K-Theory 9: 1t7-137, 1995. ~17 © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

Approximately Unitarily Equivalent Morphisms and Inductive Limit C*-Algebras*

MARIUS DADARLAT Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A.

(Received: May 1994)

Abstract. It is shown that two unital ,-homomorphisms from a commutative C*-algebra C(X) to a unital C*-algebra B are stably approximately unitarily equivalent if and only if they have the same class in the quotient of the Kasparov group KK(C(X),B) by the closure of zero. A suitable generalization of this result is used to prove a classification result for certain inductive limit C*-algebras.

Key words: *-homomorphisms, unitary equivalence, Kasparov groups, C'*-algebras, real rank z e r o .

O. Introduction

Let X be a compact metrizable space and let B be a unital C*-algebra. We prove that the .-homomorphisms from C(X) to B are classified up to stable approxi- mate unitary equivalence by / ( - t heo ry invariants. This result is used to obtain a classification theorem for certain real rank zero inductive limits of homogeneous C*-algebras. The classification of various classes of C*-algebras of real rank zero in terms of invariants based on K-theory presupposes a passage from algebraic objects to geometric objects (see jEll], [Lil], [EGLP1], [EG], [BrD], [G], [R0], [LiPh]). An underlying idea of this paper is that this passage can be done by using approximate morphisms. K-theory becomes a source of approximate morphisms thanks to the realization of K-theory in terms of asymptotic morphisms, due to A. Connes and N. Higson [CH]. The main results of the paper are Theorems A and B, below.

By the universal coefficient theorem for the Kasparov KK-groups [RS], Ext(K.(C(X)), t(._I(B)) is a subgroup of Kt f (C(X) , B). Following [R0], we let KL(C(X) , B) denote the quotient of KK(C(X) , B) by the subgroup of pure extensions in Ext(K.(C(X)), K._I (B) ) . If K.(C(X)) is isomorphic to a direct sum of cyclic groups, then KL(C(X), B) coincides with KK(C(X) , B).

THEOREM A. Let X be a compact metrizable space. Let B be a unital C*- algebra and let q~, ~: C( X ) --+ B be two unital *-homomorphisms. The following assertions are equivalent.

* This research was partially supported by NSF grant DMS-9303361.

118 MARIUS DADARLAT

(a) [~] = [¢] in K L ( C ( X ) , B). (b) For any finite subset F C C(X) and any e > 0 there exist k E N, a unitary

u ~ Uk+I(B) andpoints X l , . . . , X k in X such that

Nudiag(~(a),a(xl), . . . ,a(xk))u* - diag(¢(a),a(xl), . . . ,a(xk))tl < e

for all a C F.

Two .-homomorphisms satisfying the condition (b) of Theorem A are said to be stably unitarily equivalent. If condition (b) is satisfied without the ,-homomorphism rl(a ) = diag(a(xl), . . . , a(xk)), then we say that ~p and ¢ are approximately uni- tarily equivalent. Theorem A generalizes certain results in [Lil-3] and [EGLP]. If ~ and ¢ are .-monomorphisms and B is a purely infinite, simple C*-algebra, then one can drop the ,-homomorphism 71 from condition (b), see Theorem 1.7. The apparition of pure extensions in this context is related to the phenomenon of quasidiagonality, see [Br], [Sa], ~rD]. The author believes that future generaliza- tions of Theorem A should be based on Voiculescu's noncommutative Weyl-Von Neumann Theorem IV].

We use a suitable version of Theorem A for asymptotic morphisms to prove the following theorem.

THEOREM B. Let A, B be two simple C*-algebras of real rank zero. Suppose that A and B are inductive limits

kr,.

A = li___mAn and B = li__mBn,An = ( ~ M[n,i](C(Xn,i)), i=l

in

i= l

Suppose that X~,i,Ya,i are finite C W complexes, K°(X~,;) and K°(Y~,i) are torsion free and supn,i { dim( Xn,i )~ dim(Yn,i ) ) < cv. Then a is isomorphic to B if and only if

(I¢.(a), K.(A)+, e.(A)) u (K.(B), U.(B)+, r,.(B)).

Theorem B is proved by showing that the C*-algebras A and B are isomorphic to certain inductive limits of subhomogeneous C*-algebras with one-dimensional spectrum that are known to be classified by ordered K-theory [Ell]. In the pro- cess we use the semiprojectivity of the dimension-drop C*-algebras proved by T. Lofing [Lo-1]. In a remarkable recent paper [EG], Elliott and Gong classify the simple C*-algebras of real rank zero that are inductive limits of homogeneous C*- algebras with spectrum 3-dimensional finite C W complexes. Roughly speaking, this classification result is achieved in two stages. The first stage corresponds to

APPROXIMATELY UNtTARILY EQUIVALENT MORPHISMS ~_ 19

passing from K-theory to homotopy and uses the connective KK-theory intro- duced by A. Nemethi and the author [DN]. The second stage corresponds to showing that homotopic ,-homomorphisms are stably approximately unitarily equivalent. A similar approach is taken in [G]. In this paper, we show that these techniques can be combined with techniques of approximate morphisms to produce classification results for inductive limits of homogeneous C*-algebras with higher dimensional spectra. A key tool of our approach is a suspension theorem in E-theory due to T. Loring and the author [DL], (see also [D2]). The idea of using homotopies of asymptotic morphisms in the study of approximate unitary equivalence of , - homomorphisms appears also in [LiPh]. The study of inductive limits of matrix algebras of continuous functions was proposed by E. G. Effros [Eft].

1. Approximately Unitarily Equivalent Morphisms and K-Theory

In this section we prove Theorem A and give an application for ,-monomorphisms from C ( X ) to purely infinite, simple C*-algebras,

We need the following result from [EGLP].

THEOREM 1.1. [EGLP] Let X be the cone over a compact metrizable space and let B be a unital C*-algebra. For any finite subset G C C ( X ), any ~ > 0 and any unital ,-homomorphism p: C( X ) -~ B, there are two unital ,-homomorphisms with finite-dimensional image, r/:C(X) --+ Ms-I (B) and ~:C(X) ---+ Ms(B), such that

[[diag(p(a), ~/(a))- ~(a)[[ < e

for all a E G.

The following result is implicitly contained in [EG].

THEOREM 1.2, Let X be a finite, connected C W complex. For any finite subset F C C(X)andanye > O, thereexistr E N, aunital*-homomorphismr:C(X) --+ MT_I(C(X)) and a unital ,-homomorphism #: C ( X ) --+ Mr(C(X) ) with finite dimensional image, such that

Ildiag(a,r(a))-/*(a)[[ < e

for all a E F. Proof. By [DN] there exist/~ C ~ and a unital *-homomorphism cr: C ( X ) -+

M R - I ( C ( X ) ) such that ~b: C ( X ) -+ MR(C(X)),definedby ~b(a) = diag(a, a(a)) is homotopic to an evaluation map a ~-+ a(zo)lR. Let ~: C ( X ) --+ M R ( C ( X x [0, 1])) be a ,-homomorphism implementing a homotopy from ~b to the eval- uation map a ~ a(x0)lR. Since ~(a)(a:, 1) = a(z0)lR for all a E C(X) , one can identify ~I~ with a ,-homomorphism ~: C(X) -+ MR(C(f() ) where

= X × [0, 1]/X x {1} is the cone over X. Moreover, the natural inclusion

120 MARIUS DADARLAT

X -~ X x {0} ~ A? induces a restriction ,-homomorphism p: M R ( C ( f ( ) ) ---+ M n ( C ( X ) ) and ¢ factors as ¢ = p~/. Set G = ~ ( F ) . Since J~ is contractible we can use Theorem 1.1 to produce unital ,-homomorphisms

~ M(s_I )R(C(X) ) , (:Mn(C(2)) ~ M s R ( C ( X ) )

with finite-dimensional image such that

[[diag(p(d),zl(d))- ~(d)[[ < E

for all d E G. Hence

[Idiag(p (a), 7¢l(a)) - (a)ll <

for all a E F. Set r = sR, r (a ) = diag(a(a), 7/krl(a)) and #(a) = ~9(a) . Then

[[diag(a, r(a)) - ~(a)l t < e

for all a E F. []

DEFINITION 1.3. A C*-algebra A is said to have property (H) , if for any finite sub- set F C A and any e > 0, there exist r E N, a .-homomorphism 7-: A --+ M r - 1 (A) and a .-homomorphism #: A --+ Mr(A) with finite-dimensional image such that

I l d i a g ( 7 - ( a ) , a ) - (a)ll < E

for all a E F.

It is clear that any finite dimensional C*-algebra has property (H). Theorem 1.2 shows that if X is a finite C W complex, then C ( X ) has property (H). It is not hard to see that the class of C*-algebras with property (H) is closed under direct sums and tensor products.

For C*-algebras C, D let Map(C, D) denote the set of all linear, contractive, completely positive maps from C to D. If G is a finite subset of C and/~ > 0 we say that ~p E Map(C, D) is 6-multiplicative on G if [l~(ab) - ~(a)~(b)l I < 6 for all a, b E G.

LEMMA 1.4. Let A be a C*-algebra with property ( H ). Let e > 0 and let F C A

be a finite set. There are ~ > 0 and a finite subset F C A such that i f B is any unital C*-algebra and qa0, ~Pl, • • •, ¢Pn is a sequence o f maps in Map(A, B) such that ~oj is 6-multiplicative on [" for j = 0 , . . . , n, then there exist k E N, a ,-homomorphism 7: A --+ Mk( B) with finite dimensional image and a unitary u E Uk+l( B) such that

APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS 121

]]u diag(q~o(a), zl(a))u* - diag(q~n(a), ~/(a))ll

< c + max max l[~j+~(a) - ~j(a)t i a ~ F O<~j<~ n - 1

for all a E F. Proof. For given F C A and E > 0, let r E 1~, 7- and # be as in Definition 1.3.

Then D = #(A) is a finite dimensional C*-subalgebra of Mr(A). By elementary perturbation theory (see [BraD, there is a finite subset G of D containing # ( F ) and there is ~ > 0 such that whenever E is a C*-algebra and • E Map(D, E ) is ~-multiplicative on G, there exists a , -homomorphism k~: D --+ E satisfying I[~'(d) - ~(d)i I < e for all d E G.

For s E N set qo~,j = ~j ® ida: M~(A) --+ Ms(B). It is easily seen that one can find ~ > 0 and _P C A finite such that if Tj E Map(A, B) is ~-multiplicative on /~ then ~ , j is ~-multiplicative on G. Since ~ , j is ~5-multiplicative on G, there is a • -homomorphism ¢j : D ~ M~.(B) such that l l ~ , j ( d ) - ¢ j ( d ) l l < c for all d E G a n d j = O , . . . , n .

Define L, L': A --+ Mnr(B) by

L = diag(~_l,oT-, ~0, ~ - 1 , 1 r , ~ 1 , . . . , ~-1 , ,~-1r , cpn-1),

L t = diag(~o, ~ - l , o r , ~1, ~-1,17", • • . , ~n-1, ~ r - I , n - l T ) •

Note that L is unitarily equivalent to U. Thus, there is a permutation unitary u E Unr+I(B) such that

Let

u d i a g ( r ' , ~,~)u* = d i a g ( ~ , L).

A = m a x max II~j+~(a)-~j(a)l l aEF O<<.j<~ n - 1

Since II~j+l(a) - ~ ( a ) t l ~ ~ for an a E F

l l d i a g ( ~ o ( a ) , L ( a ) ) - diag(L'(a), ~ ( a ) ) l l

~< rnax I I~ j+ l (a ) - ~j (a ) l l = A. 3

Using (1) and (2), we obtain

Iludiag(~yo(a),L(a))u* - diag(~n(a),L(a))ll <~ A

for all a E F.

(1)

(2)

(3)

122 MARIUS DADARLAT

On the other hand,

IlL(a) - d iag(~ , ,0#(a ) , . . . , qa~,~_l#(a))[I

= [[diag(~,0(r(a) • a - Iz (a) ) , . . . , qa,,~-l(r(a) • a - ~(a)))ll

~< l iT(a) • a - ~ (a ) l l < E (4)

for all a E F , since ~ , j are norm decreasing. Note that I I ~ r , j / 4 a ) - C j ~ ( a ) f l < for all a E F since # ( F ) C G and II~,,j(d) - CAd)ll < : for all d E G. This implies

Ildiag(qo~,o/~(a),..., qo~,~_l/z(a)) - d i ag (¢0#(a ) , . . . , ¢~-1~(a))11 < ¢ (5)

for all a E F. The , -homomorphism defined by ~ = d i ag (¢0# , . . . , ¢,~-1#) has finite-dimensional image. Using (4) and (5) we obtain

I l L ( a ) - ~(a)ll < 2e (6)

for all a E F. Combining (3) and (6), we find

[[u diag(qo0(a), o(a))u* - diag(qa,~(a), ~(a))l[ < 4e + A

for all a E F. []

Suppose that the C*-algebra A is unitai. Suppose that the , -homomorphism # from Definition 1.3 and the maps ~j are unital. Then it easily seen that one can arrange for the , -homomorphism r/to be unital.

The notion of asymptotic morphism due to Connes and Higson led to a geo- metric realization of E-theory [CH]. Let A, B be separable C*-algebras. Roughly speaking, an asymptotic morphism from A to B is a continuous family of maps ~t: A ~ B , t E T = [1,oo), which satisfies asymptotically the axioms for , - homomorphisms. A homotopy of asymptotic morphisms ~t, ¢~: A --~ B is given

by an asymptotic morphism ~t: A ~ B[0, 1] such that ~0 ) = ~t and ~ l ) = ¢~. Here B[0, 1] denotes the C*-algebra of continuous functions from the unit interval to B. The homotopy classes of asymptotic morphisms from A to B are denoted by [[A, B]] and the class of ~t by [[~]]. Two asymptotic morphisms ~t, ~bt are said equivalent if ~ t ( a ) -- Ct(a) ~ 0, as t ~ oo for all a E A. Equivalent asymptotic morphisms are homotopic. In this paper we deal exclusively with asymptotic mor- phisms from nuclear C*-algebras. It was observed in [D] that if A is nuclear then any asymptotic morphism from A to B is equivalent to a completely positive linear asymptotic morphism. This is a consequence of the Choi-Effros theorem [CE], and it applies for homotopies of asymptotic morphisms as well. Henceforth, throughout the paper, by an asymptotic morphism we will mean a contractive completely posi- tive linear asymptotic morphism unless stated otherwise. Let Moo denote the dense

APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS 123

• -subalgebra of the compact operators E obtained as the union of the C*-algebras M,~. Using approximate units it is not hard to see that any asymptotic morphism from A to B ® E is equivalent to an asymptotic morphism ~t: A ~ B ® Moo for which there is a function a: T ~ N such that (pt(A) C B ® M~(t). The map ct is called a dominating function for 7~t. This applies also to homotopies and yields a bijection

[[A,B ® M~]] ~ [[A, B ®/C]].

We consider here only asymptotic morphisms that are dominated by functions a as above. Recall that if A is nuclear, then the Kasparov group K K ( A , B) is isomorphic to [[SA, SB ®/C]] (see [CH]).

Let X be a finite connected C W complex with base point x0 and let B be a unital C*-algebra. Then by the suspension theorem of [DL], [[Co(X\xo), B ® Mc~]] KK(Co(X\xo) , B). Let ~t: Co(X\xo) ~ B ® M ~ be an asymptotic morphism and let a be a dominating function for ~t. For each t E T we let ~ ' : C ( X ) --, B ® M~(t) denote the unital extension of Tt: Co(X\xo) ~ B ® Mc~(~) with

~p~(1) = 1B ® 1~(~). Note that i f a ( t ) ~</3(t) then p~ = 1B ® l~(t)qot~lB ® l~(t).

THEOREM 1.5. Let X be a finite connected C W complex with base point xo and let B be a unital C*-algebra. Let ~t , Zbt: Co( X \ xo) ~ B ® M ~ be two asymptotic morphisms. Suppose that [[~t]] = [[g3t]] in K K ( Co( X \ xo ) , B ). Then for any finite set F C C ( X ) and any e > 0 there are to >>. 1 and maps a,/3: [1, c~) ~ N with dominating both qo t and ~bt such that for any t >1 to there exist a unitary u E U ( B ® M~(t) ® Mp(t)+l) anda unital *-homomorphism rl: C( X ) ~ B ® M~(t) ® M~(t) with finite dimensional image such that

I ludiag(~(a) ,~(a))u*- diag(¢~(a), ~/(a))ll < e

for all a E F. Proof. Using the suspension theorem in E-theory of [DL], we find an asymptotic

morphism ~t:Co(X\xo) ~ B[0, 1] ® M ~ such that ¢~I °) = 7~t and ~}1) = Ct. Let a: [1, oc) ---> N be a function dominating 7~t, Ct, and ~t. We are going to use Lemma 1.4 and the notation from there. We find to such that if ~t: C(X) --+ (B[0, 1] ® M ~ ) ~ is the unital extension of ~t, then ~t is (~-multiplicative on P for all t ) to. Since the image of ~ commutes with i s ® la(t), it follows that

• ~: C ( X ) --. B[0, 1] ® M~(t) is ~-multiplicative on P for all t/> to. Let t ) to be fixed. By uniform continuity we can find a sequence of points so = 0 , . . . , Sn = t in the unit interval such that

max max H~s3) '~ (a ) - < ~. aEF O<~j<~ n - 1

By applying Lemma 1.4 for the sequence of unital maps ~I "~)'~ we find a unital ,-homomorphism ~/with finite dimensional image and a unitary u such that

[lu d i a g ( ~ ( a ) , 'q(a))u* - diag(~b~(a), da) ) t l < 2E

124 MARIUS DADARLAT

for all a E F. [:3

In the proof of Theorem A we are going to apply Theorem 1.5 for ,-homomor- phisms. Note that if ~p: C(X) ~ B is a ,-homomorphism, whose restriction to Co(X\xo) is denoted by ~ too, then ~ ( a ) = ~(a) + a(xo)(la(t) - ~(1)).

1.6. THE PROOF OF THEOREM A

Recall from [F] that an extension of Abelian groups

O ~ K -+ G-L, H -+ O

is called pure if its restriction to any finitely generated subgroup of H is trivial. The isomorphism classes of pure extensions form a subgroup of Ext(//, K). The universal coefficient theorem of [RS] gives an exact sequence

0--+ Ext(K.(A) ,K(B)._I)-+ KK(A,B)--* Hom(K.(A),K.(B))-+ 0

for A in a large class of separable nuclear C*-algebras and any C*-algebra B that has a countable approximate unit. The quotient of KK(A, B) by the subgroup of pure extensions in Ext(K.(A), K._I(B)) is denoted by KL(A, B) [Re]. The subgroup of pure extensions was studied and characterized in the setting of [BDF] in [KS].

(a) =~ (b) Suppose first that X is a finite C W complex. Since K . (C (X)) is finite- ly generated any pure extension of K.(C(X)) is trivial, hence KL(C(X) , B) = K K ( C ( X ) , B). Let X1, . . . , Xm be the connected components of X. Let ei be the unit of C(Xi). Using the definition of the K0 group we find R E N, a ,- homomorphism ( : C ( X ) ~ MR(C1B) and a unitary v E UR+I(B.) such that vdiag(~(ei),~(ei))v* = diag(~b(ei),~(ei)) for i = 1 , . . . , r a . Note that since and ¢ are unital, ~ can be chosen to be unital. After replacing ~o, ¢ by v diag(~y, ~)v* and diag(¢, ~), we may assume that ~(ei) = ~b(ei), i = 1,. . . , m. Let ~oi, ¢i denote the restrictions of !P and ¢ to C(Xi). Then [~i] = [~bi] E KK(C(Xi) , B). Let

> 0 and let Fi = F N C(Xi). Using Theorem 1.5 for each i we find k(i) E N, a ,-homomorphism r/i: C(Xi) ~ Mk(o(B) with finite dimensional image and a unitary ui E Uk(i)+l(B) such that

Ilu~ diag(Ti(a), 71i(a ))u~ - diag(~bi(a), ~(a))ll <

for all a E Fi. Let k = k(1) + . . . + k(ra) and define r/': C(X) = @iC(Xi) -+ Mk(B) by r / ' (a l , . . . ,a ,~) = diag@l(al), . . . ,r/m(am)). By conjugating (ul, 1 , . . . , 1) by suitable permutation unitaries we find unitaries vl E Uk+l (B) such that

Ilvi diag(~vi(a), ~'(a))v~ - diag(¢i(a), ~/(a))ll < 6

APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS 125

for all a E Fi. Let Pi = diag(~i(ei), 7/'(ei)) = diag(¢i(ei), rl'(ei)). By choosing small enough we can perturb v~ to unitaries wi E Uk+I(B) such that wipiw* = Pi and

liwi d iag (~ (a ) , ~f(a))w~ - diag(~bi(a), ~/'(a))ll < e

for all a E F / , i = 1 , . . . ,m . Letp = lk --~'(1). Let u E Uk+I(B)be the unitary w~pl + . . . + Wmpm + diag(O, p). Fix x0 E X and define r/: C ( X ) --~ Mk(B) by ~?(a) = ~'(a)+a(xo)p. Then ~7 is a unital ,-homomorphism with finite-dimensional image and

llu diag( (a), rl(a))u* - diag(¢(a) , (a))ll < e

for all a E F. In the general case, we embed X into the Hilbert cube I ~° and write X =

MX~, X~ = P~ × I "~n where R, are finite C W complexes with P~+I C P~ × I and w~ = {n + 1, n + 2, . . .} . Let gi: I ~ ~ 1 be the ith-coordinate map. By abuse of notation we let gi also denote the restrictions of gi to X and Pn. Without loss of generality we may assume that the set F in the statement of the Theorem is equal to {gl ,g2, . . . ,g~} for some n. Let now n be fixed and choose m /> n such that for any y E P~ there is x E X such that lgi(x) - gi(Y)] < e for i = 1 , . . . , n. Let Pro: C(Pm) ---, C ( X ) be induced by the X ~ P m × I ~m -+ Pm with the second arrow standing for the canonical projection. Consider the , - homomorphisms ~PPm, ePm: C(Pm) --+ B, the finite set {91,... ,gn} C C(Pm) and ¢ > 0. Since the Kasparov product induces a product at the level of the KL-groups [R0] and KL(C(Pm) , B) = KK(C(Pm) , B), it follows that the , - homomorphisms ~Pm, zbpra give rise to the same KK-element. By the first part of the proof we find a unital ,-homomorphism r/0: C(Pm) --+ Ma(B) with finite- dimensional image and a unitary u E Uk+I(B) such that

lludiag(~pm(g~), 'qo(g~))u* - diag(¢p,~(gi), ~o(g~))ll < c

for i = 1 , . . . , n . The map V0 can be expressed as ~]0(a) = E~=la(yj)qj with yj E Pm and mutually orthogonal projections qj. The integer m was chosen so that we can find points xj E X with lgi(Yj) - gi(xj)l < c for i = 1 , . . . , n. Define r/: C ( X ) ---+ Mk(B) by ~(a) = E~=la(xj)qj. Then

diag( (g ), - diag(¢(gl) , ?(gi))ll < 2v

f o r i = 1 , . . .~n . So far we have proven that if [~2] = [¢] E K L ( C ( X ) , B) , thenforanye > 0and

F C C ( X ) finite, there exist k E H, aunital ,-homomorphism r/: C ( X ) --, Mk(B) with finite dimensional image and a unitary u E Uk+l (B) such that

Nu diag(~p(a), rl(a))u* - diag(¢(a) , ~/(a))ll <

126 MARIUS DADARLAT

for all a E F . Next we show that r/ can be chosen of the form r/(a) = d i a g ( a ( x l ) , . . . , a(xk)) with x l , . . . , xk E X. This is done in two steps.

First we note that if w E Uk+~(B), (: C(X) --+ M~(B) is a . -homomorphism with finite-dimensional image, ¢ /= w diag(r/, ~)w*, and ¢t = 1 @ w(u ® 1~)1 @ w*, then

/l~ diag(~(a) , ¢/(a))z2* - diag(¢(a) , ~a)) l l < c

for all a E F. This remark shows that we have the freedom to change ~ by taking direct sums and conjugating by unitaries. The proof of (a) =~ (b) is completed by using the following elementary Lemma:

LEMMA. Let ~/: C ( X ) --+ Mk( B ) be a unital ,-homomorphism with finite dimen- sional image. Then there are a unital ,-homomorphism with finite-dimensional image ~: C( X ) -+ Mr(B) and a unitary w E Uk+r( B ) such that w diag(~, ~)w* is equal to a .-homomorphism of the form ~)(a) = diag(a(x 1), • • •, a(x k+r)).

Proof There are distinct points Xl , . . . ,Xn in X and a partition of lk into mutually orthogonat projections P l , - . . , P~ such that r/(a) = E~=la(xi)p~ for all a E C(X) . The proof is done by induction after n. Define ~(a) = a(xl)p2 + a(x2)Pl + ~=3a(xi)Pi, ~t(a) = a(xl)(Pl +/92)+ ~n=3a(xi)Pi,~t(a) = a(x2)(Pl+ P2) + E~=3a(xi)Pi. Then it is easily seen that diagQ/, ( ) is unitarily equivalent to diagQf, ~'). The formulae for 7/' and ~t involves n - 1 distinct points each.

(b) =~ (a) This is a generalization of Proposition 5.4 in [R¢]. The proof is based on an adaptation of the argument given in [Re] and it remains valid if we replace C ( X ) by any C*-algebra in the class A/" of [RS]. To save notation set A = C(X) . The first step is to show that K,(¢p) = K,(~b). This is standard and follows easily from (b) by using functional calculus and the definition of K-theory. Therefore by the universal coefficient theorem of [RS] the difference element z = [qD] - [¢] lies in the image of Ext (K, (A) , K , _ I ( B ) ) in K K ( A , B). It is shown in [Re] that the element z is given by

0 ~ K . ( S B ) -+ K . (E ) ~r, K . (A) --+ 0 (7)

where E is the C*-algebra of all pairs (.f, a), where a E A, f E B[0, 1], f (0 ) = ~(a) , f (1 ) = ~b(a). The map 7r:E -+ A is given by ~r(f,a) = a. Our aim is to show that z corresponds to a pure extension. By assumption there exist a sequence of . -homomorphisms ~/i: C ( X ) --+ Mn(i) with finite dimensional image and a sequence of unitaries ui E Ura(i)(B) where m(i) = 1 + n(1) + - . . + n(i), such that if

qoi = diag(~, ~ /h . . . , r/i), ~bi = diag(¢, 7h , . . . , ~/i)

then

lim Ilui~i(a)u7 - ~b~(a)N = 0 (8) i--+oo

APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS 127

for all a C C ( X ) . Note that K,(cpi) = K.(~bi), As above we have a 'difference' extension

~(~) o ~ M.~(i)(SB) ~ Ei ---+ A ~ 0

corresponding to the pair ~0i, ~'i. There is a natural embedding 7i: Ei -+ Ei+I, 7 i ( f , a) = ( f ® ~li+ l( a), a). This yields a commutative diagram

,(i) , 0 0 - , M~( i ) (SB ) -

0 '~ Mm(i+I)(SB) ~(i+1) o, 0

• Ei-,,

' ~ i + 1

, A

ida

. A

Taking the inductive limit of these extensions we obtain an extension

O -+ IC ® S B ~ Eoo ~% A --+ O

whose KK-theory class is equal to z (by the continuity of K-theory and the five lemma). With this construction in hands, by using (8) and reasoning as in the proof of Proposition 5.4 in IRe] one shows that if g E K , ( A ) and n9 = 0 for some n then there is i >/ 1 and h E K , ( E i ) with nh = 0 and 7r(i),(h) = g. This shows that the group extension

0 ~ K . ( B ) - + K . ( E o o ) ~ K.(A)- -+ 0

is pure. We conclude that the extension (7) is pure. O

Another proof of (b) ~ (a) can be obtained by constructing a sequence Xi E Map(A, E ~ ) such that Ir~xi = idA and I I x~(ab) - x d a ) ~ d b ) l l ~ 0, as i ~ ~ ,

for all a, b C A. This goes as follows. We may assume that there is a path ofunitaries ui(s), s E [0, 1], in U~(1)(B) with ud0 ) = I and u d l ) = ul. Let al E Map(A, E~) be defined by

ai(a)(s) = ~ (ui(28)~i(a)ui(2s)*,a) for0 ~< 8 ~< 1/2 [ ( ( 2 - 2~)ui~oi(a)u7 + (2~ - 1)¢;(a), a) for 1/2 ~< ~ ~< 1

Let 0i denote the embedding of Ei into E ~ . It is not hard to see that the sequence of maps Xi = 0iai has the desired properties. The sequence X~ gives an approximate splitting of the extension E ~ . Then one shows as in [BrD] that the corresponding extension in K-theory is pure.

Suppose that the C*-algebra B is purely infinite and simple. Then Theorem A can be modified as follows.

128 MARIUS DADARLAT

THEOREM 1.7. Let X be a compact metrizable space. Let B be a purely infinite, simple, unital C*-algebra and let qo, ~b: C ( X ) --+ B be two unital ,-monomor- phisms. Then [[~]] = [[4]] in K L ( C ( X ) , B ) i f and only i f ~ is approximately unitarily equivalent to 4. That is, i f and only i f there is a sequence o f unitaries un E U( B) such that flu~(a)u: - ¢(a)ll -~ O f or all a E C ( X ) .

Proof Fix e > 0 and F C C ( X ) finite. Using Theorem A all we have to do is to find a unitary wo E U ( B ) such that ilwo (a)wG - ¢(a)ll < 5c for all a E F. Let k E N,~7:C(X) --, M k ( B ) and u E Uk+I(B) be provided by Theorem A. Since r/is unital and has finite dimensional image there exist L E N, distinct points x l , . . . , XL in X and a partition of lk into mutually orthogonal nonzero projections P l , . . . ,PL E M~:(B) such that

L

rl(a) = ~ a(xl)Pi i = l

for all a E C(X) . D e f i n e 3`: C(X) --+ Mk+I(B) by 7 = u~u*. Set q = 3 ' (1) and qi = upiu*. Then

L L

url(a)u* = E a(xi)qi and q + E qi = lk+l. i=1 i= l

(9)

We are going to use repeatedly a result of Zhang [Zhl] asserting that a simple, purely infinite C*-algebra has real rank zero. Since 3' is a monomorphism and B has real rank zero we can apply Lemma 4.1 in [EGLP] to get nonzero, mutually orthogonal projections d, d l , . . . , d L in q M~ + l ( B )q such that d + d l + . . . + d L = q and

L a(xi)di 7(a) - d T ( a ) d - E < ~ i=1

(lO)

for all a E F. Similarly, there are nonzero, mutually orthogonal projections e, el, . . . , eL in B such that c + el + . . . + eL = l a n d

¢ ( a ) L ,

i=I

Off)

for all a E F. Since B is simple and purely infinite, di is equivalent to a subpro- jection ei of ci. Recall that two projections e and f in a C*-algebra A are called equivalent if there is a partial isometry v E A such that v*v = e and vv* = f . The proof of Lemma 4.1 in [EGLP] shows that (10 ~) remains true if we replace el by nonzero subprojections ei <~ ci and c by e = 1 - el . . . . . eL. Therefore we may assume that di is equivalent to c~ in Mk+l (B). Since Mk+l (B) is simple and purely infinite, d i+ qi is equivalent to a subprojection of di. Similarly ci ® Pi is equivalent

APPROXIMATELY UNITARILY EQUIVALENT MORPHtSMS 1 2 9

to a subprojection of ci. Actually, we can find partial isometries v, w C Mk+I(B) ! such that v(di + qi)v* + d~ = di and w(ci ® pi)w* + c i = ci for some nonzero

projections d~, e~, i = 1 , . . . , L. Define T.y: C(X) --+ Mzk+z(B) by

( L L L ~) Tv(a ) = diag d7(a)d* + ~ a(z~)di + ~ a(xi)q,:, ~ a(x;)d .

i=1 i=1 i=1 /

Using v one obtains a partial isometry V C m2k+z(B) with V*V = lk+l + EL=ld~, VV* = q and such that

L

VT, v(a)V* = dT(a)d + E a(xi)di. (11) i = l

Similarly, if T0: C(X) ~ M2k+2(B) is defined by

Te(a) = diag c¢(a)c* + ~ a(z~)c~ ® ~ a(xJpi, ~., a(xg)c , i=1 i = ! i=1

L ; * then there is a partial isometry W E M2k+2(B), W*W = lk+l + ~i=lCi, W W = 1B such that

L

WT~(a)W* = ce(a)c + ~ a(x~)c~. (11') i=1

Nx a C A. From (10), (11) and Off), (11') we obtain

llv*7(a)V - ~:~(a)]l < E, (12)

t l W * ¢ ( a ) W - T¢(a)ll < e. (12')

On the other hand, using (10) and (t0 t) we obtain

IIT.v(a)-diag(u(cp(a)®'rl(a))u*, L~=l a(xi)d~) l < e, (13)

11 ( :) Ti~(a) - diag ~(a) ® r~(a),~_a(xi)c < e. (13') i=1

By construction [di] = [ci] and [qi] = [Pi] in K0(B). This implies [d~] = [c}] in Ko(B). Since these are proper projections in Mk+I(B) we can find a unitary

130 MARIUS DADARLAT

* ~ for i 1, , L (see [Cu], [Zh2]). Let uo E Uk+I(B) such that uodiu o = c i = ... U = lk+l ® u0 E U2k+2(B). Then

L

-diag(¢(a)®rl(a),~=la(xi)c~) l < e, (14)

since Ilu(qo(a) ® rl(a))u* - ~(a) ® (a)ll < ~. Using (13), (13') and (14) we obtain

llUT (a)U*- T ,(a)ll < 3e. (15)

Then from (12), (12') and (15)

llfW*u (a)u*Wf* - W* (a)Wll <

Finally if we set w0 = WUV*u, then

II 0 (a)w;- ¢(a)ll <

for all a E F. [3

Certain special cases of Theorem 1.7 have been previously proven in [Lil-3] and [EGLP] and the above proof uses ideas from those papers. For other applications of the asymptotic morphisms in the study of ,-homomorphisms we refer the reader to [DL], [D1] and [LiPh].

Remarks. 1.8. (a) By a result of T. Loring [Lo2], if X is a finite connected CW complex with base point x0 and O,~ is a Cuntz algebra with n >/ 4 even, then all the elements of KK(Co(X\xo), 0,~) are induced by ,-homomorphisms Co(X\xo) --+ On. By combining this result with Theorem 1.7 we obtain a bijection between KK(Co(X\xo), 0~) and the set of all approximate unitary equivalence classes of unital ,-monomorphisms from C(X) to On.

(b) Let Y be the dyadic solenoid and let X be the one point compactifi- cation of (Y\Yo) × IR. Let Q(tI) be the Calkin algebra. There are infinitely many unitary equivalence classes of unital ,-monomorphisms from C(X) to Q(11). These classes are parametrized by the Brown-Douglass-Fillmore group Ext(C(X)) ~ KK(Co(X\xo), Q(H)) ~ Ext,(Z[1/2], Z). However, since all the extensions in Ext~(Z[1/2], Z) are pure, it follows that KL(Co(X\xo), Q(H)) = 0 and any two unital ,-monomorphisms from C(X) to the Calkin algebra are approx- imately unitarily equivalent (see [Br]).

APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS 131

2. Inductive Limit C*-Algebras

Consider the following list of C*-algebras: the scalars, C; the circle algebra, C(T); the unital dimension-drop interval, defined below; and all C*-algebras arising from these by forming matrix algebras and taking finite direct sums. The collection of all these C*-algebras will be denoted D. The collection of all quotients of C*- algebras in D will be denoted D. A C*-algebra will be called an AD-algebra if it is isomorphic to an inductive limit of C*-algebras in D.

By the nonunital dimension-drop interval we mean

~,~ = { f E C([0,1],M,~) { f(O) = O, f(1) is scalar}.

The unital dimension-drop interval ~,~ is the unitization of ~,~. The K-theory of lI,~ is easily computed, K0(]i,~) = 0, Kl(lIm) ~ Z/ra.

LEMMA 2.1. Let X, Y be finite, connected C W complexes and let k, m, r E N. Let 7: Mk(C(X)) --+ P M m ( C ( Y ) ) P be a unital *-homomorphism where P is a projection in Mm(C(Y)). Let u: Mk(C(X)) --, Mkr(C(X)) be a uni- tal ,-homomorphism of the form u(a) = diag(a,a(xl) , . . . ,a(xr_l)) ,xi E X. Suppose that rank(P) > lOkrdim(Y). Then there are *-homomorphisms 70, 71: )dk(C(X)) --+ PMm(C(Y) )P with orthogonal images such that 70 has finite dimensional image, 71 factors through u and 7 is homotopic to 7o + 71.

Proof The result is a consequence of Theorems 4.2,8 and 4,2.11 in [DN]. []

LEMMA 2.2. Let H be a finite subset of C( S2), let ~ > 0 and let X be a finite, connected C W complex. There is mo such that if m ) too, then for any unital *-homomorphism a': C(S 2) ~ Mm( C( X) ) there exist a unital *-homomorphism a: C (S 2) --+ Mm(C(X) )anda C*-subalgebra D of Mm(C(X)) isomorphictoa quotient of a direct sum of matrix algebras over C( S 1 ), such that (r ~ is homotopic to cr and dist(cr(a),D) < E for all a E 11.

Proof. If X is a product of spheres, the result appears in [EGLP1]. In particular for X = S 2 we find n and such that the unital .-homomorphism # : C(S 2) --+ M,~(C(S2)), u'(a) = (a, a (x0) , . . . , a(xo)) is homotopic to a *-homomorphism u with u(11) approximately contained to within z in a circle algebra. The more general situation we consider here is reduced to the case X = S 2 by using Lem- ma 2.1. Indeed, by Lemma 2.1 there is m0 such that any unital *-homomorphism a': C(S 2) -+ Mm(C(X)) , m ) rao, is homotopic to a direct sum between a , - homomorphism with finite-dimensional image and a ,-homomorphism that factors through u. []

PROPOSITION 2.3. Let X be a finite, connected C W complex. Let F be a finite subset of C ( X ) and let ~ > O. Suppose that the K-theory group Ko( C ( X ) ) is tor- sion free. Then there exist r C N, a *-homomorphism r/: C ( X ) ~ M~_I(C(X)) with finite dimensional image and a C* -subal g ebra D C M~ ( C ( X ) ) with D E such that

dist(a ® r/(a), D) <

132 MARIUS DADARLAT

for all a E F. The ,-homomorphism 77 can be chosen of the form

rf( a ) = diag( a( x l ) , . . . , a( Xr_ l ).

Proof There are c, d /> 0, re(i) /> 2 with Ko(C(X)) TM Z c+l, KI(C(X) ) -~ Z d @ Z / m ( 1 ) ® - - - ® Z/m(k) . Let N = c + d + k and set

Bj = Co(S2\pt) if 1 ~< j ~< c,

Bj = Co(S1\pt) i f c < j ~< c + d,

Bj = ]Im(j_c_d ) if c + d < j ~ N,

B = B1 @ ' " ® B:v.

If K1(C(X)) is torsion free, then we consider only Bj for 1 ~< j ~< c + d. By construction, K.(Co(X\xo)) is isomorphic to K.(B) . The universal coefficient theorem of [RS] shows that Co(X \:co) is KK-equivalent to B. Using the E-theory description of KK-theory of [CH] and the suspension theorem of [DL], we find an asymptotic morphism qPt: Co(X\:co) ~ B @ Moo yielding a KK-equivalence. Let X C K K ( B , Co(X\xo)) be such that X[[~¢]] = [[idco(x\x0)]]. The element X can be written as X = X~ + "" "XN with Xj E KK(Bj,Co(X\:co)) . It is known that every element in K K ( B j , Co(X\:co)) can be realized as a , -homomorphism Bj --. Co(X\:co) ® ML for a suitable integer L.

[Co( SI\pt) , Co(Xkxo) ® Moo] ~- KK(Co(SI\Pt) , Co(X\xo)) [Ros],

[Co( S2\pt), Co(Xkxo)® Moo] -~ KK(Co( S2kPt), Co(X\xo)) [Se], [DN],

[~m, Co( X \ xo) ® Moo] ~- g g (]Im, Co( X \ :co) ) [DL].

It follows that there are , -homomorphisms 0. ¢j. Bj --+ Co(Xkxo) ® ML with

[~ b°] = Xj. Let ~)j: f3j -+ C ( X ) ® ML be the unital extension of ¢o and let ¢ j

denote the , -homomorphism ~)j = ~j ® idoo " B j ® Moo -+ C ( X ) ® ML ® Moo. Define

N ¢: (~(J[3j @ Moo) -+ C(X) ® ML ® MN® M~o

j= l

¢ ( b l , . . . , bN) = d i a g ( ~ b l ( b l ) , . . . , ~)N(bN)).

Form the diagram

Co(X\xo) "~ ,, C (X) ® ML ® MN ® Moo

®j(By ® Moo)

APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS ~ 33

where 7(a) = a ® e for a fixed one-dimensional projection e. By construction, the above diagram commutes at the level of KK-theory. Therefore there is a hornotopy of asymptotic morphisms ff~: Co(X\xo) -+ C(X) ® ML[O, 1] ® MN ® m ~ with

= a n d =

Let F C C(X) and ~ be as in the statement of Proposition 2.3. Let F and be given by Lemma 1.4. As in the proof of Theorem 1.5, let ~ be a dominating function for ~ and find to such that ~ is $-multiplicative on _P for all t ~> to. Now let t >/to be fixed. By increasing ~(t), we may assume that the image of~ t is contained in ®~Y=I/)J ® M~(~). Moreover we may assume that ~2~ is ~-multiplicative

on _~. We obtain the following diagram

C ( X ) -- ~ • C ( X ) @ ML (~ MN ~ Ms(t)

where the superscript ~ indicates that the corresponding maps have been unital- ized as described before Theorem 1.5. The restriction of ¢ to @j (/~j ® Ms(t) )

is denoted by ¢ too. Note that ~0),~ = 7~ and ~1),~ = ¢~p~,. The next step is to modify the above diagram so that ¢ will become homotopic to a , - homomorphism which factors approximately through an algebra in 79. Set C = ~j=IN (/~j Q m~(~)), E = C(X) ® ML ® MN ® Ms(t) and H = ~p~(F) (recall that t is fixed). By using Lemma 2.2 we find m, such that ifT0: E --+ Mm(E) is defined by 7o(c) = diag(e, c(xo),.. . , c(xo)), with x0 6 X, then the ,-homomorphism 70¢ is homotopic to a ,-homomorphism a: C -+ Mm(E) such that a(H) is approxi- mately contained, to within e, in a subalgebra D of M,d.(E) with D E/5. Actually, one applies Lemma 2.2 for the partial ,-homomorphism 70¢j, 1 ~< j ~< c. Let el: C --+ ~fm(E)[O, 1] be a homotopy of ,-homomorphisms with k9 (°) = 7o¢ and ~(1) = a. Next we consider the path of maps F(s) in Map(C(X) , Mm(E)) obtained

by the juxtaposition o f T 0 ~ ~)'~ with ~ ( s ) ~ . It is clear that 7o7 ̀~ is the initial point of F (~) and a ~ is the terminal point. Since ~ and ¢p~ are $-multiplicative on /~, it follows that [(~) is ~-multiplicative on/~ for each s. This enables us to use Lemma 1.4 to show that 7o7 ~ is stably approximately unitarily equivalent to t r ~ . Namely, there exist S 6 I~, a ,-homomorphism ~/0: C(X) ~ MS-I(Mm(E)) with finite-dimensional image and a unitary u E Us(Mm(E)) such that

ltu d i a g ( a ~ ( a ) , 7/o(a))u* - diag(707~(a), ~/0(a))tl < 2e

for all a E F. Note that 707 ~ is a unital *-homomorphism of the form

a ~ diag(a, a(xo),. . . , a(xo)) = diag(a, ~1(a)).

134 MARIUS DADARLAT

Since ~ ( F ) C H and a(11) is approximately contained to within c in D, it follows that, for all a C /; ,diag(a, r/l(a), r/0(a)) is approximately contained

to within 3E in u(D @ image(r/0))u* E /). By using the Lemma that appears in the proof of Theorem A, we see that ~0 can be taken of the form r/0(a) = diag(a(xl) , . . . ,a(xr_l)) . [~

THEOREM 2.4. Let A be a C*-algebra of real rank zero. Suppose that A is an inductive limit

k~

A : li_m (A~, 7m,~), A,, : ( ~ M[n,i](C(X,~,i)). 4=1

Suppose that each Xn,i is a finite connected C W complex, Ko( C( Xn,i) ) is torsion free and d = supn,i { dim( Xn,i ) ) < 0o. Then A is isomorphic to an AT) algebra.

Proof. Let e,~,i denote the unit of A~,~ = M[~,i](C(X,~,i)) and let 7,~i,n: A~,~ --+ Am,j be the partial ,-homomorphisms of 7,~,~. The C*-algebras in 79 are semipro- jective [Lo]. By Proposition 1.2 in [DL1], (see also Theorem 3.8 in [Lo]) it suffices to show that for any finite subset F C An,i and e > 0, there is m/> n, such that for any 1 <~ j <~ km there exist a C*-algebra D E D, and a .-homomorphism a: D -+ f jA,~,j f j , f j = .~j,i (e ~ such that dist(7~i,~(a), a(D)) < e for all a E F. By Theorem 1.4.14 in [EG] we can assume that F is weakly approximately constant to within ~. To simplify notation say A~,i = Mk(C(X)) . By Proposition 2.3 there exist r E N and a unital .-homomorphism u: Mk(C(X)) --+ Mkr(C(X)) of the form u(a) = diag(a, a(xl ) , . . , a(x~-l)), and there is a unital .-homomorphism ~: D --+ Mk~(C(X)) with D E 79 such that dist(u(a), ( (D) ) < e for all a E F. By Theorem 2.5 in [Su] and Lemma 2.3 in [EG] there is ra >i n so that for each j , 1 <~ j <~ km, one of the following conditions are satisfied.

(i) There is a ,-homomorphism #: A~,i --* f jAm,j f j with finite dimensional image such that - (a)ll < E for all a E F.

(ii) rank(7~i,~(e~,i)) = rank(fj) = kq where q > lOrd.

If we are in the first case, then we are done. Thus we may assume that (ii) holds. Define uj: Mk(C(X) ) ~ Mk~(C(X)) @ Mk, uj(a) = u(a) ® a(xo). We apply Lemma 2.1 for 7~,r~ and uj. Hence we find a .-homomorphism ~bj: Mkr(C(X)) ® Mk --+ f jA,~,j f j such that Cjuj is homotopic to .m,,~'v~" as .-homomorphisms from M~(C(X)) to f jAm,j f j . Since F is approximately weakly constant to within e and A has real rank zero, by Theorem 2.29 in lEG], there exist s ) m and a unitary u E 7~m(fj)A~Ts,m(fj) such that

IluTs,m~pjuj(a)u* - %,,~7~i,~(a)I1 < 70e

APPROXIMATELY UNITARILY EQUIVALENT MORPHISMS 135

for all a • F . Define ~/: D O Mk --* M k r ( C ( X ) ) O M k , { j ( d , A ) = { ( d ) ® A. The various maps that are involved here can be visualized on the diagram

"[rn,n ""/s,m F ( . . . . . . . . . . An,i "" f j A m , j f j ~ " As

D ® M ~

Since dist(uj(a), { j (D ® Mk)) < s for all a E F , if we set cr = u(%,mCj~j)u*, then

dist(%,mT~*n(a),c~(D)) < 100&

This concludes the proof. []

In Theorem 2.4 one can allow d = ec under the additional assumption that A has slow dimension growth. A similar remark applies to Theorem B.

2.5 THE PROOF OF THEOREM B

Let A and B be as in the statement of Theorem B. Theorem 2.4 shows that A and B are A D algebras. By Theorem 7.1 in [Ell], ordered, scaled K- theory is a complete invariant for the simple A D algebras of real rank zero.

References

[B1] Blackadar, B.: K-theory for Operator Algebras, MSRI Monographs No. 5, Springer- Verlag, Berlin and New York, 1986.

[BDR] Blackadar, B., Dadarlat, M. and RCrdam, M.: The real rank of inductive limit C*-algebras, Math. Scand. 69 (1991), 211-216.

[Bra] Bratteli, O.: Inductive limits of finite-dimensional (7*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195-234.

[B r ] Brown, L. G.: The universal coefficient theorem for Ext and quasidiagonality, in Operator Algebras and Group Representations, Pitman Press, Boston, London, Melbourne, 1983, pp. 60-64. Brown, L., Douglas, R. and Fillmore, R: Extensions of C*-algebras K-homology, Ann. of Math. 105 (1977), 265-324. Brown, L. G. and Pedersen, G. K.: C*-algebras of real rank zero, J. Funet. Anal. 99 (1991), 131-149. Brown, L. G. and Dadarlat, M.: Extensions of C*-algebras and quasidiagonality, to appear in J. London Math. Soc. Choi, M.-D. and Effros, E. G.: The completely positive lifting problem for C*-algebras, Ann. of Math. 104 (1976), 585-609. Connes, A. and Higson, N.: Deformations, morphismes asymptotiques et K-thtofie bivari- ante, C. R. Acad. Sci. Paris 313 (1990), 101-106.

[BDF]

[BP]

[BrD]

[CE]

[CH]

136 MARIUS DADARLAT

[Cu] [D]

[D1]

[D2]

[DN]

[DL]

[DL1]

[Eft]

JEll]

lEG]

lEG1]

[EGLP]

[EGLP1]

[G]

[KS]

[Lil]

[Li2]

[Li3] [LiPh]

[Lo]

[Lol]

[Lo2]

[Re] [Pal

[Ros]

[RS]

[Se]

[Sa]

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APPROXIMATELY UNITAR1LY EQUIVALENT MORPHISMS 137

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[Zhll

[Zh2]

Su, H.: On the classification of real rank zero C*-algebras: inductive limits of matrix algebras over non-Hausdorff graphs, Thesis, University of Toronto. Voiculescu, D.: A non-commutative Weyl-Von Neumann theorem, Rev. Roumaine Math. PuresAppl. 21 (1976), 97-I 13. Zhang, S.: Certain C*-algebras with reak rank zero and their corona and multiplier alge- bras, Part I, Pacific J. Math. 155 (1992), 169-197. Zhang, S.: Certain C*-algebras with real rank zero and their corona and multiplier algebras, Part II, K-Theory 6 (1992), 1-27.