iii) x ′ ( t ) , y ′ ( t ) , x ′ j,i ( t ) , y ′ j,i ( t ) ∈ A ⊗ M for t ∈ (0 , δ ].(iv) x ′ , y ′ , x ′ j,i , y ′ j,i ∈ U ( A ⊗ Z p , q ) .(v) x ′ , y ′ , x ′ ◦ h, y ′ ◦ h ∈ CU (1 A ⊗ Z p , q ) .(Note that the determinants of x and y are both one.) Next, we replace x , y , x j,i , y j,i with x ′ , y ′ , x ′ j,i , y ′ j,i respectively in the expressions in (2.7) and (2.8) – except at one occurenceof y . We now proceed with the details.By (i)–(v), we have that x ′ ∈ CU (1 A ⊗ Z p , q ) ⊆ G . Since G is a normal subgroup of U ( A ⊗ Z p , q ) , z i = x ′ ,i x ′ ,i x ′ ,i ( x ′ ,i , x ′ ) x ′ ,i ∗ ( x ′ ,i ∗ , x ′ ) x ′ ,i x ′ ,i ∗ x ′ ,i ∗ ∈ G. Note also by (2.7) and (i)–(v), that ( u i ( t ) , v i ( t )) 0 00 ( u i ( t ) , v i ( t )) 00 0 1 = z i ( t )for t ∈ [ δ, y ′ ∈ G and since G is a normal subgroup of U ( A ⊗ Z p , q ) , z ′ i = y ′ ,i y ′ ,i z i ( y ′ , y ′∗ ,i )[ y ′ ,i ∗ ( y ′ ,i ∗ , y ′ ) y ′ ,i ] y ′∗ ,i y ′∗ ,i ∈ G. Again, by (2.8) and (i)–(v), ( u i ( t ) , v i ( t )) 0 00 ( u i ( t ) , v i ( t )) 00 0 ( u i ( t ) , v i ( t )) = z ′ i ( t )for t ∈ [ δ, w ( t ) = n Y i =1 ( u i ( t ) , v i ( t ))for t ∈ [0 , δ ]. To obtain this, we replace the last occurrence of y ′ in z ′ n by another element y ′′ ∈ G . We define y ′′ as follows: y ′′ ( t ) = y ( t )for t ∈ [ δ, t ∈ [0 , δ ], set y ′′ ( t ) to be y ′∗ ,n ( t ) y ′∗ ( t ) y ′ ,n ( t ) y ′ ,n ( t )( y ′ ( t ) , y ′∗ ,n ( t )) ∗ z n ( t ) ∗ y ′∗ ,n ( t ) y ′∗ ,n ( t ) hQ n − i =1 z ′ i ( t ) i ∗ y ′ ,n ( t ) y ′ ,n ( t ) y ′ ,n ( t ) ∗ . (The complicated definition over [0 , δ ] is to ensure that hQ n − i =1 z ′ i ( t ) i y ′ ,n ( t ) y ′ ,n ( t ) z n ( t )( y ′ ( t ) , y ′∗ ,n ( t ))[ y ′∗ ,n ( t )[ y ′∗ ,n ( t ) y ′ ( t ) y ′ ,n ( t ) y ′′ ( t )] y ′ ,n ( t )] y ′∗ ,n ( t ) y ′∗ ,n ( t ) = 1for t ∈ [0 , δ ].) From (2.8) and (i.)–(v.), we have that y ′′ ∈ C [0 , ⊗ A ⊗ M p ⊗ M q , y ′′ ( t ) ∈ A ⊗ M for t ∈ (0 ,
1] and y ′′ (0) = 1. Thus, y ′′ ∈ U ( A ⊗ Z p , q ) . We now prove that y ′′ ∈ G . Note that by (i)–(v) and by an argument similar to thatused to show that z i , z ′ i ∈ G , we have that z i ◦ h, z ′ i ◦ h, x ′ ◦ h, y ′ ◦ h ∈ G . Note that y ′′ isequal to( y ′∗ ,n ◦ h )( y ′∗ ◦ h )( y ′ ,n ◦ h )( y ′ ,n ◦ h )( y ′ ◦ h, y ′∗ ,n ◦ h ) ∗ ( z ∗ n ◦ h )( y ′∗ ,n ◦ h )( y ′∗ ,n ◦ h ) hQ n − i =1 ( z ′ i ◦ h ) i ∗ ( y ′ ,n ◦ h )( y ′ ,n ◦ h )( y ′∗ ,n ◦ h ).Since G is a normal subgroup of U ( A ⊗ Z p , q ) , we have that y ′′ ∈ G . Since G is a normalsubgroup of U ( A ⊗ Z p , q ) and w = " n − Y i =1 z ′ i y ′ ,n y ′ ,n z n ( y ′ , y ′∗ ,n )[ y ′∗ ,n [ y ′∗ ,n y ′ y ′ ,n y ′′ ] y ′ ,n ] y ′∗ ,n y ′∗ ,n we have that w ∈ G as required. (cid:3) Lemma 2.8.
Consider the supernatural numbers p = 2 ∞ and q = 3 ∞ . Let A be a simpleunital C ∗ -algebra and let G be a closed normal subgroup of U ( A ⊗ Z p , q ) that contains CU (1 A ⊗ Z p , q ) . Let u i , v i : [0 , → U ( A ⊗ Z p , q ) ( ≤ i ≤ n ) be norm-continuous pathsand let w = n Y i =1 ( u i , v i ) = ( u , v )( u , v ) ... ( u n , v n ) . Note that w ∈ CU ( C [0 , ⊗ A ⊗ M p ⊗ M q ) ⊆ CU ( A ⊗ Z p , q ) . If w (1) = 1 , then w ∈ G .Proof. The proof is similar (actually slightly easier) to that of Lemma 2.7. (cid:3)
Lemma 2.9.
Consider the supernatural numbers p = 2 ∞ and q = 3 ∞ . Let A be a simpleunital C ∗ -algebra and let G be a closed normal subgroup of U ( A ⊗ Z p , q ) that contains CU (1 A ⊗ Z p , q ) . Let u i , v i : [0 , → U ( A ⊗ Z p , q ) ( ≤ i ≤ n ) be norm-continuous pathsand let w = n Y i =1 ( u i , v i ) = ( u , v )( u , v ) ... ( u n , v n ) . Note that w ∈ CU ( C [0 , ⊗ A ⊗ M p ⊗ M q ) ⊆ CU ( A ⊗ Z p , q ) . Then w ∈ G .Proof. Decompose w into a product of unitaries w = w ′ w ′′ such that w ′ satisfies the hypothe-ses of Lemma 2.7 and w ′′ satisfies the hypotheses of Lemma 2.8. (Sketch of argument: For all i , since u i (1) ∈ U ( A ⊗ Z p , q ) , there exists a norm continuous path u ′ i : [0 , → U ( A ⊗ Z p , q ) such that u ′ i (0) = 1 and u ′ i (1) = u i (1). Let w ′ = Q ni =1 ( u ′ i , v i ) and w ′′ = w ′∗ w .) Now applyLemmas 2.7 and 2.8. (cid:3) Lemma 2.10.
Consider the supernatural numbers p = 2 ∞ and q = 3 ∞ . Let A be a simpleunital C ∗ -algebra and let G be a closed normal subgroup of U ( A ⊗ Z p , q ) that contains CU (1 A ⊗ Z p , q ) . Then CU ( A ⊗ Z p , q ) ⊆ G . HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 13 Proof.
Say that u, v ∈ U ( A ⊗ Z p , q ) . Thus, u, v are constant functions from [0 ,
1] to U ( A ⊗ M p ⊗ M q ) (always taking the constant values u (0), v (0) respectively). It followsfrom Lemma 2.9 that ( u, v ) ∈ G . Thus, since G is a group, the commutator subgroup of U ( A ⊗ Z p , q ) is contained in G . Hence, since G is closed, CU ( A ⊗ Z p , q ) ⊆ G . (cid:3) Theorem 2.11.
Let A be an exact, separable, simple, unital, Z -stable C ∗ -algebra. Supposethat G is a closed normal subgroup of U ( A ) that properly contains T . Then CU ( A ) ⊆ G .Proof. Since A is Z -stable, A ∼ = A ⊗ Z ∼ = A ⊗ Z ⊗ Z . Hence, it is enough to prove thetheorem with A replaced by A ⊗ Z ⊗ Z . By Lemma 2.6, we have that CU (1 A ⊗Z ⊗ Z ) ⊆ G .It suffices to prove the following: Let u, v ∈ U ( A ⊗ Z ⊗ Z ) be given. Then ( u, v ) ∈ G .By Lemma 2.5, there is a ∗ -isomorphism Φ : A ⊗ Z → A ⊗ Z ⊗ Z such that Φ isapproximately unitarily equivalent to the map A ⊗ Z → A ⊗ Z ⊗ Z : a a ⊗ Z , where theunitaries come from U ( A ⊗ Z ⊗ Z ) . Let u ′ , v ′ ∈ U ( A ⊗ Z ) be unitaries such that Φ( u ′ ) = u and Φ( v ′ ) = v . Hence, u ′ ⊗ Z p , q , v ′ ⊗ Z p , q are approximately unitarily equivalent to u , v respectively, with implementing unitaries in U ( A ⊗ Z ⊗ Z ) .Let p , q be the supernatural numbers that are given by p = 2 ∞ and q = 3 ∞ . ByTheorem 3.4 of [28], Z is a C ∗ -inductive limit Z = S ∞ n =1 Z n where Z n ∼ = Z p , q for all n ≥ u ′ ⊗ Z p , q , v ′ ⊗ Z p , q ) ∈ G . Since ( u ′ ⊗ Z p , q , v ′ ⊗ Z p , q ) is approximately unitarily equivalentto ( u, v ), with unitaries coming from U ( A ⊗Z ⊗Z ) and since G is a closed normal subgroupof U ( A ⊗ Z ⊗ Z ) , ( u, v ) ∈ G as required. (cid:3) Lemma 2.12.
Let A be an exact, unital, stably finite, Z -stable C ∗ -algebra. For every ǫ > ,there exists δ > such that for every self-adjoint element a ∈ A such that | τ ( a ) | < δ for all τ ∈ T ( A ) , dist( e ia , CU ( A ) ) = inf {k e ia − u k : u ∈ CU ( A ) } < ǫ. Proof.
This follows from [31] which gives a topological group isomorphism:∆ : U ( A ) /CU ( A ) → Aff( T ( A )) /K ( A )where ∆ is the determinant map. Note that for a self-adjoint a ∈ A , ∆([ e i πa ]) = a + K ( A ). (cid:3) Lemma 2.13.
Let A be an exact, simple, unital, stably finite C ∗ -algebra and let C be aUHF-algebra. Then for every ǫ > , for every N ≥ , for every nonzero selfadjoint element a ∈ A ⊗ C and nonzero projection p ∈ A ⊗ C , there exists a self-adjoint element c ∈ p A p such that | τ ( a ) − N τ ( c ) | < ǫ for all τ ∈ T ( A ⊗ C ) .Proof. We have that C can be realized as an inductive limit C = ∞ [ k =1 M n k where the connecting maps are diagonal maps M n k → M n k +1 : c nk +1 nk M c. Note that n k → ∞ . Hence, A ⊗ C can be realized as an inductive limit A ⊗ C = ∞ [ k =1 M n k ( A )where the connecting maps are diagonal maps M n k ( A ) → M n k +1 ( A ) : c nk +1 nk M c. Note that the connecting map divides c up into n k +1 n k pairwise orthogonal Cuntz equivalentpieces, each with n k n k +1 th trace of c for any tracial state on A ⊗ C .The result follows from the nature of the connecting maps for the inductive limit decom-position of A ⊗ C and since A ⊗ C has strict comparison for positive elements (see Lemma2.2). (cid:3) Lemma 2.14.
Let A be a unital C ∗ -algebra and let a, b ∈ A be self-adjoint elements. Then e ia e ib e − i ( a + b ) ∈ CU ( A ) . Proof.
This follows immediately from the formulaexp( ia ) exp( ib ) exp( − i ( a + b )) = lim n →∞ exp( ia ) exp( ib ) (cid:20) exp (cid:18) − ian (cid:19) exp (cid:18) − ibn (cid:19)(cid:21) n . (cid:3) The next lemma is motivated by the main result of [23]. However, we note that unlike[23], we need to assume that our C ∗ -algebra is nuclear. In fact, it is not clear to us whetherif A is an arbitrary unital simple separable quasidiagonal C ∗ -algebra and C is a UHF-algebrathen A ⊗ C has the “sufficiently many projections” condition of Popa. (See [23] (1.1), (2.1),(2.1’) and (2.1”).) Lemma 2.15.
Let A be a nuclear, separable, simple, unital C ∗ -algebra, and let C be aUHF-algebra. Then A is quasidiagonal if and only if A ⊗ C has the Popa property; i.e., forevery ǫ > and for every finite subset F ⊆ A ⊗ C , there exists a nonzero finite dimensionalsub- C ∗ -algebra D of A ⊗ C with unit p = 1 D such that for every a ∈ F , the following hold:i. k pa − ap k < ǫ andii. pap is within ǫ of an element of D .Proof. By Theorem 1 of [32], if A ⊗ C has the Popa property then A ⊗ C is quasidiagonal(see, for example, [3] the argument after Theorem 12.1.); and hence, A is quasidiagonal.This completes the proof of the “if” direction. HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 15 We now prove the “only if” direction. Suppose that A is quasidiagonal. Since C is aUHF-algebra, it can be expressed as an inductive limit C = ∞ [ k =1 M n k where the connecting maps are defined by c diag( c, c, ..., c ) (each c being repeated n k +1 n k times). Hence, A ⊗ C is an inductive limit A ⊗ C = ∞ [ k =1 M n k ( A )where the connecting maps are diagonal maps. It suffices to prove the Popa property forfinite subsets of the building blocks M n k ( A ). Let K ≥ ǫ > F ⊆ M n K ( A ) be a finite set. We may assume that the elements of F have norm lessthan or equal to one. Let M ( M n K ( A ) ⊗ K ) be the multiplier algebra of the stabilizationof M n K ( A ). (Note that M n K ( A ) ⊗ K ∼ = A ⊗ K .) Define φ : M n K ( A ) → M ( M n K ( A ) ⊗ K )by φ ( a ) = a ⊗ M ( K ) and let ψ ′ : M n K ( A ) → M ( K ) = B ( H ) be any unital (and henceessential) ∗ -homomorphism. Set ψ = 1 M nK ( A ) ⊗ ψ ′ : M n K ( A ) → M ( M n K ( A ) ⊗ K ). Hence, φ, ψ : M n K ⊗ A → M ( M n K ( A ) ⊗ K ) are injective, unital ∗ -homomorphisms. Note that φ and ψ are both full ∗ -homomorphisms. Hence, since A is nuclear, it follows, by [6], thatthere exists a unitary u ∈ M ( M n K ( A ) ⊗ K ) such that for every a ∈ M n K ⊗ A ,(2.10) φ ( a ) − uψ ( a ) u ∗ ∈ M n K ( A ) ⊗ K . Since M n K ( A ) is quasidiagonal, ψ ′ ( M n K ( A )) is a quasidiagonal collection of operators on H ; i.e., there exists an increasing sequence { p n } ∞ n =1 of finite rank operators on H such thati. p n → B ( H ) in the strong operator topology andii. k p n ψ ′ ( a ) − ψ ′ ( a ) p n k → a ∈ M n K ⊗ A .This and (2.10) implies that there exists an integer N ≥ k u ( p m − p n ) u ∗ φ ( a ) − φ ( a ) u ( p m − p n ) u ∗ k < ǫ for all m > n ≥ N and for all a ∈ F (2) u ( p m − p n ) u ∗ φ ( a ) u ( p m − p n ) u ∗ is within ǫ of an element of the finite dimensional C ∗ -algebra u ( p m − p n ) B ( H )( p m − p n ) u ∗ for all a ∈ F and all m > n ≥ N .Now let { e i,j : 1 ≤ i, j < ∞} be a system of matrix units for K . Note that 1 M ( M nK ( A ) ⊗ K ) = P ∞ i =1 ⊗ e i,i where the sum converges in the strict topology on M ( M n K ( A ) ⊗ K ). Hence,for all a ∈ M n K ( A ), φ ( a ) = a ⊗ ∞ X i =1 a ⊗ e i,i where the sum converges in the strict topology on M ( M n K ( A ) ⊗ K ). Also, note that forevery n ≥
1, lim M →∞ M X i =1 ⊗ e i,i ! up n u ∗ = up n u ∗ . Hence, (1)–(2) and the definition of φ implies that there exists an M ≥ r ∈ M M ( M n K ( A )) (which, inside M ( M n K ( A ) ⊗ K ), is close to u ( p m − p n ) u ∗ for some integers m > n ) such that the following hold: (i) M = n L n K for some L ≥ K .(ii) k ( L M a ) r − r ( L M a ) k < ǫ for all a ∈ F .(iii) There exists a finite dimensional sub- C ∗ -algebra D ⊆ M M ( M n K ( A )) such that theunit of D is 1 D = r . ( D will be “close to” u ( p m − p n ) B ( H )( p m − p n ) u ∗ for somepositive integers m > n .)(iv) For every a ∈ F , r (cid:16)L M a (cid:17) r is within ǫ of an element of D .Now since, in the inductive limit decomposition of A ⊗ C , the connecting map M n K ( A ) → M n L ( A ) has the form c L M c (a diagonal map), we are done. (cid:3) Lemma 2.16.
Let A be a quasidiagonal, nuclear, separable, simple, unital C ∗ -algebra. Let a ∈ A be a self-adjoint element and F ⊆ A be a finite set. Then for every ǫ > , there existsa unitary u ∈ CU ( A ⊗ Z ) such that k ( e ia ⊗ Z )( b ⊗ Z )( e − ia ⊗ Z ) − u ( b ⊗ Z ) u ∗ k < ǫ for all b ∈ F .Proof. We may assume that every element of F has norm less than or equal to one. Let p , q be the relatively prime supernatural numbers given by p = 2 ∞ and q = 3 ∞ . By Theorem3.4 of [28], Z is a C ∗ -inductive limit Z = S ∞ n =1 Z n where Z n ∼ = Z p , q for all n ≥ A ⊗ M p and A ⊗ M q have the Popa property. Hence, let D ⊆ A ⊗ M p and D ⊆ A ⊗ M q be nonzero finite dimensional simple sub- C ∗ -algebras with units e = 1 D and e = 1 D such that the following statements hold:(1) k ce − e c k < ǫ for all c ∈ ( F ⊗ M p ) ∪ { a ⊗ M p } ∪ { e ita ⊗ M p : t ∈ [0 , } .(2) e ce is within ǫ of an element of D for all c ∈ ( F ⊗ M p ) ∪ { a ⊗ M p } ∪ { e ita ⊗ M p : t ∈ [0 , } .(3) k ce − e c k < ǫ for all c ∈ ( F ⊗ M q ) ∪ { a ⊗ M q } ∪ { e ita ⊗ M q : t ∈ [0 , } .(4) e ce is within ǫ of an element of D for all c ∈ ( F ⊗ M q ) ∪ { a ⊗ M q } ∪ { e ita ⊗ M q : t ∈ [0 , } .Plug ǫ into Lemma 2.12 to get a positive real number δ >
0. Now let { e i,j } ≤ i,j ≤ m be asystem of matrix units for D . By Lemma 2.13, let d ∈ e , ( A ⊗ M p ) e , be a self-adjointelement such that | τ ( a ) − mτ ( d ) | < δ τ ∈ T ( A ⊗ Z ). Consider the element d ∈ e ( A ⊗ M p ) e given by d = L m d . (The i th copy of d sits inside e i,i .) Then d is a self-adjoint element of A ⊗ M p such that thefollowing statements hold:(a) d commutes with every element of D and e − isd commutes with every element of D ⊕ (1 − e )( A ⊗ M p )(1 − e ) for all s ∈ [0 , k e − isd c − ce − isd k < ǫ for all s ∈ [0 ,
1] and for all c ∈ ( F ⊗ M p ) ∪ { e ita ⊗ M p : t ∈ [0 , } .(b) | τ ( a ) − τ ( d ) | < δ for all τ ∈ T ( A ). Hence, | τ ((1 − t ) a ) − τ ((1 − t ) d ) | < δ forall τ ∈ T ( A ) and for all t ∈ [0 , δ , by Lemma 2.14 andLemma 2.12, the map [0 , → U ( A ⊗ Z ) defined by t ( e i (1 − t ) a ⊗ e − i (1 − t ) d is HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 17 an element of U ( A ⊗ Z p , q ) ⊆ U ( A ⊗ Z ) which is within ǫ of an element u of CU ( A ⊗ Z ) .By a similar argument, we can find a self-adjoint element d ∈ e ( A ⊗ M q ) e satisfyingthe following statements:(i) k e − isd c − ce − isd k < ǫ for all s ∈ [0 ,
1] and for all c ∈ ( F ⊗ M q ) ∪ { e ita ⊗ M q : t ∈ [0 , } .(ii) The map [0 , → U ( A ⊗ Z ) defined by t ( e ita ⊗ e − itd is an element of U ( A ⊗ Z p , q ) ⊆ U ( A ⊗ Z ) which is within ǫ of an element u of CU ( A ⊗ Z ) .By (a)–(b) and (i)–(ii), we have that k ( e ia ⊗ Z )( b ⊗ Z )( e − ia ⊗ Z ) − u u ( b ⊗ Z ) u ∗ u ∗ k = k ( e i (1 − t ) a e ita ⊗ Z )( e − i (1 − t ) d e − itd e itd e i (1 − t ) d )( b ⊗ Z )( e − ita e − i (1 − t ) a ⊗ Z ) − u u ( b ⊗ Z ) u ∗ u ∗ k≤ k ( e i (1 − t ) a e ita ⊗ Z )( e − i (1 − t ) d e − itd e itd e i (1 − t ) d )( b ⊗ Z )( e − ita e − i (1 − t ) a ⊗ Z ) − ( e i (1 − t ) a ⊗ e − i (1 − t ) d ( e ita ⊗ e − itd ( b ⊗ Z ) e itd ( e − ita ⊗ e i (1 − t ) d ( e − i (1 − t ) a ⊗ k + k ( e i (1 − t ) a ⊗ e − i (1 − t ) d ( e ita ⊗ e − itd ( b ⊗ Z ) e itd ( e − ita ⊗ e i (1 − t ) d ( e − i (1 − t ) a ⊗ − u u ( b ⊗ Z ) u ∗ u ∗ k < ǫ/
100 + 4 ǫ/ < ǫ for all b ∈ F . Hence, taking u = u u , we are done. (cid:3) Lemma 2.17.
Let A be a quasidiagonal, nuclear, separable, simple, unital C ∗ -algebra. Let v be a unitary in U ( A ) and let F ⊆ A be a finite set. Then for every ǫ > , there exists aunitary u ∈ CU ( A ⊗ Z ) such that k ( v ⊗ Z )( b ⊗ Z )( v ∗ ⊗ Z ) − u ( b ⊗ Z ) u ∗ k < ǫ for all b ∈ F .Proof. Since v ∈ U ( A ) , v has the form v = e ia e ia ...e ia n where a , a , .., a n are self-adjointelements of A . The rest of the proof is the same as that of Lemma 2.16. (cid:3) Lemma 2.18.
Let A be a quasidiagonal, nuclear, separable, simple, unital Z -stable C ∗ -algebra. Let v be a unitary in U ( A ) and let F ⊆ A be a finite set. Then for every ǫ > ,there exists a unitary u ∈ CU ( A ) such that k vcv ∗ − ucu ∗ k < ǫ for all c ∈ F .Proof. Since A is Z -stable, A ∼ = A ⊗ Z ∼ = A ⊗ Z ⊗ Z . Hence, it is enough to prove thetheorem with A replaced by A ⊗ Z ⊗ Z . By Lemma 2.5 and Lemma 4.1 of [20], there existsa ∗ -isomorphism Φ : A ⊗ Z → A ⊗ Z ⊗ Z which is approximately unitarily equivalent to thenatural inclusion map A ⊗ Z → A ⊗ Z ⊗ Z defined by b b ⊗ Z , where we can choose the unitaries to be in CU ( A ⊗ Z ⊗ Z ) . Hence, we may assume that the elements of F ∪ { v } are all inside A ⊗ Z ⊗ Z . The result then follows from Lemma 2.17. (cid:3) Lemma 2.19.
Let A be an exact, separable, simple, unital, Z -stable C ∗ -algebra with uniquetracial state. Let v be a unitary in U ( A ) . Then there exists a unitary u ∈ CU ( A ) suchthat vav ∗ = uau ∗ for all a ∈ A .Proof. Let τ be the unique tracial state of A . Since v ∈ U ( A ) , v has the form v = e ia e ia ...e ia n where a , a , ..., a n ∈ A are self-adjoint elements. By Lemma 2.12 and Lemma2.14, and since e iτ ( a + a + ... + a n ) A , theunitary u = e − iτ ( a + a + ... + a n ) v satisfies the requirements of the lemma. (cid:3) Theorem 2.20.
Let A be an exact, separable, simple, unital Z -stable C ∗ -algebra. Supposethat either(1) A is nuclear and quasidiagonal, or(2) A has unique tracial state.Then we have the following:(a) CU ( A ) / T is a simple topological group.(b) Every automorphism in Inn ( A ) can be realized using unitaries in CU ( A ) .(c) Inn ( A ) is a simple topological group.Proof. (a) and (b) follow from Lemmas 2.18, 2.19 and Theorem 2.11.(c) follows from (a) and (b), arguing as in the proof of Theorem 3.2(b) of [22] (which isa modification of the argument of Corollary 2.5 in [7]). For the convenience of the reader,we provide the (short) argument: Let G be a nontrivial closed normal subgroup of Inn ( A ).Let H = { u ∈ CU ( A ) : Ad( u ) ∈ G } . Then H is a closed normal subgroup of CU ( A ) such that H contains all scalar unitaries. Since G is nontrivial, let β ∈ G be differentfrom the identity automorphism. Hence, there exists v ∈ CU ( A ) such that v ∗ β ( v ) / ∈ C v )) − β Ad( v ) β − = Ad( v ∗ β ( v )), it follows that v ∗ β ( v ) ∈ H . Therefore, by (a), H = CU ( A ) . Hence, by (b), G = Inn ( A ). (cid:3) Bott Maps and Continuous path of unitaries K -theory with Z n coefficient. For n ∈ N , define θ n : C ( S ) → C ( S ) by θ n ( x ) = x n .By identifying C (0 ,
1) with (cid:8) f ∈ C ( S ) : f (1) = 0 (cid:9) ⊆ C ( S )then θ n | C (0 , is a homomorphism from C (0 ,
1) to C (0 , C n the mappingcone of θ n | C (0 , . Set C = C . For a C ∗ -algebra A , define K ( A ) by K ( A ) = M i =0 ∞ M n =0 K i ( A ; Z n ) HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 19 where K i ( A ; Z n ) = K i ( A ⊗ C n ). For C ∗ -algebras A and B , a homomorphism from K ( A )to K ( B ) is a collection of group homomorphisms ( φ n , φ n ) ∞ n =0 , where φ in : K i ( A ; Z n ) → K i ( B ; Z n ), satisfying the Bockstein operations of [5].3.2. Embeddings. (1) For each n ∈ N , let j n : A ⊗ C ((0 , ) → A ⊗ C n be the natural inclusion.(2) For a C ∗ -algebra A , let j A : A ⊗ C (0 , → A ⊗ C ( S ) be the canonical embeddingthat sends a ⊗ f , to a ⊗ f , using the identification of C (0 ,
1) as a sub- C ∗ -algebraof C ( S ).(3) Let A be a C ∗ -algebra. Define ι A : A → A ⊗ C ( S ) by ι A ( a ) = a ⊗ C ( S ) .(4) Let m, n ∈ N with 0 < m ≤ n ∈ N , let A , . . . , A m , B , . . . , B n be unital C ∗ -algebras, let ι , . . . , ι m ∈ { , . . . , n } be pairwise distinct numbers, and let α j : A j → B ι j be a ∗ -homomorphism for each j . Then these ∗ -homomorphisms induce a ∗ -homomorphism α = α ⊗ α ⊗ · · · ⊗ α m : A ⊗ · · · ⊗ A m → B ι ⊗ · · · ⊗ B ι m . The composition of this map with the canonical unital embedding ι : B ι ⊗ · · · ⊗ B ι m → B ⊗ · · · ⊗ B n will be denoted by α [ ι ...ι m ] . Note that ι may be expressed as id [ ι ...ι m ] .3.3. Partial maps on K ( A ) . Let A be a C ∗ -algebra. Let A † be A if A is unital and theunitization of A if A is not unital. We will denote the set of projections and unitaries in S ∞ m =1 ( A ⊗ M m ) † by P ( A ) and we will denote the set of projections and unitaries in ∞ [ m =1 ∞ [ n =0 ( A ⊗ M m ⊗ C n ) † by P ( A ).Let P be a subset of P ( A ) and ψ : A → B be a contractive, completely positive linearmap, we now define K ( ψ ) | P as in Section 2.3 of [15]. We use K ( ψ ) | P instead of the notation[ ψ ] | P used in [15]. Definition 3.1.
Let A and B be C ∗ -algebras and let ψ : A → B be a linear map. Let ǫ > S ⊆ A . Then ψ is S - ǫ -multiplicative if k ψ ( ab ) − ψ ( a ) ψ ( b ) k < ǫ for all a, b ∈ S .The following lemmas are well-known and can be found in Section 2.5 of [10] Lemma 3.2.
For each ǫ > , there exists δ > such that the following holds: Suppose A is a unital C ∗ -algebra. (1) If x ∈ A with k x ∗ x − A k < δ and k xx ∗ − A k < δ , then there exists a unitary u ∈ A such that k x − u k < ǫ . Moreover, if ǫ < and if u and u are unitaries such that k x − u i k < ǫ , then [ u ] = [ u ] in K ( A ) . (2) If x ∈ A is a self-adjoint element with (cid:13)(cid:13) x − x (cid:13)(cid:13) < δ , then there exists a projection p in A such that k x − p k < ǫ . Moreover, if ǫ < and if p and p are projectionssuch that k x − p i k < ǫ , then [ p ] = [ p ] in K ( A ) . Lemma 3.3.
Let A be a C ∗ -algebra and let C be a nuclear C ∗ -algebra. Let ǫ > and let F be a finite subset of A ⊗ C . Then there exist δ > and a finite subset G of A such that if ψ : A → B is a contractive, linear map that is G - δ -multiplicative, then ψ ⊗ id C : A ⊗ C → B ⊗ C is F - ǫ -multiplicative.Proof. Set m = max {k x k : x ∈ F } . Note that each element of A ⊗ C can be approximatedby a finite linear combinations of elementary tensors, a ⊗ c , where a ∈ A and c ∈ C . Thus,there exists N ∈ N and there exist a , . . . , a N ∈ A and c , . . . c N ∈ C such that for each x ∈ F , there exist a finite subset S x ⊆ { k ∈ N : k ≤ N } and functions k x , n x : S →{ k ∈ N : k ≤ N } such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x − X i ∈S x a k x ( i ) ⊗ c n x ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < min (cid:26) ǫ m + 1) , (cid:27) . Note that (cid:13)(cid:13)P i ∈S x a k x ( i ) ⊗ c n x ( i ) (cid:13)(cid:13) ≤ m .Define M, m ∈ R and G ⊆ A as follows: M = max {|S x | : x ∈ F } m = {k c i k : i ∈ N , ≤ i ≤ N }G = { a i : i ∈ N , ≤ i ≤ N } . Set δ = ǫ M ( m +1) . Suppose ψ : A → B is a contractive, linear map such that ψ is G - δ -multiplicative. If x, y ∈ { a i : i ∈ N , ≤ i ≤ N } and s, t ∈ { c i : i ∈ N , ≤ i ≤ n } , then k ( ψ ⊗ id C )( x ⊗ s )( ψ ⊗ id C )( y ⊗ t ) − ( ψ ⊗ id C )( xy ⊗ st ) k = k ψ ( x ) ψ ( y ) ⊗ st − ψ ( xy ) ⊗ st k≤ k ψ ( x ) ψ ( y ) − ψ ( xy ) k m ≤ δm . Let S , S be subsets of { i ∈ N : i ≤ N } and k i , n i : S i → { i ∈ N : i ≤ N } be functions.Set y = P i ∈S a k ( i ) ⊗ c n ( i ) and y = P i ∈S a k ( i ) ⊗ c n ( i ) k ( ψ ⊗ id C ) ( y ) ( ψ ⊗ id C ) ( y ) − ( ψ ⊗ id C ) ( y y ) k < X i ∈S ,j ∈S δm < ǫ . Let x , x ∈ F . Then there exist subsets S , S of { i ∈ N : i ≤ N } and functions k i , n i : S i → { i ∈ N : i ≤ N } such that k x i − y i k < min (cid:26) ǫ m + 1) , (cid:27) HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 21 where y j = P i ∈S j a k j ( i ) ⊗ c n j ( i ) . Therefore, k ( ψ ⊗ id C )( x )( ψ ⊗ id C )( x ) − ( ψ ⊗ id C )( x x ) k≤ k ( ψ ⊗ id C )( x − y ) k k ( ψ ⊗ id C )( x ) k + k ( ψ ⊗ id C )( y ) k k ( ψ ⊗ id C )( x − y ) k + k ( ψ ⊗ id C )( y )( ψ ⊗ id C )( y ) − ( ψ ⊗ id C )( y y ) k + k ( ψ ⊗ id C )( y ( y − x )) k + k ( ψ ⊗ id C )(( y − x ) x ) k≤ k x − y k k x k + k y k k x − y k + k ( ψ ⊗ id C )( y )( ψ ⊗ id C )( y ) − ( ψ ⊗ id C )( y y ) k + k y k k y − x k + k y − x k k x k < ǫ. (cid:3) Using Lemma 3.3 and arguing as in Remark 4.5.1 and 6.1.1 of [10] we get the followinglemma. See also Section 2.3 of [15].
Lemma 3.4.
Let A be a C ∗ -algebra. Let P be a finite set of projections in S ∞ m =1 ( A ⊗ M m ) † ,let P be a finite set of unitaries in S ∞ m =1 ( A ⊗ M m ) † . Then there exist δ > and a finitesubset F of A such that the following holds: Suppose B is a C ∗ -algebra and ψ : A → B isa contractive, completely positive, linear map such that ψ is F - δ -multiplicative. Then thereexist a finite set G of projections in S ∞ m =1 ( B ⊗ M m ) † and a finite set of unitaries G of S ∞ m =1 ( B ⊗ M m ) † such that (1) for each p ∈ P , there exists e ( p ) ∈ G such that k ψ ( p ) − e ( p ) k < and if p , p ∈ P such that [ p ] = [ p ] in K ( A ) , then [ e ( p )] = [ e ( p )] in K ( B ) and if p , p , p ⊕ p ∈ P , then [ e ( p ⊕ p )] = [ e ( p )] + [ e ( p )] in K ( B ) . (2) for each u ∈ P , there exists v ( u ) ∈ G such that k ψ ( u ) − v ( u ) k < and if u , u ∈ P such that [ u ] = [ u ] in K ( A ) , then [ v ( u )] = [ v ( u )] in K ( B ) and if u , u , u ⊕ u ∈ P , then [ v ( u ⊕ u )] = [ v ( u )] + [ v ( u )] in K ( B ) Definition 3.5.
Let A be a C ∗ -algebra and let P be a finite subset of P ( A ). By Lemma3.4, there exist a finite subset F of A and δ > B isa C ∗ -algebra and ψ : A → B is a contractive, complete positive, linear map such that ψ is F - δ -multiplicative. Then there exists a finite subset Q of P ( A ) such that for each projection p ∈ P ∩ ( A ⊗ M m ⊗ C n ) † and unitary u ∈ P ∩ ( A ⊗ M m ⊗ C n ) † , there exist a projection e ( p ) ∈ Q ∩ ( B ⊗ M m ⊗ C n ) † and a unitary in v ( u ) ∈ Q ∩ ( A ⊗ M m ⊗ C n ) † such that k ( ψ ⊗ id M m ⊗ id C n )( p ) − e ( p ) k <
12 and k ( ψ ⊗ id M m ⊗ id C n )( u ) − v ( u ) k < . Moreover,(1) If p , p are projections in P and [ p ] = [ p ] in K ( A ), then [ e ( p )] = [ e ( p )] in K ( B ).(2) If u , u are unitaries in P and [ u ] = [ u ] in K ( A ), then [ v ( u )] = [ v ( u )] in K ( B ).(3) If p , p , p ⊕ p are projections in P , then [ e ( p ⊕ p )] = [ e ( p )] + [ e ( p )] in K ( B )(4) If u , u , u ⊕ u are unitaries in P , then [ v ( u ⊕ u )] = [ v ( u )] + [ v ( u )] in K ( B ). Let P be the image of P in K ( A ). Define the partial map on K ( A ) induced by ψ , K ( ψ ) | P : P → K ( B ) , by K ( ψ ) | P ( x ) = ( [ e ( p )] , if x = [ p ] for some projection p in P [ v ( u )] , if x = [ u ] for some unitary u in P . By Lemma 3.4 and the above remarks, we have that K ( ψ ) | P is a well-defined map on P .Moreover, as in Section 6.1.1 of [10], by enlarging F and decreasing the size of δ if necessary, K ( ψ ) | P can be extended to a well-defined group homomorphism from the sub-group G ( P )of K ( A ) generated by P .Throughout the paper, we will abuse notation and denote this extension by K ( ψ ) | P . Remark 3.6. (1) Note that if p ∈ P and [ p ] ∈ K i ( A ; Z n ), then K ( ψ ) | P ([ p ]) is an ele-ment of K i ( B ; Z n ). Hence, K ( ψ ) | P is group homomorphism from G ( P ) ∩ K ∗ ( A ; Z n )to K ∗ ( B ; Z n ). We will denote this homomorphism by K ∗ ( ψ ; Z n ) | P . When n = 0,we simply write K ∗ ( ψ ) | P .(2) Throughout the paper, for P ⊆ P ( A ), when we write K ( ψ ) | P we mean that ψ : A → B is a contractive, complete positive, linear map such that ψ is F - δ -multiplicativefor some δ > F of P ( A ), so that we have a well-definedhomomorphism from G ( P ) to K ( B ). Lemma 3.7.
Let A be a nuclear C ∗ -algebra. Let P be a finite subset of P ( A ) . Then thereexist a finite subset F of A and δ > such that if B is a C ∗ -algebra and ψ , ψ : A → B are contractive, completely positive, linear maps such that ψ i is F - δ -multiplicative and k ψ ( a ) − ψ ( a ) k < δ for all a ∈ F , then K ( ψ i ) | P is well-defined and K ( ψ ) | P = K ( ψ ) | P . Proof.
Choose a finite subset F of A and δ > ψ : A → B that is also F - δ -multiplicative, then K ( ψ ) | P is well-defined. Note that we can choose δ small enough with δ < δ and a finite subset F of A large enough with F ⊆ F such that if ψ : A → B is a contractive, completely positive,linear map that is F - δ -multiplicative, then for each p ∈ P , then k ( ψ ⊗ id M m ⊗ id C n )( p ) − x ( p ) k < x ( p ) is element of P ( B ) given in Lemma 3.4.Let ψ , ψ : A → B be contractive, completely positive, linear maps such that ψ i is F - δ -multiplicative. Let p ∈ P . Let x i ( p ) be the element in P ( B ) given by Lemma 3.4. Bythe definition of K ( ψ i ) | P , K ( ψ i ) | P ([ p ]) = [ x i ( p )]. Since k ( ψ i ⊗ id M m ⊗ id C n )( p ) − x i ( p ) k < k x ( p ) − x ( p ) k < . Therefore, [ x ( p )] = [ x ( p )] in K ( B ). Hence, K ( ψ ) | P ([ p ]) = K ( ψ ) | P ([ p ]). (cid:3) HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 23 Bott Maps.
We now define the Bott maps as in [16]. Let ǫ > F be a finitesubset of A ⊗ C ( S ). Then there exist δ > G of A such that thefollowing holds: if h : A → B is a ∗ -homomorphism and u is a unitary in B such that k h ( a ) u − uh ( a ) k < δ for all a ∈ G , then the contractive, completely positive, linear map ϕ h,u : A ⊗ C ( S ) → B defined by ϕ h,u ( a ⊗ f ) = h ( a ) f ( u ) is F - ǫ -multiplicative.Note that for every finite subset F of C and ǫ >
0, there exist δ > G of A such that the following holds: if ψ : C → A ⊗ C ( S ) and h : A → B are homomorphismsand u is unitary in B such that k h ( a ) u − uh ( a ) k < δ for all a ∈ G , then the contractive, completely positive, linear map ϕ h,u ◦ ψ : C → B is F - ǫ -multiplicative. Definition 3.8.
Note that by the K¨unneth Formula [29], the embedding j A : A ⊗ C (0 , → A ⊗ C ( S ) induces injective group homomorphisms K ( j A ) : K ( A ⊗ C (0 , → K ( A ⊗ C ( S )) K ( j A ) : K ( A ⊗ C (0 , → K ( A ⊗ C ( S )) . Using Bott periodicity to identify K ( A ⊗ C (0 , K ( A ) and K ( A ⊗ C (0 , K ( A ), we obtain injective group homomorphisms β (0) A : K ( A ) → K ( A ⊗ C ( S )) β (1) A : K ( A ) → K ( A ⊗ C ( S )) . Using Bott periodicity again, we obtain injective group homomorphisms β (0) A ,k : K ( A ; Z k ) → K ( A ⊗ C ( S ); Z k ) β (1) A ,k : K ( A ; Z k ) → K ( A ⊗ C ( S ); Z k ) . Let P be a finite subset of P ( A ). By the above remarks and Section 3.3, there exist δ > F of A such that if h : A → B is a ∗ -homomorphism and u is a unitaryin B with k h ( a ) u − uh ( a ) k < δ for all a ∈ F , then K ( ϕ h,u ◦ j A ) | P is well-defined, which we will denote asBott( h, u ) | P . In particular, bott ( h, v ) | P = K ( ϕ h,u ) ◦ β (1) A | P and bott ( h, v ) | P = K ( ϕ h,u ) ◦ β (0) A | P . If uh ( a ) = h ( a ) u for all a ∈ A , then Bott( h, u ) is well-defined on K ( A ). Lemma 3.9.
Let A be a unital C ∗ -algebra and let P be a finite subset of P ( A ) . Then thereexist δ > and a finite subset F of A such that the following holds: if h : A → B is a ∗ -homomorphism and u , . . . , u n are unitaries in B with k h ( a ) u i − u i h ( a ) k < δn for all a ∈ F and for all i , then Bott( h, u i ) | P and Bott( h, u · · · u n ) | P are well-defined and Bott( h, u · · · u n ) | P = n X i =1 Bott( h, u i ) | P . Moreover, if
Bott( h, u ) | P = Bott( h, u ) | P , then Bott( h, u u ∗ ) | P = 0 .Proof. Let δ > F be a finite subset of A such that if h : A → B is a ∗ -homomorphismand u is a unitary in B such that k h ( a ) u − uh ( a ) k < δ for all a ∈ F , then Bott( h, u ) | P is well-defined.Suppose h is a ∗ -homomorphism and u , . . . , u n are unitaries in B such that k h ( a ) u i − u i h ( a ) k < δn for all a ∈ F and for all i . Then k h ( a ) u · · · u n − u · · · u n h ( a ) k≤ k h ( a ) u · · · u n − u h ( a ) u . . . u n k + n − X i =2 k u u · · · u i − h ( a ) u i u i +1 · · · u n − u · · · u i h ( a ) u i +1 · · · u n k + k u · · · u n − h ( a ) u n − u · · · u n h ( a ) k = n X i =1 k h ( a ) u i − u i h ( a ) k < δ for all a ∈ F . Hence, by the choice of δ and F , Bott( h, u i ) | P and Bott( h, u · · · u n ) | P arewell-defined. Moreover, enlarging F and decreasing the size of δ if necessary, we get thatBott( h, u · · · u n ) | P = n X i =1 Bott( h, u i ) | P . (cid:3) Some technical results.Lemma 3.10.
Let A be a unital C ∗ -algebra. Let ǫ > , F be a finite subset of A , and F be a finite subset of A ⊗ C ( S ) . Then there exist δ > , a finite subset G of A , and a finitesubset G of A ⊗ C ( S ) such that the following holds: if ψ : A ⊗ C ( S ) → A is a contractive,completely positive, linear map and u is a unitary in A such that (1) ψ is G - δ -multiplicative; (2) (cid:13)(cid:13) ψ ( a ⊗ C ( S ) ) − a (cid:13)(cid:13) < δ for all a ∈ G ; and (3) k ψ (1 A ⊗ z ) − u k < δ where z is the function on the circle that sends ξ to ξ , then (i) k ua − au k < ǫ for all a ∈ F and (ii) k ψ ( x ) − ϕ id A ,u ( x ) k < ǫ for all x ∈ F . HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 25 Proof.
Set m = max {k a k : a ∈ F } and set G = F ∪{ A } . Note that (cid:8) a ⊗ C ( S ) : a ∈ A (cid:9) and (cid:8) A ⊗ z k : k ∈ Z (cid:9) generate A ⊗ C ( S ). Therefore, there exist N ∈ N and a finite subsetof A , { a i ∈ A : i ∈ Z , | i | ≤ N } , such that for each x ∈ F , there exist a finite subset S x of Z and functions k x , m x : S x → { k ∈ Z : | k | ≤ N } such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x − X i ∈S x a k x ( i ) ⊗ z m x ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ǫ . Define M and m as follows: M = max {| S x | : x ∈ F } and m = max {k a i k : i ∈ Z , | i | ≤ N } . Set δ = min n ǫ m +4 , ǫ M ((2 N − m +2) o . Let G = (cid:0)(cid:8) a ⊗ C ( S ) : a ∈ F (cid:9) ∪ (cid:8) a i ⊗ C ( S ) : i ∈ Z , | i | ≤ N (cid:9)(cid:1) ∪ (cid:16)(cid:8) a i ⊗ z j : i, j ∈ Z , | i | , | j | ≤ N (cid:9) ∪ n A ⊗ z k : k ∈ Z , | k | ≤ N o(cid:17) . Note that for each a ∈ F (3.1) k ua − au k ≤ k ( u − ψ (1 A ⊗ z )) a k + (cid:13)(cid:13) ψ (1 A ⊗ z )( a − ψ ( a ⊗ C ( S ) )) (cid:13)(cid:13) + (cid:13)(cid:13) ψ (1 A ⊗ z ) ψ ( a ⊗ C ( S ) ) − ψ ( a ⊗ z ) (cid:13)(cid:13) + (cid:13)(cid:13) ψ ( a ⊗ z ) − ψ ( a ⊗ C ( S ) ) ψ (1 A ⊗ z ) (cid:13)(cid:13) + (cid:13)(cid:13) ψ ( a ⊗ C ( S ) )( ψ (1 A ⊗ z ) − u ) (cid:13)(cid:13) + (cid:13)(cid:13) ( ψ ( a ⊗ C ( S ) ) − a ) u (cid:13)(cid:13) < (2 m + 4) δ< ǫ. Since for each k ∈ Z \ { } with | k | ≤ N , (cid:13)(cid:13)(cid:13) ψ (1 A ⊗ z k ) − u k (cid:13)(cid:13)(cid:13) ≤ (2 N − δ (3.2)and (cid:13)(cid:13) ψ (1 A ⊗ z ) − u (cid:13)(cid:13) = (cid:13)(cid:13) ψ (1 A ⊗ C ( S ) ) − A (cid:13)(cid:13) < δ (3.3)we have that for each i, k ∈ Z with | i | ≤ N and | k | ≤ N ,(3.4) (cid:13)(cid:13)(cid:13) ψ ( a i ⊗ z k ) − ϕ id A ,u ( a i ⊗ z k ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ψ ( a i ⊗ z k ) − a i u k (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ψ ( a i ⊗ z k ) − ψ ( a i ⊗ C ( S ) ) ψ (1 A ⊗ z k ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ψ ( a i ⊗ C ( S ) )( ψ (1 A ⊗ z k ) − u k ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ( ψ ( a i ⊗ C ( S ) ) − a i ) u k (cid:13)(cid:13)(cid:13) < ((2 N − m + 2) δ. Therefore, for each x ∈ F , k ψ ( x ) − ϕ id A ,u ( x ) k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ψ ( x ) − ψ X i ∈S x a k x ( i ) ⊗ z m x ( i ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ψ X i ∈S x a k x ( i ) ⊗ z m x ( i ) ! − ϕ id A ,u X i ∈S x a k x ( i ) ⊗ z m x ( i ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ϕ id A ,u X i ∈S x a m x ( i ) ⊗ z k x ( i ) ! − ϕ id A ,u ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ǫ M ((2 N − m + 2) δ< ǫ. (cid:3) The following lemma is the result of the proof of Lemma 4 of [21]. In [21], we did notstate the lemma as below since we were not concerned with the bott maps.
Lemma 3.11.
Let A and B be unital C ∗ -algebras satisfying the UCT. Suppose ϕ : K ( A ) → K ( B ) is a homomorphism such that ϕ | K ( A ) is positive. Suppose γ : K ( A ) → K ( B ) is agroup homomorphism. Then there exists α : K ( A ⊗ C ( S )) → K ( B ) such that (1) α | K ( A ⊗ C ( S )) is positive; (2) α ◦ K ( ι A ) = ϕ ; (3) α ◦ β (1) A = 0 ; and (4) α ◦ β (0) A = γ . Lemma 3.12.
Let A be an infinite dimensional, simple, unital C ∗ -algebra satisfying theUCT. Suppose A is a tracially AI algebra. Then for each ǫ > , for each finite subset P of P ( A ) , and for each finite subset F of A ⊗ C ( S ) , there exists a finitely generatedsubgroup G of K ( A ) containing [1 A ] such that the following holds: for every unital, simple,tracially AI algebra B , if γ : G → K ( B ) is a homomorphism and ϕ : A → B is aunital ∗ -homomorphism, then there exists a contractive, completely positive, linear map ψ : A ⊗ C ( S ) → B such that (1) ψ is F - ǫ -multiplicative; (2) K ( ψ ◦ ι A ) | P = K ( ϕ ) | P ; (3) K ( ψ ) ◦ β (1) A | P = 0 ; and (4) K ( ψ ) ◦ β (0) A | P = γ | P .Proof. Since A is an infinite dimensional tracially AI algebra satisfying the UCT, we mayassume that A = lim −→ ( A n , ϕ n,n +1 )where A n = L k ( n ) i =1 P [ n,i ] M [ n,i ] ( C ( X [ n,i ] )) P [ n,i ] , each X [ n,i ] is a connected finite CW-complex,and ϕ n,n +1 is a unital, ∗ -monomorphism.Choose n ∈ N large enough and choose finite subsets, F n of A n and P n of P ( A n ),such that every element of F is within ǫ to an element of ( ϕ n ⊗ id C ( S ) )( F n ) and forevery p ∈ P , there exists e p ∈ P n such that [ p ] = K ( ϕ n , ∞ )([ e p ]). HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 27 Set G = K ( ϕ n , ∞ )( K ( A n )). Let B be a simple, unital, tracially AI algebra. Suppose γ : G → K ( B ) is a homomorphism and suppose ϕ : A → B is a unital ∗ -monomorphism. Then,by Lemma 3.12, there exists α ∈ Hom Λ ( K ( A n ⊗ C ( S )) , K ( B )) such that α | K ( A n ⊗ C ( S )) is a positive homomorphism, α ◦ K ( ι A n ) = K ( ϕ ◦ ϕ n , ∞ ), α ◦ β (1) A n | K ( A n ) = 0 and α ◦ β (0) A n | K ( A n ) = γ ◦ K ( ϕ n ).By Theorem 9.12 of [12] and Theorem 5.4 of [16], there exists a nuclear, separable,simple, unital, tracially AF algebra B ′ and an embedding ϕ : B ′ → B such that K ( ϕ ) isan isomorphism. Composing the maps obtained from Proposition 9.10 of [12] and Theorem6.2.9 of [10] with ϕ , we get a sequence of unital, contractive, completely positive, linearmaps { L n ,k : A n ⊗ C ( S ) → B } ∞ k =1 such thatlim k →∞ k L n ,k ( xy ) − L n ,k ( x ) L n ,k ( y ) k = 0for all x, y ∈ A n ⊗ C ( S ), K ( L n ,k ◦ ι A n ) | P n = α ◦ K ( ι A n ) | P n ,K ( L n ,k ) = α | K ( A n ⊗ C ( S )) , and K ( L n ,k ) = α | K ( A n ⊗ C ( S )) .Since A n ⊗ C ( S ) is nuclear, there exists a sequence of contractive, completely positive,linear maps, { ψ n ,k : A ⊗ C ( S ) → A n ⊗ C ( S ) } ∞ k =1 such thatlim k →∞ (cid:13)(cid:13) ( ψ n ,k ◦ ( ϕ n , ∞ ⊗ id C ( S ) ))( x ) − x (cid:13)(cid:13) = 0for all x ∈ A n ⊗ C ( S ). Set β n ,k = L n ,k ◦ ψ n ,k for large enough k , β n ,k satisfies thedesired property. Hence, set ψ = β n ,k . (cid:3) Definition 3.13.
Let A be a unital C ∗ -algebra such that T ( A ) = ∅ . Denote the state spaceof K ( A ) by S ( K ( A )). The canonical map from T ( A ) to S ( K ( A )) which sends τ to thefunction { [ p ] τ ( p ) } will be denoted r A ( τ )([ p ]) = τ ( p ).Let A and B be unital C ∗ -algebras such that T ( A ) and T ( B ) are nonempty sets. Let κ ∈ Hom Λ ( K ( A ) , K ( B )) such that κ ([1 A ]) = [1 B ] in K ( B ). Then an affine map∆ : T ( B ) → T ( A )is said to be compatible to κ if the diagram T ( B ) r B / / ∆ (cid:15) (cid:15) S ( K ( B )) κ S (cid:15) (cid:15) T ( A ) r A / / S ( K ( A ))is commutative, where κ is the homomorphism from K ( A ) to K ( B ) induced by κ and κ S ( f ) = f ◦ κ . Lemma 3.14.
Let C = M n ( C ( X )) ⊗ C ( S ) where X is either [0 , or a space with onepoint and let A be a simple, tracially AI algebra. Suppose κ ∈ Hom Λ ( K ( C ) , K ( A )) such that κ ( K ( C ) + \ { } ) ⊆ K ( A ) + \ { } and κ ([1 C ]) = [1 A ] and γ : T ( A ) → T ( C ) is a continuousaffine map that is compatible to κ .Let ǫ > and H be a finite subset of C s.a. . Then there exists a unital ∗ -homomorphism h : C → A such that (1) K ( h ) = κ and (2) sup {| τ ◦ h ( x ) − γ ( τ )( x ) | : τ ∈ T ( A ) } < ǫ for all x ∈ H .Proof. By Theorem 9.12 of [12] and Theorem 5.4 of [16], there exists a unital, separable,nuclear, simple, tracially AF algebra B and an embedding ϕ : B → A such that K ( ϕ ) is anisomorphism. Hence, there exists κ ∈ Hom Λ ( K ( C ) , K ( B )) such that κ ( K ( C ) + \ { } ) ⊆ K ( A ) + \ { } , κ ([1 C ]) = [1 B ], and κ = K ( ϕ ) ◦ κ .Using Lemma 6.2 of [16] and applying ϕ , there exist a projection p in A , a finite dimen-sional sub- C ∗ -algebra F of A with 1 F = 1 A − p and unital ∗ -homomorphisms h : C → p A p and h : C → F such that K ( h + h ) = κ and τ ( p ) < ǫ τ ∈ T ( A ).Since K ( h ) = 0, by Lemma 9.5 of [12], there is a sub- C ∗ -algebra B of (1 A − p ) A (1 A − p ) where B is a finite direct sum of C ∗ -algebras of the form M k and M n ( C ([0 , ∗ -homomorphism h : C → B such that K ( h ) = K ( h ) and | τ ◦ h ( f ) − τ (1 A − p ) γ ( τ ( f )) | < ǫ f ∈ H and for all τ ∈ T ( A ).Set h = h + h . Then h : C → A is a unital ∗ -homomorphism such that K ( h ) = K ( h + h ) = κ and | τ ◦ h ( f ) − γ ( τ )( f ) | < | τ ◦ h ( f ) − τ (1 A − p ) γ ( τ )( f ) | + ǫ < ǫ | τ ◦ h ( f ) − τ (1 A − p ) γ ( τ )( f ) | + ǫ < ǫ for all τ ∈ T ( A ) and for all f ∈ H . (cid:3) Lemma 3.15.
Let A be as in Lemma 3.12. For each ǫ > and for each finite subsets P , F ,and F of P ( A ) , A s.a. , and A ⊗ C ( S ) respectively, there exists a finitely generated subgroup G of K ( A ) containing [1 A ] such that the following holds: for every homomorphism γ : G → K ( A ) there exists a unital, contractive, completely positive, linear map ψ : A ⊗ C ( S ) → A such that (1) ψ is F - ǫ -multiplicative; (2) K ( ψ ) ◦ K ( ι A ) | P = K (id A ) | P ; (3) sup {| τ ◦ ψ ◦ ι A ( a ) − τ ( a ) | : τ ∈ T ( A ) } < ǫ for all a ∈ F ; (4) K ( ψ ) ◦ β (1) A | P = 0 ; and (5) K ( ψ ) ◦ β (0) A | P = γ | P .Proof. Let {H n } ∞ n =1 be an increasing sequence of finite subsets of A whose union is densein A . Now, for each n ∈ N , there exist a projection p n ∈ A , a sub- C ∗ -algebra D n = L k ( n ) i =1 M m ( i,n ) ( C ( X [ i,n ] )) of A , where X [ i,n ] is either [0 ,
1] or a space with one point with1 D n = p n , and a sequence of contractive, completely positive, linear maps { L n : A → D n } ∞ n =1 such that HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 29 (a) k p n x − xp n k < n for all x ∈ H n ;(b) k p n xp n − L n ( x ) k < n for all x ∈ H n ;(c) k x − (1 A − p n ) x (1 A − p n ) − L n ( x ) k < n for all x ∈ H n with k x k ≤
1; and(d) τ (1 A − p n ) < n for all τ ∈ T ( A ).Note that lim n →∞ k L n ( xy ) − L n ( x ) L n ( y ) k = 0for all x, y ∈ A .Denote the i th summand of D n by D [ n,i ] and let d [ n,i ] = 1 D [ n,i ] . Choose n large enoughsuch that n < ǫ . Let P be a finite subset of P ( A ) such that P contains P , d [ n,i ] , p n .Choose a finite subset F of A ⊗ C ( S ) such that F contains ι A ( F ) ∪ F and the set (cid:8)(cid:0) (1 A − p n ) ⊗ C ( S ) (cid:1) x (cid:0) (1 A − p n ) ⊗ C ( S ) (cid:1) : x ∈ F (cid:9) . Let G be the finitely generated subgroup of K ( A ) in Lemma 3.12 which corresponds to n , P , and F .Suppose γ : G → K ( A ) is a homomorphism. Then, by Lemma 3.12, there exists acontractive, completely positive, linear map L : A ⊗ C ( S ) → A such that(a) L is F - n -multiplicative;(b) K ( L ◦ ι A ) | P = K (id A ) | P ;(c) K ( L ) ◦ β (1) A | P = 0; and(d) K ( L ) ◦ β (0) A | P = γ | P Choose a projection q n ∈ A such that [ q n ] = K ( L ) (cid:16)P k ( n ) i =1 [ d [ n,i ] ⊗ C ( S ) ] (cid:17) . Let G n bea finite subset of D n such that G n contains the generators of D n . Define η : T ( A ) → T ( D n ⊗ C ( S )) by η ( τ ) = 1 τ ( p n ) τ ◦ (id A ⊗ ev) | D n ⊗ C ( S ) . Since K ( L ) = K (id A ⊗ ev), we have that K (cid:0) L | D n ⊗ C ( S ) (cid:1) and γ are compatible. Also,note that K ( D n ⊗ C ( S )) = K ( ι D n )( K ( D n )). Therefore K (cid:0) L | D n ⊗ C ( S ) (cid:1) sends K ( D n ⊗ C ( S )) + \ { } to K ( q n A q n ) + \ { } . Hence, by Lemma 3.14, there exists a unital ∗ -homomorphism h : D n ⊗ C ( S ) → q n A q n such that K ( h ) = K (cid:0) L | D n ⊗ C ( S ) (cid:1) andsup {| ( τ ◦ h )( g ) − η ( τ )( g ) | : τ ∈ T ( A ) } < n for all h ∈ G n .Define ψ : A ⊗ C ( S ) → A by ψ ( x ) = L (cid:0) [(1 A − p n ) ⊗ C ( S ) ] x [(1 A − p n ) ⊗ C ( S ) ] (cid:1) + (cid:0) h ◦ ( L n ⊗ id C ( S ) ) (cid:1) ( x ) . By construction, ψ satisfies the desired properties of the lemma. (cid:3) Theorem 3.16.
Let A be as in Lemma 3.12. For each ǫ > and for each finite subsets P , F , and F of P ( A ) , A , and A ⊗ C ( S ) respectively, there exists a finitely generated subgroup G of K ( A ) containing [1 A ] such that the following holds: for every homomorphism γ : G → K ( A ) there exists a unital, contractive, completely positive, linear map ψ : A ⊗ C ( S ) → A such that (1) ψ is F - ǫ -multiplicative; (2) K ( ψ ) ◦ K ( ι A ) | P = K (id A ) | P ; (3) k ( ψ ◦ ι A )( a ) − a k < ǫ for all a ∈ F ; (4) K ( ψ ) ◦ β (1) A | P = 0 ; and (5) K ( ψ ) ◦ β (0) A | P = γ | P .Proof. Arguing as in the proof of Theorem 4: The tracially AI case pp. 439 of [21] and usingLemma 3.15 instead of Lemma 7 of [21], we get the desired result. (cid:3)
Corollary 3.17.
Let A be as in Lemma 3.12. Let ǫ > and let F and P be finite subsetsof A and P ( A ) respectively. Then there exists a finitely generated subgroup G of K ( A ) containing [1 A ] such that the following holds: if γ : G → K ( A ) , then there exists a unitary w ∈ A such that (1) k aw − wa k < ǫ for all a ∈ F ; (2) bott (id A , w ) | P = 0 and bott (id A , w ) | P = γ | P ; and (3) γ ([1 A ]) = [ w ] .Proof. Let ǫ > P be a finite subset of P ( A ), and F be a finite subset of A . Set Q = j A ( P ).By Lemma 3.7, there exist δ > H of A ⊗ C ( S ) corresponding to Q . By Lemma 3.10, there exist δ >
0, a finite subset G of A , and finite subset G of A ⊗ C ( S ) corresponding to min { δ , ǫ } , H , and F . By Lemma 3.2, there exists δ corresponding min { δ , δ } . By Theorem 3.16, there exists a finitely generated subgroup G of K ( A ) containing [1 A ] corresponding to P , G ∪ F , G ∪ H , and δ = min { ǫ, δ , δ , δ } .Suppose γ : G → K ( A ) is a group homomorphism. Then there exists a unital, contrac-tive, completely positive, linear map ψ : A ⊗ C ( S ) → A such that(a) ψ is ( G ∪ H )- δ -multiplicative;(b) (cid:13)(cid:13) ψ ( a ⊗ C ( S ) ) − a (cid:13)(cid:13) < δ for all a ∈ G ∪ F ;(c) K ( ψ ) ◦ β (1) A | P = 0; and(d) K ( ψ ) ◦ β (0) A | P = γ | P .Since k L (1 A ⊗ z ) ∗ L (1 A ⊗ z ) − A k < δ < δ and k L (1 A ⊗ z ) L (1 A ⊗ z ) ∗ − A k < δ < δ ,there exists a unitary w ∈ A such that k ψ (1 A ⊗ z ) − w k < δ . Therefore, ψ is ( G ∪ H )- δ -multiplicative with (cid:13)(cid:13) ψ ( a ⊗ C ( S ) ) − a (cid:13)(cid:13) < δ and k ψ (1 A ⊗ z ) − w k < δ for all a ∈ G ∪ F . Hence, k ψ ( x ) − ϕ id A ,w ( x ) k < min { ǫ, δ } for all x ∈ H and k wa − aw k < min { ǫ, δ } for all a ∈ F . Thus, K ( ψ ) | Q = K ( ϕ id A ,w ) | Q . Sobott (id A , w ) | P = K ( ϕ id A ,w ) ◦ β (1) A | P = K ( ψ ) ◦ β (1) A | P = 0and bott (id A , w ) | P = K ( ϕ id A ,w ) ◦ β (0) A | P = K ( ψ ) ◦ β (0) A | P = γ | P . Moreover, γ ([1 A ]) = K ( ψ ) ◦ β (0) A ([1 A ]) = K ( ψ )([1 A ⊗ z ]) = [ w ] . (cid:3) HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 31 Definition 3.18.
Let A denote the class of all nuclear, separable, simple, unital C ∗ -algebras A such that A ⊗ M p is a tracially AI algebra that satisfies the UCT for all supernaturalnumbers p of infinite type. By Theorem 2.11 of [18], A ⊗ M p is a tracially AI algebra thatsatisfies the UCT for all supernatural numbers p of infinite type if and only if A ⊗ M p is atracially AI algebra that satisfies the UCT for some supernatural number p of infinite type.Let A Z denote the class of C ∗ -algebras A in A such that A is Z -stable. Note that if A ∈ A Z , then A ⊗ M p ∈ A Z for all supernatural numbers p . Notation 3.19.
Let Q be the UHF algebra such that K ( Q ) = Q and [1 Q ] = 1. Definition 3.20.
Let A be a unital C ∗ -algebra and let G be a subgroup of K ( A ). Then H [1 A ] ( G , K ( A )) denotes the subgroup of all x ∈ K ( A ) such that there exists a homomor-phism α from G to K ( A ) with α ([1 A ]) = x . Lemma 3.21.
Let A be nuclear, separable C ∗ -algebra satisfying the UCT such that K ∗ ( A ) is torsion free. Let P be a finite subset of P ( A ) . Then there exist a finite subset Q of P ( A ) , δ > and a finite subset of F such that if B is a nuclear, simple C ∗ -algebra satisfyingthe UCT and ψ : A → B is a contractive, completely positive, linear map with ψ a F - δ -multiplicative map, then K ∗ ( ψ ) | Q = 0 implies that K ( ψ ) | P = 0 .Proof. Let α : K ∗ ( A ) → K ( A ) be the natural homomorphism given in [5]. Since K ∗ ( A ) istorsion free, by [5], there exists a finite subset P of P ( A ) such that for each p ∈ P , thereexists q ∈ P such that α ([ q ]) = [ p ] in K ( A ).Set Q = P ∪P . Choose δ > F of A such that if B is a nuclear, simple C ∗ -algebra satisfying the UCT and ψ : A → B is a contractive, completely positive, linearmap with ψ being a F - δ -multiplicative map, then K ( ψ ) | P and K ( ψ ) | Q are well-defined.Suppose B is a nuclear, simple C ∗ -algebra satisfying the UCT and ψ : A → B is acontractive, completely positive, linear map with ψ being a F - δ -multiplicative map. Define ι : A → A ⊗ O ∞ by ι ( a ) = a ⊗ O ∞ and ι : B → B ⊗ O ∞ by ι ( b ) = b ⊗ O ∞ . By theK¨unneth Formula [29], K ( ι i ) is an isomorphism. Note that ι ◦ ψ = ( ψ ⊗ id O ∞ ) ◦ ι is acontractive, completely positive, linear map that is F - δ -multiplicative.By Theorem 6.7 of [11], there exists a homomorphism ϕ : A ⊗ O ∞ → B ⊗ O ∞ such that K ( ϕ ) ◦ K ( ι ) | Q = K (( ψ ⊗ id O ∞ ) ◦ ι ) | Q = K ( ι ) ◦ K ( ψ ) | Q . Let β : K ∗ ( A ⊗ O ∞ ) → K ( A ⊗ O ∞ ) and β : K ∗ ( B ⊗ O ∞ ) → K ( B ⊗ O ∞ ) be the naturalhomomorphisms given in [5]. Suppose K ∗ ( ψ ) | Q = 0. Then K ∗ ( ϕ ) ◦ K ∗ ( ι ) | Q = 0. Let p ∈ P . Then there exists q ∈ P such that α ([ q ]) = [ p ] in K ( A ). Then K ( ι )( K ( ψ ) | P ([ p ])) = K ( ϕ ◦ ι )([ p ])= K ( ϕ ◦ ι )( α ([ q ]))= β ( K ∗ ( ϕ ◦ ι )([ q ]))= 0 . Since K ( ι ) is an isomorphism, K ( ψ ) | P ([ p ]) = 0. (cid:3) Theorem 3.22.
Let A be in A Z . Let ǫ > and F be a finite subset of A . Then thereexists a finitely generated subgroup G of K ( A ) such that the following holds: if [ u ] ∈ H [1 A ] ( G , K ( A )) , then there exists a continuous path of unitaries w ( t ) in A ⊗ M p ⊗ M q such that (1) w (0) ∈ A ⊗ M p ⊗ M q and w (1) ∈ A ⊗ M p ⊗ M q ; (2) [ w (0)] = [ u ⊗ M p ⊗ M q ] in K ( A ⊗ M p ⊗ M q ) and [ w (1)] = [ u ⊗ M p ⊗ M q ] in K ( A ⊗ M p ⊗ M q ) ; and (3) (cid:13)(cid:13) w ( t )( a ⊗ M p ⊗ M q ) − ( a ⊗ M p ⊗ M q ) w ( t ) (cid:13)(cid:13) < ǫ for all a ∈ F and t ∈ [0 , ,where p and q are supernatural numbers of infinite type with M p ⊗ M q isomorphic Q .Proof. Let {F ,n } ∞ n =1 be an increasing sequence of finite subsets of A such that S ∞ n =1 F ,n isdense in A , {F ,n } ∞ n =1 be an increasing sequence of finite subsets of M p such that S ∞ n =1 F ,n isdense in M p , {F ,n } ∞ n =1 be an increasing sequence of finite subsets of M q such that S ∞ n =1 F ,n is dense of M q . Let {P ,n } ∞ n =1 be an increasing sequence of finite subsets of P ( A ), {P ,n } ∞ n =1 be an increasing sequence of finite subsets of P ( M p ), and {P ,n } ∞ n =1 be an increasing sequenceof finite subsets of P ( M q ) such that ∞ [ n =1 P ,n = P ( A ) , ∞ [ n =1 P ,n = P ( M p ) , and ∞ [ n =1 P ,n = P ( M q ) . For each n , let G ,n be the finitely generated subgroup of K ( A ⊗ M p ) provided by Corollary3.17 that corresponds to F ,n ⊗F ,n , P ,n ⊗P ,n , and n and let G ,n be the finitely generatedsubgroup of K ( A ⊗ M q ) provided by Corollary 3.17 that corresponds to F ,n ⊗ F ,n , P ,n ⊗P ,n , and n . Note that we may assume that G ,n ⊆ G ,n +1 , G ,n ⊆ G ,n +1 , S ∞ n =1 G ,n = K ( A ⊗ M p ), and S ∞ n =1 G ,n = K ( A ⊗ M q ).Let {G n } ∞ n =1 be an increasing sequence of finitely generated subgroups of K ( A ) contain-ing [1 A ] such that if P ni =1 x i ⊗ y i is a generator of G ,n , then x i ∈ G n and if P ni =1 x i ⊗ y i is a generator of G ,n , then x i ∈ G n . (Note that we are identifying K ( A ⊗ M p ) with K ( A ) ⊗ K ( M q ) and K ( A ⊗ M q ) with K ( A ) ⊗ K ( M q ) . )Since K ( A ⊗ M p ⊗ M q ) and K ( A ⊗ M p ⊗ M q ) are torsion free groups, by Theorem 8.4of [14] and Lemma 3.21, there exist δ >
0, a finite subset G of A ⊗ M p ⊗ M q , and a finitesubset H of P ( A ⊗ M p ⊗ M q ) such that if v is a unitary in A ⊗ M p ⊗ M q with k xv − vx k < δ for all x ∈ G and bott (id A ⊗ M p ⊗ M q , v ) | H = 0 and bott (id A ⊗ M p ⊗ M q , v ) | H = 0then there exists a continuous path of unitaries v ( t ) in A ⊗ M p ⊗ M q such that v (0) =1 A ⊗ M p ⊗ M q , v (1) = v , and k xv ( t ) − v ( t ) x k < ǫ x ∈ F ⊗ M p ⊗ M q and t ∈ [0 , n large enough such that n − < ǫ and if v is a unitary in A ⊗ M p ⊗ M q with k vx − xv k < n − for all x ∈ F ,n ⊗ F ,n ⊗ F ,n , then k vx − xv k < δ x ∈ G and ifbott (id A ⊗ M p ⊗ M q , v ) | P ,n ⊗P ,n ⊗P ,n = 0 and bott (id A ⊗ M p ⊗ M q , v ) | P ,n ⊗P ,n ⊗P ,n = 0 HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 33 then bott (id A ⊗ M p ⊗ M q , v ) | H = 0 and bott (id A ⊗ M p ⊗ M q , v ) | H = 0 . Note that the last part of the statement can be done since we are identifying K i ( A ⊗ M p ⊗ M q )with K i ( A ) ⊗ K ( M p ) ⊗ K ( M q ).Let [ u ] ∈ H [1 A ] ( G n , K ( A )). Then there exists γ : G n → K ( A ) such that γ ([1 A ]) = [ u ].By Corollary 3.17, there exists a unitary v ∈ A ⊗ M p such that k v ( x ⊗ x ) − ( x ⊗ x ) v k < n for all x ∈ F ,n and x ∈ F ,n ,bott (id A ⊗ M p , v ) | P ,n ⊗P ,n = 0 , and bott (id A ⊗ M p , v ) | P ,n ⊗P ,n = γ ⊗ K (id M p ) | P ,n ⊗P ,n . By Corollary 3.17, there exists a unitary v ∈ A ⊗ M q such that k v ( x ⊗ x ) − ( x ⊗ x ) v k < n for all x ∈ F ,n and x ∈ F ,n ,bott (id A ⊗ M q , v ) | P ,n ⊗P ,n = 0 , and bott (id A ⊗ M q , v ) | P ,n ⊗P ,n = γ ⊗ K (id M q ) | P ,n ⊗P ,n . Set w = v ⊗ M q ∈ A ⊗ M p ⊗ M q and set w = id [1 , , ( v ⊗ M p ) ∈ A ⊗ M p ⊗ M q . Then k xw i − w i x k < n for all x ∈ F ,n ⊗ F ,n ⊗ F ,n , for each i = 0 , (id A ⊗ M p ⊗ M q , w i ) | P ,n , P ,n ⊗P ,n = 0and bott (id A ⊗ M p ⊗ M q , w i ) | P ,n ⊗P ,n ⊗P ,n = γ ⊗ K (id M p ) ⊗ K (id M q ) | P ,n ⊗P ,n ⊗P ,n . Note that k xw ∗ w − w ∗ w x k < n − for all x ∈ F ,n ⊗ F ,n ⊗ F ,n . By Lemma 3.11,bott (id A ⊗ M p ⊗ M q , w ∗ w ) | P ,n ⊗P ,n ⊗P ,n = − bott (id A ⊗ M p ⊗ M q , w ) | P ,n ⊗P ,n ⊗P ,n + bott (id A ⊗ M p ⊗ M q , w ) | P ,n ⊗P ,n ⊗P ,n = 0bott (id A ⊗ M p ⊗ M q , w ∗ w ) | P ,n ⊗P ,n ⊗P ,n = − bott (id A ⊗ M p ⊗ M q , w ) | P ,n ⊗P ,n ⊗P ,n + bott (id A ⊗ M p ⊗ M q , w ) | P ,n ⊗P ,n ⊗P ,n = 0 . Therefore, k xw ∗ w − w ∗ w x k < δ x ∈ G , bott (id A ⊗ M p ⊗ M q , w ∗ w ) | H = 0 , and bott (id A ⊗ M p ⊗ M q , w ∗ w ) | H = 0 . Hence, there exists a continuous path of unitaries v ( t ) in A ⊗ M p ⊗ M q such that (cid:13)(cid:13) v ( t )( a ⊗ M p ⊗ M q ) − ( a ⊗ M p ⊗ M q ) v ( t ) (cid:13)(cid:13) < ǫ a ∈ F and for all t ∈ [0 , v (0) = 1 A ⊗ M p ⊗ M q and v (1) = w ∗ w .Set w ( t ) = w v ( t ). Then w ( t ) is a continuous path of unitaries in A ⊗ M p ⊗ M q such that w (0) = w , w (1) = w , and (cid:13)(cid:13) w ( t )( a ⊗ M p ⊗ M q ) − ( a ⊗ M p ⊗ M q ) w ( t ) (cid:13)(cid:13) < ǫ for all a ∈ F .Note that [ v ] = γ ⊗ K (id M p )([1 A ⊗ M p ]) = γ ([1 A ]) ⊗ [1 M p ] = [ u ⊗ M p ] in K ( A ⊗ M p ) and[ v ] = γ ⊗ K (id M q )([1 A ⊗ M q ]) = γ ([1 A ]) ⊗ [1 M q ] = [ u ⊗ M q ] in K ( A ⊗ M q ). Hence, [ w (0)] =[ u ⊗ M p ⊗ M q ] in K ( A ⊗ M p ⊗ M q ) and [ w (1)] = [ u ⊗ M p ⊗ M q ] in K ( A ⊗ M p ⊗ M p ). (cid:3) The automorphism group of a Z -stable C ∗ -algebra Let A be a separable, unital C ∗ -algebra and let { G n } ∞ n =1 be an increasing sequence offinitely generated subgroups of K ( A ) containing [1 A ]. Note that there exists a sequence ofsurjective homomorphisms K ( A ) H [1 A ] ( G , K ( A )) ← K ( A ) H [1 A ] ( G , K ( A )) ← · · · which gives an inverse limit group lim ← K ( A ) H [1 A ] ( G n , K ( A )) . For each n ∈ N , equip K ( A ) H [1 A ] ( G n ,K ( A )) with the discrete topology and give the inverse limit the inverse limit topology.For each g ∈ K ( A ), denote the image of g in the inverse limit by ˇ g . For each g ∈ U ( A ),denote the image of g in Inn( A )Inn ( A ) by ˆ g = Ad( g ) + Inn ( A ). The authors in [21] showed thatthe map µ : ˆ g ˇ g extends uniquely to a continuous group homomorphism from Inn( A )Inn ( A ) tolim ← K ( A ) H [1 A ] ( G n , K ( A )) . With an abuse of notation, we denote this unique extension by µ .Moreover, by Theorem 2 of [21], if A satisfies Property (C) (see Definition 4.1), then µ issurjective and if, in addition, the natural map from U ( A ) /U ( A ) to K ( A ) is an isomorphism,then µ is an isomorphism of topological abelian groups. Consequently, Inn( A )Inn ( A ) is a totallydisconnected group. Moreover, the authors showed that if A is a unital C ∗ -algebra satisfyingthe UCT and if A is either a Kirchberg algebra or simple tracially AI algebra, then A satisfiesProperty (C). We use the results from the previous section to show that every element in A Z satisfies Property (C). HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 35 Z -stable C ∗ -algebras and Property (C).Definition 4.1. Let A be a unital C ∗ -algebra. A is said to satisfy Property (C) if for each ǫ > F of A , there exists a finitely generated subgroup G of K ( A )containing [1 A ] such that the following holds: for every u ∈ U ( A ) with [ u ] ∈ H [1 A ] ( G , K ( A )),there exists w ∈ U ( A ) such that k wa − aw k < ǫ for all a ∈ F and [ w ] = [ u ] in U ( A ) /U ( A ) .The following is an equivalent definition of Property (C) which will be easier to check inour context. Lemma 4.2.
Let A be a unital C ∗ -algebra. Then the following are equivalent: (1) A satisfies Property (C). (2) For each ǫ > and for each finite subset F of A , there exists a finitely generatedsubgroup G of K ( A ) containing [1 A ] such that the following holds: for every u ∈ U ( A ) with [ u ] ∈ H [1 A ] ( G , K ( A )) , there exists w ∈ U ( A ) such that k uau ∗ − waw ∗ k < ǫ for all a ∈ F .Proof. (1) = ⇒ (2) : Let ǫ > F be a finite subset of A . Let G be the finitely generatedsubgroup of K ( A ) given in Property (C) that corresponds to ǫ and F . Let u ∈ U ( A ) with[ u ] ∈ H [1 A ] ( G , K ( A )). Then, by our assumption, there exists v ∈ U ( A ) such that k va − av k < ǫ for all a ∈ F and uv ∗ ∈ U ( A ) . Set w = uv ∗ . Then w ∈ U ( A ) and k uau ∗ − waw ∗ k = k a − v ∗ av k = k va − av k < ǫ for all a ∈ F .(2) = ⇒ (1) : Let ǫ > F be a finite subset of A . Let G be the finitely generatedsubgroup of K ( A ) given in (2) that corresponds to ǫ and F . Suppose u ∈ U ( A ) such that[ u ] ∈ H [1 A ] ( G , K ( A )). Then, by our assumption, there exists v ∈ U ( A ) such that k uau ∗ − vav ∗ k < ǫ for all a ∈ F . Set w = v ∗ u . Since [ v ] ∈ U ( A ) , [ w ] = [ u ] in U ( A ) /U ( A ) . Moreover, k wa − aw k = k v ∗ ua − av ∗ u k = k uau ∗ − vav ∗ k < ǫ for all a ∈ F . (cid:3) Lemma 4.3.
Let A and B be separable C ∗ -algebras, with B unital and Z -stable and let p and q be supernatural numbers which are relatively prime. Suppose ϕ , ϕ : A → B are ∗ -homomorphisms such that ϕ ⊗ Z p , q , ϕ ⊗ Z p , q : A → B ⊗ Z p , q are approximately unitarily equivalent via unitaries { w n } n ∈ N with [ w n ] = 0 in K ( B ⊗ Z p , q ) .Then ϕ , ϕ are approximately unitarily equivalent via unitaries { v n } n ∈ N in B such that [ v n ] = 0 in K ( B ) . Consequently, for every ǫ > and for every finite subset F of A , there exists δ > suchthat if w is a unitary in B ⊗ Z p , q such that (cid:13)(cid:13) ( ϕ ⊗ Z p , q )( a ) − w ( ϕ ⊗ Z p , q )( a ) w ∗ (cid:13)(cid:13) < δ for all a ∈ F and [ w ] = 0 in K ( B ⊗ Z p , q ) , then there exists a unitary v in B with [ v ] = 0 in K ( B ) such that k ϕ ( a ) − vϕ ( a ) v ∗ k < ǫ for all a ∈ F .Proof. Since B ∼ = B ⊗ Z ∼ = B ⊗ Z ⊗ Z , by Lemma 2.5, there exist a ∗ -isomorphism ν B : B → B ⊗ Z and a sequence of unitaries { u n } n ∈ N in B ⊗ Z such thatlim n →∞ k ν B ( b ) − u n ( b ⊗ Z ) u ∗ n k = 0for all b ∈ B . Let σ p , q : Z p , q → Z be the unital embedding in Proposition 3.4 of [33]. Notethat lim n →∞ (cid:13)(cid:13) ϕ i ( a ) − ν − B ( u n ) ν − B ( ϕ i ( a ) ⊗ Z ) ν − B ( u ∗ n ) (cid:13)(cid:13) = 0and ϕ ( a ) ⊗ Z = ϕ ( a ) ⊗ σ p , q (1 Z p , q )= (id B ⊗ σ p , q )( ϕ ( a ) ⊗ Z p , q ) ϕ ( a ) ⊗ Z = ϕ ( a ) ⊗ σ p , q (1 Z p , q )= (id B ⊗ σ p , q )( ϕ ( a ) ⊗ Z p , q )for all a ∈ A . Therefore,lim n →∞ k ϕ ( a ) ⊗ Z − (id B ⊗ σ p , q )( w n )( ϕ ( a ) ⊗ Z )(id B ⊗ σ p , q )( w ∗ n ) k = lim n →∞ (cid:13)(cid:13) (id B ⊗ σ p , q )( ϕ ( a ) ⊗ Z p , q ) − (id B ⊗ σ p , q )( w n ( ϕ ( a ) ⊗ Z p , q ) w ∗ n ) (cid:13)(cid:13) = 0for all a ∈ A . Set v n = ν − B ( u n [id B ⊗ σ p , q ] ( w n ) u ∗ n ). Let a ∈ A . Then k ϕ ( a ) − v n ϕ ( a ) v ∗ n k≤ (cid:13)(cid:13) ϕ ( a ) − ν − B ( u n ) ν − B ( ϕ ( a ) ⊗ Z ) ν − B ( u ∗ n ) (cid:13)(cid:13) + k ν − B ( u n ) ν − B ( ϕ ( a ) ⊗ Z ) ν − B ( u ∗ n ) − ν − B ( u n (id B ⊗ σ p , q )( w n )) ν − B ( ϕ ( a ) ⊗ Z ) ν − B ((id B ⊗ σ p , q )( w ∗ n ) u ∗ n ) k + (cid:13)(cid:13) ν − B ( u n (id B ⊗ σ p , q )( w n )) ν − B ( ϕ ( a ) ⊗ Z ) ν − B ((id B ⊗ σ p , q )( w ∗ n ) u ∗ n ) − v n ϕ ( a ) v ∗ n (cid:13)(cid:13) ≤ (cid:13)(cid:13) ϕ ( a ) − ν − B ( u n ) ν − B ( ϕ ( a ) ⊗ Z ) ν − B ( u ∗ n ) (cid:13)(cid:13) + k ϕ ( a ) ⊗ Z − (id B ⊗ σ p , q )( w n )( ϕ ( a ) ⊗ Z )(id B ⊗ σ p , q )( w ∗ n ) k + (cid:13)(cid:13) ϕ ( a ) − ν − B ( u n ) ν − B ( ϕ ( a ) ⊗ Z ) ν − B ( u ∗ n ) (cid:13)(cid:13) . Therefore, for each a ∈ A , lim n →∞ k ϕ ( a ) − v n ϕ ( a ) v ∗ n k = 0 . HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 37 Since [ w n ] = 0 in K ( B ⊗ Z p , q ), then[ v n ] = (cid:0) K ( ν − B ) ◦ K (Ad( u n )) ◦ K (id B ⊗ σ p , q ) (cid:1) ([ w n ])= 0in K ( B ). (cid:3) Lemma 4.4.
Let A be in A Z . Then the canonical homomorphism from U ( A ) /U ( A ) to K ( A ) is an isomorphism.Proof. Let p be a supernatural number of infinite type. Then A ⊗ M p is a tracially AI algebra.Thus A is a finite Z -stable C ∗ -algebra. Hence, by [27], A has stable rank one. Since A is asimple unital C ∗ -algebra with stable rank one, by Theorem 10.12 of [24] and Corollary 7.14of [1], the canonical homomorphism from U ( A ) /U ( A ) to K ( A ) is an isomorphism. (cid:3) Theorem 4.5.
Let A be in A Z . Then A satisfies Property (C).Proof. By Lemma 4.2, it is enough to prove (2) of Lemma 4.2. Let ǫ > F be afinite subset of A . Let δ > F and ǫ . Let G be the finitely generated subgroup of K ( A ) given in Theorem 3.22 correspondingto F and δ .Let u ∈ U ( A ) with [ u ] ∈ H [1 A ] ( G , K ( A )). Let p and q be supernatural numbers of infinitetype such that M p ⊗ M q is isomorphic to Q . By Theorem 3.22, then there exists a continuouspath of unitaries w ( t ) in A ⊗ M p ⊗ M q such that(1) w (0) ∈ A ⊗ M p ⊗ M q and w (1) ∈ A ⊗ M p ⊗ M q ;(2) [ w (0)] = [ u ⊗ M p ⊗ M q ] in K ( A ⊗ M p ⊗ M q ) and [ w (1)] = [ u ⊗ M p ⊗ M q ] in K ( A ⊗ M p ⊗ M q ); and(3) (cid:13)(cid:13) w ( t )( a ⊗ M p ⊗ M q ) − ( a ⊗ M p ⊗ M q ) w ( t ) (cid:13)(cid:13) < δ for all a ∈ F and t ∈ [0 , Z p , q = (cid:8) f ∈ C ([0 , , M p ⊗ M q ) : f (0) ∈ M p ⊗ M q and f (1) ∈ M p ⊗ M q (cid:9) , A ⊗ Z p , q can be identified with the C ∗ -algebra (cid:8) f ∈ C ([0 , , A ⊗ M p ⊗ M q ) : f (0) ∈ A ⊗ M p ⊗ M q and f (1) ∈ A ⊗ M p ⊗ M q (cid:9) . With this identification, w is an element of U ( A ⊗ Z p , q ) such that (cid:13)(cid:13) w ( a ⊗ Z p , q ) − ( a ⊗ Z p , q ) w (cid:13)(cid:13) < δ for all a ∈ F . Since [ w (0)] = [ u ⊗ M p ⊗ M q ] in K ( A ⊗ M p ⊗ M q ) and [ w (1)] = [ u ⊗ M p ⊗ M q ]in K ( A ⊗ M p ⊗ M q ), we have that[ (cid:0) ( u ⊗ Z p , q ) w ∗ (cid:1) (0)] = 0in K ( A ⊗ M p ⊗ M q ) and [ (cid:0) ( u ⊗ Z p , q ) w ∗ (cid:1) (1)] = 0in K ( A ⊗ M p ⊗ M p ). By Proposition 5.2 of [33], [( u ⊗ Z p , q ) w ∗ ] = 0 in K ( A ⊗ Z p , q ). Let a ∈ F . Then (cid:13)(cid:13) (Ad( u ) ⊗ Z p , q )( a ) − Ad (cid:0) ( u ⊗ Z p , q ) w ∗ (cid:1) ( a ⊗ Z p , q ) (cid:13)(cid:13) < δ. Hence, by Lemma 4.3, there exists a unitary v ∈ U ( A ) such that [ v ] = 0 in K ( A ) and k uau ∗ − vav ∗ k < ǫ for all a ∈ F . By Lemma 4.4, v ∈ U ( A ) . We have just shown that (2) of Lemma 4.2 holds.Therefore, A satisfies Property (C). (cid:3) The structure of the automorphism group of a C ∗ -algebra in A Z . Recall from[5], there is a natural order structure on K ( A ). Let Aut( K ( A )) be the topological groupof all ordered group isomorphisms of K ( A ), which preserve the grading and the Bocksteinoperations in [5]. For a unital C ∗ -algebra A , let Aut( K ( A )) be the subgroup of Aut( K ( A ))which sends [1 A ] to [1 A ]. Set J ( A ) = ( K ( A ) , T ( A ) , U ( A ) /CU ( A )) . Then an element of Aut( J ( A )) is an ordered tuple ( α, λ T , λ U ) where α ∈ Aut( K ( A )) , λ T : T ( A ) → T ( A ) is an affine homeomorphism, and λ U : U ( A ) /CU ( A ) → U ( A ) /CU ( A ) isa homeomorphism such that α , λ T , and λ U are compatible is the sense of Section 2, pp. 5of [19]. Theorem 4.6.
Let A be a C ∗ -algebra in A Z satisfying the UCT. Then (a) Inn ( A ) is a simple topological group; (b) Inn( A )Inn ( A ) is totally disconnected; and (c) the sequence → Inn( A ) → Aut( A ) → Aut( J ( A )) is exact.In addition, if K ( A ) is torsion free, then the sequence in (c) becomes a short exact sequence → Inn( A ) → Aut( A ) → Aut( J ( A )) → . Proof. (a) follows from Theorem 2.20. (b) follows from Theorem 2 of [21] and Theorem 4.5and Lemma 4.4. (c) follows from Theorem 4.8 of [19].Suppose K ( A ) is torsion free. Let ( α, λ T , λ U ) be an element of Aut( J ( A )) . By Theorem5.9 of [19], there exist unital homomorphisms φ, ψ : A → A such that(i) ( α, λ T , λ U ) is induced by φ and(ii) ( α − , λ − T , λ − U ) is induced by ψ .Let {F n } ∞ n =1 be an increase sequence of finite subsets of A whose union is dense in A . UsingTheorem 4.8 of [19], we get a sequence of unital homomorphisms φ n , ψ n : A → A such that k ( ψ n ◦ φ n )( x ) − x k < n and k ( φ n +1 ◦ ψ n )( a ) − a k < n for all a ∈ F n , and φ n is unitarily equivalent to φ for each n ∈ N and ψ n is unitarilyequivalent to ψ for each n ∈ N . Therefore, we get an isomorphism β : A → A such that β and φ induce the same element in Aut( J ( A )) , i.e., β induces ( α, λ T , λ U ). (cid:3) Remark 4.7.
Note that if A is a nuclear, separable, unital, tracially AI algebra satisfyingthe UCT, then by [12], the sequence0 → Inn( A ) → Aut( A ) → Aut( J ( A )) → K ( A ) has an element with finite order. Moreover, A is an element of A Z . The authors believe that the above sequence should be exact for an arbitrary A in A Z satisfying the UCT but the proof has eluded the authors. HE AUTOMORPHISM GROUP OF A SIMPLE C ∗ -ALGEBRA 39 Acknowledgement
The authors are grateful to the referee for a careful reading of the paper and usefulsuggestions.
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