Be a Normal Operator. the Weyl-Von Neumann-Berg Theorem Says That ∈

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 2, February 1997, Pages 529–545 S 0002-9947(97)01851-5

THE GENERALIZED BERG THEOREM AND BDF-THEOREM

HUAXIN LIN

Abstract. Let A be a separable simple AF -algebra with ﬁnitely many extreme traces. We give a necessary and suﬃcient condition for an essentially normal element x M(A), i.e., π(x)isnormal(π:M(A) M(A)/A is the quotient map), having∈ the form y + a for some normal element→ y M(A) and a A. We also show that a normal element x M(A) can be∈ quasidiagonalized∈ if and only if the Fredholm index ind(λ∈ x) = 0 for all λ − 6∈ sp(π(x)). InthecasethatAis a simple C∗-algebra of real rank zero, with stable rank one and with continuous scale, K1(A)=0,and K0(A) has countable rank, we show that a normal element x M(A) with zero Fredholm index can be written as ∈ ∞ x = λn(en en 1)+a, − − n=1 X where en is an (increasing) approximate identity for A consisting of projec- { } tions, λn is a bounded sequence of numbers and a A with a <for any given >{ 0}. ∈ k k

1. Introduction Let x B(l2) be a normal operator. The Weyl-von Neumann-Berg theorem says that ∈

∞ x = λn(en en 1)+a, − n=1 − X where en is an (increasing) approximate identity for consisting of projections, { } K λn is a bounded sequence, a and a is smaller than any given posi- { } ∈K k k 2 tive number, where is the C∗-algebra of compact operators on l . Note that B(l2)=M( ),the multiplierK algebra of . Recently, Terry Loring and Jack Spiel- berg ([LS])K show that there are normal elementsK in M(A) which are not quasidi- agonalizable for some separable simple AF -algebra A with a unique trace (up to scalar multiplication). Their counterexamples are based on the fact that a normal element in M(A) may have nonzero Fredholm index. We will consider those separable simple AF -algebras with ﬁnite many (extreme) traces. We found that the Fredholm index is the only obstruction for quasidiagonality. In the case that A has continuous scale and K0(A) has countable rank, we show that a normal element

Received by the editors July 27, 1993. 1991 Mathematics Subject Classiﬁcation. Primary 46L05. Key words and phrases. The Berg theorem, the BDF-theorem, weak (FN), C∗-algebra with real rank zero.

c 1997 American Mathematical Society

529

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x M(A) can be written as ∈

∞ x = λn(en en 1)+a, − n=1 − X where en is an approximate identity for A consisting of projections, λn is a bounded{ sequence} of numbers and a A with a <for any given {>} 0,if and only if the Fredholm index ind(λ ∈ x) = 0 fork k all λ sp(π(x)(π:M(A) M(A)/A is the quotient map). We also− found that this6∈ obstruction is small,→ in fact, it is “infinitesimal”, in the sense that ind(λ x) must be an infinitesimal − element in K0(A), if x is a normal element in M(A). Therefore if A is a separable simple AF -algebra with finite extreme traces and with no infinitesimal elements in K0(A), then every normal element in M(A) can be quasidiagonalized. A related problem is when an essentially normal element (i.e. π(x)isnormalinM(A)/A, where π : M(A) M(A)/A is the quotient map) x M(A) can be written as x = y +a, where y→ M(A) is a normal element and a ∈A. One might imagine that ind(λ x) = 0 for∈ all λ sp(π(x)) would be necessary∈ and sufficient for x = y + a. However,− we found that6∈ ind(λ x) = 0 is neither necessary nor sufficient. It turns out, with some surprise, that− an essentially normal element x M(A)canbe ∈ written as x = y+a if and only if ind(λ x) is an infinitesimal element in K0(A)for every λ sp(π(x)) and the remaining Γ(−x) is zero, where Γ is an index which will be defined6∈ in section 2. This implies that every essentially normal element in M(A) has the form y + a, where y M(A)isnormalanda A, if A is a finite matroid algebra. One may view these∈ results as some version of∈ a generalized BDF-theorem for essentially normal elements.

Acknowledgements This work was done while the author was visiting the State University of New York at Buﬀalo in 1992. The author would like to thank Lewis Coburn and Cather- ine Olsen for their eﬀorts to make this visit possible. He is grateful to Terry Loring for sending him the preprint [LS].

2. C∗-algebras with weak (FN)

Recall that a C∗-algebra A is said to have real rank zero if the set of selfadjoint elements with finite spectrum is dense in As.a.. This class of C∗-algebras is cur- rently under intensive study (see [BP], [BBEK], [BDR], [BKR], [Ell2]-[Ell3], [EE], [G], [R], [Zh1]-[Zh6], [EGLP]and [Ln3]-[Ln13], etc.). In this section, we consider the problem when a normal element in a C∗-algebra of real rank zero can be approximated by normal elements in the algebra with finite spectrum. It is clear that the unitaries which are not connected with the identity in the unitary group can not be approximated by normal elments with finite spectra. The notion of weak (FU) (a C∗-algebra A is said to have weak (FU) if the set of unitaries with finite spectra is dense in the connected component of a unitary group of A˜ containing the identity) was introduced by N. C. Phillips ([Ph]). It has been shown ([Ln7]) that every C∗-algebra with real rank zero has weak (FU). The problem for general normal elements is much more complicated. We begin with the following definition.

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Deﬁnition 2.1. Let φ : C(X) A be a monomorphism. The map φ induces a homomorphism →

φ : K1(C(X)) K1(A). ∗ → In particular, if x is a normal element in A, then x gives a monomorphism φ : C(sp(x)) A. We define γ(x)=φ .Let I be a (closed and two sided) ideal of A. → ∗ Suppose that J is the kernel of the composition πI φ. Then ◦ C(X)/J ∼= C(XI ) for some compact subset XI of X. This induces a monomorphism I φ : C(XI ) A/I. → We define Γ(φ)= (φI) :I is an ideal of A { ∗ } and Γ(x)=Γ(φ),γI(x)=φI,where φ : C(sp(x)) A is the map induced by the normal element x A. Notice∗ I could be 0 . → ∈ { } Definition 2.2 (cf. [Ln11]). A C∗-algebra A is said to have weak (FN) if every normal element x A with Γ(x) = 0 can be approximated by normal elements in A with finite spectra.∈ It has been shown in [Ln11] that all simple AF -algebras with countably many extreme traces, purely infinite simple C∗-algebras and separable simple C∗-algebras with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank have weak (FN). These algebras include UHF-algebras, matroid algebras, C∗-algebras of On, the Bunce-Deddens algebras and the irrational rotation algebras. In the case of AF -algebras and On, since the K1-groups are trivial, these algebras in fact have (FN). In this section, we will show that some non-simple C∗-algebras have also weak (FN). Added in proof (April 1996): It has been shown by the author in 1994 that every AF-algebra has (FN) and in 1996 by Friis and Rørdam that every C∗-algebra of real rank zero with connected unitary group and with property (IR) has (FN), and by the author that every simple C∗-algebra of real rank zero and with property (IR) has weak (FN). These include all simple C∗-algebras of real rank zero, stable rank one as well as all purely infinite simple C∗-algebras. Remark 2.3. We are grateful to Terry Loring who pointed out that our original definition is not appropriate for non-simple C∗-algebra. Let A = B C, where B ⊕ is any (unital) C∗-algebra with real rank zero but nontrivial K1-group and C is any (unital) C∗-algebra with real rank zero. Let u be a unitary in B which is not connected to the identity in the unitary group of B. Set x = u y, where y C is a normal element with spectrum sp(y)=D¯ (D¯ is the closed unit⊕ disk). Clearly∈ γ(x)=0,but x can not be approximated by normal elements in A with finite spectra. So maps (φI ) are needed. It should also be noted that when A is simple, Γ=γ. ∗

Proposition 2.4. Let A be a C ∗-algebra with weak (FN). Then A˜ has weak (FN). Proof. Let x A˜ be a normal element with Γ(x)=0.Then we may write x = λ+a for some a ∈A and λ C. Clearly, a is a normal element with Γ(a)=0.The conclusion follows∈ easily.∈ The following two lemmas are contained in [Ln9, 4], and [Ln11, 2.7].

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Lemma 2.5. Let A be a C ∗ -algebra of real rank zero and x A be a normal element with γ(x)=0.For any >0,there is a δ>0such that∈ if p A is a projection and ∈ px xp <δ/2, and pxp y <δ/2, k − k k − k where y pAp, and λ(1 p) (1 p)x(1 p) Inv0((1 p)A(1 p)) if dist(λ, sp(x)) ∈ − − − − ∈ − − δ, then λp y Inv0(pAp),ifdist(λ, sp(x)) δ. ≥ − ∈ ≥ Proof. Suppose that dist(λ, sp(x)) δ. We notice that ≥ (λ [(1 p)x(1 p)+pxp]) (λ x) <δ. k − − − − − k Therefore, by [Ln11, 2.3 and 2.7], λp pxp Inv0(pAp). Now − ∈ (λp y) (λp pxp) <δ/2. k − − − k Then, by [Ln9, 4] and [Ln11, 2.7], λp y Inv0(pAp). − ∈ The following is proved in [Ln11, 3.12]. Theorem 2.6 ([Ln11, 3.12]). Let Ω be a compact subset of the plane. For any >0,there exist δ>0and an integer L such that for any (unital) C∗-algebra A of real rank zero and a normal element x in a C∗-algebra B A with sp(x) Ω, if p A is a projection and if pf(x),f(x)p Afor any f C⊃(sp(x)), ⊂ ∈ ∈ ∈ px xp <δ and λp pxp Inv0(pAp) k − k − ∈ for λ Ω, then there are normal elements y ML(pAp) and z ML+1(pAp) with finite6∈ spectrum sp(y),sp(z) Ω such that ∈ ∈ ⊂ pxp y z <. k ⊕ − k Lemma 2.7. Let A be a C ∗-algebra of real rank zero and I be an ideal of A. Suppose that A/I has weak (FN) and I is a purely infinite simple C∗-algebra. Then A has weak (FN). Proof. Let x A be a normal element with Γ(x)=0.For any η>0,by [Ln11, 4.3] ∈ as in the first few lines in the proof of [Ln11, 4.4], there are λi sp(x)=Xand ∈ nonzero mutually orthogonal projections p0 A such that i ∈ n x λip0 p0xp0 <η/4and p0x xp0 <η/4, k − i − k k − k i=1 X n where p0 =(1 p0 )and λi is η/16-dense in X. Set − i=1 i { } P I⊥ = a : ai = ia =0, for i I . { ∈ }

From the proof of [Ln11, 4.3], if pi0 I⊥, by taking a subprojection, one may assume 6∈ k that p0 I. Therefore, we may assume that pi = p0 I,i =1,2, ..., k,and λi i ∈ i ∈ { }i=1 is η/16-dense in sp(cx), where c is the central open projection in A∗∗ corresponding k to the ideal I. Set p =1 pi. From the proof of [Ln11, 3.4]we have − i=1 k P x λipi pxp <η/4and px xp <η/4. k − − k k − k i=1 X

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Set y = pxp. Let π : A A/I be the quotient map. Then π(y)=π(x).By our assumption, we may assume→ that n π(y) αj q¯j <η/8, k − k j=1 X whereq ¯j are mutually orthogonal projections in pAp/I ∼= pAp/pIp and αj sp(π(x)). Since pAp has real rank zero, by [Zh2, 2.3], every projection in pAp/pIp∈ lifts to a projection in pAp. Therefore there is a pIp such that ∈ n y αj qj a <η/8, k − − k j=1 X where qj are mutually orthogonal projections in pAp. Since I has real rank zero, I has an approximate identity consisting of projections. Therefore there are projections ej qj Iqj such that ∈ eae a <η/16, ey ye < 3η/16 k − k k − k and n eye αj ej eae <η/4, k − − k j=1 X n where e = j=1 ej. This also implies that P ex xe <η/2. k − k n Set z = αj ej eae. Then j=1 − P z exe <η/2. k − k Notice that cx is a normal element (in A∗∗)andexe = ecxe. Moreover, ef(cx)= ef(x) A and f(cx)e = f(x)ce = f(x)e A. It follows from 2.5 that for any ∈ ∈ λ C Xη,λ z Inv0(I˜), where Xη = ξ : dist(ξ,sp(cx)) <η.It then ∈ \ − ∈ { } follows 2.6, if η is small enough, there are normal elements y1 ML(eIe)) and ∈ y2 ML+1(eIe)) with ﬁnite spectra contained in sp(cx) such that ∈ z y1 y2 </4 k ⊕ − k for some integer L. Without loss of generality, we may further assume that y1 = k i=1 λidi, where di are mutually orthogonal projections in ML(eIe) such that k d =theidentityofM (eIe). Pi=1 i k Now we will “absorb” y1. Since I is purely inﬁnite simple C∗-algebra, there is a Ppartial isometry

v (p e e)ML+1(I)(p e e) ∈ ⊕ ⊕···⊕ ⊕ ⊕···⊕ (there are L copies of e) such that v∗div pi,i=1,2,...,k, ≤ k v∗v = v∗div and vv∗ = e e e ⊕ ⊕···⊕ i=1 X (there are L copies of e). There is then a partial isometry u such that k u∗(z y1)u = λiq0 z ⊕ i ⊕ i=1 X

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and u∗y2u is normal and has ﬁnite spectrum, where pi0 = v∗div. So k n n λipi y [ λi(pi p0 ) αj (qj ej ) u∗y2u] </2. k ⊕ − − i ⊕ − ⊕ k i=1 i=1 j=1 X X X Therefore k n x [ λi(pi p0 ) αj (qj ej ) u∗y2u] <. k − − i ⊕ − ⊕ k i=1 j=1 X X

Theorem 2.8. Let A be a C ∗ -algebra of real rank zero. Suppose that there is a sequence of ideals:

I1 I2 In In+1 ⊂ ⊂···⊂ ⊂ ⊂···

such that A is the closure of n∞=1 In and each In+1/In is a purely inﬁnite simple C∗-algebra. Then A has weak (FN). S Proof. Let x A be a normal element with Γ(x)=0,and let πn : A A/In be a quotient map.∈ Since →

dist(x, In) 0, as n , → →∞

πn(x) 0. k k→ Let Xn = sp(πn(x)). For any >0,there is an integer N such that if n>N,

λ </2 for all λ Xn. | | ∈ Set a positive continuous function f defined on sp(x) such that f vanishes on XN and f(ξ) ξ </2 for all ξ sp(x). Then we have k − k ∈ f(x) x </2. k − k Clearly, f(x) IN . So it is enough to prove the theorem for finitely many In’s. ∈ Suppose that I1 I2 = A and I1 and I2/I1 are purely infinite simple C∗-algebras ⊂ and A has real rank zero. Since purely infinite simple C∗-algebras have weak (FN) (see [Ln11, 4.4]), we can apply 2.7. Suppose that we show that 2.8 holds for n ideals. Now if I1 I2 In+1 = A, Ik+1/Ik is a purely infinite simple C∗- ⊂ ⊂ ··· ⊂ algebra for each k and A has real rank zero. By induction assumption, A/I1 has weak (FN). So 2.7 applies.

Remark 2.9. Let I A B 0 → → → be a short exact sequence of C∗-algebras, where B is a Bunce-Deddens algebra and I is isomorphic to On . Since K1(On )=0,by [Zh4, 3.2], A has real rank zero. It follows from 2.7⊗K that A has weak⊗K (FN).

Corollary 2.10. Let A be a σ-unital purely inﬁnite simple C∗-algebra with K1(A) =0.Then M(A) has weak (FN). Proof. By [Zh6, 1.3], M(A)/A is purely inﬁnite simple and M(A) has real rank zero.

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2.11. Let A be a non-elementary separable simple C∗-algebras with real rank zero and stable rank one. Fix a nonzero projection e A. Let ∆ be the set of positive ∈ homomorphisms τ from K0(A)intoRsuch that τ(e)=1andQT (A)bethesetof (lower semicontinuous) quasitraces τ such that τ(e)=1.It follows from [BH, III] that there is an (affine) homeomorphism χ : QT (A) ∆. We will identify these two compact sets. In fact ∆ is a Choquet simplex ([BH,→ II. 4.4]). We will use the notation Aff(∆) for the set of all real affine continuous functions defined on ∆. The map χ gives a homomorphism θ : K0(A) Aff(∆) (see [BH, III. 1.3]). If K0(A) → is weakly unperforated, then K0(A) has the strict order induced from θ. Moreover, since A has stable rank one, for any two projections in M (A)ifτ(p)<τ(q) for all τ QT (A), then q p, i.e., there is a partial isometry∞ v M (A)such ∈ ∈ ∞ that v∗v = p, vv∗ q. Furthermore, θ(K0(A)) is dense in Aff(∆).K0(A)issaidto have finite rank (countable≤ rank) if there are only finitely (countably) many extreme n points in ∆. If K0(A)hasrankn, then Aff(∆) = R . We say that x K 0 (A)is ∼ ∈ infinitesimal if e x e for all 0 < Q(see [Eff, 4]). If K0(A) is weakly unperforated, by− the≤ proof≤ of [Eff, 4.2], ∈

ker θ = x K0(A):xis inﬁnitesimal . { ∈ } Let 1(ˆ τ)=τ(1) = sup τ(p):p A .We set { ∈ } S = τ ∆:1(ˆτ) < , { ∈ ∞}

J = f θ(K0(A)) : f(τ)=0,if τ S { ∈ ∈ } and

SJ = τ ∆:f(τ)=0,if f J . { ∈ ∈ } Then SJ is closed and S SJ . One can easily check that SJ is a face of ∆. In ⊂ [Ln10], we show that if A is a nonunital separable simple C∗-algebra with real rank zero, stable rank one and weakly unperforated K0(A), then (1) K1(M(A)) = 0 (this holds without assuming that K0(A) is weakly unperforated); { } 1 (2) U(M(A)/A)/U0(M(A)/A)=K1(M(A)/A)=θ− (J); Notice that (2) implies that ker θ K1(M(A)/A). If the function 1(ˆ τ)=τ(1) is continuous on ∆, then we say that A⊂has continuous scale (see [Ln10]and [Ln1]). If A has continuous scale, then K1(M(A)/A)=ker θ. The notations and results stated here will be used frequently without warning.

Proposition 2.12. Let A be a separable simple C∗-algebra with real rank zero, stable rank one, trivial K1(A) and weakly unperforated K0(A) of rank n. Then M(A)/A has a (ﬁnite) sequence of ideals

0=I0 I1 Ik Jk+1 = M(A)/A (k n) ⊂ ⊂···⊂ ⊂ ≤ such that Ii+1/Ii is purely inﬁnite simple for each i.

Proof. Fix a nonzero projection e A. Let τ1,τ2, ..., τn be extreme points of ∈ ∆= τ:τ(e)=1 , { } where τ are quasitraces on A. Suppose that τi(1) = ,i=1,2, ..., k. Set ∞

+ J = a M(A)+ : τm(a) < ,m=1,2, ..., k i +1 ,i=1,2, ..., k +1. i { ∈ ∞ − }

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+ Let Ji be the ideal generated by Ji . It is enough to show that Ji+1/Ji is purely infinite and simple. We also let J0 = A. The fact that we have finite tower A = I0 J1 Jk+1 = M(A)andJi/Ji 1 is simple follows from the proof of Theorem⊂ ⊂ 2 ···in [Ln1]. ⊂ We now use induction− on k. If k =0,then A is finite, it follows from [Ln1] and [Zh2] that M(A)/A is purely infinite simple. Suppose that 2.12 holds for k. Note, by [Ln4, 2.8] (see also [Ln7, 9]), K1(Ji 1)=0.So every − projection in Ji/Ji 1 lifts to a projection in Ji. Let e Ji/Ji 1 be a projection. − ∈ − Then there is a projection E M(A) such that φi(E)=e, where φi : Ji/Ji 1 is ∈ − the quotient map. Thus, by induction assumption, eJi/Ji 1e is a purely infinite − n simple C∗-algebra for i =1,2, ..., k. Since θ(K0(A)) is dense in R , it is easy to see that there are projections pi Ji Ji 1,i=2,3, ..., k. So, there is at least ∈ \ − one nonzero e Ji/Ji 1. Since Ji/Ji 1 is simple and has nonzero projections, it is ∈ − − the linear span of projections ([Pd3]). It follows [Zh7, 1.1] that (Ji/Ji 1) has − ⊗K an approximate identity consisting of projections. Thus (Ji/Ji 1) is a purely − ⊗K infinite simple C∗-algebra for i =1,2, ..., k, whence Ji/Ji 1 is a purely infinite − simple C∗-algebra for i =1,2, ..., k. It remains to show that M(A)/Jk is purely infinite and simple. It suffices to show that the closure of φk+1(x)(M(A)/Jk)φk+1(x) has an infinite projection for every positive element x in M(A)/A. By [Zh6, 1.1], the closure x(M(A)/A)x is the closure of linear combinations of projections in the closure x(M(A)/A)x, which implies that at leat one projection p in the closure x(M(A)/A)x such that φk+1(p) is not zero. It is easy to find a projection q p such that τi(p q)= and ≤ − ∞ τi(q)= for i =1,2, ..., k. In particular φk(q) =0andφk(p q)=0.By [Ln10, ∞ 6 − 6 3], one shows that φk(p) is equivalent to φk(q). This implies that φk(p) is infinite.

Corollary 2.13. Let A be a separable simple C∗-algebra with real rank zero, stable rank one, trivial K1(A) and weakly unperforated K0(A) of ﬁnite rank. Then M(A)/A has weak (FN).

Proof. As in the proof of 2.12, K1(Ji)=0foreachJi.Then, by repeated application of [Zh2, 2.3] (also [BP, 3.14]), M(A) has real rank zero. Then 2.13 follows from 2.12, 2.8 and [Ln11] immediately.

3. The generalized Berg theorem

Definition 3.1 (cf. [Zh1, 1.3] and [LS]). Let A be a C∗-algebra. We say that the Weyl-von Neumann-Berg theorem holds for A if every normal element of M(A)is quasidiagonal; i.e. given any normal element x M(A) there is an approximate ∈ identity en for A consisting of projections such that { } ∞ x (en en 1)x(en en 1) A, − − − n=1 − − ∈ X where the sum is taken in the strict topology. Loring and Spielberg [LS] show that a normal element x M(A)maynotbe quasidiagonalized. It happens when the normal element x has∈ γ(x) =0.Suppose that A is a separable simple AF -algebra with finitely many extreme traces.6 We will show that a normal element x M(A) is quasidiagonal if and only if γ(x)=0. ∈ Lemma 3.2 (cf. [Zh1, 3.9]). Let A be a σ-unital C∗-algebra. Suppose that M(A) has real rank zero. If y M (A),π(y)is a normal element in M(A)/A and π(y) can be approximated by normal∈ elements in M(A)/A with finite spectra, then, for

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any >0,there is an approximate identity en for A consisting of projections and a A such that { } ∈ ∞ y = (en en 1)y(en en 1)+a, − − n=1 − − X a </2,and k k n (en en 1)y y(en en 1) </2 . k − − − − − k Proof. The proof is contained in [Zh1, 3.9]. For any >0,there exist mutually orthogonal projections q1,q2, ..., qn M(A)andλ1,λ2, ..., λn sp(y) such that ∈ n ∈ π(y) λiqi <. k − k i=1 X Then, since A has real rank zero, each qi = j∞=1 eij , where eij are mutually orthogonal projections in qiAqi and the sum converges in the strict topology. So we may write P n ∞ λiqi = αmpm, i=1 m=1 X X where pm are mutually orthogonal projections in A and m∞=1 pm converges to the identity{ } 1 in the strict topology. Then we can apply [Zh1, 3.9] ((b) implies (e)). To have exactly the same inequalities required in theP lemma, one can take a subsequence of en which is still an approximate identity. { } Theorem 3.3. Let A be a separable simple C∗-algebra with real rank zero, stable rank one, trivial K1(A), weakly unperforated K0(A) of ﬁnite rank. Suppose that every quasitrace is a trace. Then a normal element x M(A) is quasidiagonal if and only if γ(x)=0. ∈ Proof. Since x is normal, it follows from [Ln13, 1.13] that γI (x) = 0 for every ideal I of M(A) which contains A but is not A itself. If γ(x)=0,then Γ(π(x)) = 0. Since M(A)/A has weak (FN), by 2.13, π(x)can be approximated by normal elements (in M(A)/A) with ﬁnite spectrum. Then we can apply 3.2. Now we suppose that x is quasidiagonal. We will show that γ(x)=0.Let en be an approximate identity for A consisting of projections such that { }

∞ x (en en 1)x(en en 1) A. − − − n=1 − − ∈ X Let f C(sp(x)) such that π(f(x)) is a unitary in M(A)/A. Since en is an approximate∈ identity for A, { }

(en en 1)x x(en en 1) 0asn . k − − − − − k→ →∞ It follows that ∞ π[ (en en 1)f(x)(en en 1)] = π(f(x)). − − n=1 − − X Therefore

(en en 1) (en en 1)f(x)(en en 1)∗(en en 1)f(x)(en en 1) 0 k − − − − − − − − − − − k→

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and

(en en 1) (en en 1)f(u)(en en 1)(en en 1)f(x)(en en 1)∗ 0 k − − − − − − − − − − − k→ as n .Thus, by a standard argument involving the polar decomposition, there →∞ are unitaries un (en en 1)A(en en 1) such that ∈ − − − − un (en en 1)f(x)(en en 1) 0 k − − − − − k→ as n .Notice that →∞ ∞ ∞ (en en 1)f(x)(en en 1) un A. − − n=1 − − − n=1 ∈ X X Therefore ∞ π( un)=π(f(u)). n=1 X Moreover, n∞=1 un is a unitary in M(A). It follows from [Ln7, 9] that K1(M(A)) = 0. This implies that γ(x)=0. P Lemma 3.4. Let A be as in 3.2. If x M(A) is a normal element, then im[γ(π(x))] ∈ ker θ(K0(A)). ⊂ Proof. Let f : sp(π(x)) S1 be a continuous function. Then f(π(x)) is a unitary in M(A)/A. So there is→ a partial isometry v M(A) such that π(v)=f(π(x)). Let g C(sp(x)) such that π(g(x)) = f(π(x∈)). Set y = g(x). Then there is an ∈ element a A such that v = y + a. Let en be an approximate identity for A. ∈ { } Then a∗y = limn a∗eny and ay∗ = limn a(eny)∗. Therefore, for any trace τ, →∞ →∞ τ(a∗y)=τ(ay∗). Similarly, τ(y∗a)=τ(ya∗). Since y∗y = yy∗, we have

τ(v∗v vv∗)=τ(a∗a aa∗)+τ(a∗y ay∗)+τ(y∗a ya∗)=0. − − − − This implies that [v∗v] [vv∗] kerθ(K0(A)). It follows from [Ln7, 9] that K1(M(A)) =0.From the six-term− exact∈ sequence in K-theory, the map

K1(M(A)/A) K0(A) → is injective. Therefore im[γ(x)] ker θ(K0(A)). ⊂ Corollary 3.5. Let A be as in 3.3. Suppose that K0(A) has no inﬁnitesimal elements (i.e. ker θ =0). Then every normal element x M(A) is quasidiagonal. ∈ We would now like to present a stronger version of the Berg theorem for separable simple AF -algebras with continuous scale and K0(A) of countable rank. The following lemma is proved in [EGLP, 4.4], even though it stated diﬀerently.

Lemma 3.6 (cf. [EGLP, 4.4], [Ln11, 4.8], and [Ln6, 2]). Let A be a unital C ∗ - algebra of real rank zero. Let ∆ be a set of countably many (ﬁnite) traces on A. Suppose that ∆ is w∗-compact and the identity of A is a continuous function on ∆. Suppose that φ : C(X) A is a monomorphism, where X is a compact metric → space. For any >0,any ﬁnitely many f1,f2, ..., fm C(X), and integer K>0 ∈ there are mutually orthogonal projections p1,p2, ..., pn in A and λ1,λ2, ..., λn X such that ∈ n φ(fs) (ys + fs(λi)pi) <, s=1,2, ..., m, k − k i=1 X

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n n where ys =(1 pi)φ(fs)(1 pi), − i=1 − i=1 P n P n (1 pi)φ(fs) φ(fs)(1 pi) <, k − − − k i=1 i=1 X X n t(pk)>Kt(1 pi) − i=1 X for all t ∆,s=1,2, ..., m, k =1,2, ..., n, j =1,2, ..., l, and for any λ X, there ∈ ∈ is λi such that

dist(λ, λi) <.

Theorem 3.7. Let A be a separable simple C∗-algebra with real rank zero, stable rank one and continuous scale. Suppose that K1(A)=0,every quasitrace of A is a trace and K0(A) has countable rank. Then every normal element x M(A) with ∈ γ(x)=0,and >0,there exist an approximate identity en for A consisting of projections and a A such that { } ∈ ∞ x = λn(en en 1)+a − n=1 − X where λn is a bounded sequence of complex numbers and a <. { } k k Proof. Since A has continuous scale, by [Ln2], M(A)/A is simple, and hence purely inﬁnite simple ([Zh2, 5.1]). So M(A)andM(A)/A has real rank zero. Let ∆ be as in 2.11. It follows from 3.6 that for any η>0andintegerL>0,there are mutually orthogonal projections p0 M(A)andλi sp(x) such that i ∈ ∈ n x λip0 p0xp0 <η/4 k − i − k i=1 X and

t(pi0 ) > (L +1)t(p0) n for all i and t ∆, where p0 =1 p0 and λi is η/16-dense in sp(x). Since A ∈ − i=1 i { } has real rank zero and p0 Ap0 has continuous scale for each i, there is a projection i i P pi p0 Ap0 for each i such that ∈ i i n t(pi) > (L +1)t(p0 + (p0 pi)) i − i=1 X for each i and t ∆. From the proof of [Ln11, 3.4], we have ∈ n x λipi pxp <η/4, k − − k i=1 X n n where p =1 i=1 pi = p0 + i=1(pi0 pi). Set z = pxp. Then π(x)=π(z).It follows from [Ln12,− 7.1] that there is an− element a pAp such that P P ∈ ∞ z = λn(qn qn 1)+a − − k=1 X

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where λn sp(x)and qn is an approximate identity for pAp consisting of projections. There∈ is an integer{ }N such that

(1 qN )a <η/8and (a(1 qN ) <η/8. k − k k − k We may write

∞ z = λn(qn qn 1)+qNzqN + b, − − n=N +1 X where b A and b <η/4.We also have ∈ k k qN z zqN <η/4. k − k Notice that z = pxp. So qN zqN = qNxqN . By Lemma 2.6, if η is small enough, there are normal elements z1 ML(qN AqN )andz2 ML+1(qN AqN ) with ﬁnite ∈ ∈ spectra sp(zi) sp(x) such that ∈ qN zqN z1 z2 </8. k ⊕ − k n Notice that L depends only in . We may also assume that z1 = i=1 λidi for some mutually orthogonal projections di ML(qN AqN ). Since t(pi) >t(di) for all t ∆ ∈ P ∈ and i =1,2, ..., n, There is a unitary u ML+1((1 p)+qN)A(1 p)+qN)such that ∈ − − n λipi + qN zqN U∗z2U </8. k − k i=1 X Furthermore, z3 = U ∗z2U ((1 p)+qN)A((1 p)+qN)andsp(z3) is ﬁnite. If η is small enough, we then∈ have − −

∞ x = λn(qn qn 1)+z3 +b1 +b, − − n=N+1 X where b1 A and b1 + b <. ∈ k k Corollary 3.8. Theorem 3.7 holds for σ-unital purely inﬁnite simple C∗-algebras with trivial K1-group. Proof. The proof is the same as that of 3.7. Notice that in a purely inﬁnite simple C∗-algebra A any projection is “larger” than any others and in M(A) A any two projections are equivalent. \

4. The generalized BDF-theorem

In this section, A is always a separable simple C∗-algebra with real rank zero, stable rank one, trivial K1(A) and weakly unperforated K0(A) of ﬁnite rank. Fur- thermore, we assume that all quasitraces are traces. 4.1. After we discussed the Berg theorem, it would be unnatural not to consider the BDF-theorem for essentially normal elements. Essentially normal elements are those elements x M(A) such that π(x) is normal. The question, as in the operator theory, is when∈ can an essentially normal element x M(A) be written as x = y + a, where y M(A)isnormalanda A? A natural∈ ﬁrst thought might ∈ ∈ be : “index must be zero.” Well, by 3.7, γI (x) = 0 for any ideal I of M(A)which strictly contains A, if x is a normal element in M(A). How about the Fredholm index? Mingo and Spielberg’s examples show that even normal elements can have

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nonzero Fredholm index. So, the second thought is that it would be hopeless to give a definite answer. However, recent results in extension theory [Ln12] encourage us to have the third thought. Lemma 3.4 said that normal elements can only have an infinitesimal nonzero Fredholm index. We discover, by using results in [Ln12], that an essentially normal element x M(A) with infinitesimal Γ(x)hasformy+a. ∈ Lemma 4.2 ([Ln14, 3]). Let p and q be two open projections in A∗∗ both not in A. If t(p)=t(q)for all t ∆, where ∆ is the same as in 2.8, then Her(p) = Her(q), ∈ ∼ where Her(p) and Her(q) are hereditary C∗-subalgebras corresponding to the open projection p and q, respectively. Proof. The proof is exactly the same as that of [Ln14, 3].

Lemma 4.3 (cf. [MS]). For any ζ ker θ, there is a normal element x M(A) 1 ∈ ∈ such that sp(x)=D,¯ sp(π(x)) = S and [π(x)] = ζ in K1(M(A)/A). Proof. Take a nonzero projection e A. It follows from [EL, 7.3] that there is a homomorphism φ : C(S2) eAe ∈such that φ (b)=ζ, where b is the “Bott” → ∗ element. Let us denote the restriction of φ on C0(D)byφitself. Then, as in [MS], φ (b)=ζ. Let h be a strict positive element of C0(D)andB=Her(φ(h)), the ∗ hereditary C∗-subalgebra of eAe generated by φ(h). Combining [MS, Proposition] and [MS, Theorem 2], we obtain a normal element y M(B) such that sp(y)=D¯ ∈ and [π(y)] = ζ in K1(M(B)/B). n Since θ(K0(A)) is dense in R (see 2.11), there is a projection p M(A)such ∈ that t(p)=t(q) for all t ∆, where q is the open projection in A∗∗ corresponding ∈ to B. It follows from 4.2 that B ∼= pAp. Therefore M(B) ∼= pM(A)p. We may assume that y pM(A)p. Let 1 be the identity of M(A). Set x = y u, where u ∈ ⊕ is a unitary in (1 p)M(A)(1 p). Notice that K1((1 p)M(A)(1 p)) = 0. So sp(x)=D¯ and − − − −

[π(x)] = ζ in K1((M(A)/A).

Lemma 4.4. Let X be a compact subset of the closed unit disk D,¯ and O1,O2, ..., On, ... be the sequence of bounded components of C X. Set \ Xi = λi ξ : ξ X { − ∈ } 1 for some λi Oi. Let gi : Xi S be a generator of K1(C(X)) corresponding to ∈ → Oi . If ζ ker θ, then for each i, there is a normal element xi M(A) such that ∈ ∈ sp(xi) D,¯ sp(π(xi)) X and ⊂ ⊂ [gi(λi π(xi))] = ζ and [gj (λj π(xi))] = 0 − − in K1(M(A)/A) if j = i. 6 Proof. Notice that we identify S1 with the unit circle on the plane. There is a 1 compact subset Fi Xi such that gi : Fi S is a homeomorphism. Let x M(A) ⊂ → ∈ be the normal element in 4.3. There is fi C(sp(x)) such that λi fi 1and ∈ k − k≤ 1 π(fi(x)) = gi− (π(x)).

Deﬁne xi = λi fi(x). −

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Lemma 4.5. Let X be a compact subset of the plane, and let ψ : K1(C(X)) ker θ be a homomorphism. Then there is a normal element x M(A) satisfying→ the following: ∈ (1) sp(π(x)) = X, (2) φ = ψ, where φ : sp(π(x)) M(A)/A is the isomorphism induced by the element ∗π(x). → Proof. Without loss of generality, we may assume that X D.¯ It is easy to con- ⊂ struct a sequence of mutually orthogonal projections pi M(A) A such that ∈ \ ∞ pn converges to a projection p in the strict topology, where p M(A)and i=1 ∈ 1 p A. For each i, by applying 4.3, there is xi piM(A)pi satisfying P− 6∈ ∈ [gi(λi xi)] = ψ(g0)and[gj(λj xi)] = 0 − i − for j = i and g0(z)=gi(λi z).Set 6 i − ∞ y = xi. i=1 X n Note that i=1 xi converges to y in the strict topology. Then y is a normal element in M(A). Take a dense sequence of numbers in X such that each isolated point P occurs inﬁnitely often and an approximate identity en for (1 p)A(1 p) consisting of projections. Set { } − −

∞ z = λn(en en 1). − n=1 − X Deﬁne x = y+z. It is easy to see that the element x satisﬁes the required conditions.

The following result is basically contained in [Ln12].

Lemma 4.6. Let A be a separable simple C∗-algebra with real rank zero, stable rank one, K1(A)=0,weakly unperforated K0(A), and let σ : C(X) M(A)/A be an extension, where X is a compact metric space. Suppose that→p M(A) ∈ is a projection such that [π(p)] = 0, i.e., τ(p) θ(K0(A)) for all τ ∆, and ∈ ∈ σ0 : C(F ) M(pAp)/pAp is a null extension (see [Ln12, 2.2] or [Ln12, 2.9]), → where F is a compact subset of X. Then σ σ0 is unitarily equivalent to σ. ⊕ Proof. By [Ln12, 1.5], there is a nonzero projection q M(A) and injective maps ∈ σ0 : C(X) M((1 q)A(1 p))/(1 q)A(1 q) → − − − − and σ1 : C(F ) M(qAq)/qAq such that σ1 is totally trivial and σ = σ0 +σ1. Since → σ1 is totally trivial, there exist an approximate identity en for qAq consisting of { } projections and a dense sequence ξn in F with isolated points repeated inﬁnitely often such that { }

∞ σ1(f)=π[ f(ξn)(en en 1)]. − n=1 − X One can ﬁnd nonzero projections dn (en en 1) such that n∞=1 τ(dn)converges ≤ − − uniformly on ∆. Let q0 = ∞ dn Then q0Aq0 has continuous scale. Furthermore n=1 P P

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the map

∞ σ10 (f)=π[ f(ξn)(en en 1)] − n=1 − X is an extension of C(F )byq0Aq0. Fix any point ξ F. Let h C(F ) be such that ∈ ∈ h(ξ)=0andh>0elsewhereonF. Denote by B the hereditary C∗-subalgebra of M(q0Aq0)/q0Aq0 generated by σ10 (h). Then B is σ-unital. It follows from [Pd2, 15] that B⊥ is a nonzero hereditary C∗-subalgebra of M(q0Aq0)/q0Aq0. Since M(q0Aq0)/q0Aq0 has real rank zero ([Ln7]), by [BP, 2.6], B⊥ has real rank zero. Therefore there is a nonzero projection d B⊥. It is easy to verify that d com- ∈ mutes with im σ10 . Moreover,

σ0 (f)=f(ξ)d+(1 d)σ0(f)(1 d) 1 − 1 − for all f C(F ). Since M(q0Aq0)/q0Aq0 is simple ([Ln2]), it is in fact purely in- ∈ ﬁnite simple. There is a nonzero projection d0 d such that [π(q0) d0]=0in ≤ − K0(M(A)/A). Let p0 M(A) be a projection such that π(p0)=π(q0) d0.Set ∈ − σ2(f)=f(ξ)d0 +(1 d)σ0(f)(1 d) − 1 − for all f C(F ). Then σ2 : C(F ) M(p0Ap0)/p0Ap0 is a (totally trivial) extension. ∈ → Clearly p0 commutes with im σ. Let σ00 = p0σ. Then σ = σ00 σ2. It follows from ⊕ [Ln12, 2.3] that σ2 σ0 is unitarily equivalent to σ2. Therefore, σ σ0 is equivalent to σ. ⊕ ⊕ Theorem 4.7. Let x M (A) be an essentially normal element, i.e., π(x) is normal in M(A)/A. Then∈x canbewrittenasx=y+a, where y M(A) is a normal ∈ element and a A if and only if γ(π(x)) ker θ and γI (x)=0for all ideals I of M(A) which strictly∈ contain A. ∈ Proof. The “only if” part follows from 3.6 and [Ln13, 1.13]. Suppose now that x M(A),π(x)isnormal,γ(π(x)) ker θ and γI (x)=0for all ideals I which strictly∈ contain A. Suppose that X = sp∈(π(x)). Let σ : C(X) M(A)/A be the extension induced by the normal element π(x). Let p M(A→) ∈ such that [π(p)] = 0, i.e., τ(p) θ(K0(A)) for all τ ∆. So pAp has continuous ∈ ∈ scale and K1(M(pAp)/pAp)=ker θ (see [Ln10]). By [Ln12, 7.1], there is an extension σ1 : C(X) M(pAp)/pAp such that γ1(σ1)= γ(π(x)). Set B = → − (1 p)M2(A)(1 p). Let z(ξ)=ξfor ξ X. Then Γ(π(x) σ1(z)) = 0. Therefore, ⊕ ⊕ ∈ ⊕ by 3.5, there is a normal element y1 M(A) such that π(y1)=π(x) σ1(z).Now ∈ ⊕ we use the notation y2 for a normal element in M(pAp) satisfying the conditions in 4.5, with ψ = γ(π(x)). Let σ2 : C(X) M(pAp)/pAp be the extension induced → by the normal element π(y2). Then γ1(σ1 σ2)=0.By [Ln12, 7.1], σ1 σ2 is totally trivial. It follows from [Ln12, 4.3] that⊕ there is a totally trivial extension⊕ σ3 : C(X) M(pAp)/pAp such that σ1 σ2 σ3 is a null extension. It follows → ⊕ ⊕ from 4.4 that σ is unitarily equivalent to σ σ1 σ2 σ3. There is a (diagonal) ⊕ ⊕ ⊕ normal element y3 M(A) such that π(y3)=σ3(z).Since ∈ σ σ1 σ2 σ3(z)=π(y1) π(y2) π(y3), ⊕ ⊕ ⊕ ⊕ ⊕ there is a partial isometry u M2(A) such that u∗u =1,uu∗=1 p p pand ∈ ⊕ ⊕ ⊕ π(u∗xu)=π(y1) π(y2) π(y3). ⊕ ⊕ We conclude that x = y + a, where y = u(y1 y2 y3)u∗ and a A. ⊕ ⊕ ∈

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Corollary 4.8. Suppose that K0(A) has no infinitesimal elements. Then an essentially normal element x M(A) has the form x = y + a, where y M(A) is normal and a A if and only∈ if Γ(x)=0. ∈ ∈ Corollary 4.9 (cf. [Ln6, 11.7]). If A is a finite matroid algebra, then every essentially normal element x M(A) canbewrittenasx=y+a, where y M(A) is normal and a A. ∈ ∈ ∈ Proof. We notice the fact that if A is a finite matroid algebra, K1(M(A)/A)=0 (see [Ell1]).

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Department of Mathematics, East China Normal University, Shanghai 20062, China Current address: Department of Mathematics, University of Oregon, Eugene, Oregon 97403- 1222 E-mail address: [email protected]

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