Inductive Limits of Subhomogeneous C -algebras with Hausdorff Spectrum

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1 Inductive Limits of Subhomogeneous C -algebras with Hausdorff Spectrum Huaxin Lin arxiv: v2 [math.oa] 21 Nov 2008 Abstract We consider unital simple inductive limits of generalized dimension drop C -algebras They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop C -algebras. Suppose that A is one of these C -algebras. We show that A Q has tracial rank no more than one, where Q is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let A and B be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then A = B if and only if they have the same Elliott invariant. 1 Introduction A unital AH-algebra is an inductive limit of finite direct sums of C -algebras with the form PC(X,M k )P, where X is a finite dimensional compact metric space, k 1 is an integer and P C(X,M k ) is a projection. One of the successful stories in the program of classification of amenable C -algebras, or otherwise known as the Elliott program is the classification of unital simple AH-algebras with no dimension growth (see [3]) by their K-theoretical invariant, or otherwise called the Elliott invariant. Other classes of inductive limits of building blocks studied in the program include so-called simple inductive limits of dimension drop interval (or circle) algebras (see [16], [2], [17], [4] and [15], for example). One reason to study the simple inductive limits of dimension drop interval algebras is to construct unital simple C -algebras with the Elliott invariant which can not be realized by unital simple AH-algebras. Notably, the so-called Jiang Su-algebras of unital projectionless simple ASH-algebra Z whose Elliott invariant is the same as that of complex field. However, until very recently, they were classified as separate classes of simple C -algebras. In particular, it was not known that one C -algebra in one of these classes should be isomorphic to ones in a different class with the same Elliott invariant in general. Furthermore, methods used in classification of unital simple AH-algebras and unital simple dimension drop algebras dealt with maps from dimension drop algebras to dimension drop algebras and maps between some homogeneous C -algebras. These methods do not work for maps from dimension drop algebras into homogeneous C -algebras or maps from homogeneous C -algebras into dimension drop algebras in general. Therefore most classification theorems for simple inductive limits of basic building blocks do not deal with mixed building blocks. In this paper we consider a general class of inductive limits of sub-homogenous C -algebras with Hausdorff spectrum which includes both AH-algebras and all known unital simple inductive limits of dimension drop algebras as well as inductive limits with mixed basic building blocks. We will give a classification theorem for this class of unital simple C -algebras. Let X be a compact metric space. Let k 1 be an integer, let ξ 1,ξ 2,...,ξ n X and let m 1,m 2,...,m n 1 be integers such that m j k, j = 1,2,...,n. One may write M k = M mj M k/m(j). Let B j = Mmj 1 k/m(j) be a unital C -subalgebra of M k (1 Bj = 1 Mk ). 1

2 Definition 1.1. Define D k (X,ξ 1,ξ 2,...,ξ n,m 1,m 2,...,m n ) = {f C(X,M k ) : f(ξ j ) B j, 1 j n}. In the above form, if X is an interval, it is known as (general) dimension drop interval algebra, and if X = T, it is known as (general) dimension drop circle algebra. Here we allow high dimensional dimension drop algebras. So when X is a connected finite CW complex, algebras D k (X,ξ 1,ξ 2,...,ξ n,m 1,m 2,...,m n ) and finite direct sums of their unital hereditary C - subalgebras may be called general dimension drop algebras. When m j = k, 1 j d(j), j = 1,2,...,N, it is a homogenous C -algebra. Thus AH-algebras, inductive limits of dimension drop interval algebras and inductive limits of dimension drop circle algebras are inductive limit of general dimension drop algebras. A classification theorem for unital simple inductive limits of those general dimension drop algebras would not only generalize many previous known classification theorems but more importantly it would unify these classification results. However, one may consider more general C -algebras. Let X 1,X 2,...,X n X be disjoint compact subsets. Let k, m 1,m 2,...,m n and B 1,B 2,...,B n be as above. Define Definition 1.2. D X,k,{Xj },n = {f C(X,M k ) : f(x) B j for all x X j, 1 j n}. (e 1.1) By a generalized dimension drop algebra, we mean C -algebras of the form: N P j DXj,r(j),{X j,i },d(j)p j j=1 where P j D Xj,r(j),{X i,j },d(j) is a projection, X j is a compact metric space with finite dimension and with property (A) (see 4.1 bellow) and {X j,i } is a collection of finitely many disjoint compact subsets of X j. Note that, in the following, A Z and A have exactly the same Elliott invariant whenever K 0 (A) is weakly unperforated. We prove the following theorem: Theorem 1.3. Let A and let B be two unital simple inductive limits of generalized dimension drop algebras. Then A Z = B Z if and only if (K 0 (A),K 0 (A) +,[1 A ],K 1 (A),T(A),r A ) = (K 0 (B),K 0 (B) +,[1 B ],K 1 (B),T(B),r B ). (see 2.10). Denote by D the class of inductive limits of generalized dimension drop algebras. By a recent result of W. Winter ([24]), unital simple ASH-algebras with no dimension growth are Z-stable. Therefore we also have the following: Theorem 1.4. Let A, B D be two unital simple C -algebras with no dimension growth. Then A = B, if and only if (K 0 (A),K 0 (A) +,[1 A ],K 1 (A),T(A),r A ) = (K 0 (B),K 0 (B) +,[1 B ],K 1 (B),T(B),r B ). This unifies the previous known classification theorems for unital simple ASH-algebras with Hausdorff spectrum such as [3], [2], [4], [17] and [15], etc. In fact, we show that, if A D is a unital simple C -algebra, then the tracial rank of A Q is no more than one, where Q is the UHF-algebra with (K 0 (Q),K 0 (Q) +,[1 Q ]) = (Q, Q +,1). We then obtain the above mentioned 2

3 classification theorems by applying a recent classification theorem in [11] for unital simple C - algebras A with the tracial rank of A Q no more than one. Therefore one may view our result here as an application of the classification result in [11]. Acknowledgments: This work was mostly done when the author was in East China Normal University during the summer of It was partially supported by an NSF grant (DMS074813), the Chang-Jiang Professorship from East China Normal University and Shanghai Priority Academic Disciplines. 2 Preliminaries 2.1. Let {A n } be a sequence of C -algebras and let ϕ n : A n A n+1 be homomorphisms. Denote by A = lim n (A n,ϕ n ) the C -algebra of the inductive limit. Put ϕ n,m = ϕ m ϕ m 1 ϕ n : A n A m for m > n. Denote by ϕ n, : A n A the homomorphism induced by the inductive limit system Let X be a compact metric space, let ξ X and let d > 0. Set Suppose that 0 < d 2 < d 1. Set B(ξ,d) = {x X : dist(x,ξ) < d}. A(ξ,d 1,d 2 ) = {x X : d 1 > dist(x,ξ) > d 2 }. Definition 2.3. Let X be a compact metric space, let A be a C -algebra and let B C(X,A) be a C -subalgebra. For each x X, denote by π x : C(X,A) A the homomorphism defined by π x (f) = f(x) for all f C(X,A). Suppose that C x = π x (B). Then π x : B C is a homomorphism Let A be a C -algebra, let x, y A be two elements and let ǫ > 0. We write if x y < ǫ. Suppose that S A is a subset. We write x ǫ y, x ǫ S, if x ǫ y for some y S. Let B be another C -algebra, let ϕ,ψ : A B be two maps, let S A be a subset and let ǫ > 0. We write ϕ ǫ ψ on S, if ψ(x) ǫ ϕ(x) for all x S Let A be a unital C -algebra. Denote by U(A) the unitary group of A. Denote by U 0 (A) the connected component of U(A) containing the identity Let A be a unital C -algebra. We say B is a unital C -subalgebra of A if B A is a unital C -algebra and 1 B = 1 A. 3

4 Definition 2.7. Let A be a unital C -algebra and let C be another C -algebra. Suppose that ϕ 1,ϕ 2 : C A are two homomorphisms. We say that ϕ 1 and ϕ 2 are asymptotically unitarily equivalent if there exists a continuous path of unitaries {u(t) : t [0, )} A such that lim t u(t) ϕ 2 (c)u(t) = ϕ 1 (c) for all c C. (e 2.2) We say that ϕ 1 and ϕ 2 are strongly asymptotically unitarily equivalent if in (e 2.2) u(0) = 1 A. Of course, ϕ 1 and ϕ 2 are strongly asymptotically unitarily equivalent if they are asymptotically unitarily equivalent and if U(A) = U 0 (A). Definition 2.8. Denote by I the class of those C -algebras with the form n i=1 M r i (C(X i )), where each X i is a finite CW complex with (covering) dimension no more than one. A unital simple C -algebra A is said to have tracial rank no more than one (we write T R(A) 1) if for any finite subset F A, ǫ > 0, any nonzero positive element a A, there is a C -subalgebra C I such that if denote by p the unit of C, then for any x F, one has (1) xp px ǫ, (2) there is b C such that b pxp ǫ and (3) 1 p is Murray-von Neumann equivalent to a projection in aaa. Definition 2.9. For a supernatural number p, denote by M p the UHF algebra associated with p (see [1]). We use Q throughout the paper for the UHF-algebra with (K 0 (Q),K 0 (Q) +,[1 Q ]) = (Q, Q +,1). We may identify Q with the inductive limit lim n (M n!,h n ), where h n : M n! M (n+1)! is defined by a a 1 n+1 for a M n!. Let A be the class of all unital separable simple amenable C -algebras A in N for which TR(A M p ) 1 for some supernatural number p of infinite type. Definition Let A be a unital stably finite separable simple amenable C -algebra. Denote by T(A) the tracial state space of A. We also use τ for τ Tr on A M k for any integer k 1, where Tr is the standard trace on M k. By Ell(A) we mean the following: (K 0 (A),K 0 (A) +,[1 A ],K 1 (A),T(A),r A ), where r A : T(A) S [1A ](K 0 (A)) is a surjective continuous affine map such that r A (τ)([p]) = τ(p) for all projections p A M k, k = 1,2,..., where S [1A ](K 0 (A)) is the state space of K 0 (A). Suppose that B is another stably finite unital separable simple C -algebra. A map Λ : Ell(A) Ell(B) is said to be a homomorphism if Λ gives an order homomorphism λ 0 : K 0 (A) K 0 (B) such that λ 0 ([1 A ]) = [1 B ], a homomorphism λ 1 : K 1 (A) K 1 (B), a continuous affine map λ ρ : T(B) T(A) such that λ ρ (τ)(p) = r B(τ)(λ 0 ([p])) for all projection in A M k, k = 1,2,..., and for all τ T(B). We say that such Λ is an isomorphism, if λ 0 and λ 1 are isomorphisms and λ ρ is a affine homeomorphism. In this case, there is an affine homeomorphism λ ρ : T(A) T(B) such that λ 1 ρ = λ ρ. Theorem 2.11 (Corollary 11.9 of [11]). Let A, B A. Then A Z = B Z if Ell(A Z) = Ell(B Z). 4

5 3 Approximately unitary equivalence The following is known. Lemma 3.1. Let ϕ,ψ : Q Q be two unital homomorphisms. Then they are strongly asymptotically unitarily equivalent. Proof. Since ϕ and ψ are unital, one computes that ϕ and ψ induce the same identity map on K 0 (Q). Since K 0 (Q) is divisible and K 1 (Q) = {0}, one then checks that [ϕ] = [ψ] in KK(Q,Q). With K 1 (Q) = {0}, it follows from Corollary of [10] that ϕ and ψ are strongly asymptotically unitarily equivalent. Lemma 3.2. For any ǫ > 0 and any finite subset F Q, there exists δ > 0 and a finite subset G Q satisfying the following: suppose that u U(Q) and ϕ : Q Q is a unital homomorphism such that [u, ϕ(b)] < δ for all b G. (e 3.3) Then there exists a continuous path of unitaries {u(t) : t [0,1]} in Q such that u(0) = u, u(1) = 1, [u(t), ϕ(a)] < ǫ for all a F. (e 3.4) Moreover, Length({u(t)}) 2π. Proof. Note that K 1 (Q) = {0} and K i (Q C(T)) = Q is torsion free divisible group, i = 0,1. One computes that Bott(ϕ,u) = 0. It follows from [9] that such {u(t)} exists. Theorem 3.3. Let X be a compact subset of [0,1] with a = inf{t : t X} and b = sup{t : t X}. Suppose that ϕ,ψ : Q C(X,Q) are two unital monomorphisms. Then ϕ and ψ are approximately unitarily equivalent. Moreover, if π a ϕ = π a ψ (or π b ϕ = π b ψ, or both hold) then there exists a sequence of unitaries {u n } C(X,Q) such that π a (u n ) = 1 (or π b (u n ) = 1, or π a (u n ) = π b (u n ) = 1) and lim adu n ϕ(x) = ψ(x) for all x Q. n Furthermore, if F Q is a full matrix algebra as a unital C -subalgebra, π a ϕ F = π a ψ F and π b ϕ F = π b ψ F, one can find, for each ǫ > 0, a unitary u such that for all f F and π a (u) = π b (u) = 1. ad u ϕ(f) ψ(f) < ǫ 5

6 Proof. To simplify the notation, without loss of generality, we may assume that a = 0 and b = 1. Fix ǫ > 0 and a finite subset F. Without loss of generality, we may assume that f 1 for all f F. Let δ > 0 and G be the finite subset required by 3.2 corresponding to ǫ and F. Without loss of generality, we may assume that ǫ < δ and F G. Let ϕ,ψ : Q C(X,Q) be two unital homomorphisms. Denote by π t : C(X,Q) Q the point-evaluation map at the point t ( X). There is a partition: 0 = t 0 < t 1 < t n = 1 with t i X such that either (t i 1,t i ) X =, or π ti ϕ(b) π t ϕ(b) < δ/16 and π ti ψ(b) π t ψ(b) < δ/16 (e 3.5) for all t [t i 1,t i ] X, i = 1,2,...,n and all b G. Since K 0 (Q) = Q, [1 Q ] = 1 and K 1 (Q) = {0}, one computes that [π t ϕ] = [π t ψ] in KK(Q,Q) for all t X. It follows from 3.1 that there is, for each i, a unitary v i Q such that ad v i π ti ϕ(b) π ti ψ(b) < δ/16, (e 3.6) i = 0,1,...,n. Put w i = u i 1 u i, if (t i 1,t i ) X. Then, if (t i 1,t i ) X, w i (π ti 1 ϕ(b))w i δ/16 u i (π ti 1 ψ(b))u i (e 3.7) δ/16 u i (π ti ψ(b))u i (e 3.8) δ/16 π ti ϕ(b) (e 3.9) δ/16 π ti 1 ϕ(b) (e 3.10) for all b G. It follows from 3.2 that there exists a continuous path of unitaries {w i (t) : t [t i 1,t i ]} in Q such that w i (t i 1 ) = w i, w i (t i ) = 1 and [w i (t), π π ti 1 (a)] < ǫ/16 (e 3.11) for all t [t i 1,t i ] X and a F, i = 1,2,...,n. Define u(t i ) = u i and (if (t i 1,t i ) X ) u(t) = w i (t)u i for all t [t i 1,t i ] X (e 3.12) Note that u(t) is continuous. Moreover, when (t i 1,t i ) X, for t [t i 1,t i ] X, ad u(t) π t ϕ(a) ǫ/16 ad u(t) π ti ϕ(a) (e 3.13) ǫ/16 ad u i π ti ϕ(a) (e 3.14) ǫ/16 π ti ψ(a) (e 3.15) ǫ/16 π t ψ(a) (e 3.16) for all a F, where the first estimate follows from (e 3.5), the second estimate follows from (e 3.11), the third follows from (e3.6) and the last one follows from (e 3.5). Note that u(t) C(X,Q). It follows that ϕ and ψ are approximately unitarily equivalent. To prove the second part of the statement, we use what we have just proved. Let ǫ > 0 and F Q be a finite subset. Let δ and G be required by 3.2 for ǫ/4 and F. Without loss of generality, we may assume that F G. Put ǫ 1 = min{δ/4,ǫ/4}. There exists η > 0 such that π t ϕ(a) π t ϕ(a) < ǫ 1 /2 and (e 3.17) π t ψ(a) π t ψ(a) < ǫ 1 /2 (e 3.18) 6

7 for all a G, whenever t t η and t,t X. From what we have shown, there exists a unitary w C(X,Q) such that ad w ψ(a) ϕ(a) < ǫ 1 /2 for all a G. If 0 is an isolated point in X, then we may assume that π 0 (w) = 1. Otherwise, let η 1 = sup{t : t (0,η] X}. Since π 0 ϕ = π 0 ψ and π 1 ϕ = π 1 ψ, we compute that π η1 (w)π η1 ψ(a) π η1 ψ(a)π η1 (w) < ǫ 1 for all a G. It follows from 3.2 that there exists a continuous path of unitaries {W(t) : t [0,η 1 ]} Q such that W(0) = 1, W(η 1 ) = w(η 1 ). [W(t), ψ(x)] < ǫ/4 for all x F and t [0,1]. Define v C(X,Q) by π t (v) = W(t) if t [0,η 1 ] X and π t (v) = π t w for t (η 1,1] X. Then v C(X,Q) ad v ψ(a) ϕ(a) < ǫ/2 for all a F and v(0) = 1. We can deploy the same argument for the end point 1. Thus we obtain a unitary u C(X,Q) such that u(0) = 1 = u(1) and ad u ψ(a) ϕ(a) < ǫ for all a F. Let F Q be a full matrix algebra as a unital C -subalgebra. Let {e i,j : 1 i,j k} be a system of matrix unit. From the above, there is a unitary W C(X,Q) such that W ψ(e i,j )W ϕ(e i,j ) < ǫ 1 i,j k. 8k One has a unitary v C(X,Q) such that Define Then W 1 is a unitary such that ϕ(e 1,1 ) = v ψ(e 1,1 )v. W 1 = k ψ(e i,1 )vϕ(e 1,i ). i=1 For any ǫ > 0, there exists 1 > δ > 0 such that W 1 ψ(f)w 1 = ϕ(f) for all f F. W 1 (t) W 1 (t ) < ǫ/4 (e 3.19) for all t,t X and t t < δ. Since U(ϕ(e i,i )Qϕ(e i,i )) = U 0 (ϕ(e i,i )Qϕ(e i,i )), it is easy to obtain a continuous path of unitaries {v(t) : t [0,δ]} U(Q) such that v(δ) = W 1 (0), v(0) = 1 and u(t)π a ϕ(f) = π a ϕ(f)u(t) for all f F and all t [0,δ]. Define now u(t) C(X,Q) by u(t) = v(t) if t [0,δ) X and u(t) = W 1 ( t δ 1 δ ) if t [δ,1] X. By (e 3.19), ad u ψ(f) ϕ(f) < ǫ for all f F. One may arrange the other end point b similarly One can avoid to using Lemma 3.2 by modifying the last part of the above proof to obtain

8 4 Approximate factorizations Definition 4.1. Let X be a connected compact metric space. We say that X has the property (A), if, for any ξ X and any δ > 0, there exists δ > δ 1 > δ 2 > δ 3 > 0 satisfying the following: there exists a homeomorphism χ : B(ξ,δ 1 ) \ {ξ} A(ξ,δ 1,δ 3 ) such that χ A(ξ,δ1,δ 2 ) = id A(ξ,δ1,δ 2 ). Every locally Euclidean connected compact metric space has the property (A). All connected simplicial complex has the property (A). Definition 4.2. Let X be a compact metric space and let k 1 be an integer. Fix ξ 1,ξ 2,...,ξ n X and fix positive integers m 1,m 2,...,m n for which m j k, j = 1,2,...,n. Denote by D k,n = D k (X,ξ 1,ξ 2,...,ξ n,m 1,m 2,...,m n ). If X 1,X 2,...,X n X are disjoint compact subsets, put D k,{xj },n = D k (X,X 1,X 2,...,X n,m 1,m 2,...,m n ), when {X j : 1 j n} and {m j : 1 j n} are understood. Let P D k,{xj },n be a projection. Then f(x) P(x) P D k,n,{xj }P for all f C(X). Similarly, if P D k,n is a projection, then f(x) P(x) PD k,n P. Denote by C k,{xj },n the subalgebra {f 1 D k,n,{xj } : f C(X)}. Then C k,n,{xj }P is the center of P D k,n,{xj }P. Similarly, set C k,n = {f 1 Dk,n : f C(X)}. Then C k,n P is the center of PD k,n P. Lemma 4.3. Let X be a connected compact metric space with property (A), let ξ 1,ξ 2,...,ξ n X, let k,m 1,m 2,...,m n 1 be integers with m j k, j = 1,2,...,n. Let P D k,n be a projection. Then, for any ǫ > 0, any finite subset F PD k,n P Q there exists a unital monomorphism such that where ψ : PD k,n 1 P Q PD k,n P Q id PDk,n P Q ǫ ψ ı on F, ı : PD k,n P Q PD k,n 1 P Q is the embedding. Moreover, ψ maps (C k,n 1 P) 1 Q into (C k,n P) 1 Q. Proof. We may assume that f 1 for all f F. Fix ǫ > 0 and a finite subset F PD k,n P Q. We may assume that P F. We may assume that f 1 for all f F and F PD k,n P M m!, where M m! is a unital C -subalgebra of Q. There is δ > 0 such that whenever dist(x,x ) δ. f(x) f(x ) < ǫ/16 for all f F 8

9 Without loss of generality, we may also assume that dist(ξ j,ξ n ) > δ, j = 1,2,...,n 1. There are δ > δ 1 > δ 2 > δ 3 > 0 such that there exists a homeomorphism χ : B(ξ n,δ 1 ) \ {ξ n } A(ξ n,δ 1,δ 3 ) such that χ A(ξn,δ 1,δ 2 ) = id A(ξn,δ 1,δ 2 ). For each f C(X,M k ) Q define f(x) = f(x) if dist(x,ξ n ) > δ 2, f(x) = f(ξ n ) if dist(x,ξ n ) δ 3 and f(x) = f(χ 1 (x)) if x A(ξ n,δ 2,δ 3 ). Note that, for each f F, f f < ǫ/16. (e 4.20) Moreover P is also a projection. Note that, for each x X, π x ( PD k,m P Q) = Q for m 1. There are δ 3 > δ 4 > δ 5 > 0 such that there is a homeomorphism χ 1 : B(ξ n,δ 3 ) \ {ξ n } A(ξ n,δ 3,δ 5 ) such that (χ 1 ) A(ξn,δ 3,δ 4 ) = id A(ξn,δ 3,δ 4 ). Let 0 < δ 6 < δ 5. There is a unital isomorphism λ : π ξn ( PD k,n 1 P Q) πξn ( PD k,n P Q). Note that π ξn ( PD k,n P Q) is a unital C -subalgebra. In particular. (λ πξn ( PD k,n P Q) ) 0 = id K0 (π ξn ( PD k,n P Q)). (e 4.21) Therefore, there is a unitary U 0 π ξn ( PD k,n P Q) such that ad U 0 λ(a) = a for all a π ξn ( PD k,n P Mm! ). To simplify the notation, we may assume that λ(a) = a for all a π ξn ( PD k,n P Mm! ). (e 4.22) On the other hand, by embedding π ξn ( PD k,n P Q) into πξn ( PD k,n 1 P Q) unitally, we may view λ maps π ξn ( PD k,n 1 P Q) into πξn ( PD k,n 1 P Q). Then (ı λ) 0 = id K0 (π ξn ( PD k,n 1 P Q)). It then follows from 3.1, there is a continuous path of unitaries such that u(0) = 1 and {u(t) : t [0,δ 6 )} π ξn ( PD k,n 1 P Q) lim t δ 6 ad u(t) λ(f) = f for all f π ξn ( PD k,n 1 P Q). Define a projection P 1 C(X,M k ) Q as follows: P 1 (x) = P(x) 1 Q for all dist(x,ξ n ) δ 4, 1 P 1 (x) = P(χ 1 (x)) 1 Q, if x A(ξ n,δ 4,δ 5 ) and P 1 (x) = u (t)( P(ξ n ) 1 Q )u(t) = P(ξ n ) 1 Q, if dist(x,ξ n ) = t for 0 t δ 6, and P 1 (x) = P(ξ n ) 1 Q if δ 5 > dist(x,ξ n ) δ 6. Thus P 1 = P 1 Q. Define Φ : PDk,n 1 P Q = P1 (D k,n 1 Q)P 1 P 1 (D k,n Q)P 1 as follows: Φ(f)(x) = f(x), if dist(x,ξ n ) δ 4, Φ(f)(x) = f(χ 1 1 (x)), if x A(ξ n,δ 4,δ 5 ), Φ(f)(x) = f(ξ n ), if δ 6 dist(x,ξ n ) < δ 5 and Φ(f)(x) = u (t)λ(f)u(t) if dist(x,ξ n ) = t for 0 t < δ 6. It follows that for all f F and for those x X with dist(x,ξ n ) δ 6. Φ( f)(x) f(x) < ǫ/16 (e 4.23) 9

10 Define a continuous map Γ : B(ξ n,δ 5 ) [0,δ 5 ] by Γ(x) = dist(x,ξ n ) for x B(ξ n,δ 5 ). Put Y = Γ(B(ξ n,δ 6 )) and Y 1 = Γ(B(ξ n,δ 5 ). Define a projection P 0 C(Y,π ξn (P 1 (D k,n 1 Q)P 1 )) by P 0 (t) = u (t)( P(ξ n ) 1 Q )u(t). We note that P 0 (t) = P(ξ n ) 1 Q for all t Y. Define ψ 0 : π ξn ( PD k,n P Q) C(Y,P1 (ξ n )M k (Q)P 1 (ξ n ))( = C(Y,Q)) by ψ 0 (a)(t) = a for all a π ξn ( PD k,n P Q). Define ψ1 : π ξn ( PD k,n P Q) C(Y,P1 (ξ n )M k (Q)P 1 (ξ n )) by ψ 1 (a)(t) = u (t)λ(a)u(t) for a π ξn ( PD k,n P Q), t Y \ {1} and ψ(a)(δ6 ) = a for all a π ξn ( PD k,n P Q). It follows from 3.3 that there exists a unitary such that U 1 (0) = U 1 (δ 6 ) = P 1 (ξ n ) and U 1 C(Y,P 1 (ξ n )M k (Q)P 1 (ξ n )) (U 1 ) ψ 1 U 1 ǫ/16 ψ 0 on {f(ξ n ) : f F}. (e 4.24) Put U 2 = U 1 +(1 Mk (Q) P 1 (ξ n )). Let U 3 C(X,M k (Q)) be defined by U 3 (x) = 1 if dist(x,ξ n ) δ 6 and U 3 (x) = U 2 (t) if dist(x,ξ n ) = t for t [0,δ 6 ). We have, by (e 4.23), U 3(x) Φ( f)u 3 (t) f(x) < ǫ/16 (e 4.25) for all f F and x X with dist(x,ξ n ) δ 6, and, by (e 4.24), for all f F and x X with dist(x,ξ n ) < δ 6. Note that P P < ǫ/16. There is a unitary U 4 C(X,M k) such that and U 3(x)Φ( f)u 3 (x) f(x) < ǫ/16 (e 4.26) U 4 1 < 2ǫ/16 (U 4) PU 4 = P. (see Lemma of [14] for example). Then U 4 (x) = P(x) if dist(x,ξ n) δ 3 and U 4 (ξ n) = P(ξ n ). Define U 4 = (1 C(X,Mk ) Q 1 C(X,Mk )) + U 4 C(X,M k) 1 Q. Define ψ : PD k,n 1 P Q PD k,n P Q by ψ(f) = ad U 4 U 3 Φ(f) for all f PD k,n 1 P Q. Now we estimate (by applying (e 4.25) and e 4.26)) that f ψ(f) f f + f ψ( f) (e 4.27) < ǫ/16 + 2ǫ/16 + f U 3 Φ( f)u 3 (e 4.28) < (1 + 2)ǫ + 2ǫ/16 < ǫ (e 4.29) 16 for all f F. Note that ψ is a unital monomorphism. This proves the first part of the lemma. The last part follows from the construction that ψ maps (C k,n 1 P) 1 Q into (C k,n P) 1 Q. 10

11 Lemma 4.4. Let X be a connected compact metric space with property (A), let ξ 1,ξ 2,...,ξ n X, let k,m 1,m 2,...,m n 1 be integers with m j k, j = 1,2,...,n. Let P D k (X,ξ 1,ξ 2,...,ξ n,m 1,m 2,...,m n ) be a projection. Then, for any ǫ > 0, any finite subset F PC(X,M k )P Q there exists a unital monomorphism ψ : PC(X,M k )P Q PD k,n P Q such that where id PDk,n P Q ǫ ψ ı on F, ı : PD k,n P Q PC(X,M k )P Q is the embedding. Moreover, ψ maps (C(X) P) 1 Q into (C k,n P) 1 Q. Proof. We prove this by induction on n. If n = 1, then the lemma follows from 4.3 immediately. Suppose that the lemma holds for integers 1 n N. Denote by D k,l = D k (X,ξ 1,ξ 2,...,ξ l,m 1,m 2,...,m l ), l = 1,2,...,. Let ǫ > 0, let F PC(X,M k )P Q and let F Q be a unital finite dimensional C - subalgebra. By the inductive assumption, there exists a unital monomorphism ψ 1 : PC(X,M k )P Q PD k,n P Q such that ψ 1 (f) f < ǫ/2 (e 4.30) for all f F. Moreover there exists a unital finite dimensional C -subalgebra F 0 Q such that ψ 1 (PC(X,M k )P F) C F. By Lemma 4.3, there exists a unital monomorphism ψ 2 : PD k,n P Q PD k,n+1 P Q such that ψ 2 (f) f < ǫ/2 (e 4.31) for all f W 1 ψ 1(F)W 1 and a unital finite dimensional C -subalgebra F 1 Q such that ψ 2 (PD k,n P F 0 ) C F0. Define ψ : PC(X,M k )P Q PD k,n+1 P Q by ψ = ad W 1 ψ 2 ad W 1 ψ 1. Then ψ(f) f = W 1 ψ 2 (W1 ψ 1(f)W 1 )W1 f (e 4.32) W 1 ψ 2 (W1 ψ 1 (f)w 1 )W1 ψ 1 (f) + ψ 1 (f) f (e 4.33) ψ 2 (W 1 ψ 1 (f)w 1 ) W 1 ψ 1 (f)w 1 + ǫ/2 < ǫ/2 + ǫ/2 = ǫ (e 4.34) for all f F. Moreover, by 4.3, ψ maps (C(X) P) 1 Q into (D k,n P) 1 Q. This completes the induction. Lemma 4.5. Let X be a connected compact metric space with property (A), let X 1,X 2,...,X n X be compact subsets, let k,m 1,m 2,...,m n 1 be integers with m j k, j = 1,2,...,n. Let P D k,{xj },n = D k (X,X 1,X 2,...,X n,m 1,m 2,...,m n ) be a projection. Then, for any ǫ > 0, any finite subset F PC(X,M k )P Q there exists a unital monomorphism ψ : PC(X,M k )P Q P D k,{xj },np Q 11

12 such that id P D k,{xj },np Q ǫ ψ ı on F, where ı : P D k,{xj },np Q PC(X,M k )P Q is the embedding. Moreover, ψ maps (C(X) P) 1 Q into (C k,{xj },np) 1 Q. Furthermore, if F Q is a unital finite dimensional C -subalgebra, we may choose ψ so that there exists another unital finite dimensional C -subalgebra F 1 Q and a unital C -subalgebra such that C F PD k,n P Q ψ(pc(x,m k ))P F) C F and there is a unitary W M 2 (P D k,{xj },np Q) such that W C F W = P 1 M 2 (P D k,{xj },np F 1 )P 1 for some projection P 1 M 2 (P D k,{xj },np F 1 ). Proof. Let {ξ i : i = 1,2,...} be a dense sequence of X n. Put D k,{xj } n 1 j=1,{ξ i},l = D {}}{ k (X,X 1,X 2,...,X n 1,ξ 1,ξ 2,...,ξ l,m 1,m 2,...,m l 1, m n,m n,...,m n ). Note that PD k,{xj } n 1 j=1,{ξ i},l+1 P PD k,{x j } n 1 j=1,{ξ i},l P and l=1 PD k,{xj } n 1 j=1,{ξ i},l P = P D k,{xj },np. Let ǫ > 0 and let a finite subset F P D k,{xj } n 1 j=1,n 1P Q be given. Let {F 0,m } be an increasing sequence of finite subsets of P D k,{xj } n 1 j=1,n 1P Q whose union is dense in P D k,{xj } n 1,n 1P Q. Let {F,m} be an increasing sequence of finite subsets of j=1 P D k,{xj } n j=1,n P Q whose union is dense in P D k,{xj } n j=1,n P Q. Let {F l,m } be an an increasing sequence of finite subsets of PD k,{xj } n 1 j=1,{ξ i},l P Q whose union is dense in PD k,{x j } n 1 j=1,{ξ i},l P Q. By replacing F j,m by s j F s,m, we may assume that F j+1,m F j,m, j = 0,1,2,..., and F,m F j,m, j = 1,2,... Let {ǫ n } be a sequence of decreasing positive numbers such that n=1 ǫ n < ǫ/2. We may also assume that F = F 0,1. By applying 4.4, one obtains, for each l, a unital monomorphism ψ l : PD k,{xj } n 1 j=1,{ξ i},l 1 P Q PD k,{xj } n 1 j=1,{ξ i},l P Q which maps C k,{x j } n 1 j=1,{ξ i},l 1 P into C k,{x j } n 1 j=1,{ξ P such that i},l id PDk,{Xj } n 1 j=1,{ξ i },lp Q ǫl /2 ψ n ı on G l 1,l, (e 4.35) where G 0,1 = F 0,1 = F and G l 1,l = F l 1,l ψ l 1 (G l 1,l ), l = 2,3,... Define A = lim l (PD k,{xj } n 1 j=1,{ξ i},l 1 P Q,ψ l). We claim that A = P D k,{xj },np Q. Put B l = PD k,{xj } n 1 j=1,{ξ i},l 1 P Q, l = 1,2,... and put C = C(X,M k) Q. Note that B 1 = P D k,{xj } n 1 j=1,n 1P Q. l 12

13 Consider the following diagram: ψ 1 ψ 2 ψ 2 B 1 B2 B3 A ı ı ı C id C id C id C. It follows from (e 4.35) that the diagram is one-sided approximately intertwining. Therefore, by an argument of Elliott (see Theorem of [6], for example) there exists a homomorphism Φ : A C = C(X,M k ) Q such that Φ ψ l, P j=l ǫ ı on G l l 1,l and (e 4.36) Φ ψ l, (b) = lim ı ψ l,s (b) s (e 4.37) for all b B l 1. Combining (e 4.35) and (e 4.37), we conclude that for all b B l. Since C m C l, if m l, we conclude that for all b C m, m = l,l + 1,... It follows that Therefore dist(φ ψ l, (b),c l ) = 0 (e 4.38) dist(φ ψ m, (b),c l ) = 0 (e 4.39) dist(φ(a),c l ) = 0 for all l. (e 4.40) Φ(A) l=1 C l = PD k,{xj },np Q. (e 4.41) Put D = PD k,{xj },np Q. Then ı : D B l for each l. Thus, as above, we obtain the following one-sided approximately intertwining: D id D id D id D ı ı ı ψ 1 ψ 2 ψ 3 B 1 B2 B3 A. From this we obtain a homomorphism J : D A such that for all d D. We then compute (by e 4.37) and (e 4.35)) that J(d) = lim l ψ l, ı(d) (e 4.42) Φ J = id D. (e 4.43) This, in particular, implies that Φ maps A onto D. We then conclude that Φ gives an isomorphism from A to D and Φ 1 = J. Put Ψ = Φ ψ 1, : PD k,{xj } n 1 j=1,n 1P Q D. Then, by (e 4.36), we have id D ǫ/2 Ψ ı on F = F 0,1. (e 4.44) Moreover, Ψ is a unital monomorphism. Furthermore, we also note that Ψ maps C k,{xj } n 1 j=1,n 1P 1 Q into C k,{xj },np 1 Q. 13

14 We now use the induction. The same argument used in the proof of 4.4 proves that there exists a unital monomorphism ψ : PC(X,M k )P Q PD k,{xj },np Q such that id D ǫ ψ ı on F (e 4.45) and ψ maps (C(X,M k ) P) 1 Q ) into (C k,{xj },n P) 1 Q ). This proves the first two parts of the statement. To prove the last part of the statement, let F 0 Q be a unital finite dimensional C - subalgebra. We may assume, without loss of generality, that F 0 M m!. We may also assume that m > 2(dim(X) + 1) and 2(rank(P) m! dim(x)) rank(p) m!. Put r = rank(p) m! dim(x). There is a system of matrix units {e i,j } r i,j=1 PC(X,M k)p M m! such that e i,i is a rank one trivial projections and r P 1 Mm! = q 0 e i,i. where q 0 is a projection of rank dim(x). Let E be the C -subalgebra generated by (C(X) P 1 Mm! ) e 1,1 and {e i,j : 1 i,j r}. There is a projection q 00 E with rank dim(x) and a unitary w 0 PC(X,M k )P M m! such that i=1 w 0 q 00w 0 = q 0. It follows that there exists a projection P 1 M 2 (E) and a unitary W 0 M 2 (PC(X,M k )P M m! ) such that W 0 P 1M 2 (E)P 1 W 0 = PC(X,M k )P M m!, (e 4.46) For any δ > 0, there is an integer N m such that dist(ψ(e i,j ),D M N! ) < δ for some integer N m, 1 i,j r. It follows from Lemma of [6] that, by choosing small δ, there exists a unitary W 1 D M N! such that W 1 1 < ǫ/2 and W 1 ψ(e i,j )W 1 D M N!, i,j = 1,2,...,r. Since ψ(c(x) P 1 Q ) is in the center of D Q, ψ(fw 1 e i,jw 1 ) = ψ(f)ψ(w 1 ) ψ(e i,j )ψ(w 1 ) (e 4.47) for all f C(X) P 1 Q and 1 i,j r. It follows that = ψ(w 1 ) ψ(f)ψ(e i,j )ψ(w 1 ) = ψ(w 1 ) ψ(fe i,j )ψ(w 1 ) (e 4.48) ψ(w 1 ) ψ(e)ψ(w 1 ) PD k,{xj },np M N!. Put P 2 = (ψ id M2 )(P 1 ). Then P 2 M 2 (PD k,{xj },np M N! ). Define W 3 = By (e 4.46), ( ) ψ(w1 ) 0. 0 ψ(w 1 ) ψ(pc(x,m k )P M m! ) = (ψ id M2 )(PC(X,M k )P M m! ) (e 4.49) Put W = W 3 (ψ id M 2 (W 0 )). = (ψ id M2 )(W 0 P 1M 2 (E)P 1 W 0 ) (e 4.50) = (ψ id M2 )(W 0 ) (P 2 M 2 (ψ(e))p 2 )(ψ id M2 )(W 0 ) (e 4.51) = (ψ id M2 )(W 0 ) (W 3 (W 3 P 2M 2 (ψ(e))p 2 )W 3 )W 3 )(ψ id M 2 )(W 0 ) (e 4.52) ψ id M2 (W 0 ) (W 3 (P 2 M 2 (PD k,{xj },np M N! )P 2 )W 3 )(ψ id M2 )(W 0 )(e 4.53) 14

15 5 The main results Theorem 5.1. Let A = lim n (A n,ϕ n ) be a unital simple inductive limit of generalized dimension drop algebras. Then TR(A Q) 1. Proof. Write Q = n=1 F n, where each F n is a unital simple finite dimensional C -subalgebras and F n F n+1, n = 1,2,... Let Φ n = ϕ n id Q : A n Q A n+1 Q, n = 1,2,... Write A n = m(n) j=1 P n,j D(X n,j,r(n,j))p n,j, where X n,j is a connected compact metric space with finite dimension and with property (A), r(n,j) 1 is an integer, D(X n,j,r(n,j)) = D r(n,j) (X n,j,r(n,j),y n,j,1,,...,y n,j,d(n,j),m n,j,1,...,m n,j,d(n,j) ), Y n,j,1,y n,j,2,...,y n,j,d(n,j) are disjoint compact subsets of X n,j and P n,j D(X n,j,r(n,j)) is a projection, n = 1,2,... Define B n = m(n) j=1 P n,j C(X n,j,m r(n,j) )P n,j, n = 1,2,... Denote by ı n : A n B n the embedding. Denote d n = max{dim(x n,j : 1 j m(n)}. Let {F n } be an increasing sequence of finite subsets of A Q for which F n Φ n, (A n Q), n = 1,2,..., and n=1 F n is dense in A Q. Let G n A n Q be a finite subset such that Φ n, (G n ) = F n, n = 1,2,... Without loss of generality, we may assume that G n A n F n, n = 1,2,... Let {ǫ n } be a sequence of decreasing positive numbers such that n=1 ǫ n < 1/2. It follows from 4.5 that there exists a unital monomorphism Ψ 1 : B 1 Q A 1 Q such that Ψ 1 ı 1 ǫ1 id A1 Q on G 1. (e 5.54) Moreover, by 4.5, there exists a unital finite dimensional C -subalgebra E 1,1 Q, a unital C -subalgebra C F1 A 1 Q, a projection P 1 M 2 (A 1 E 1,1 ) and a unitary W 1 M 2 (A 1 Q) such that Ψ 1 (B 1 F 1 ) C F1 and W 1 C F 1 W 1 = P 1 M 2 (A 1 E 1,1 )P 1. Put U 1 = 1 B1 and C 1 = B 1 F 1. Define θ 1 = Φ 1 Ψ 1 : B 1 Q A 2 Q. Suppose that {S 1,k } is an increasing sequence of finite subsets of B 1 Q such that k=1 S 1,k is dense in B 1 Q. We may assume that S 1,k m(1) j=1 P 1,j C(X 1,j,M r(1,j) )P 1,j F k. Define S 1 = S 1,1 ı 1 (G 1 ) and G 2 = G 2 θ 1 (S 1 ). To simplify the notation, we may assume, without loss of generality, that there exists a finite subset G 2 A 2 F 2 such that G 2 ǫ 2 /2 G 2. By applying 4.5, there exists a unital monomorphism Ψ 2 : B 2 Q A 2 Q such that Ψ 2 ı 2 ǫ2 id B2 on G 2 G 2. (e 5.55) 15

16 Moreover, by 4.5, there exists a unital finite dimensional C -subalgebra E 2,1 Q, a unital C - subalgebra C F2 A 2 Q, a projection P 2 M 2 (A 2 E 2,1 ) and a unitary W 2 M 2 (P 2 (A 2 Q)P 2 ) such that Ψ 2 (B 2 E 1,1 ) C F2 and (e 5.56) W 2 C F2 W 2 = P 2 M 2 (A 2 E 2,1 )P 2. (e 5.57) Define ψ 1 = ı 2 Φ 1 Ψ 1 : B 1 Q B 2 Q. Define θ 2 = Φ 2 Ψ 2. We have Also, by (e5.54) and (e5.55), we have ı 2 θ 1 = ψ 1. (e 5.58) θ 1 ı 1 ǫ1 Φ 1 on G 1 and (e 5.59) θ 2 ı 2 ǫ2 Φ 2 on G 2 G 2. (e 5.60) Let {S 2,k } be an increasing sequence of finite subsets of B 2 Q such that its union is dense in B 2 Q. We may assume that ψ 1 (S 1,2 ),ı 2 (G 2 G 2 ) ǫ 3 /2 S 2,2. (e 5.61) We may also assume that (Φ 1 id M2 )(W 1 )S 2,k(Φ 1 id M2 )(W 1 ) (Φ 1 id M2 )(P 1 )M 2 (B 2 E 2,k )(Φ 1 id M2 )(P 1 ), where {E 2,k } is an increasing sequence of finite dimensional C -subalgebras of Q whose union is dense in Q. We assume that E 1,1 E 2,2 and E 2,2 is simple and d 3 2 rank(e 2,2 ) < 1/22. Define C 2 = W 1 (Φ 1 id M2 )(P 1 )M 2 (B 2 E 2,2 )(Φ 1 id M2 )(P 1 )W1 B 2 Q. Define U 2 = W1. Then C 2 = U2 (Φ 1 id M2 )(P 1 )M 2 (B 2 E 2,2 )(Φ 1 id M2 )(P 1 )U 2. Moreover, ψ 1 (C 1 ) C 2. Define S 2 = S 2,2 ı 2 (G 2 ) C 2 and G 3 = G 3 θ 2 (S 2 ). We may assume that there is a finite subset G 3 A 3 F 3 such that G 3 ǫ 3 /2 G 3. We will use induction. Suppose that unital monomorphism Ψ n : B n Q A n Q, ψ n = ı n+1 Φ n Ψ n : B n Q B n+1 Q, θ n = Ψ n Φ n : B n Q A n+1 Q (n = 1,2,...,N), a unital finite dimensional C -subalgebra E n,1 Q (n = 1,2,...,N 1), a projection P n M 2 (A n E n,1 ) and a unitary W n M 2 (A n E n,1 ) (n = 1,2,...,N) have been constructed such that θ n ı n ǫn Φ n on G n G n, (e 5.62) ψ n = ı n+1 θ n, (e 5.63) W nψ n (B n E n 1,n 1 )W n P n M 2 (A n E n,1 )P n, (e 5.64) where {E n,k } is an increasing sequence of unital finite dimensional C -subalgebras of Q whose union is dense in Q, with E n 1,n 1 E n,1, E n,n is a full matrix algebra and d 3 n rank(e n,n ) < 1/2n, (e 5.65) where {S n,k } is an increasing sequence of finite subsets of B n Q whose union is dense in B n Q, n 1 j=1 ψ j,n 1(S j,n ) ı n (G n G n) ǫn+1 /2 S n,n, (e 5.66) 16

17 U n S n,k U n P (n 1) M 2 n 1(B n E n,k )P (n 1), (e 5.67) where U n = (Φ n 1 id M2 n 1 )(W n 1 1 M 2 )ψ (n 1) n 1 (U n 1), n = 3,4,...,N + 1, where P (n 1) M 2 n 1(B n E n,n ) with U n P (n 1) U n = 1 Bn Q = 1 Bn E n,n, where G n+1 = G n+1 θ n (S n ), G n+1 ǫ n+1 /2 G n+1, G n+1 A n+1 F n+1 is a finite subset, n = 1,2,...,N + 1 (with G 1 = G 1), where ψ (i) n = ψ n id M2 i, i = 1,2,... Furthermore, C n = U n P (n 1) M 2 n 1(B n E n,n )P (n 1) U n and ψ n 1 (C n 1 ) C n, n = 2,3,...,N+ 1. By applying 4.5, we obtain a unital monomorphism Ψ N+1 : B N+1 Q A N+1 Q such that Ψ N+1 ı N+1 ǫn+1 id BN+1 on G N+1 G N+1. (e 5.68) Moreover, there exists a unital finite dimensional C -subalgebra E N+1,1 Q, a projection P N+1 M 2 (A N+1 E N+1,1 ) and a unitary W N+1 M 2 (A N+1 E N+1,1 ) such that W N+1(Ψ N+1 (B N+1 E N+1,N+1 ))W N+1 P N+1 M 2 (A N+1 E N+1,1 )P N+1. (e 5.69) Define θ N+1 = Φ N+1 Ψ N+1 : B N+1 Q A N+2 Q and ψ N+1 = ı N+2 θ N+1 : B N+1 Q B N+2 Q. Then, by(e5.62) θ N+1 ı N+1 ǫn+1 Φ N+1 on G N+1. (e 5.70) Let {E N+1,k } be an increasing sequence of unital finite dimensional C -subalgebras of Q whose union is dense in Q and E N,N E N+1,1, E N+1,N+1 is a full matrix algebra with d N+1 rank(e N+1,N+1 ) < 1/2N+1. (e 5.71) Let {S N+1,k } be an increasing sequence of finite subsets of B N+1 Q whose union is dense in B N+1 Q. We may assume that N j=1 ψ j,n (S j,n ) ı n (G n G n) ǫn+1 /2 S N+1,N+1. Since U N+1 P (N) U N+1 = 1 BN+1 1 Q, we may further assume that U N+1 S N+1,k U N+1 (Φ N+1 id M2 )(P (N) )M 2 (B N+1 E N+1,k )(Φ N+1 id M2 )(P (N) ). Define U N+2 = (Φ N+1 id )(W M2 N N+1 1 M 2 N )ψ(n+1) N+1 (U N+1). Put ŪN+i = (ψ N+i id M2 )(U N+i ), i = 1,2, and Φ (i) m = Φ m id,m,i = 1,2,..., and P (N) M2 i = ψ (N) N+1 (P (N) ). Then (using (e 5.64)) ψ N+1 (C N+1 ) = ψ (N) N+1 (U N+1P (N) M 2 N(B N+1 E N+1,N+1 )P (N) U N+1 ) = Ū N+1 P (N) ψ (N) N+1 (M 2 N(B N+1 E N+1,N+1 )) P (N) Ū N+1 Ū N+1 P (N) M 2 N(Φ (2) N+1 (W N+1P N+1 )M 2 (B N+2 E N+2,N+2 )Φ (2) N+1 (P N+1W N+1)) P (N) Ū N+1 Note that Φ (2) N+1 (W N+1P N+1 WN+1 ) has the form ( ) 1BN+2 1 EN+2,N

18 Therefore P (N) Φ (2) N+1 (W N+1P N+1 W N+1) 1 M2 N = P (N). (e 5.72) Put W N+1 = Φ (N+1) N+1 (W N+1 1 M2 N ) and P (N+1) = W N+1 P (N) WN+1. (e 5.73) One has U N+2P (N+1) U N+2 = ψ (N+1) N+1 (U N+1) WN+1 P (N+1) W N+1 ψ (N+1) N+1 (U N+1) (e 5.74) = ψ (N+1) N+1 (U N+1) P (N) ψ (N+1) N+1 (U N+1) (e 5.75) = ψ (N+1) N+1 (U N+1P (N) U N+1 ) = ψ (N+1) N+1 (1 B N+1 Q) = 1 BN+2 Q. (e 5.76) It follows that (see (e 5.73)) Define Therefore ψ N+1 (C N+1 ) (e 5.77) Ū N+1 P (N) WN+1 M 2 N+1(B N+2 E N+2,N+2 ) W N+1 P (N) Ū N+1 (e 5.78) = Ū N+1 W N+1 P (N+1) M 2 N+1(B N+2 E N+2,N+2 )P (N+1) W N+1 Ū N+1 (e 5.79) = U N+2 P (N+1) M 2 N+2(B N+2 E N+2,N+2 )P (N+1) U N+2. (e 5.80) C N+1 = U N+2 P (N+1) M 2 N+2(B N+2 E N+2,N+2 )P (N+1) U N+2. ψ N+1 (C N+1 ) C N+2. Put E n = A n Q and B n = B n Q and define C = lim n (B n,ψ n). It follows from (e 5.62) and (e 5.63) that the following diagram Φ E 1 Φ 1 2 Φ E2 3 E3 ց ı 1 ր θ1 ց ı 2 ր θ2 ց ı 3 ψ 1 B ψ 2 2 B 3 B 1 is approximately intertwining in the sense of Elliott. It follows that A Q = C. In particular, C is a unital simple C -algebra. By (e 5.67), S n,n C n, n = 1,2,... It follows from this and (e 5.66) that Thus n=1 ψ n, (C n ) = C. C = lim n (C n,(ψ n ) Cn ). It follows from (e 5.65) that C is a unital simple AH-algebra with very slow dimension growth. It follows from a theorem of Guihua Gong (see Theorem 2.5 of [8]) that TR(C) 1. Now we are ready to prove Theorem The proof of Theorem 1.3: 18

19 Proof. It follows from 5.1 that A, B A. Thus the theorem follows from 2.11 (see [11]) A unital simple inductive limit of generalized dimension drop algebra A is said to have no dimension growth, if A = lim n (A n,ϕ n ), where A n = m(n) j=1 P n,j D n,j P n,j, D n,j = D k(n) (X n,j,y n,1,y n,2,...,y n,s(n,j),m n,1,m n,2,...,m n,s(n,j) ), X n,j is a finite dimensional connected compact metric space with property (A), {Y n,i : 1 i s(n,j)} is a collection of finitely many disjoint compact subsets of X n,j, P n,j D n,j is a projection and {dim(x n,j )} is bounded The proof of Theorem 1.4: Proof. By [22], ASH-algebras with no dimension growth are of finite decomposition rank. It follows from another recent result of W. Winter ([24]) that A and B are Z-stable. Therefore the theorem follows from 1.3 immediately. References [1] J. Dixmier, On some C -algebras considered by Glimm, J. Funct. Anal. 1, [2] G. A. Elliott, G. Gong, Guihua, X. Jiang, H. Su, A classification of simple limits of dimension drop C -algebras, Operator algebras and their applications (Waterloo, ON, 1994/1995), , Fields Inst. Commun., 13, Amer. Math. Soc., Providence, RI, 1997 [3] G. A. Elliott, G. Gong and L. Li On the classification of simple inductive limit C - algebras, II: The isomorphism theorem, Invent. Math. 168 (2007), no. 2, [4] X. Jiang and H. Su, On a simple unital projectionless C*-algebra, Amer. J. Math 121 (1999), no. 2, [5] H. Lin, Tracial topological ranks of C -algebras, Proc. London Math. Soc., 83 (2001), [6] H. Lin, An introduction to the classification of amenable C*-algebras, World Scientific Publishing Co., River Edge, NJ, [7] H. Lin, Traces and simple C*-algebras with tracial topological rank zero, J. Reine Angew. Math. 568 (2004), [8] H. Lin, Simple nuclear C*-algebras of tracial topological rank one, J. Funct. Anal. 251 (2007), [9] H. Lin, Approximate homotopy of homomorphisms from C(X) into a simple C -algebra, preprint, arxiv.org/ OA/ [10] H. Lin, Asymptotic unitary equivalence and asymptotically inner automorphisms, arxiv:math/ (2007). 19

20 [11] H. Lin, Asymptotically unitary equivalence and classification of simple amenable C*- algebras, arxiv: (2008). [12] H. Lin, Localizing the Elliott conjecture at strongly self-absorbing C*-algebras, II an appendix, arxiv: v3 (2007). [13] H. Lin and Z. Niu, The range of a class of classifiable separable simple amenable C*- algebras, Adv. Math. 219 (2008) [14] G. J. Murphy, C -Algebras and Operator Theory,. Academic Press, Inc., Boston, MA, x+286 pp. ISBN: [15] J. Mygind, Classification of certain simple C*-algebras with torsion K 1, Canad. J. Math. 53 (2001), no. 6, [16] H. Su, On the classification of C -algebras of real rank zero: inductive limits of matrix algebras over non-hausdorff graphs, Mem. Amer. Math. Soc. 114 (1995), no. 547, viii+83 pp. [17] K. Thomsen, Limits of certain subhomogeneous C -algebras, Mm. Soc. Math. Fr. (N.S.) 71 (1997), vi+125 pp. [18] A. Toms, On the classification problem for nuclear C*-algebras, Ann. of Math., to appear (arxiv:math/ ). [19] A. Toms and W. Winter, Z-stable ASH algebras, Canad. J. Math. 60 (2008), [20] J. Villadsen, The range of the Elliott invariant of the simple AH-algebras with slow dimension growth. K-Theory 15 (1998), [21] J. Villadsen, On the stable rank of simple C -algebras, J. Amer. Math. Soc. 12 (1999), [22] W. Winter, Decomposition rank of subhomogeneous C -algebras, Proc. London Math. Soc. 89 (2004), [23] W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C*-algebras, arxiv: v3 (2007). [24] W. Winter, Decomposition rank and Z-stability, preprint (arxiv: ) (2008). 20