Proper asymptotic unitary equivalence in $\KK$-theory and projection lifting from the corona algebra
aa r X i v : . [ m a t h . OA ] S e p PROPER ASYMPTOTIC UNITARY EQUIVALENCE IN KK -THEORY AND PROJECTION LIFTING FROM THECORONA ALGEBRA HYUN HO LEE
Abstract.
In this paper we generalize the notion of essentialcodimension of Brown, Douglas, and Fillmore using KK-theoryand prove a result which asserts that there is a unitary of theform ‘identity + compact’ which gives the unitary equivalence oftwo projections if the ‘essential codimension’ of two projectionsvanishes for certain C ∗ -algebras employing the proper asymptoticunitary equivalence of KK-theory found by M. Dadarlat and S.Eilers. We also apply our result to study the projections in thecorona algebra of C ( X ) ⊗ B where X is [0 , −∞ , ∞ ), [0 , ∞ ),and [0 , / { , } . Introduction
When two projections p and q in B ( H ), whose difference is com-pact, are given, an integer [ p : q ] is defined as the Fredholm index of v ∗ w where v, w are isometries on H with vv ∗ = p and ww ∗ = q . Thisnumber is called the essential codimension because it gives the codi-mension of p in q if p ≤ q [BDF]. A modern interpretation of thisessential codimension is provided using the Kasparov group KK( C , C ).Indeed, a ∗ -homomorphism from C to B ( H ) is determined by the im-age of 1 which is a projection. Thus we can associate to the essentialcodimension a Cuntz pair. An important result of the essential codi-mension is the following: [ p : q ] = 0 if and only if there is a unitary u of the form identity + compact such that upu ∗ = q . Motivated bythis result, Dadarlat and Eilers defined a new equivalence relation onKK-group [DE]. When π, σ : A → L ( E ) are two representations, with E is a Hilbert B -module, we say π and σ are properly asymptoticallyunitarily equivalent and write π ≅ σ if there is a continuous path ofunitaries u : [0 , ∞ ) → U ( K ( E ) + C E ), u = ( u t ) t ∈ [0 , ∞ ) , such that • lim t →∞ k σ ( a ) − u t π ( a ) u ∗ t k = 0 for all a ∈ A , Mathematics Subject Classification.
Primary:46L35.
Key words and phrases.
KK-theory, proper asymtotic unitary equivalence, ab-sorbing representation, the essential codimension. • σ ( a ) − u t π ( a ) u ∗ t ∈ K ( E ) for all t ∈ [0 , ∞ ), and a ∈ A .Note that the word ‘proper’ reflects the fact that implementing uni-taries are of the form ‘identity+compact’. The main result of them is[DE, Theorem 3.8] which asserts that if φ, ψ : A → M ( B ⊗ K ( H ))is a Cuntz pair of representations, then the class [ φ, ψ ] vanishes inKK( A, B ) if and only if there is another representation γ : A → M ( B ⊗ K ( H )) such that φ ⊕ γ ≅ ψ ⊕ γ . When B = C , which corre-sponds to K -homology, the result is improved as a non-stable version.In fact, if ( φ, ψ ) is a Cuntz pair of faithful, non-degenerate representa-tions from A to B ( H ) such that both images do not contain any non-trivial compact operator, then the cycle [ φ, ψ ] = 0 in KK( A, C ) if andonly if φ ≅ ψ [DE, Theorem 3.12]. This fits nicely with the above as-pect of the essential codimension. An abstract version of this is provedgiven a Cuntz pair of absorbing representations (See Theorem 2.11).Thus the proper asymptotic unitary equivalence must be the right no-tion and tool for further developments of the non-stable K-theory. Ourintrinsic interest lies in when this non-stable version of proper asymp-totic unitary equivalence happens as shown in K -homology case. Weshow a similar result for K -theory. In fact, we prove that if ( φ, ψ ) isa Cuntz pair of faithful representations from C → M ( B ⊗ K ) whoseimages are not in B ⊗ K , then [ φ, ψ ] = 0 in K ( B ) if and only if φ ≅ ψ provided that B is non-unital, separable, purely infinite simple C ∗ -algebra such that M ( B ) has real rank zero (See Theorem 2.14).Besides our intrinsic interest, Theorem 2.14 was motivated by theprojection lifting problem from the corona algebra to the multiplieralgebra of a C ∗ -algebra of the form C ( X ) ⊗ B . To lift a projectionfrom a quotient algebra to a projection has been a fundamental questionrelated to K-theory (See [Da]). We show that a projection in the coronaalgebra is ‘locally’ liftable to a projection in the multilpler algebra butnot ‘globally’ in general. In other words, it can be represented byfinitely many projection valued functions so that its discontinuities aredescribed in terms of Cuntz pairs. They give rise to K -theoreticalobstructions. We show that these discontinuities can be resolved ifcorresponding K -theoretical terms are vanishing. In this process, thecrucial point of proper asymptotic unitary equivalence is exploited asa key step (See Theorem 3.3).2. Proper asymptotic unitary equivalence
Let E be a (right) Hilbert B -module. We denote L ( E, F ) by the C ∗ -algebra of adjointable, bounded operators from E to F . The idealof ‘compact’ operators from E to F is denoted by K ( E, F ). When
ROPER ASYMPTOTIC UNITARY EQUIV. AND PROJECTION LIFTING 3 E = F , we write L ( E ) and K ( E ) instead of L ( E, E ) and K ( E, E ).Throughout the paper, A is a separable C ∗ -algebra, and all Hilbertmodules are assumed to be countably generated over a separable C ∗ -algebra. We use the term representation for a ∗ -homomorphism from A to L ( E ). We let H B be the standard Hilbert module over B whichis H ⊗ B where H is a separable infinite dimensional Hilbert space.We denote M ( B ) by the multiplier algebra of B . It is well-known that L ( H B ) = M ( B ⊗ K ) and K ( H B ) = B ⊗ K where K is the C ∗ -algebraof the compact operators on H [Kas80]. Definition 2.1. [DE, Definition 2.1] Let π, σ be two representationsfrom A to E and F respectively. We say π and σ are approximatelyunitarily equivalent and write π ∼ σ , if there exists a sequence ofunitaries u n ∈ L ( E, F ) such that for any a ∈ A (i) lim n →∞ k σ ( a ) − u n π ( a ) u ∗ n k = 0,(ii) σ ( a ) − u n π ( a ) u ∗ n ∈ K ( F ) for all n . Definition 2.2. [DE, Definition 2.5] A representation π : A → L ( E )is called absorbing if π ⊕ σ ∼ π for any representation σ : A → L ( F ).We say that π and σ are asymptotically unitarily equivalent, andwrite π ∼ asym σ if there is a unitary valued norm continuous map u :[0 , ∞ ) → L ( E, F ) such that t → σ ( a ) − u t π ( a ) u ∗ t lies in C ([0 , ∞ )) ⊗K ( E ) for any a ∈ A , or if(i) lim t →∞ k σ ( a ) − u t π ( a ) u ∗ t k = 0,(ii) σ ( a ) − u t π ( a ) u ∗ t ∈ K ( F ) for all t ∈ [0 , ∞ ).If π : A → L ( E ) is a representation, we define π ( ∞ ) : A → L ( E ( ∞ ) ) byas π ( ∞ ) = π ⊕ π ⊕ · · · where E ( ∞ ) = E ⊕ E ⊕ · · · . Lemma 2.3.
Let ψ be an absorbing representation, and φ be a repre-sentation, of a separable C ∗ -algebra A on the standard Hilbert B -module H B . Then there exists a sequence of isometries { v n } ⊂ L ( H ( ∞ ) B , H B ) such that for each a ∈ Av n φ ( ∞ ) ( a ) − ψ ( a ) v n ∈ K ( H ( ∞ ) B , H B ) , k v n φ ( ∞ ) ( a ) − ψ ( a ) v n k → as n → ∞ ,v ∗ j v i = 0 for i = j. Proof.
Let S i , i = 1 , , · · · , be a sequence of isometries of L ( H B )such that S ∗ i S j = 0 , i = j , and P i S i S ∗ i = 1 in the strict topology. HYUN HO LEE
Let φ ∞ ( a ) = P i S i φ ( a ) S ∗ i . Since ψ is absorbing, there is a unitary U ∈ L ( H B , H B ) such that(1) U ∗ ψ ( a ) U − φ ∞ ( a ) ∈ K ( H B ) a ∈ A. Define T : H ( ∞ ) B → H B by T = ( S , S , · · · ). Then φ ∞ ( a ) = T φ ( ∞ ) ( a ) T ∗ . Thus equation (1) is rewritten as(2) T ∗ U ∗ ψ ( a ) U T − φ ( ∞ ) ( a ) ∈ K ( H ( ∞ ) B ) a ∈ A. If we identify φ ( ∞ ) as ( φ ( ∞ ) ) ( ∞ ) , there is a partition N i , i = 1 , , , · · · ,of N so that we generate a sequence of isometries v i ∈ L ( H ( ∞ ) B , H B )from U T = (
U S , U S , · · · , ). More concretely, if we let ν i : N i → N be bijections, we can define v i = ( U S ν − i (1) , U S ν − i (2) , · · · ). It is easilychecked that v i v ∗ j = 0 for i = j . Equation (2) implies that v ∗ i ψ ( a ) v i − φ ( ∞ ) ( a ) ∈ K ( H ( ∞ ) B ) , (cid:13)(cid:13) v ∗ i ψ ( a ) v i − φ ( ∞ ) ( a ) (cid:13)(cid:13) → i → ∞ . Finally, our claim follows from( v n φ ( ∞ ) ( a ) − ψ ( a ) v n ) ∗ ( v n φ ( ∞ ) ( a ) − ψ ( a ) v n ) = φ ( ∞ ) ( a ∗ )( φ ( ∞ ) ( a ) − v ∗ n ψ ( a ) v n )+ ( φ ( ∞ ) ( a ∗ ) − v ∗ n ψ ( a ) v n ) φ ( ∞ ) ( a ) − ( φ ( ∞ ) ( a ∗ a ) − v ∗ n ψ ( a ∗ a ) v n ) . (cid:3) Lemma 2.4. [DE, Lemma 2.6]
Let π : A → L ( E ) and σ : A → L ( F ) be two representations. Suppose that there is a sequence of isometries v i : F ( ∞ ) → E such that for a ∈ Av i σ ( ∞ ) ( a ) − π ( a ) v i ∈ K ( F ( ∞ ) , E ) , lim i →∞ k v i σ ( ∞ ) ( a ) − π ( a ) v i k → , and v ∗ j v i = 0 for i = j . Then π ⊕ σ ∼ asym π . We say φ : A → B ( H ) is admissible if φ is faithful, non-degenerate,and φ ( A ) ∩ K = { } . The main result in [Voi] states that any pairof admissible representations φ and ψ satisfies that φ ∼ ψ . Dadarlatand Eilers proved a much stronger version which states that any pairof admissible representations φ and ψ satisfies φ ∼ asym ψ [DE, Theorem3.11]. Since the admissible representation is absorbing, the followingresult is the appropriate generalization of Voiculescu’s result. Theorem 2.5.
If two representations ψ , φ of a separable C ∗ -algebra A on the standard Hilbert B -module H B are absorbing, then we have φ ∼ asym ψ . ROPER ASYMPTOTIC UNITARY EQUIV. AND PROJECTION LIFTING 5
Proof.
By Lemma 2.3 and Lemma 2.4, we have ψ ⊕ φ ∼ asym ψ , and theproof is complete by symmetry. (cid:3) Definition 2.6.
Let φ be a representation from A to M ( B ⊗ K ). Thenwe define a C ∗ -algebra by D φ ( A, B ) = { x ∈ M ( B ⊗ K ) | xφ ( a ) − φ ( a ) x ∈ B ⊗ K, a ∈ A } . Lemma 2.7. If M ( B ⊗ K ) has real rank zero, then D φ ( C , B ) has realrank zero for any representation φ : C → M ( B ⊗ K ) .Proof. The proof of the lemma is essentially based on the argumentdue to Brown and Pedersen [BP].Note that any representation φ : C → M ( B ⊗ K ) is determined by φ (1), which is a projection in M ( B ⊗ K ). Say φ (1) = p . Then we seethat D φ ( C , B ) = { x ∈ M ( B ⊗ K ) | xp − px ∈ B ⊗ K } .To show D φ ( C , B ) has real rank zero, it is enough to show any self-adjoint element in D φ ( C , B ) is approximated by a self-adjoint, invert-ible element. Let x be a self-adjoint element. Using the obvious matrixnotation x = (cid:18) a cc ∗ b (cid:19) ,xp − px ∈ B ⊗ K implies that c is ‘compact’, i.e., it is in B ⊗ K .Since M ( B ⊗ K ) has real rank zero, pM ( B ⊗ K ) p and (1 − p ) M ( B ⊗ K )(1 − p ) have real rank zero. Given ǫ > b invertiblein (1 − p ) M ( B ⊗ K )(1 − p ) with b = b ∗ and k b − b k < ǫ . Thenconsidering a − cb − c ∗ , we can find a in pM ( B ⊗ K ) p with a = a ∗ and k a − a k < ǫ , such that a − cb − c ∗ is invertible in pM ( B ⊗ K ) p . Then (cid:18) p cb − − p (cid:19) , (cid:18) p b − c ∗ − p (cid:19) are in D φ ( C , B ) since cb − is ‘compact’.Thus x = (cid:18) a cc ∗ b (cid:19) = (cid:18) p cb − − p (cid:19) (cid:18) a − cb − c ∗ b (cid:19) (cid:18) p b − c ∗ − p (cid:19) is invertible in D φ ( C , B ). Evidently k x − x k < ǫ , so we are done. (cid:3) Let us recall the definition of Kasparov group KK(
A, B ). We referthe reader to [Kas81] for the general introduction of the subject. A KK-cycle is a triple ( φ , φ , u ), where φ i : A → L ( E i ) are representationsand u ∈ L ( E , E ) satisfies that(i) uφ ( a ) − φ ( a ) u ∈ K ( E , E ),(ii) φ ( a )( u ∗ u − ∈ K ( E ), φ ( a )( uu ∗ − ∈ K ( E ). HYUN HO LEE
The set of all KK -cycles will be denoted by E ( A, B ). A cycle is de-generate if uφ ( a ) − φ ( a ) u = 0 , φ ( a )( u ∗ u −
1) = 0 , φ ( a )( uu ∗ −
1) = 0 . An operator homotopy through KK -cycles is a homotopy ( φ , φ , u t ),where the map t → u t is norm continuous. The equivalence relation ∼ oh is generated by operator homotopy and addition of degenerate cyclesup to unitary equivalence. Then KK( A, B ) is defined as the quotientof E ( A, B ) by ∼ oh . When we consider non-trivially graded C ∗ -algebras,we define a triple ( E, φ, F ), where φ : A → L ( E ) is a graded repre-sentation, and F ∈ L ( E ) is of odd degree such that F φ ( a ) − φ ( a ) F ,( F − φ ( a ), and ( F − F ∗ ) φ ( a ) are all in K ( E ) and call it a Kas-parov ( A, B )-module. Other definitions like degenerate cycle and op-erator homotopy are defined in similar ways. Let v be a unitary in M n ( D φ ( A, B )). Define φ n : A → L B ( B n ) by φ n ( a )( b , b , · · · , b n ) =( φ ( a ) b , φ ( a ) b , · · · , φ ( a ) b n ). Let B n ⊕ B n be graded by ( x, y ) ( x, − y ).Then (cid:18) B n ⊕ B n , (cid:18) φ n φ n (cid:19) , (cid:18) vv ∗ (cid:19)(cid:19) is a Kasparov ( A, B )-module. The class of this module depends onlyon the class of v in K ( D φ ( A, B )) so that the construction gives rise toa group homomorphism Ω : K ( D φ ( A, B )) → KK(
A, B ). Lemma 2.8.
Let φ an absorbing representation from A to L ( H B ) = M ( B ) where B is a stable C ∗ -algebra. Then Ω : K ( D φ ( A, B )) → KK(
A, B ) is an isomorphism.Proof. See [Th, Theorem 3.2]. In fact, Thomsen proved K ( D φ ( A,B )( D φ ( A,A,B )) )is isomorphic to KK( A, B ) via a map Θ where D φ ( A, A, B ) = { x ∈ D φ ( A, B ) | xφ ( A ) ⊂ B } is the ideal of D φ ( A, B ). However, the sameproof shows Ω is an isomorphism. Alternatively we can show that K i ( D φ ( A, A, B )) = 0 for i = 0 , K ∗ ( M ( B )) = 0. Thus, using the six term exactsequence, K ∗ ( D φ ( A, B )) is isomorphic to K ∗ (cid:16) D φ ( A,B )( D φ ( A,A,B )) (cid:17) . This impliesthe map Ω which is the composition with Θ and q is an isomorphism.Here q is the induced map between K-groups from the quotient mapfrom D φ ( A, B ) onto D φ ( A,B )( D φ ( A,A,B )) . (cid:3) Definition 2.9. [DE, Definition 3.2] If π, σ : A → L ( E ) are repre-sentations, we say that π and σ are properly asymptotically unitarilyequivalent and write π ≅ σ if there is a continuous path of unitaries u : [0 , ∞ ) → U ( K ( E ) + C I E ), u = ( u t ) t ∈ [0 , ∞ ) such that for all a ∈ A ROPER ASYMPTOTIC UNITARY EQUIV. AND PROJECTION LIFTING 7 (i) lim t →∞ k σ ( a ) − u t π ( a ) u ∗ t k = 0,(ii) σ ( a ) − u t π ( a ) u ∗ t ∈ K ( E ) for all t ∈ [0 , ∞ ) . In the above, we introduced the Fredholm picture of KK-group.There is an alternative way to describe the element of KK -group. TheCuntz picture is described by a pair of representations φ, ψ : A →L ( H B ) = M ( B ⊗ K ) such that φ ( a ) − ψ ( a ) ∈ K ( H B ) = B ⊗ K . Sucha pair is called a Cuntz pair. They form a set denoted by E h ( A, B ).A homotopy of Cuntz pairs consists of a Cuntz pair (Φ , Ψ) : A → M ( C ([0 , ⊗ ( B ⊗ K )). The quotient of E h ( A, B ) by homotopy equiv-alence is a group KK h ( A, B ) which is isomorphic to KK(
A, B ) via themapping sending [ φ, ψ ] to [ φ, ψ, φ, ψ ] = 0 in KK h ( A, B ) if and onlyif there is a representation γ : A → M ( B ⊗ K ) = L ( H B ) such that φ ⊕ γ ≅ ψ ⊕ γ [DE, Proposition 3.6]. The point is that the equivalence isimplemented by unitaries of the form compact + identity. Sometimes,we can have a non-stable equivalence keeping this useful point. Definition 2.10.
Let A be a C ∗ -algebra. Denote by e A its unitization.We say that A has K -injectivity if the map from U ( e A ) / U ( e A ) to K ( A )is injective where U ( e A ) is the unitary group and U ( e A ) is the connectedcomponent of the identity. We note that H. Lin proved in [Lin96,Lemma 2.2] that real rank zero implies K -injectivity. Theorem 2.11.
Let A be a separable C ∗ -algebra and let ψ, φ : A → H B be a Cuntz pair of absorbing representations. Suppose that thecomposition of φ with the natural quotient map π : M ( B ⊗ K ) → M ( B ⊗ K ) /B ⊗ K , which will be denoted by ˙ φ , is faithful. Further, we supposethat D φ ( A, B ) satisfies K -injectivity. If [ φ, ψ ] = 0 in KK ( A, B ) , then φ ≅ ψ .Proof. The proof of this theorem is almost identical to the one givenin [DE, Theorem 3.12]. We just give the proof to illustrate how ourassumptions play the roles.By Theorem 2.5, we get a continuous family of unitaries ( u t ) t ∈ [0 , ∞ ) in M ( B ⊗ K ) such that(3) u t φ ( a ) u ∗ t − ψ ( a ) ∈ C ([0 , ∞ )) ⊗ ( B ⊗ K ) . Note that (3) implies [ φ, ψ ] = [ φ, u φu ∗ ] (See [DE, Lemma 3.1]). Weassume that [ φ, ψ ] = 0 and we conclude that [ φ, u φu ∗ ] = 0. Since( φ, φ, u ∗ ) is unitarily equivalent to ( φ, u φu ∗ , φ, φ, u )] = [( φ, φ, u ∗ )] = 0 . HYUN HO LEE
Since the isomorphism Ω : K ( D φ ( A, B )) → KK(
A, B ) sends [ u ] to[ φ, φ, u ] by Lemma 2.8, K -injectivity implies that u is homotopic to1 in D φ ( A, B ). Thus we may assume that u = 1 in (3).Let E φ be a C ∗ -algebra φ ( A ) + B ⊗ K . We define ( α t ) t ∈ [0 , ∞ ) inAut ( E φ ) by Ad( u t ). Note that α = id and ( α t ) is a uniform continu-ous family of automorphisms. Thus we apply Proposition 2.15 in [DE]and get a continuous family ( v t ) [0 , ∞ ) of unitaries in E φ such that(4) lim t →∞ k α t ( x ) − Ad v t ( x ) k = 0for any x ∈ E φ .Combining (4) with (3), we obtain ( v t ) [0 , ∞ ) of unitaries in E φ suchthat lim t →∞ k v t φ ( a ) v ∗ t − ψ ( a ) k = 0for any a ∈ A . Since ˙ φ is faithful, we can replace ( v t ) [0 , ∞ ) by a familyof unitaries in B ⊗ K + C (cid:3) Recall the definition of the esssential codimension of Brown, Douglas,and Fillmore defined by two projections p, q in B ( H ) whose differenceis compact as we have defined in Introduction. Using KK-theory, or K-theory, we generalize this notion as follows, keeping the same notation. Definition 2.12.
Given two projections p, q ∈ M ( B ⊗ K ) such that p − q ∈ B ⊗ K , we consider representations φ, ψ from C to M ( B ⊗ K )such that φ (1) = p, ψ (1) = q . Then ( φ, ψ ) is a Cuntz pair so that wedefine [ p : q ] as the class [ φ, ψ ] ∈ KK( C , B ) ≃ K ( B ). Lemma 2.13. [Lin1]
Let B be a non-unital ( σ -unital) purely infinitesimple C ∗ -algebra. Let φ, ψ be two monomorphisms from C ( X ) to M ( B ⊗ K ) where X is a compact metrizable space. If ˙ φ, ˙ ψ are stillinjective, then they are approximately unitarily equivalent. The following theorem is a sort of generalization of BDF’s resultabout the essential codimension.
Theorem 2.14.
Let B be a non-unital ( σ -unital) purely infinite sim-ple C ∗ -algebra such that M ( B ⊗ K ) has real rank zero. Suppose twoprojections p and q in M ( B ⊗ K ) = L ( H B ) such that p − q ∈ B ⊗ K and neither of them are in B ⊗ K . If [ p, q ] ∈ K ( B ) vanishes, thenthere is a unitary u in id + B ⊗ K such that upu ∗ = q .Proof. Step1: Let φ, ψ : C → M ( B ⊗ K ) be representations from p and q respectively. Evidently φ is injective. Moreover, it does not containany “compacts” since p does not belong to B ⊗ K . Thus ˙ φ is faithful. ROPER ASYMPTOTIC UNITARY EQUIV. AND PROJECTION LIFTING 9
Recall ψ ∞ is defined by ψ ∞ ( a ) = P S i ψ ( a ) S ∗ i where { S i } is a sequenceof isometries in M ( B ⊗ K ) such that S i S ∗ j = 0 for i = j . Supposethat ψ ∞ ( λ ) = 0 for λ ∈ C . Then S ∗ i ψ ∞ ( λ ) S i = ψ ( λ ) = 0 or λq = 0.Thus λ = 0. Similarly, ˙ ψ ∞ is injective. Then they are approximatelyunitarily equivalent by applying Lemma 2.13 to X = { x } . Thus wehave a unitary U in L ( H B ) such that(5) U ∗ φ ( a ) U − ψ ∞ ( a )for a ∈ C .Note that to get a sequnce of isometries { v i } ∈ L ( H ( ∞ ) B , H B ) satis-fying the conditions of Lemma 2.3, what we needed was the equation(5). Following the same argument in the proof of Theorem 2.5, weget φ ∼ asym ψ . In other words, we have a continuous family of unitaries( u t ) t ∈ [0 , ∞ ) in M ( B ⊗ K ) such that u t φ ( a ) u ∗ t − ψ ( a ) ∈ C ([0 , ∞ )) ⊗ ( B ⊗ K ) for any a in A . Since D φ ( C , B ) has real rank zero, it satisfies K -injectivity. Thus itfollows that φ ≅ ψ as in the proof of Theorem 2.11.Step2: For large enough t , we can take u t = u of the form ‘identity+ compact’ such that k upu ∗ − q k <
1. For the moment we write upu ∗ as p . Thus k p − q k <
1. Note that p − q ∈ B ⊗ K . Then z = pq + (1 − p )(1 − q ) ∈ B ⊗ K is invertible and pz = zq . If weconsider the polar decomposition of z as z = v | z | . It is easy to checkthat v ∈ B ⊗ K and vpv ∗ = q . Now w = vu is also a unitary of theform ‘identity + compact’ such that wpw ∗ = q. (cid:3) Application: projection lifting
In this section, we show an application of proper asymptotic unitaryequivalence of two projections. In this application, with an additionalreal rank zero property, the unitary of the form ‘identity + compact’plays a crucial role as we shall see.Let B be a stable C ∗ -algebra such that the multiplier algebra M ( B )has real rank zero. Let X be [0 , , [0 , ∞ ), ( −∞ , ∞ ) or T = [0 , / { , } .When X is compact, let I = C ( X ) ⊗ B which is the C ∗ -algebra of(norm continuous) functions from X to B . When X is not compact,let I = C ( X ) ⊗ B which is the C ∗ -algebra of continuous functions from X to B vanishing at infinity. Then M ( I ) is given by C b ( X, M ( B ) s ),which is the set of bounded functions from X to B ( H ), where M ( B ) isgiven the strict topology. Let C ( I ) = M ( I ) /I be the corona algebra of I and also let π : M ( I ) → C ( I ) be the natural quotient map. Then anelement f of the corona algebra can be represented as follows: Considera finite partition of X , or X r { , } when X = T given by partitionpoints x < x < · · · < x n all of which are in the interior of X anddivide X into n + 1 (closed) subintervals X , X , · · · , X n . We can take f i ∈ C b ( X i , M ( B ) s ) such that f i ( x i ) − f i − ( x i ) ∈ B for i = 1 , , · · · , n and f ( x ) − f n ( x ) ∈ B where x = 0 = 1 if X is T . Lemma 3.1.
The coset in C ( I ) represented by ( f , · · · , f n ) consists offunctions f in M ( I ) such that f − f i ∈ C ( X i ) ⊗ B for every i and f − f i vanishes (in norm) at any infinite end point of X i .Proof. If X is compact, then we set x = 0, x n +1 = 1. Otherwise, weset x = x − X contains −∞ , and x n +1 = x n + 1 when X contaions + ∞ . Then we define a function in C ( X ) ⊗ B by m i ( x ) = x − x i − x i − x i − ( f i ( x i ) − f i − ( x i )) , if x i − ≤ x ≤ x ix − x i +1 x i − x i +1 ( f i ( x i ) − f i − ( x i )) , if x i ≤ x ≤ x i +1 , otherwisefor each i = 1 , · · · n . In addition, we set m = m n +1 = 0. Then wedefine a function e f from f i ’s by e f ( x ) = f i ( x ) − m i ( x ) / m i +1 ( x ) / X i . It follows that f i ( x i ) − m i ( x i ) / m i +1 ( x i ) / f i − ( x i ) − m i − ( x i ) / m i ( x i ) /
2. Thus e f is well defined. The conditions f − f i ∈ C ( X i ) ⊗ B for each i imply that f − e f is norm continuous functionfrom X to B since f | X i ( x i ) − e f | X i ( x i ) = f | X i − ( x i ) − e f | X i − ( x i ). (cid:3) Similarly ( f , · · · , f n ) and ( g , · · · , g n ) define the same element of C ( I ) if and only if f i − g i ∈ C ( X i ) ⊗ B for i = 0 , · · · , n if X is compact.( f , · · · , f n ) and ( g , · · · , g n ) define the same element of C ( I ) if and onlyif f i − g i ∈ C ( X i ) ⊗ B for i = 0 , · · · , n − f n − g n ∈ C ([ x n , ∞ )) ⊗ B if X is [0 . ∞ ). ( f , · · · , f n ) and ( g , · · · , g n ) define the same elementof C ( I ) if and only if f i − g i ∈ C ( X i ) ⊗ B for i = 1 , · · · , n − f n − g n ∈ C ([ x n , ∞ )) ⊗ B , f − g ∈ C (( −∞ , x ]) ⊗ B if X = ( −∞ , ∞ ).The following theorem says that any projection in the corona algebraof C ( X ) ⊗ B for some C ∗ -algebras B is described by a “locally trivialfiber bundle” with the fibre H B in the sense of Dixmier and Duady[DixDua]. ROPER ASYMPTOTIC UNITARY EQUIV. AND PROJECTION LIFTING 11
Theorem 3.2.
Let I be C ( X ) ⊗ B or C ( X ) ⊗ B where B is a stable C ∗ -algebra such that M ( B ) has real rank zero. Then a projection f in M ( I ) /I can be represented by ( f , f , · · · , f n ) as above where f i is aprojection valued function in C ( X i ) ⊗ M ( B ) s for each i .Proof. Let f be the element of M ( I ) such that π ( f ) = f . Without lossof generality, we can assume f is self-adjoint and 0 ≤ f ≤ X does not contain any infinite point. Choose a point t ∈ X . Then there is a self-adjoint element T ∈ M ( B ) suchthat T − f ( t ) ∈ B and the spectrum of T has a gap around 1 / f ( t ) + T − f ( t ) whichis still self-adjoint whose image is f . Thus we may assume f ( t )is a self-adjoint element whose spectrum has a gap around 1 / r ( f ( t )) : t → f ( t ) − f ( t ) is norm continuous where r ( x ) = x − x , if we pick a point z in (cid:0) , (cid:1) such that z / ∈ σ ( f ( t ) − f ( t ) ), then σ ( f ( s )) omits r − ( J ) for s sufficientlyclose to t where J is an interval containing z . In other words,there is δ > b > a > | t − s | < δ , then σ ( f ( s )) ⊂ [0 , a ) ∪ ( b, f t ( s ) = χ ( b, ( f ( s )) for s in ( t − δ, t + δ ) where χ ( b, is the characteristic function on ( b, f t − f ∈ C ( t − δ, t + δ ) ⊗ B .By repeating the above procedure, since X is compact, wecan find n + 1 points t , · · · , t n , n + 1 functions f t , · · · , f t n ,and an open covering { O i } such that t i ∈ O i , O i ∩ O i − = ∅ ,and f t i is projection valued function on O i . Now let f i = f t i as above. Take the point x i ∈ O i − ∩ O i for i = 1 , · · · , n .Then f i ( x i ) − f i − ( x i ) = f i ( x i ) − f ( x i ) + f ( x i ) − f i − ( x i ) ∈ B and f ( x ) − f n ( x ) ∈ B if applicable. Let X i = [ x i , x i +1 ] for i = 1 , · · · , n − X = [0 , x ], and X n = [ x n , f i is also defined on X i , ( f , · · · , f n ) is what we want.(ii) let X be [0 , ∞ ). Since f ( t ) − f ( t ) → t goes to ∞ , for given δ in (0 , / M > t ≥ M then k f ( t ) − f ( t ) k < δ − δ . It follows that σ ( f ( t )) ⊂ [0 , δ ) ∪ (1 − δ, t ≥ M . Then again χ (1 − δ, ( f ( t )) is a continuous projectionvalued function for t ≥ M such that f ( t ) − χ (1 − δ, ( f ( t )) vanishesin norm as t goes to ∞ . By applying the argument in ( i ) to[0 , M ], we get a closed sub-intervals X i for i = 0 , · · · , n − , M ] and f i ∈ C b ( X i , B ( H )). Now if we let X n = [ M, ∞ ) and f n ( t ) = χ (1 − δ, ( f ( t )), we are done.(iii) The case X = ( −∞ , ∞ ) is similar to (ii). (cid:3) When a projection f ∈ C ( I ) is represented by ( f , f , · · · , f n ) byTheorem 3.2, we note that f i ( x ) is a projection in M ( B ⊗ K ) foreach x ∈ X i and f i ( x i ) − f i − ( x i ) ∈ B . Applying Definition 2.12 wehave K -theoretical terms k i = [ f i ( x i ) : f i − ( x i )] ∈ KK( C , B ) for i =1 , , · · · , n . The following theorem shows that if all k i ’s are vanishing,then a projection f in C ( I ) lifts to a projection in M ( I ). Theorem 3.3.
Let I be C ( X ) ⊗ B where B is a σ -unital, non-unital,purely infinite simple C ∗ -algebra such that M ( B ) has real rank zero or K ( B ) = 0 (See [Zhang] ). Let a projection f in M ( I ) /I be representedby ( f , f , · · · , f n ) , where f i is a projection valued function in C ( X i ) ⊗ M ( B ) s for each i , as in Theorem 3.2. If k i = [ f i ( x i ) : f i − ( x i )] = 0 forall i , then the projection f in M ( I ) /I lifts.Proof. Note that, by Zhang’s dichotomy, B is stable [Zhang, Theorem1.2]. By induction, assume that f j ( x j ) = f j − ( x j ) for j = 1 , , · · · , i − f i ( x i ) = p i , f i − ( x i ) = p i − . Since [ p i : p i − ] = 0, we have aunitary u of the form ‘ identity + compact ’ such that k p i − u ∗ p i − u k < / B has real rank zero, given 0 < ǫ < / v ∈ C B with finite spectrum such that k u − v k < ǫ [Lin93], [Lin96]. Then k p i − vp i − v ∗ k ≤ k p i − up i − u ∗ k + k up i − u ∗ − vp i − v ∗ k < . Note that p i − vp i − v ∗ ∈ B . Thus we have wp i w ∗ = vp i − v ∗ for someunitary w ∈ id + B . (Recall that Step 2 of the proof of Theorem 2.14.)Let g i = wf i w ∗ , then f i − g i ∈ C ( X i ) ⊗ B since w is of the form ‘identity + compact’.On the other hand, we can write v as e ih where h is a self-adjointelement in B since v has the finite spectrum. A homotopy of unitaries t → e ith , which are of the form “ identity + compact”, connects 1 to v . Now we define g i − ( t ) asexp (cid:18) i t − x i − x i − x i − h (cid:19) f i − ( t )exp (cid:18) i t − x i − x i − x i − h (cid:19) for t ∈ [ x i − , x i ]. Then we see that g i − ( x i ) = g i ( x i ), g i − − f i − ∈ C ( X i − ) ⊗ K , and g i − ( x i − ) = f i − ( x i − ). Moreover, if we let g i +1 = wf i +1 w ∗ , then f i +1 − g i +1 ∈ C ( X i +1 ) ⊗ B , and[ g i +1 ( x i +1 ) : g i ( x i +1 )] = [ wf i +1 ( x i +1 ) w ∗ : wf i ( x i + 1) w ∗ ]= [ f i +1 ( x i +1 ) : f i ( x i +1 )] = 0 . Then ( f , f , · · · , f n ) and ( f , f , · · · , g i − , g i , g i +1 , f i +2 , · · · , f n ) definethe same element f while the k i ’s are unchanged and i -th discontinuityis resolved. So we take the latter as ( f , · · · , f n ) such that f j ( x j ) = ROPER ASYMPTOTIC UNITARY EQUIV. AND PROJECTION LIFTING 13 f j − ( x j ) for j = 1 , . . . , i . We can repeat the same procedure untilwe have f i ( x i ) = f i − ( x i ) for all i . It follows that ( f , · · · , f n ) is aprojection in M ( C ( X ) ⊗ B ) which lifts f . (cid:3) Remark 3.4.
When I = C ( X ) ⊗ B where X is [0 , ∞ ) or ( −∞ , ∞ ),the similar result holds replacing C ( X i ) ⊗ B with C ( −∞ , x ] ⊗ B or C [ x n , ∞ ) ⊗ B for i = 0 or i = n respectively.4. Acknowledgements
Although this work was not carried out at Purdue, a significant in-fluence on the auther was made by Larry Brown and Marius Dadarlatwho have aquainted him with geometric ideas in operator algebras. Healso would like to thank Huaxin Lin for answering the question relatedto Lemma 2.14.
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