A class of simple C -algebras with stable rank one

Transcription

1 Journal of Functional Analysis ) wwwelseviercom/locate/jfa A class of simple C -algebras with stable rank one George A Elliott a,toanmho b, Andrew S Toms b, a Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4 b Department of Mathematics and Statistics, York University, 4700 Keele St, Toronto, Ontario, Canada M3J 1P3 Received 13 March 2007; accepted 4 August 2008 Available online 8 November 2008 Communicated by Alain Connes Abstract We study the limits of inductive sequences A i,φ i ) where each A i is a direct sum of full matrix algebras over compact metric spaces and each partial map of φ i is diagonal We give a new characterisation of simplicity for such algebras, and apply it to prove that the said algebras have stable rank one whenever they are simple and unital Significantly, our results do not require any dimension growth assumption 2008 Elsevier Inc All rights reserved Keywords: AH algebras; Stable rank; Dimension growth 1 Introduction Let X and Y be compact Hausdorff spaces A -homomorphism φ :M m CX) ) Mnm CY ) ) is called diagonal if there are n continuous maps λ i : Y X such that The authors were supported by the Natural Science and Engineering Council of Canada * Corresponding author Fax: addresses: elliott@mathtorontoedu GA Elliott), toan@mathstatyorkuca TM Ho), atoms@mathstatyorkuca AS Toms) /$ see front matter 2008 Elsevier Inc All rights reserved doi:101016/jjfa

2 308 GA Elliott et al / Journal of Functional Analysis ) f λ f λ 2 0 φf)= 0 0 f λ n The λ i are called the eigenvalue maps or simply eigenvalues of φ The multiset {λ 1,λ 2,,λ n } is called the eigenvalue pattern of φ and is denoted by epφ) This definition can be extended to -homomorphisms φ : n M ni CXi ) ) i=1 m M mj CYj ) ) j=1 by requiring, roughly, that each partial map φ ij :M ni CXi ) ) M mj CYj ) ) induced by φ be diagonal A precise definition can be found in Section 2) C -algebras obtained as limits of inductive systems A i,φ i ) where n i A i = M ni,j CXi,j ) ) j=1 and each φ i is diagonal form a rich class They include AF algebras, simple unital AT algebras and hence the irrational rotation algebras) [6], Goodearl algebras [10], and some interesting examples of Villadsen and the third named author connected to Elliott s program to classify amenable C -algebras via K-theory [15,17] The structure of these algebras is only well understood when they satisfy some additional conditions such as very) slow dimension growth or the combination of real rank zero, stable rank one, and weak unperforation of the K 0 -group situations in which the strong form of Elliott s classification conjecture can be verified [3,5,7 9] In this paper we give a new characterisation of simplicity for AH algebras with diagonal connecting maps As a consequence we are able to prove that such algebras have stable rank one whenever they are unital and simple The significance of our result derives from the fact that we make no assumptions on the dimension growth of the algebras; we obtain a general theorem on the structure of algebras heretofore considered wild As suggested by M Rørdam in his recent ICM address, it is high time we became friends with such algebras, as opposed to treating them simply as a source of pathological examples 2 Preliminaries 21 Basic notation We use M n to denote the set of n n complex matrices Given a closed subset E of a compact metric space X, d) and δ>0weset and make the convention that B δ ) = B δ E) = { x X de,x) < δ },

3 GA Elliott et al / Journal of Functional Analysis ) AH systems with diagonal maps Definition 21 We will say that a unital -homomorphism n φ : M ni CXi ) ) ) M k CY ) = Mk CY ) i=1 is diagonal if there exist natural numbers k 1,,k n such that i k i = k and n i k i, an embedding ι : and diagonal maps n M ki M k, φ i : M ni CXi ) ) M ki CY ) such that n φ = φ i Notice that k i = 0 is allowed) We will say that a unital -homomorphism φ : is diagonal if each restriction i=1 φ j : is diagonal i=1 i=1 n M ni CXi ) ) m M mj CYj ) ) j=1 n M ni CXi ) ) M mj CYj ) ) i=1 Let A be the limit of the inductive sequence A i,φ i ), where k i A i = M ni,t CXi,t ) ), 1) t=1 X i,t is a connected compact metric space, and n i,t and k i are natural numbers Define A i,t := M ni,t CXi,t ) ), X i := X i,1 X i,2 X i,ki, and

4 310 GA Elliott et al / Journal of Functional Analysis ) φ i,j := φ j 1 φ i Let φ t,l i,j :M n i,t CXi,t ) ) M nj,l CXj,l ) ) and k i φi,j l : M ni,t CXi,t ) ) M nj,l CYj,l ) ) t=1 denote the appropriate restrictions of φ If each φ i is unital and diagonal, then we refer to A i,φ i ) as an AH system with diagonal maps The limit algebra A will be called a diagonal AH algebra, and we will refer to A i,φ i ) as a decomposition of A Assume that A as above is diagonal We will view φ t,l i,j as a diagonal map from M n i,t CX i,t )) into the cut-down of M nj,l CX j,l )) by φ t,l i,j 1) For fixed i and j, we will denote by ep ij the multiset which is the union, counting multiplicity, of the eigenvalue patterns of each φ t,l i,j ;ep ij is the eigenvalue pattern of φ i,j ; an element of ep ij is an eigenvalue map of φ i,j For fixed i, j, and l, we will denote by ep l ij the multiset which is the union of the eigenvalue patters of each φt,l i,j Let us now show that the bonding maps φ i may be assumed to be injective Let A i,φ i ) be a decomposition for a diagonal AH algebra A as above Fix i N and 1 t k i For each j>i and 1 l k j,letx j,l i,t denote the closed subset of X i,t which is the union of the images of the eigenvalue maps of φ t,l i,j Put Xj i,t = l Xj,l i,t, and X i,t = j Xj i,t Since Xj i,t Xj+1 i,t,wehave that X i,t is closed subset of X i,t Define and à i,t = M ni,t C X i,t ) ) k i à i = à i,t t=1 Define diagonal maps φ t,l i,i+1 : à i,t à i+1,l by replacing the eigenvalue maps of φ t,l i,i+1 with their restrictions to X i+1,l Define φ i : à i à i+1 in a manner analogous to the definition of φ i It follows that à i, φ i ) is a diagonal AH system with limit A, and φ i is injective by construction We assume from here on that all bonding maps in diagonal AH systems are injective One way to construct a simple diagonal AH algebra is to ensure that for each i N and x in a specified dense subset of X i there is some j i such that for each l {1,,k j } the diagonal map φ t,l i,j contains the eigenvalue map ev x : X j,l X i,t given by ev x y) = x The next definition gives and approximate version of this situation Definition 22 Say that a diagonal AH algebra A with decomposition A i,φ i ) has the property P if for any i N, element f in A i, ɛ>0, and x X i there exist j i and unitaries u l A j,l, l {1,,k j } such that

5 GA Elliott et al / Journal of Functional Analysis ) ) fx) 0 u lφ t,l i,j f )u l <ɛ 0 b l for some appropriately sized b l Note that diagf x 0 ), b l ) A j,l ) We will prove in the sequel that A as in Definition 22 is simple if and only if it has property P 23 A characterisation of simplicity Proposition 21 of [4] gives some necessary and sufficient conditions for the simplicity of an AH algebra We will have occasion to apply these in the proof of our main result, and so restate the said proposition in the particular case of an AH system with diagonal injective maps Proposition 23 Let A i,φ i ) be an AH system with diagonal injective maps, and set A = lim i A i,φ i ) The following conditions are equivalent: i) A is simple; ii) For any positive integer i and any nonempty open subset U of X i, there is a j 0 i such that for every j j 0 and l {1,,k j } we have ep l ij ) 1U) = Xj,l, where ep l ij ) 1 U) denotes the union of the sets λ 1 U), λ ep l ij ; iii) For any nonzero element a in A i, there is a j 0 i such that for every j j 0, φ ij a)x) is not zero, for every x in X j 24 Paths between permutation matrices Given any permutation π S n,letu[π] denote the permutation matrix in M n corresponding to π, that is, U[π] is obtained from the identity of M n by moving the ith row to the πi)th row, for i {1, 2,,n} Any two permutation matrices are homotopic inside the unitary group UM n ) of M n, but we want to define some particular homotopies for use in the sequel Let π and σ be elements of S n, viewed as permutations of the canonical basis vectors e 1,,e n of C n Let R ={e W,1,,e W,dimW ) } be the set of basis vectors upon which π and σ agree, and choose γ S n be such that γσv)= γπv)= v, v R Then U[γ ]U[σ ] and U[γ ]U[π] fix e 1,,e R Put W = span{e 1,,e R }, and let V be the orthogonal complement of W There are a canonical unital embedding of M dimw ) M dimv ) into M n and unitaries u, v UM dimv ) ) such that U[γ ]U[π]=1 MdimW ) v; U[γ ]U[σ ]=1 MdimW ) u Choose a homotopy gt) between u and v inside UM dimv ) ) g0) = v and g1) = u and put ut) = U[γ ] 1 1 MdimV ) gt) )

6 312 GA Elliott et al / Journal of Functional Analysis ) Then ut) is a homotopy of unitaries between U[π] and U[σ ] such that 25 Applications of Urysohn s Lemma ut)v) = U[π]v) = U[σ ]v), v R, t [0, 1] Lemma 24 Let σ S n be given There is a homotopy u :[0, 1] UM n ) between u0) = 1 n and u1) = U[σ ] which moreover has the following property: for any complex numbers λ 1,λ 2,,λ n such that λ i = λ σi) for every i = 1, 2,,n, we have λ λ λ λ 2 0 ut) u t) =, t [0, 1] 0 0 λ n 0 0 λ n Proof Let us first consider the case that σ is a k-cycle The hypotheses of the lemma guarantee that the desired conclusion holds already for t {0, 1} Choose the homotopy ut) as in Section 24 by using our given value of σ and setting π equal to the identity element of S n The hypothesis λ i = λ σi), i {1,,n}, implies that there is a λ C such that for each i {1,,n} which is not fixed by σ we have λ i = λ In other words, if one decomposes diagλ 1,,λ n ) into a direct sum of two diagonal matrices using the decomposition C n = W V V and W as in Section 24 then the direct summand corresponding to V is scalar k k matrix By construction, ut) = vt) 1 n k, with vt) UM k ) It follows that ut) commutes with diagλ 1,,λ n ) for each t 0, 1) Now suppose that σ is any permutation on n letters, and write σ as a product of disjoint cycles: σ = σ 1 σ 2 σ l For each j {1,,l}, letu i j) denote the unitary path between U[id] and U[σ j ], constructed as in Section 24 Now ut) := u 1 t)u 2 t) u l t) is a path of unitaries with u0) = U[id] and u1) = U[σ ], and ut) commutes with diagλ 1,,λ n ) since each u j t) does Lemma 25 Let σ be any permutation in S n, and A,B disjoint nonempty closed subsets of a metric space X Let λ 1,,λ n : X C be continuous Then, there exists a unitary v M n CX)) such that i) vx) = 1 n, x A, ii) vx) = U[σ ], x B, and iii) vx) commutes with diagλ 1 x),,λ n x)) whenever λ i x) = λ σi) x) for each i {1,,n} Proof Find a unitary path ut) connecting U[id] to U[σ ] using Lemma 24, so that ut) commutes with diagλ 1 x),,λ n x)) for each t 0, 1) and each x X for which λ i x) = λ σi) x), i {1,,n} By Urysohn s Lemma, there is a continuous map f : X [0, 1] which is equal to

7 GA Elliott et al / Journal of Functional Analysis ) zero on A and equal to one on B It is straightforward to check that vx) := uf x)) satisfies the conclusion of the lemma Lemma 26 Let Y be a closed subset of a normal space X, and let f : Y S n be continuous Then, there is a continuous map f from X to the n + 1 disk D n+1 which extends f Proof View D n+1 as the cube [0, 1] n+1, with S n as its boundary Then fy)= f 1 y),, f n+1 y) ), where each f i : Y [0, 1] is continuous Extend each f i to a continuous map f i : X [0, 1], and put fy)= f 1 y),, f n+1 y) ) 3 The main theorem Let A be a diagonal AH algebra with decomposition A i,φ i ), and assume that the φ i are injective In this section we will prove that if A is simple then it has the property P cf Section 2) The converse also holds, but is easier by far) Let us begin with an outline of our strategy, before plunging headlong into the proof Assume first that A is simple and A i = M ni CX i )), so that there are no partial maps to contend with Let there be given a natural number i, an element f of A i, a point x 0 X i, and some ɛ>0 Put U = B ɛ x 0 ) By Proposition 23 there is a j 0 with j 0 i such that for any j j 0, X j = λ 1 1 U) λ 1U) λ 1 U), 2 n where f λ φ i,j f ) = 0 0 f λ n On each closed subset λ 1 t U), the range of the eigenvalue map λ t is within ɛ of x 0 Toshow that A has the property P, we require a unitary u in A j and an element b f M nj n i CX j )) such that ) uφ i,j f )u fx0 ) 0 <ɛ 0 b f We would like uy) to exchange the first and tth diagonal entries of φ i,j f ) whenever y λ 1 t U), but this operation is unlikely to be well-defined the sets λ 1 1 U),,λ 1 n U) need not be mutually disjoint The remainder of this section is devoted to overcoming this complication

8 314 GA Elliott et al / Journal of Functional Analysis ) Given positive integers m n, let us denote by M m CY )) 1 n m the set of all n n matrices of the form ) a 0, 0 1 where a M m CY )) and 1 n m M n m Theorem 31 Let there be given a diagonal -homomorphism φ : CX) M n CY )) with the eigenvalue pattern {λ 1,λ 2,,λ n }, a point x 0 in X, an element f of CX), and a tolerance ɛ>0 Choose η>0 such that fx) fy) <ɛwhenever dx,y) < 2η d is the metric on X) Suppose that F 1,,F m are nonempty closed subsets of Ym n) such that dλ i y), x 0 )<η whenever y F i Then, there is a unitary u in M m CY )) 1 n m and an element b M n m 1 CY )) such that for each y m i=1 F i we have fx 0 ) by) 0 0 uy)φf )y)u y) 0 0 λ m+1 y) 0 < 2ɛ 2) λ n y) Note that if n m 1 = 0, then there is no b in the matrix in 2)) Proof Let ρ denote the metric on Y Choose δ>0 such that dλ i x), λ i y)) < η whenever ρx,y) δ, i {1,,n} For each 1 i m, f λ i y) fx 0 ) <ɛ, for all y B δ F i ) Set ε i y) = f λ i y) fx 0 ) for all y in B δ F i ) Then, ε i is a continuous map from B δ F i ) to the disk of radius ɛ in the complex plane By Lemma 26, ε i can be extended to a continuous function from Y to the complex plane such that ε i ɛ let us also denote this extension map by ε i ) For m<i n,setε i = 0 For each i {1,,n}, put g i = f λ i ɛ i, so that g i CX) Set g = diagg 1,g 2,,g n ) Then, for each i {1,,m} and y B δ F i ),wehave g i y) = fx 0 ); if i {m + 1,,n}, then g i = f λ i For any unitary u M n CY )) we have uφf )u ugu = diagε1,,ε n ) < 2ɛ We have therefore reduced our problem to proving the following claim

9 GA Elliott et al / Journal of Functional Analysis ) Claim There is a unitary u in M m CY )) 1 n m such that fx 0 ) bx) 0 0 ugu m = 0 0 g m+1 x) 0, x F i, i= g n x) where b M n m 1 CY )) Proof We will assume that for some 1 k<mthere is a unitary u k M k CY )) 1 n k such that fx 0 ) bx) 0 0 u k gu k = k 0 0 g k+1 x) 0, x F i, 3) i= g n x) and then prove that the same statement holds with k replaced by k + 1 Since 3) clearly holds when k = 1 just take u 1 to be the identity matrix of M n CY )) this recursive argument will prove our claim Assume that 3) holds for some k<m Put B = F k+1 and A = Y \B δ B) Apply Lemma 25 with these choices of A and B and with σ = 1 k + 1) to obtain a unitary v M n CY )) We then have that v = 1 n on A and v = U[1 k + 1)] on B Inspecting the construction of v, we find that it has the following form: vy) = U [ 2 k + 1) ] v ) y) 0 U [ 2 k + 1) ], y Y, 4) 0 1 where v y) is a unitary matrix in M 2 CY )) equal to 1 2 on A and equal to U[12)] on B Define u k+1 := vu k Let us show that u k+1 satisfies the requirements of the claim It is clear that u k+1 is an element of M k+1 CY )) 1 n k 1 First suppose that y B, so that g k+1 y) = fx 0 ) Since u k+1 = vu k we have cy) 0 0 u k+1 y)gy)u k+1 y) = U[ 1 k + 1) ] 0 fx 0 ) 0 U[ 1 k + 1) ] 0 0 g n y)

10 316 GA Elliott et al / Journal of Functional Analysis ) fx 0 ) by) 0 0 = 0 0 g k+2 y) g n y) for some cy),by) M k Now suppose that y m i=1 F i \B δ B) Y \B δ B) In this case vy) = 1 n and u k+1 y) = u k y) and there is nothing to prove Finally, suppose that y B δ B)\B) m i=1 F i ) As in the case y B, wehaveg k+1 y) = fx 0 ) From 3) and this last fact we have fx 0 ) fx 0 ) U [ 2 k + 1) ] u k y)gy)u k y)u[ 2 k + 1) ] 0 0 dy) 0 0 = g k+2 y) g n y) for some dy) M k 1 Since the upper left 2 2 corner of the matrix above is scalar, the entire matrix commutes with v y) 1 n 2 It follows that u k+1 y)gy)u k+1 y) is equal to fx 0 ) fx 0 ) U [ 2 k + 1) ] 0 0 b 1 y) g k+2 y) 0 U [ 2 k + 1) ] g n y) Computing this product yields a matrix of the form fx 0 ) by) g k+2 y) 0, g n y) as required For any C -algebra B there is an isomorphism π : B M n M n B)

11 GA Elliott et al / Journal of Functional Analysis ) given by π b a ij ) ) ba 11 ba 1n = 5) ba n1 ba nn Proposition 32 Suppose that φ : M m CX)) M nm CY )) is a diagonal -homomorphism with epφ) ={λ 1,λ 2,,λ n } Let φ : CX) M n CY )) be the diagonal -homomorphism given by f λ f λ 2 0 φf) = 0 0 f λ n Then, φ id Mm :CX) M m M n CY )) M m is unitarily equivalent to φ Proof On the one hand we have f λ φ id Mm f cij ) ) 0 f λ 2 0 = c ij ), 0 0 f λ n while on the other we have f λ 1 ) c ij ) 0 0 φ f c ij ) ) 0 f λ 2 ) c ij ) 0 = 0 0 f λ n ) c ij ) With the identifications CX) M n = Mn CX)) and M n CY )) M m = Mnm CY )) given by 5) in mind, one sees that Ad U[π] ) φ id Mm ) = φ, where π is the permutation in S nm and given by πkn + i) = i 1)m + k + 1fork = 0, 1, 2,,m 1 and i = 1, 2,,n Corollary 33 Let φ : M m CX)) M nm CY )) be a diagonal -homomorphism with eigenvalue pattern {λ 1,λ 2,,λ n }, and let ɛ>0 be given Let f be any element of M m CX)) and choose η>0 such that fx) fy) <ɛ whenever dx,y) < 2η

12 318 GA Elliott et al / Journal of Functional Analysis ) d is the metric on X) Let U be the open ball centered at x 0 X with radius η, and suppose that Y = λ 1 1 U) λ 1U) λ 1 U) Then there is a unitary u in M nm CY )) and element b M nm m CY )) such that ) fx0 uφf ) 0 )u <ɛ 0 b 2 Proof By Proposition 32, we can assume that m = 1 Let F i be the closure of λ 1 i U) for each i Now, apply Theorem 31 Since Y = n i=1 F i, we are done Now, we are ready to prove the main theorem of this section Theorem 34 Let A = lim A i,φ i ) be a unital diagonal AH algebra Then, A is simple if and only if A has the property P of Definition 22 n Proof Suppose that A has property P Letf A i be nonzero, so that there is a point x 0 in X i such that fx 0 ) 0 By the definition of property P, there are an integer j>iand unitaries u l A j,l, l {1,,k j } such that ) u lφ t,l fx0 i,j f ) 0 )u l <ɛ 0 b l for some appropriately sized b l We may assume that ɛ< fx 0 ), so that φ ij f ) is nowhere zero This implies that the ideal of A j generated by φ ij f ) is all of A j, and that the ideal of A generated by the image of f is all of A Since f was arbitrary, A is simple Now assume that A is simple, and let f A i,t be nonzero Recall that the φ i maybetaken to be injective By Proposition 23 there exists, for each x 0 X i,t and ɛ>0, a j 0 >i with the following property: for every j j 0 and every l {1,,k j } we have where δ is some positive number such that ep 1 φ t,l i,j ) Bδ x 0 ) ) = X j,l, d fx),fy) ) <ɛ whenever dx,y) < 2δ As pointed out in Section 2, the map φ t,l i,j may be viewed as a diagonal map from A i,t into the cutdown of A j,l by the projection φ t,l i,j 1); any unitary u in this corner of A j,l gives rise to a unitary ũ in A j,l by setting ũ = u 1 Aj,l φ t,l i,j 1) Combining this observation with Corollary 33 we see that there exists, for each l {1,,k j }, a unitary u l A j,l such that ) u lφ t,l fx0 i,j f ) 0 )u l <ɛ 0 b l Thus, A has property P, as desired

13 GA Elliott et al / Journal of Functional Analysis ) Stable rank Theorem 41 Let A = lim A i,φ i ) be a simple unital diagonal AH algebra Then, A has stable rank one Before proving Theorem 41, we situate it relative to other results on the stable rank of general approximately homogeneous AH) algebras Recall that an AH algebra is an inductive limit C - algebra A = lim i A i,φ i ), where n i A i = p i,l CXi,l ) K ) p i,l 6) l=1 for compact connected Hausdorff spaces X i,l, projections p i,l CX i,l ) K, and natural numbers n i IfA is separable, then one may assume that the X i,l are finite CW-complexes [1,11] The inductive system A i,φ i ) is referred to as a decomposition for A All AH algebras in this paper are assumed to be separable If an AH algebra A admits a decomposition as in 6) for which { dimxi,1 ) max 1 l n i rankp i,1 ),, dimx } i,n i ) i 0, rankp i,ni ) then we say that A has slow dimension growth Theorem 1 of [2] states that every simple unital AH algebra with slow dimension growth has stable rank one Villadsen in [17] constructed simple diagonal AH algebras which do not have slow dimension growth, but which do have stable rank one; the converse of [2, Theorem 1] does not hold There are in fact a wealth of simple diagonal AH algebras without slow dimension growth which exhibit all sorts of interesting behaviour cf [14 16]), whence Theorem 41 is widely applicable Simple AH algebras may have stable rank strictly greater than one, and there is reason to believe that Theorem 41 is quite close to being best possible One might be able to generalise our result to the setting of AH algebras where the projections φ i,j p i,l ) appearing in 6) can be decomposed into a direct sum of a trivial projection θ j and a second projection q j such that τq j ) 0asj for any trace τ Otherwise, one finds oneself in a situation very similar to the construction of [18], where the stable rank is always strictly greater than one Let us now prepare for the proof of Theorem 41 Lemma 42 Let a M m CX)) be block diagonal, ie, a 1 0 a =, 0 0 a n where a i M ki CX)) for natural numbers k 1,,k n If the size of the matrix 0 in the upper lefthand corner of a is strictly greater than max 1 i n k i, then a can be approximated arbitrarily closely by invertible elements in M m CX))

14 320 GA Elliott et al / Journal of Functional Analysis ) Proof Let ɛ>0 be given, and let k denote the size of the matrix 0 in the upper left-hand corner of a Since k>k i, i {1,,n}, there is a permutation matrix U such that Ua = is nilpotent As was proved in [13], every nilpotent element in a unital C -algebra can be approximated arbitrarily closely by invertible elements We may thus find an invertible element b M m CX)) such that Ua b <ɛ, and a U 1 b = U 1 Ua U 1 b U 1 Ua b <ɛ The lemma now follows from the fact that U 1 b is invertible It easy to prove that a M n CX)) is invertible if and only if ax) is invertible for each x X The proof of the next lemma is also straightforward Lemma 43 Let p,q be orthogonal projections in a C -algebra A, and let ɛ>0 be given If elements a and b in A can be approximated to within ɛ by invertible elements in pap and qaq, respectively, then a + b can be approximated to within ɛ by an invertible element in p + q)ap + q) Now, we are ready to prove Theorem 41 Proof of Theorem 41 Since every element in A can be approximated arbitrarily closely by elements in i=1 A i, it will suffice to prove for any ɛ>0 and any a A i, there is an invertible element in A whose distance to a is less than ɛ Note that we are using the injectivity of the φ i to identify A i with its image in A) By Lemma 43, we may assume that A i = M ni CX i )) Wealsoassumethata is not invertible By the comment preceding Lemma 43, there is a point x 0 X i such that detax 0 )) = 0 There are permutation matrices u, v M ni and a matrix c M ni 1 such that ) 0 0 uax 0 )v = 0 c Let b denote the element uav Following the lines of the proof of Lemma 42, it will suffice to prove that b can be approximated to within ɛ by an invertible element of A For each j>i, φ i,j b) is a tuple of k j elements If each coordinate of φ i,j b) can be approximated to within ɛ by an invertible element in the corner of A generated by the unit of A j,l, then φ i,j b) can be approximated to within ɛ by an invertible element of A We may therefore assume that A j = M nj CY j ), and concern ourselves with proving that φ i,j b) is approximated to within ɛ by an invertible element in A

15 GA Elliott et al / Journal of Functional Analysis ) By Theorem 34, there exist an integer j>i, a unitary w A j and an element b such that We have ) wφ ij b)w bx0 ) 0 0 b <ɛ/2 ) ) bx0 ) b = 0 b, where b = diagc, b ) Put d = wφ i,j b)w, and note that it will suffice to prove that φ j,m d) is approximated to within ɛ/2 by an invertible element in A m for some m>j Since A is simple, there is an integer m>j large enough so that, for each t {1,,k m }, φ 1,t j,m either the number of the eigenvalue maps of φ 1,t j,m counted with multiplicity is strictly larger than the size of the matrix b, or the image of φ 1,t j,m is finite-dimensional In the latter case, d) is approximated to within ɛ by an invertible element in the image of φ1,t since finitedimensional C -algebras have stable rank one, so we may assume that the number of eigenvalue maps of φ 1,t j,m, counted with multiplicity, is strictly larger than the size of the matrix b Then, φ 1,t j,m d) is unitarily equivalent to b b l inside A m,t, where b k is the composition of b and the kth eigenvalue map of φ 1,t j,m, and the size of the matrix 0 at the upper left-hand corner of the above matrix is strictly bigger than the size of the matrix b k for every k By Lemma 42, the matrix above can be approximated to within ɛ/2 by an invertible in A m,t, as required Corollary 44 Let A be a simple unital diagonal AH algebra If A has real rank zero and weakly unperforated K 0 -group, then A is tracially AF Proof By Theorem 41, A has stable rank one The corollary then follows from a result of Lin [12, Theorem 21] Corollary 44 applies, for instance, to simple unital diagonal AH algebras for which the spaces X i in some diagonal decomposition for A are all contractible This contractibility hypothesis may seem strong, but it does not substantially restrict the complexity of A; if one wants to classify all such A via K-theory and traces, then the additional assumption of very slow dimension growth and the full force of [8] and [9] are required; the collection of all such cannot be classified by topological K-theory and traces alone [15] j,m

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