aa r X i v : . [ m a t h . OA ] S e p A CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA
NORIO NAWATA
Abstract.
Combing Elliott, Gong, Lin and Niu’s result and Castillejos andEvington’s result, we see that if A is a simple separable nuclear monotracialC ∗ -algebra, then A ⊗ W is isomorphic to W where W is the Razak-Jacelonalgebra. In this paper, we give another proof of this. In particular, we showthat if D is a simple separable nuclear monotracial M ∞ -stable C ∗ -algebrawhich is KK -equivalent to { } , then D is isomorphic to W without consider-ing tracial approximations of C ∗ -algebras with finite nuclear dimension. Ourproof is based on Matui and Sato’s technique, Schafhauser’s idea in his proofof the Tikuisis-White-Winter theorem and properties of Kirchberg’s centralsequence C ∗ -algebra F ( D ) of D . Note that some results for F ( D ) is based onElliott-Gong-Lin-Niu’s stable uniqueness theorem. Also, we characterize W by using properties of F ( W ). Indeed, we show that a simple separable nuclearmonotracial C ∗ -algebra D is isomorphic to W if and only if D satisfies thefollowing properties:(i) for any θ ∈ [0 , p in F ( D ) such that τ D,ω ( p ) = θ ,(ii) if p and q are projections in F ( D ) such that 0 < τ D,ω ( p ) = τ D,ω ( q ), then p is Murray-von Neumann equivalent to q ,(iii) there exists a homomorphism from D to W . Introduction
The Razak-Jacelon algebra W is a certain simple separable nuclear monotracialC ∗ -algebra which is KK -equivalent to { } . Note that such a C ∗ -algebra must bestably projectionless, that is, W ⊗ M n ( C ) has no non-zero projections for any n ∈ N .In particular, every stably projectionless C ∗ -algebra is non-unital. In [21], Jacelonconstructed W as an inductive limit C ∗ -algebra of Razak’s building blocks [39]. Wecan regard W as a stably finite analogue of the Cuntz algebra O . In particular, W is expected to play a central role in the classification theory of simple separablenuclear stably projectionless C ∗ -algebras as O played in the classification theoryof Krichberg algebras (see, for example, [44] and [18]). We refer the reader to [13],[14] and [19] for recent progress in the classification of simple separable nuclearstably projectionless C ∗ -algebras. Note that there exist many interesting examplesof simple stably projectionless C ∗ -algebras. See, for example, [7], [12], [26], [27],[28] and [40].Combing Elliott, Gong, Lin and Niu’s result [14] and Castillejos and Evington’sresult [4] (see also [5]), we see that if A is a simple separable nuclear monotracialC ∗ -algebra, then A ⊗W is isomorphic to W . This can be considered as a Kirchberg-Phillips type theorem [23] for W . In this paper, we give another proof of this.In our proof, we do not consider tracial approximations of C ∗ -algebras with finitenuclear dimension. Also, we mainly consider in abstract settings and do not use anyclassification theorem based on inductive limit structures of W other than Razak’sclassification theorem [39]. (Actually, we need Razak’s classification theorem only Mathematics Subject Classification.
Primary 46L35, Secondary 46L40; 46L80.
Key words and phrases.
Stably projectionless C ∗ -algebra; Kirchberg’s central sequence C ∗ -algebra; KK -contractible C ∗ -algebra.This work was supported by JSPS KAKENHI Grant Number 20K03630. for W ⊗ M ∞ ∼ = W .) We obtain a Kirchberg-Phillips type theorem for W as acorollary of the following theorem. Theorem A. (Theorem 6.1)Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } . Then D is isomorphic to W .Our proof of the theorem above is based on Matui and Sato’s technique [30], [31],[32], Schafhauser’s idea [48] (see also [49]) in his proof of the Tikuisis-White-Wintertheorem [51] and properties of Kirchberg’s central sequence C ∗ -algebra F ( D ) of D .Matui-Sato’s technique enables us to show that certain (relative) central sequenceC ∗ -algebras have strict comparison. Note that a key concept in their technique isproperty (SI). This concept was introduced by Sato in [46] and [47].Borrowing Schafhauser’s idea, we show that if D is a simple separable nuclearmonotracial ( M ∞ -stable) C ∗ -algebra which is KK -equivalent to { } , then thereexist “trace preserving” homomorphisms from D to ultrapowers B ω of certain C ∗ -algebras B . Combing this and a uniqueness result for approximate homomorphismsfrom D , we obtain an existence result, that is, existence of homomorphisms from D to certain C ∗ -algebras. Schafhauser’s arguments are based on the extension theory(or KK -theory) and Elliott and Kucerovsky’s result [15] with a correction by Gabe[17]. Hence Schafhauser’s arguments are suitable for our purpose, that is, a studyof C ∗ -algebras which are KK -equivalent to { } .We studied properties of F ( W ) in [34] and [35] by using Elliott-Gong-Lin-Niu’sstable uniqueness theorem in [14]. In particular, we showed that F ( W ) has manyprojections and satisfies a certain comparison theory for projections. By theseproperties and Connes’ 2 × W is approximately inner. This argument is a traditional argument in the theoryof operator algebras (see [6]). In this paper, we remark that arguments in [34] and[35] work for a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra D which is KK -equivalent to { } . Also, we characterize W by using these propertiesof F ( W ). Indeed, we show the following theorem. Theorem B. (Theorem 6.4)Let D be a simple separable nuclear monotracial C ∗ -algebra. Then D is isomorphicto W if and only if D satisfies the following properties:(i) for any θ ∈ [0 , p in F ( D ) such that τ D,ω ( p ) = θ ,(ii) if p and q are projections in F ( D ) such that 0 < τ D,ω ( p ) = τ D,ω ( q ), then p isMurray-von Neumann equivalent to q ,(iii) there exists a homomorphism from D to W .This paper is organized as follows. In Section 2, we collect notations, definitionsand some results. In particular, we recall Matui-Sato’s technique. In Section 3, weintroduce the property W, which is a key property for uniqueness results. Also,we remark that arguments in [34] and [35] work for more general settings. InSection 4, we show uniqueness results. First, we show that if D has propertyW, then every endomorphism of D is approximately inner. Secondly, we considera uniqueness theorem for approximate homomorphisms from a simple separablenuclear monotracial M ∞ -stable C ∗ -algebra D which is KK -equivalent to { } foran existence result in Section 5. In Section 5, we show an existence result byborrowing Schafhauser’s idea. In Section 6, we show the main results in this paper.2. Preliminaries
In this section we shall collect notations, definitions and some results. We referthe reader to [1] and [38] for basics of operator algebras.
CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 3
For a C ∗ -algebra A , we denote by A + the sets of positive elements of A andby A ∼ the unitization algebra of A . Note that if A is unital, then A = A ∼ . For a, b ∈ A + , we say that a is Murray-von Neumann equivalent to b , written a ∼ b ,if there exists an element z in A such that z ∗ z = a and zz ∗ = b . Note that ∼ isan equivalence relation by [37, Theorem 3.5]. For a, b ∈ A , we denote by [ a, b ] thecommutator ab − ba . For a subset F of A and ε >
0, we say that a completelypositive (c.p.) map ϕ : A → B is ( F, ε ) -multiplicative if k ϕ ( ab ) − ϕ ( a ) ϕ ( b ) k < ε for any a, b ∈ F . Let Z and M ∞ denote the Jiang-Su algebra and the CAR algebra,respectively. We say a C ∗ -algebra A is monotracial if A has a unique tracial stateand no unbounded traces. In the case where A is monotracial, we denote by τ A theunique tracial state on A unless otherwise specified.2.1. Razak-Jacelon algebra W . The
Razak-Jacelon algebra W is a certain simpleseparable nuclear monotracial C ∗ -algebra which is KK -equivalent to { } . In [21], W is constructed as an inductive limit C ∗ -algebra of Razak’s building blocks. ByRazak’s classification theorem [39], W is M ∞ -stable, and hence W is Z -stable. Inthis paper, we do not assume any classification theorem for W other than Razak’sclassification theorem.2.2. Kirchberg’s central sequence C ∗ -algebras. We shall recall the definitionof Kirchberg’s central sequence C ∗ -algebras in [22]. Fix a free ultrafilter ω on N .For a C ∗ -algebra B , put c ω ( B ) := {{ x n } n ∈ N ∈ ℓ ∞ ( N , B ) | lim n → ω k x n k = 0 } , B ω := ℓ ∞ ( N , B ) /c ω ( B ) . We denote by ( x n ) n a representative of an element in B ω . Let A be a C ∗ -subalgebraof B ω . SetAnn( A, B ω ) := { ( x n ) n ∈ B ω ∩ A ′ | ( x n ) n a = 0 for any a ∈ A } . Then Ann(
A, B ω ) is a closed ideal of B ω ∩ A ′ . Define a (relative) central sequenceC ∗ -algebra F ( A, B ) of A ⊆ B ω by F ( A, B ) := B ω ∩ A ′ / Ann(
A, B ω ) . We identify B with the C ∗ -subalgebra of B ω consisting of equivalence classes ofconstant sequences. In the case A = B , we denote F ( B, B ) by F ( B ) and call itthe central sequence C ∗ -algebra of B . If A is σ -unital, then F ( A, B ) is unital by[22, Proposition 1.9]. Indeed, let s = ( s n ) n be a strictly positive element in A ⊆ B ω . Since we have lim k →∞ s /k s = s , taking a suitable sequence { k ( n ) } n ∈ N ⊂ N ,we obtain s ′ = ( s /k ( n ) n ) n ∈ B ω such that s ′ s = s . Then it is easy to see that s ′ ∈ B ω ∩ A ′ and [ s ′ ] = 1 in F ( A, B ). Note that the inclusion B ⊂ B ∼ induces anisomorphism from F ( A, B ) onto F ( A, B ∼ ) because we have [ xs ′ ] = [ x ] in F ( A, B ∼ )for any x ∈ ( B ∼ ) ω ∩ A ′ .Let τ B be a tracial state on B . Define τ B,ω : B ω → C by τ B,ω (( x n ) n ) =lim n → ω τ B ( x n ) for any ( x n ) n ∈ B ω . Since ω is an ultrafilter, it is easy to see that τ B,ω is a well-defined tracial state on B ω . The following proposition is a relativeversion of [34, Proposition 2.1]. Proposition 2.1.
Let B be a C ∗ -algebra with a faithful tracial state τ B , and let A be a C ∗ -subalgebra of B ω . Assume that τ B,ω | A is a state. Then τ B,ω (( x n ) n ) = 0for any ( x n ) n ∈ Ann(
A, B ω ). Proof.
Let { h λ } λ ∈ Λ be an approximate unit for A . Since τ B,ω | B is a state, we havelim τ B,ω ( h λ ) = 1. The rest of proof is same as the proof of [34, Proposition 2.1]. (cid:3) NORIO NAWATA
By the proposition above, if τ B,ω | A is a state, then τ B,ω induces a tracial stateon F ( A, B ). We denote it by the same symbol τ B,ω for simplicity. Note that If A and B are unital, then τ B,ω | A is a state if and only if 1 A = 1 B .2.3. Invertible elements in unitization algebras.
Let GL( A ∼ ) denote the setof invertible elements in A ∼ . The following proposition is trivial if 1 A ∼ = 1 B ∼ . Proposition 2.2.
Let A ⊆ B be an inclusion of C ∗ -algebras. Then GL( A ∼ ) ⊂ GL( B ∼ ). Proof.
Let x ∈ GL( A ∼ ). There exists ε > ≤ ε < ε , x + ε A ∼ ∈ GL( A ∼ ) because GL( A ∼ ) is open. Since we have Sp A ( x ) ∪ { } =Sp B ( x ) ∪ { } , x + ε B ∼ ∈ GL( B ∼ ) for any 0 < ε < ε . Therefore x ∈ GL( B ∼ ). (cid:3) The following corollary is an immediate consequence of the proposition above.
Corollary 2.3.
Let { A n } n ∈ N be a sequence of C ∗ -algebras with A n ⊆ A n +1 , andlet A = S ∞ n =1 A n . If A n ⊆ GL( A ∼ n ) for any n ∈ N , then A ⊆ GL( A ∼ ).The following proposition is well-known if B is unital. See, for example, theproof of [48, Proposition 3.2]. Proposition 2.4.
Let B be a C ∗ -algebra with B ⊆ GL( B ∼ ). Then B ω ⊆ GL(( B ω ) ∼ ). Proof.
We shall show only the case where B is non-unital. Let ( x n ) n ∈ B ω . Becauseof B ⊆ GL( B ∼ ), there exists ( z n ) n ∈ ( B ∼ ) ω such that z n ∈ GL( B ∼ ) for any n ∈ N and ( x n ) n = ( z n ) n in ( B ∼ ) ω . For any n ∈ N , put u n := z n ( z ∗ n z n ) − / , then u n is aunitary element and z n = u n ( z ∗ n z n ) / . Note that we have ( x n ) n = ( u n ) n ( x ∗ n x n ) / n .For any n ∈ N , there exist y n ∈ B and λ n ∈ C such that u n = y n + λ n B ∼ and | λ n | = 1 because u n is a unitary element in B ∼ . Since ω is an ultrafilter, thereexists λ ∈ C such that lim n → ω λ n = λ . Hence( u n ) n = ( y n ) n + λ ( B ω ) ∼ ∈ ( B ω ) ∼ . Since we have (( y n ) n + λ ( B ω ) ∼ )(( x ∗ n x n ) / n + ε ( B ω ) ∼ ) → ( x n ) n as ε →
0, ( x n ) n ∈ GL(( B ω ) ∼ ). (cid:3) Matui-Sato’s technique.
We shall review Matui and Sato’s technique in[30], [31] and [32]. Let B be a monotracial C ∗ -algebra, and let A be a simpleseparable nuclear monotracial C ∗ -subalgebra of B ω . Assume that τ B is faithfuland τ B,ω | A is a state. Consider the Gelfand-Naimark-Segal (GNS) representation π τ B of B associated with τ B , and put M := ℓ ∞ ( N , π τ B ( B ) ′′ ) / {{ x n } n ∈ N | ˜ τ B,ω (( x n ) n ) := lim n → ω ˜ τ B ( x ∗ n x n ) = 0 } where ˜ τ B is the unique normal extension of τ B on π τ B ( B ) ′′ . Note that M is avon Neumann algebraic ultrapower of π τ B ( B ) ′′ and ˜ τ B,ω is a faithful normal tracialstate on M . Since B is monotracial, π τ B ( B ) ′′ is a finite factor, and hence M is alsoa finite factor. Define a homomorphism ̺ from B ω to M by ̺ (( x n ) n ) = ( π τ B ( x n )) n .Kaplansky’s density theorem implies that ̺ is surjective. Moreover, [32, Theorem3.1] (see also [25, Theorem 3.3]) implies that the restriction ̺ on B ω ∩ A ′ is asurjective homomorphism onto M ∩ ̺ ( A ) ′ . Proposition 2.5.
With notation as above, M ∩ ̺ ( A ) ′ is a finite factor. CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 5
Proof.
Note that ˜ τ B,ω is the unique tracial state on M since M is a finite factor.It is enough to show that M ∩ ̺ ( A ) ′ is monotracial. Let τ be a tracial state on M ∩ ̺ ( A ) ′ . Since we assume that τ B,ω | A is a state, we see that if A is unital,then ̺ (1 A ) = 1 M . Hence ̺ can be extended to a unital homomorphism ̺ ∼ from A ∼ to M , and M ∩ ̺ ( A ) ′ = M ∩ ̺ ∼ ( A ∼ ) ′ . By [2, Lemma 3.21], there exists apositive element a in A ∼ such that ˜ τ B,ω ( ̺ ∼ ( a )) = 1 and τ ( x ) = ˜ τ B,ω ( ̺ ∼ ( a ) x ) forany x ∈ M ∩ ̺ ( A ) ′ . Since A is monotracial, τ ( x ) = ˜ τ B,ω ( ̺ ∼ ( a ) x ) = ˜ τ B,ω ( ̺ ∼ ( a ))˜ τ B,ω ( x ) = ˜ τ B,ω ( x ) . Indeed, let x be a positive contraction in M ∩ ̺ ( A ) ′ . For any a ∈ A , define τ ′ ( a ) := ˜ τ B,ω ( ̺ ( a ) x ). Then τ ′ is a tracial positive linear functional on A . Since A is monotracial and τ B,ω | A is a tracial state on A , there exists a positive number t such that τ ′ ( a ) = tτ B,ω ( a ) for any a ∈ A . Note that if { h n } n ∈ N is an approximateunit for A , then t = lim n →∞ τ ′ ( h n ). On the other hand, we have | ˜ τ B,ω ( x ) − τ ′ ( h n ) | = | ˜ τ B,ω ((1 − ̺ ( h n )) x ) | = | ˜ τ B,ω ((1 − ̺ ( h n )) / x (1 − ̺ ( h n )) / ) |≤ | ˜ τ B,ω (1 − ̺ ( h n )) | = | − τ B,ω ( h n ) | → n → ∞ . Hence t = ˜ τ B,ω ( x ), and ˜ τ B,ω ( ̺ ( a ) x ) = ˜ τ B,ω ( ̺ ( a ))˜ τ B,ω ( x ) for any a ∈ A . It is easy to see that this implies ˜ τ B,ω ( ̺ ∼ ( a ) x ) = ˜ τ B,ω ( ̺ ∼ ( a ))˜ τ B,ω ( x )for any a ∈ A ∼ and x ∈ M ∩ ̺ ( A ) ′ . Therefore we have τ ( x ) = ˜ τ B,ω ( x ) for any x ∈ M ∩ ̺ ( A ) ′ . Consequently, M ∩ ̺ ( A ) ′ is monotracial. (cid:3) For a, b ∈ A + , we say that a is Cuntz smaller than b , written a - b , if thereexists a sequence { x n } n ∈ N of A such that k x ∗ n bx n − a k →
0. A monotracial C ∗ -algebra B is said to have strict comparison if for any k ∈ N , a, b ∈ M k ( B ) + with d τ B ⊗ Tr k ( a ) < d τ B ⊗ Tr k ( b ) implies a - b where Tr k is the unnormalized traceon M k ( C ) and d τ B ⊗ Tr k ( a ) = lim n →∞ τ B ⊗ Tr k ( a /n ). Using [33, Lemma 5.7],essentially the same proofs as [32, Lemma 3.2] and [30, Theorem 1.1] show thefollowing proposition. See also the proof of [35, Lemma 3.6]. Proposition 2.6.
Let B be a monotracial C ∗ -algebra, and let A be a simpleseparable non-type I nuclear monotracial C ∗ -subalgebra of B ω . Assume that τ B isfaithful, τ B,ω | A is state and B has strict comparison. Then B has property (SI)relative to A , that is, for any positive contractions a and b in B ω ∩ A ′ satisfying τ B,ω ( a ) = 0 and inf m ∈ N τ B,ω ( b m ) > , there exists an element s in B ω ∩ A ′ such that s ∗ s = a and bs = s .By Proposition 2.1, ̺ induces a surjective homomorphism from F ( A, B ) to M ∩ ̺ ( A ) ′ . We denote it by the same symbol ̺ for simplicity. Using Proposi-tion 2.5 and Proposition 2.6, essentially the same proofs as [32, Proposition 3.3]and [31, Proposition 4.8] show the following proposition. See also the proof of [35,Proposition 3.8]. Proposition 2.7.
Let B be a monotracial C ∗ -algebra, and let A be a simpleseparable non-type I nuclear monotracial C ∗ -subalgebra of B ω . Assume that τ B isfaithful, τ B,ω | A is state and B has strict comparison. Then F ( A, B ) is monotracialand has strict comparison. Furthermore, if a and b are positive elements in F ( A, B )satisfying d τ B,ω ( a ) < d τ B,ω ( b ), then there exists an element r in F ( A, B ) such that r ∗ br = a . 3. Property W
In this section we shall introduce the property W, which is a key property inSection 4.
NORIO NAWATA
Definition 3.1.
Let D be a simple separable nuclear monotracial C ∗ -algebra. Wesay that D has property W if F ( D ) satisfies the following properties:(i) for any θ ∈ [0 , p in F ( D ) such that τ D,ω ( p ) = θ ,(ii) if p and q are projections in F ( D ) such that 0 < τ D,ω ( p ) = τ D,ω ( q ), then p isMurray-von Neumann equivalent to q .By arguments in [34] and [35], we see that if D is a simple separable nuclearmonotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } , then D hasproperty W. We shall give a sketch of a proof for reader’s convenience and show aslight generalization (or a relative version).Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra whichis KK -equivalent to { } and B a simple monotracial C ∗ -algebra with strict com-parison and B ⊆ GL( B ∼ ), and let Φ be a homomorphism from D to B ω such that τ D = τ B,ω ◦ Φ. By the Choi-Effros lifting theorem, there exists a sequence { Φ n } n ∈ N of contractive c.p. maps from D to B such that Φ( x ) = (Φ n ( x )) n for any x ∈ D .Since we assume τ D = τ B,ω ◦ Φ, τ B,ω | Φ( D ) is a state. Hence τ B,ω is the unique tracialstate on F (Φ( D ) , B ) by Proposition 2.7. The following proposition is an analogousproposition of [34, Proposition 4.2] and [35, Proposition 2.6]. Proposition 3.2. (i) For any N ∈ N , there exists a unital homomorphism from M N ( C ) to F (Φ( D ) , B ).(ii) For any θ ∈ [0 , p in F (Φ( D ) , B ) such that τ B,ω ( p ) = θ .(iii) Let h be a positive element in F (Φ( D ) , B ) such that d τ B,ω ( h ) >
0. For any θ ∈ [0 , d τ B,ω ( h )), there exists a non-zero projection p in hF (Φ( D ) , B ) h such that τ B,ω ( p ) = θ . Proof. (i) Since D is isomorphic to D ⊗ M ∞ = D ⊗ N n ∈ N M N ( C ), a similarargument as in the proof of [34, Proposition 4.2] shows that there exists a family { ( e ij,m ) m } N i,j =1 of contractions in D ω ∩ D ′ such that ( P N ℓ =1 e ℓℓ,m x ) m = x and( e ij,m e kl,m x ) m = ( δ jk e il,m x ) m for any 1 ≤ i, j, k, l ≤ N and x ∈ D . Note thatwe havelim m → ω k ([Φ n ( e ij,m ) , Φ n ( x )]) n k = 0 , lim m → ω k ( N X ℓ =1 Φ n ( e ℓℓ,m )Φ n ( x ) − Φ n ( x )) n k = 0and lim m → ω k ((Φ n ( e ij,m )Φ n ( e kl,m ) − δ jk Φ n ( e il,m ))Φ n ( x )) n k = 0for any 1 ≤ i, j, k, l ≤ N and x ∈ D . Hence, for any finite subset F ⊂ D and ε > { (Φ n ( e ij, ( F,ε ) )) n } N i,j =1 of contractions in B ω such thatlim n → ω k [Φ n ( e ij, ( F,ε ) ) , Φ n ( x )] k < ε, lim n → ω k N X ℓ =1 Φ n ( e ℓℓ, ( F,ε ) )Φ n ( x ) − Φ n ( x ) k < ε and lim n → ω k (Φ n ( e ij, ( F,ε ) )Φ n ( e kl, ( F,ε ) ) − δ jk Φ n ( e il, ( F,ε ) ))Φ n ( x ) k < ε for any 1 ≤ i, j, k, l ≤ N and x ∈ F . Let { F m } m ∈ N be an increasing sequence offinite subsets in D such that D = S m ∈ N F m . We can find a sequence { X m } m ∈ N ofelements in ω such that X m +1 ⊂ X m and for any n ∈ X m , k [Φ n ( e ij, ( F m , m ) ) , Φ n ( x )] k < m , k N X ℓ =1 Φ n ( e ℓℓ, ( F m , m ) )Φ n ( x ) − Φ n ( x ) k < m CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 7 and k (Φ n ( e ij, ( F m , m ) )Φ n ( e kl, ( F m , m ) ) − δ jk Φ n ( e il, ( F m , m ) ))Φ n ( x ) k < m for any 1 ≤ i, j, k, l ≤ N and x ∈ F m . For any 1 ≤ i, j ≤ N , put E ij,n := (cid:26) n / ∈ X Φ n ( e ij, ( F m , m ) ) if n ∈ X m \ X m +1 ( m ∈ N ) . Then we have ( E ij,n ) n ∈ B ω ∩ Φ( D ) ′ , N X ℓ =1 [( E ℓℓ,n ) n ] = 1 and [( E ij,n ) n ][( E kl,n ) n ] = δ jk [( E il,n ) n ]in F (Φ( D ) , B ) for any 1 ≤ i, j, k, l ≤ N . Therefore there exists a unital homomor-phism from M N ( C ) to F (Φ( D ) , B ).(ii) Since D is isomorphic to D ⊗ M ∞ = D ⊗ N n ∈ N M ∞ , a similar argument asin the proof of [34, Proposition 4.2] shows that there exists a positive contraction( p m ) m in D ω ∩ D such that (( p m − p m ) x ) m = 0 for any x ∈ D and τ D ,ω (( p m ) m ) = θ .By a similar argument as above, we obtain a projection p in F (Φ( D ) , B ) such that τ B,ω ( p ) = θ .(iii) Using Proposition 2.7 instead of [34, Proposition 4.1], we obtain the conclu-sion by the same argument as in the proof of [34, Proposition 4.2]. (cid:3) The proposition above and the same arguments as in [34, Section 4] show thefollowing corollary.
Corollary 3.3. (cf. [34, Proposition 4.8]). Let p and q be projections in F (Φ( D ) , B )such that τ B,ω ( p ) <
1. Then p and q are Murray-von Neumann equivalent if andonly if p and q are unitarily equivalent.Since we assume B ⊆ GL( B ∼ ), we obtain the following proposition by the sameargument as in the proof of [34, Proposition 4.9]. Proposition 3.4.
Let u be a unitary element in F (Φ( D ) , B ). Then there exists aunitary element w in ( B ∼ ) ω ∩ Φ( D ) ′ such that u = [ w ].There exists a homomorphism ρ from F (Φ( D ) , B ) ⊗ D to B ω such that ρ ([( x n ) n ] ⊗ a ) = ( x n Φ n ( a )) n for any [( x n ) n ] ∈ F (Φ( D ) , B ) and a ∈ D . For a projection p in F (Φ( D ) , B ), put B ωp := ρ ( p ⊗ s ) B ω ρ ( p ⊗ s ) where s is a strictly positive element in D . Define ahomomorphism σ p from D to B ωp by σ p ( a ) := ρ ( p ⊗ a ) for any a ∈ D . Since B has strict comparison, we see that if p is a projection in F (Φ( D ) , B ) such that τ B,ω ( p ) >
0, then σ p is ( L, N )-full for some maps L and N . (We refer the readerto [34, Section 3] for details of the ( L, N )-fullness.) Therefore [34, Proposition3.3] implies the following theorem. We may regard this theorem as a variant ofElliott-Gong-Lin-Niu’s stable uniqueness theorem [14, Corollary 3.15](see also [16,Corollary 8.16]). Note that [34, Proposition 3.3] is also based on the results in [15],[17], [8] and [9].
Theorem 3.5.
Let Ω be a compact metrizable space. For any finite subsets F ⊂ C (Ω), F ⊂ D and ε >
0, there exist finite subsets G ⊂ C (Ω), G ⊂ D , m ∈ N and δ > p be a projection in F (Φ( D ) , B ) such that τ B,ω ( p ) >
0. For any contractive ( G ⊙ G , δ )-multiplicative maps ψ , ψ : C (Ω) ⊗ NORIO NAWATA
D → B ωp , there exist a unitary element u in M m +1 ( B ωp ) ∼ and z , z , ..., z m ∈ Ωsuch that k u ( ψ ( f ⊗ b ) ⊕ m z }| { m M k =1 f ( z k ) ρ ( p ⊗ b ) ⊕ · · · ⊕ m M k =1 f ( z k ) ρ ( p ⊗ b )) u ∗ − ψ ( f ⊗ b ) ⊕ m z }| { m M k =1 f ( z k ) ρ ( p ⊗ b ) ⊕ · · · ⊕ m M k =1 f ( z k ) ρ ( p ⊗ b ) k < ε for any f ∈ F and b ∈ F .Using Proposition 2.7, Proposition 3.2 and Corollary 3.3 instead of [34, Propo-sition 4.1], [34, Proposition 4.2] and [34, Proposition 4.8], the same proof as [34,Lemma 5.1] shows the following lemma. Lemma 3.6.
Let Ω be a compact metrizable space, and let F be a finite subsetof C (Ω) and ε >
0. Suppose that ψ and ψ are unital homomorphisms from C (Ω) to F (Φ( D ) , B ) such that τ B,ω ◦ ψ = τ B,ω ◦ ψ . Then there exist a projection p ∈ F (Φ( D ) , B ), ( F, ε )-multiplicative unital c.p. maps ψ ′ and ψ ′ from C (Ω) to pF (Φ( D ) , B ) p , a unital homomorphism σ from C (Ω) to (1 − p ) F (Φ( D ) , B )(1 − p )with finite-dimensional range and a unitary element u ∈ F (Φ( D ) , B ) such that0 < τ B,ω ( p ) < ε, k ψ ( f ) − ( ψ ′ ( f ) + σ ( f )) k < ε, k ψ ( f ) − u ( ψ ′ ( f ) + σ ( f )) u ∗ k < ε for any f ∈ F .The following lemma is essentially the same as [34, Theorem 5.2] and [35, The-orem 5.2]. Lemma 3.7.
Let Ω be a compact metrizable space, and let F be a finite subsetof C (Ω) and F a finite subset of D , and let ε >
0. Then there exist mutually or-thogonal positive elements h , h , ..., h l in C (Ω) of norm one such that the followingholds. If ψ and ψ are unital homomorphisms from C (Ω) to F (Φ( D ) , B ) such that τ B,ω ( ψ ( h i )) > , ≤ ∀ i ≤ l and τ B,ω ◦ ψ = τ B,ω ◦ ψ , then there exist a unitary elements u in ( B ω ) ∼ such that k uρ ( ψ ( f ) ⊗ a ) u ∗ − ρ ( ψ ( f ) ⊗ a ) k < ε for any f ∈ F , a ∈ F . Proof.
Take positive elements h , h , ..., h l in C (Ω) by the same way as in the proofof [34, Theorem 5.2]. Let ψ and ψ be unital homomorphisms from C (Ω) to F (Φ( D ) , B ) such that τ B,ω ( ψ ( h i )) > ≤ i ≤ l and τ B,ω ◦ ψ = τ B,ω ◦ ψ .Define homomorphisms Ψ and Ψ from C (Ω) ⊗ D to B ω byΨ := ρ ◦ ( ψ ⊗ id D ) and Ψ := ρ ◦ ( ψ ⊗ id D ) . Note that there exists ν > τ B,ω ( ψ ( h i )) ≥ ν for any 1 ≤ i ≤ l . UsingProposition 3.4, Theorem 3.5 and Lemma 3.6 instead of [34, Corollary 4.10], [34,Corollary 3.8] and [34, Lemma 5.1], the same argument as in the proof of [34,Theorem 5.2] shows that there exist a unitary elements u in ( B ω ) ∼ such that k u Ψ ( f ⊗ a ) u ∗ − Ψ ( f ⊗ a ) k < ε for any f ∈ F , a ∈ F . Therefore we obtain the conclusion. (cid:3) The following theorem is a generalization of [34, Theorem 5.3]. See also [35,Theorem 5.3].
CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 9
Theorem 3.8.
Let N and N be normal elements in F (Φ( D ) , B ) such thatSp( N ) = Sp( N ) and τ B,ω ( f ( N )) > f ∈ C (Sp( N )) + \ { } . Thenthere exists a unitary element u in F (Φ( D ) , B ) such that uN u ∗ = N if and onlyif τ B,ω ( f ( N )) = τ B,ω ( f ( N )) for any f ∈ C (Sp( N )). Proof.
It is enough to show the if part because the only if part is obvious. LetΩ := Sp( N ) = Sp( N ), and define unital homomorphisms ψ and ψ from C (Ω)to F (Φ( D ) , B ) by ψ ( f ) := f ( N ) and ψ ( f ) := f ( N ) for any f ∈ C (Ω). By theChoi-Effros lifting theorem, there exist sequences of unital c.p. maps { ψ ,n } n ∈ N and { ψ ,n } n ∈ N from C (Ω) to B ∼ such that ψ ( f ) = [( ψ ,n ( f )) n ] and ψ ( f ) =[( ψ ,n ( f )) n ] for any f ∈ C (Ω). Let F := { , ι } ⊂ C (Ω) where ι is the identityfunction on Ω, that is ι ( z ) = z for any z ∈ Ω, and let { F ,m } m ∈ N be an increasingsequence of finite subsets in D such that D = S m ∈ N F ,m . For any m ∈ N , applyingLemma 3.7 to F , F ,m and 1 /m , we obtain mutually orthogonal positive elements h ,m , h ,m ,..., h l ( m ) ,m in C (Ω) of norm one. Since we have τ B,ω ( ψ ( h i,m )) > , ≤ ∀ i ≤ l ( m ) and τ B,ω ◦ ψ = τ B,ω ◦ ψ by the assumption, Lemma 3.7 implies that there exists a unitary element ( u m,n ) n in ( B ω ) ∼ such that k ( u m,n ) n ρ ( ψ ( f ) ⊗ a )( u ∗ m,n ) n − ρ ( ψ ( f ) ⊗ a ) k < m for any f ∈ F , a ∈ F ,m . By the definition of ρ , we havelim n → ω k u m,n ψ ,n ( f )Φ n ( a ) u ∗ m,n − ψ ,n ( f )Φ n ( a ) k < m for any f ∈ F , a ∈ F ,m . Therefore we inductively obtain a decreasing sequence { X m } m ∈ N of elements in ω such that for any n ∈ X m , k u m,n ψ ,n ( f )Φ n ( a ) u ∗ m,n − ψ ,n ( f )Φ n ( a ) k < m for any f ∈ F , a ∈ F ,m . Set u n := (cid:26) n / ∈ X u m,n if n ∈ X m \ X m +1 ( m ∈ N ) . Then we havelim n → ω k u n Φ n ( a ) u ∗ n − Φ n ( a ) k = 0 , lim n → ω k u n ψ ,n ( ι )Φ n ( a ) u ∗ n − ψ ,n ( ι )Φ n ( a ) k = 0for any a ∈ D . Therefore, ( u n ) n ∈ ( B ∼ ) ω ∩ Φ( D ) ′ and [( u n ) n ] N [( u n ) n ] ∗ = N in F (Φ( D ) , B ). Since [( u n ) n ] is a unitary element in F (Φ( D ) , B ), we obtain theconclusion. (cid:3) The following corollary is an immediate consequence of the theorem above.
Corollary 3.9. (cf. [35, Corollary 5.4]) Let p and q be projections in F (Φ( D ) , B )such that 0 < τ Bω ( p ) <
1. Then p and q are unitarily equivalent if and only if τ B,ω ( p ) = τ B,ω ( q ).The corollary above and the same argument as in the proof of [35, Corollary 5.5]show the following theorem. Theorem 3.10.
Let p and q be projections in F (Φ( D ) , B ) such that 0 < τ B,ω ( p ) ≤
1. Then p and q are Murray-von Neumann equivalent if and only if τ B,ω ( p ) = τ B,ω ( q ).By Proposition 3.2 and applying the theorem above to B = D and Φ = id D , weobtain the following corollary. Corollary 3.11.
Let D be a simple separable nuclear monotracial M ∞ -stableC ∗ -algebra which is KK -equivalent to { } . Then D has property W. Uniqueness theorem
In this section we shall show that if D has property W, then every trace pre-serving endomorphism of D is approximately inner. Furthermore, we shall considera uniqueness theorem for approximate homomorphisms from a simple separablenuclear monotracial M ∞ -stable C ∗ -algebra D which is KK -equivalent to { } foran existence theorem in Section 5.Let D be a simple separable nuclear monotracial C ∗ -algebra, and let ϕ be a tracepreserving endomorphism of D . Define a homomorphism Φ from D to M ( D ) byΦ( a ) := (cid:18) a ϕ ( a ) (cid:19) for any a ∈ D . Since ϕ is trace preserving, we see that τ M ( D ) ,ω | Φ( D ) is a state.Hence τ M ( D ) ,ω is a tracial state on F (Φ( D ) , M ( D )). (See Proposition 2.1.) Definehomomorphisms ι and ι from F ( D ) to F (Φ( D ) , M ( D )) by ι ([( x n ) n ]) := (cid:20)(cid:18)(cid:18) x n
00 0 (cid:19)(cid:19) n (cid:21) and ι ([( x n ) n ]) := (cid:20)(cid:18)(cid:18) ϕ ( x n ) (cid:19)(cid:19) n (cid:21) for any [( x n ) n ] in F ( D ). It is easy to see that ι and ι are well-defined. Put p := ι (1) and q := ι (1). Note that p and q are projections in F (Φ( D ) , M ( D ))and if { h n } n ∈ N is an approximate unit for D , then p = (cid:20)(cid:18)(cid:18) h n
00 0 (cid:19)(cid:19) n (cid:21) and q = (cid:20)(cid:18)(cid:18) ϕ ( h n ) (cid:19)(cid:19) n (cid:21) . It can be easily checked that ι is an isomorphism from F ( D ) onto pF (Φ( D ) , M ( D )) p . Lemma 4.1.
Let D be a a simple separable nuclear monotracial C ∗ -algebra withproperty W. Then D is M ∞ -stable, and hence D is Z -stable. Proof.
Since D has property W, there exists a projection p in F ( D ) such that τ D,ω ( p ) = 1 /
2. Moreover, p is Murray-von Neumann equivalent to 1 − p . Hencethere exists a unital homomorphism from M ( C ) to F ( D ). By [22, Corollary 1.13]and [22, Proposition 4.11] (see also [3, Proposition 2.12]), D is M ∞ -stable. (cid:3) The lemma above implies that if D has property W, then D has strict comparisonand D ⊆ GL( D ∼ ) by [45] and [41]. Furthermore, F (Φ( D ) , M ( D )) is monotracialand has strict comparison by Proposition 2.7. The following lemma is related to[35, Lemma 6.2]. Lemma 4.2.
With notation as above, if D has property W, then p is Murray-vonNeumann equivalent to q in F (Φ( D ) , M ( D )). Proof.
For any m ∈ N , there exists a projection q m in F ( D ) such that τ D,ω ( q m ) =1 − /m because D has property W . Proposition 2.7 implies that there exists acontraction r m in F (Φ( D ) , M ( D )) such that r ∗ m pr m = ι ( q m ). By the diagonalargument, we see that there exist a projection q ′ in F ( D ) and a contraction r in F (Φ( D ) , M ( D )) such that τ D,ω ( q ′ ) = 1 and r ∗ pr = ι ( q ′ ). Note that ι ( q ′ ) isMurray-von Neumann equivalent to prr ∗ p . There exists a projection p ′ in F ( D )such that ι ( p ′ ) = prr ∗ p and τ D,ω ( p ′ ) = 1 because ι is an isomorphism from F ( D ) onto pF (Φ( D ) , M ( D )) p . Since D has property W, there exist v and v in F ( D ) such that v ∗ v = 1, v v ∗ = p ′ , v ∗ v = 1 and v v ∗ = q ′ . Therefore we have p = ι (1) ∼ ι ( p ′ ) = prr ∗ p ∼ r ∗ pr = ι ( q ′ ) ∼ ι (1) = q. (cid:3) The following theorem is one of the main theorem in this section.
CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 11
Theorem 4.3.
Let D be a simple separable nuclear monotracial C ∗ -algebra withproperty W, and let ϕ be a trace preserving endomorphism of D . Then ϕ isapproximately inner. Proof.
By Lemma 4.2, there exists a contraction V in F (Φ( D ) , M ( D )) such that V ∗ V = (cid:20)(cid:18)(cid:18) h n
00 0 (cid:19)(cid:19) n (cid:21) and V V ∗ = (cid:20)(cid:18)(cid:18) ϕ ( h n ) (cid:19)(cid:19) n (cid:21) where { h n } n ∈ N is an approximate unit for D . It can be easily checked that thereexists an element ( v n ) n in D ω such that V = (cid:20)(cid:18)(cid:18) v n (cid:19)(cid:19) n (cid:21) , and we have( v n x ) n = ( ϕ ( x ) v n ) n , ( v ∗ n v n x ) n = x and ( v n v ∗ n ϕ ( x )) n = ϕ ( x )for any x ∈ D . Since ( v n x ) n = ( ϕ ( x ) v n ) n and ( ϕ ( x ) v n v ∗ n ) n = ϕ ( x ), we have( v n xv ∗ n ) n = ϕ ( x ) for any x ∈ D . Because of D ⊆ GL( D ∼ ), we may assume that v n is an invertible element in D ∼ for any n ∈ N . (See the proof of Proposition2.4.) For any n ∈ N , let u n := v n ( v ∗ n v n ) − / . Then u n is a unitary element in D ∼ .Since ( v ∗ n v n x ) n = x , we have ( u n x ) n = ( v n ( v ∗ n v n ) − / x ) n = ( v n x ) n for any x ∈ D .Therefore ϕ ( x ) = ( v n xv ∗ n ) n = ( u n xv ∗ n ) n = ( u n ( v n x ∗ ) ∗ ) n = ( u n ( u n x ∗ ) ∗ ) n = ( u n xu ∗ n ) n for any x ∈ D . Consequently, ϕ is approximately inner. (cid:3) Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra whichis KK -equivalent to { } . In the rest of this section, we shall consider a uniquenesstheorem for approximate homomorphisms from D to certain C ∗ -algebras. Let B be a simple monotracial C ∗ -algebra with strict comparison, B ⊆ GL( B ∼ ) and M ( B ) ⊆ GL( M ( B ) ∼ ), and let ϕ and ψ be homomorphisms from D to B ω suchthat τ D = τ B,ω ◦ ϕ = τ B,ω ◦ ψ . By the Choi-Effros lifting theorem, there existsequences of contractive c.p. maps ϕ n and ψ n from D to B such that ϕ ( a ) =( ϕ n ( a )) n and ψ ( a ) = ( ψ n ( a )) n for any a ∈ D . Define a homomorphism Φ from D to M ( B ) ω by Φ( a ) := (cid:18)(cid:18) ϕ n ( a ) 00 ψ n ( a ) (cid:19)(cid:19) n for any a ∈ D . Since τ D = τ B,ω ◦ ϕ = τ B,ω ◦ ψ , τ M ( B ) ,ω | Φ( D ) is a state. Hence τ M ( B ) ,ω is a tracial state on F (Φ( D ) , M ( B )) as above. Since D is separable, thereexist elements ( s n ) n and ( t n ) n in B ω such that [( s n ) n ] = 1 in F ( ϕ ( D ) , B ) and[( t n ) n ] = 1 in F ( ψ ( D ) , B ) by arguments in Section 2.2. Put p := (cid:20)(cid:18)(cid:18) s n
00 0 (cid:19)(cid:19) n (cid:21) and q := (cid:20)(cid:18)(cid:18) t n (cid:19)(cid:19) n (cid:21) . in F (Φ( D ) , M ( B )). It is easy to see that p and q are projections in F (Φ( D ) , M ( B ))such that τ M ( B ) ,ω ( p ) = τ M ( B ) ,ω ( p ) = 1 /
2. Theorem 3.10 implies that p is Murray-von Neumann equivalent to q . Therefore we obtain the following theorem by asimilar argument as in the proof of Theorem 4.3. Theorem 4.4.
Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } and B a simple monotracial C ∗ -algebrawith strict comparison, B ⊆ GL( B ∼ ) and M ( B ) ⊆ GL( M ( B ) ∼ ). If ϕ and ψ arehomomorphisms from D to B ω such that τ D = τ B,ω ◦ ϕ = τ B,ω ◦ ψ , then there existsa unitary element u in ( B ∼ ) ω such that ϕ ( a ) = uψ ( a ) u ∗ for any a ∈ D . The following corollary is an immediate consequence of the theorem above.
Corollary 4.5.
Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } and B a simple monotracial C ∗ -algebrawith strict comparison, B ⊆ GL( B ∼ ) and M ( B ) ⊆ GL( M ( B ) ∼ ). If ϕ and ψ aretrace preserving homomorphisms from D to B , then ϕ is approximately unitarilyequivalent to ψ . Remark 4.6. If B is a simple separable exact monotracial Z -stable C ∗ -algebra,then B has strict comparison, B ⊆ GL( B ∼ ) and M ( B ) ⊆ GL( M ( B ) ∼ ) by [45]and [41].The following corollary is also an immediate consequence of Theorem 4.4. Corollary 4.7.
Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } and B a simple monotracial C ∗ -algebra withstrict comparison, B ⊆ GL( B ∼ ) and M ( B ) ⊆ GL( M ( B ) ∼ ). For any finite subset F ⊂ D and ε >
0, there exist a finite subset G ⊂ D and δ > ϕ and ψ are ( G, δ )-multiplicative maps from D to B such that | τ B ( ϕ ( a )) − τ D ( a ) | < δ and | τ B ( ψ ( a )) − τ D ( a ) | < δ for any a ∈ G , then there exists a unitary element u in B ∼ such that k ϕ ( a ) − uψ ( a ) u ∗ k < ε for any a ∈ F . 5. Existence theorem
Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra whichis KK -equivalent to { } and B a simple separable exact monotracial Z -stable C ∗ -algebra. In this section we shall show that there exists a trace preserving homomor-phism from D to B . Many arguments in this section are motivated by Schafhauser’sproof [48] (see also [49]) of the Tikuisis-White-Winter theorem [51].The following lemma is related to [23, Lemma 2.2]. Lemma 5.1.
Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } and B a simple separable exact monotracial Z -stable C ∗ -algebra. If there exists a homomorphism ϕ from D to B ω such that τ B,ω ◦ ϕ = τ D , then there exists a trace preserving homomorphism from D to B . Proof.
By the Choi-Effros lifting theorem, there exists a sequence { ϕ n } n ∈ N of con-tractive c.p. maps from D to B such that ϕ ( a ) = ( ϕ n ( a )) n for any a ∈ D . Let { F m } m ∈ N be an increasing sequence of finite subsets in D such that D = S m ∈ N F m .For any m ∈ N , applying Corollary 4.7 to F m and 1 / m , we obtain a finite subset G m of D and δ m >
0. We may assume that G m ⊂ G m +1 , δ m > δ m +1 for any m ∈ N and lim m →∞ δ m = 0. Since we havelim n → ω k ϕ n ( ab ) − ϕ n ( a ) ϕ n ( b ) k = 0 and lim n → ω | τ B ( ϕ n ( a )) − τ D ( a ) | = 0for any a, b ∈ D , there exists a subsequence { ϕ n ( m ) } m ∈ N of { ϕ n } n ∈ N such that k ϕ n ( m ) ( ab ) − ϕ n ( m ) ( a ) ϕ n ( m ) ( b ) k < δ m and | τ B ( ϕ n ( m ) ( a )) − τ D ( a ) | < δ m for any a, b ∈ G m . Corollary 4.7 implies that for any m ∈ N , there exists a unitaryelement u m in B ∼ such that k ϕ n ( m ) ( a ) − u m ϕ n ( m +1) ( a ) u ∗ m k < m CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 13 for any a ∈ F m . Therefore it can easily be checked that the limitlim m →∞ u u · · · u m − ϕ n ( m ) ( a ) u ∗ m − · · · u ∗ u ∗ exists for any a ∈ D . Define ψ ( a ) := lim m →∞ u u · · · u m − ϕ n ( m ) ( a ) u ∗ m − · · · u ∗ u ∗ for any a ∈ D , then ψ is a trace preserving homomorphism from D to B . (cid:3) By the lemma above, it is enough to show that there exists a homomorphism ϕ from D to B ω such that τ B,ω ◦ ϕ = τ D . Borrowing Schafhauser’s idea in [48], weshall show this. By arguments in Section 2.4, there exists a following extension: η : 0 / / J / / B ω ̺ / / M / / M is a von Neumann algebraic ultrapower of π τ B ( B ) ′′ and J = ker ̺ .Note that M is a II factor because B is Z -stable (or infinite-dimensional) andmonotracial. Since D is monotracial and nuclear, π τ D ( D ) ′′ is the injective II factor. Hence there exists a unital homomorphism from π τ D ( D ) ′′ to M (see, forexample, [50, XIV. Proposition 2.15]). In particular, there exists a trace preservinghomomorphism Π from D to M . Consider a pullback extensionΠ ∗ η : 0 / / J / / E ˆ ̺ / / ˆΠ (cid:15) (cid:15) D / / Π (cid:15) (cid:15) η : 0 / / J / / B ω ̺ / / M / / E = { ( a, x ) ∈ D ⊕ B ω | Π( a ) = ̺ ( x ) } , ˆ ̺ (( a, x )) = a and ˆΠ(( a, x )) = x forany ( a, x ) ∈ E . If we could show that Π ∗ η is a split extension with a cross section γ , then ˆΠ ◦ γ is a homomorphism from D to B ω such that τ B,ω ◦ ˆΠ ◦ γ = τ D . Butwe could not show this, immediately. Note that we need to consider a separableextension in order to use KK -theory and some results in [15] and [17]. We shallconstruct a suitable separable extension η by Blackadar’s technique (see [1, II.8.5]).We shall recall some definition and some results in [15] and [17]. An extension0 −→ I −→ C −→ A −→ purely large if for any x ∈ C \ I , xIx ∗ contains a stable C ∗ -subalgebra which is full in I . Note that xIx ∗ = xx ∗ Ixx ∗ = I ∩ xCx ∗ . By [17, Theorem 2.1] (see also [15, Corollary 16]), if A is non-unitaland I is stable, then a separable extension 0 −→ I −→ C −→ A −→ Lemma 5.2.
With notation as above, suppose that there exist separable C ∗ -subalgebras J ⊂ J , B ⊂ B ω and M ⊂ M such that J is stable, η : 0 / / J / / B ̺ | B / / M / / D ) ⊂ M . Then there exists a homomorphism ϕ from D to B ω such that τ B,ω ◦ ϕ = τ D . Proof.
Consider a pullback extensionΠ ∗ η : 0 / / J / / E ̺ / / ˆΠ (cid:15) (cid:15) D / / Π (cid:15) (cid:15) η : 0 / / J / / B ̺ / / M / / E = { ( a, x ) ∈ D ⊕ B | Π( a ) = ̺ ( x ) } , ˆ ̺ (( a, x )) = a and ˆΠ(( a, x )) = x forany ( a, x ) ∈ E . Since η is purely large, it can be easily checked that Π ∗ η is purelylarge. Hence Π ∗ η is nuclear absorbing by [17, Theorem 2.1]. Because D is KK -equivalent to { } and nuclear, we have Ext( D , J ) = { } , and hence [Π ∗ η ] = 0 in Ext( D , J ). Therefore there exists a (nuclear) split extension η ′ such that Π ∗ η ⊕ η ′ is a split extension. Since Π ∗ η is nuclear absorbing, Π ∗ η is strongly unitarilyequivalent to Π ∗ η ⊕ η ′ , and hence Π ∗ η is a split extension. Let γ be a crosssection of Π ∗ η , and define ϕ := ˆΠ ◦ γ . Then ϕ is a desired homomorphism. (cid:3) A key result in the proof of purely largeness is the following Hjelmborg andRørdam’s characterization of stable C ∗ -algebras in [20] and [43]. Theorem 5.3. (Hjelmborg-Rørdam cf. [43, Theorem 2.2])Let A be a σ -unital C ∗ -algebra. Then A is stable if and only if for any a ∈ A + and ε >
0, there exist positive elements a ′ and c in A such that k a − a ′ k ≤ ε , a ′ ∼ c and k ac k ≤ ε .Before we construct a separable extension η , we shall consider properties of η . Proposition 5.4.
With notation as above, let b be a positive element in B ω \ J .(i) For any positive element a in bJb , there exists a positive element c in bJb suchthat a ∼ c and ac = 0.(ii) For any positive element a in J and ε >
0, there exist a positive element d in bJb and an element r in J such that k r ∗ dr − a k < ε .(iii) For any element x in B ω and ε >
0, there exists an element y in GL(( B ω ) ∼ )such that k x − y k < ε .For the proof of the proposition above, we need some lemmas. For a positiveelement a ∈ A and ε >
0, we denote by ( a − ε ) + the element f ( a ) in A where f ( t ) = max { , t − ε } , t ∈ Sp( a ). The same proof as in [42, Proposition 2.4] showsthe following lemma. See also [36, Corollary 8]. Lemma 5.5.
Let A be a C ∗ -algebra with A ⊆ GL( A ∼ ), and let a and b be positiveelements in A . Then a is Cuntz smaller than b if and only if for any ε >
0, thereexists a unitary element u in A ∼ such that u ( a − ε ) + u ∗ ∈ bAb .The following lemma can be regarded as an application of the construction of Z . Lemma 5.6.
Let A be a monotracial Z -stable C ∗ -algebra. For any θ ∈ (0 , / d and d ′ in A such that dd ′ = 0 and d τ A (( d − ε ) + ) = d τ A (( d ′ − ε ) + ) = (1 − ε ) θ for any 0 ≤ ε ≤ Proof.
Let µ be the Lebesgue measure on [0 , τ on C ([0 , τ ( f ) := R [0 , f dµ for any f ∈ C ([0 , ψ from C ([0 , Z such that τ = τ Z ◦ ψ .Define f and g in C ([0 , f ( t ) := θ t if t ∈ [0 , θ ] − θ t + 2 if t ∈ ( θ , θ ]0 if t ∈ ( θ,
1] and g ( t ) := t ∈ [0 , θ ] θ t − t ∈ ( θ, θ ] − θ t + 4 if t ∈ ( θ , θ ]0 if t ∈ (2 θ, . Note that for any 0 ≤ ε ≤
1, we have( f − ε ) + ( t ) = t ∈ [0 , εθ ] θ t − ε if t ∈ ( εθ , θ ] − θ t + 2 − ε if t ∈ ( θ , θ − εθ ]0 if t ∈ ( θ − εθ , g − ε ) + ( t ) = t ∈ [0 , θ + εθ ] θ t − − ε if t ∈ ( θ + εθ , θ ] − θ t + 4 − ε if t ∈ ( θ , θ − εθ ]0 if t ∈ (2 θ − εθ , . CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 15
Hence d τ (( f − ε ) + ) = d τ (( g − ε ) + ) = (1 − ε ) θ . Let s be a strictly positive elementin A , and put d := s ⊗ ψ ( f ) and d ′ := s ⊗ ψ ( g )in A ⊗ Z ∼ = A . Then d and d ′ are desired positive elements in A . (cid:3) Lemma 5.7.
Let A be a simple separable exact monotracial Z -stable C ∗ -algebra,and let b be a (non-zero) positive element in A . For any θ ∈ (0 , d τ A ( b ) / e and e ′ in bAb such that ee ′ = 0 and d τ A ( e ) = d τ A ( e ′ ) > θ . Proof.
By Lemma 5.6, there exist contractions d and d ′ in A such that dd ′ = 0and θ < d τ A ( d ) = d τ A ( d ′ ) < d τ A ( b ) /
2. Furthermore, we may assume that thereexists ε > d τ A (( d − ε ) + ) = d τ A (( d ′ − ε ) + ) > θ . Since A has strictcomparison and d τ A ( d + d ′ ) = d τ A ( d ) + d τ A ( d ′ ) < d τ A ( b ), Lemma 5.5 implies thatthere exists a unitary element u in A ∼ such that u ( d + d ′ − ε ) + u ∗ ∈ bAb . Note that( d + d ′ − ε ) + = ( d − ε ) + + ( d ′ − ε ) + because of dd ′ = 0. Put e := u ( d − ε ) + u ∗ and e ′ := u ( d ′ − ε ) + u ∗ , then e and e ′ are desired positive elements. (cid:3) Proof of Proposition 5.4. (i) We may assume that k a k = 1 and k b k = 1. Since b / ∈ J , we have τ B,ω ( b ) >
0. Take a representative ( b n ) n of b such that k b n k = 1 forany n ∈ N , and choose ε > τ B,ω ( b ) − ε >
0. Since we havelim n → ω d τ B ( b n ) ≥ lim n → ω τ B ( b n ) = τ B,ω ( b ) , there exists an element X ∈ ω such that for any n ∈ X , d τ B ( b n ) > τ B,ω ( b ) − ε . By a similar argument as in the proof of [47, Lemma 3.2], we see that there existsa representative ( a n ) n of a such that a n ∈ b n Bb n and k a n k = 1 for any n ∈ N andlim n → ω d τ B ( a n ) = 0 because of a ∈ ( b n ) n J ( b n ) n . Hence there exists an element X ∈ ω such that for any n ∈ X , d τ B ( a n ) < τ B,ω ( b ) − ε . Note that we have d τ B ( a n ) < d τB ( b n )2 for any n ∈ X ∩ X . Hence Lemma 5.7implies that for any n ∈ X ∩ X , there exist positive elements e n and e ′ n in b n Bb n such that e n e ′ n = 0 and d τ B ( e n ) = d τ B ( e ′ n ) > d τ B ( a n ). Since b n Bb n has strictcomparison and b n Bb n ⊆ GL( b n Bb n ∼ ) by [45] and [41], Lemma 5.5 shows that forany n ∈ X ∩ X , there exist unitary elements u n and v n in b n Bb n ∼ such that u n ( a n − /n ) + u ∗ n ∈ e n Be n and v n ( a n − /n ) + v ∗ n ∈ e ′ n Be ′ n . Note that ( a n − /n ) + u ∗ n v n ( a n − /n ) + = 0 for any n ∈ X ∩ X . Define z = ( z n ) n and c = ( c n ) n in B ω by z n := (cid:26) n / ∈ X ∩ X u ∗ n v n ( a n − /n ) / if n ∈ X ∩ X and c n := (cid:26) n / ∈ X ∩ X u ∗ n v n ( a n − /n ) + v ∗ n u n if n ∈ X ∩ X . It is easy to see that z, c ∈ bB ω b , z ∗ z = a , zz ∗ = c and ac = 0. Since bJb is a closedideal in bB ω b and a ∈ bJb , z and c are elements in bJb . Therefore we obtain theconclusion. (ii) Note that B ω has strict comparison (see, for example, [2, Lemma 1.23]).Since a ∈ J and b / ∈ J , we have d τ B,ω ( a / ) = 0 and d τ B,ω ( b ) >
0. Hence thereexists a sequence { s N } N ∈ N in B ω such that lim N →∞ k s ∗ N bs N − a / k = 0. Let d N := bs N a / s ∗ N b and r N := s N a / for any N ∈ N . Then we have d N ∈ bJb , r N ∈ J for any N ∈ N and r ∗ N d N r N = a / s ∗ N bs N a / s ∗ N bs N a / → a as N → ∞ . Therefore we obtain the conclusion.(iii) Since B is a simple monotracial Z -stable C ∗ -algebra, B ⊆ GL( B ∼ ) by [45]and [41]. Therefore we obtain the conclusion by Proposition 2.4. (cid:3) If B is unital, then the following lemma is a well-known consequence of Propo-sition 2.4 and Blackadar’s technique (see [1, II.8.5.4]). Lemma 5.8.
With notation as above, let S be a separable subset of B ω . Thenthere exists a separable C ∗ -algebra A such that S ⊆ A ⊂ B ω and A ⊆ GL( A ∼ ). Proof.
We shall show only the case where B is non-unital. Let A be a C ∗ -subalgebra of B ω generated by S . Since A is separable, there exists a countabledense subset { x k | k ∈ N } of A . By Proposition 5.4.(iii), for any k, m ∈ N , thereexist y k,m ∈ B ω and λ k,m ∈ C \ { } such that k x k − ( y k,m + λ k,m ( B ω ) ∼ ) k < m and y k,m + λ k,m ( B ω ) ∼ ∈ GL(( B ω ) ∼ ). Let A be a C ∗ -subalgebra of B ω generatedby A and { y k,m | k, m ∈ N } . Then we have A ⊆ GL( A ∼ ). Indeed, we have y k,m + λ k,m A ∼ ∈ GL( A ∼ ) for any k, m ∈ N because of Sp A ( y k,m ) ∪ { } = Sp B ω ( y k,m ) ∪{ } and λ k,m = 0. Since we have A = { x k | k ∈ N } and k x k − ( y k,m + λ k,m ( A ) ∼ ) k = k A ∼ x k − A ∼ ( y k,m + λ k,m ( B ω ) ∼ ) k≤ k x k − ( y k,m + λ k,m ( B ω ) ∼ ) k < m for any k, m ∈ N , we have A ⊆ GL( A ∼ ). Repeating this process, we obtaina sequence { A n } n ∈ N of separable C ∗ -subalgebras of B ω such that A n ⊆ A n +1 and A n ⊆ GL( A ∼ n +1 ) for any n ∈ N . Put A := S ∞ n =1 A n . Since we have A n ⊆ GL( A ∼ n +1 ) ⊆ GL( A ∼ ) for any n ∈ N by Proposition 2.2, we have A ⊆ GL( A ∼ ).Therefore A is a desired separable C ∗ -algebra. (cid:3) The following lemma is also based on Blackadar’s technique.
Lemma 5.9.
With notation as above, let { b k | k ∈ N } be a countable subset of B ω \ J and S a separable subset of B ω . Then there exists a separable C ∗ -algebra A such that { b k | k ∈ N } ∪ S ⊆ A ⊂ B ω and b k ( A ∩ J ) b k is full in A ∩ J for any k ∈ N . Proof.
Let A be a C ∗ -subalgebra of B ω generated by { b k | k ∈ N } and S . Since A is separable, there exists a countable dense subset { a l | l ∈ N } of ( A ∩ J ) + . ByProposition 5.4.(ii), for any k, l, m ∈ N , there exist d k,l,m ∈ b k Jb k + and r k,l,m ∈ J such that k r ∗ k,l,m d k,l,m r k,l,m − a l k < m . Let A be a C ∗ -subalgebra of B ω generated by A and { d k,l,m , r k,l,m | k, l, m ∈ N } .Then we have A ∩ J ⊆ ( A ∩ J ) b k ( A ∩ J ) b k ( A ∩ J ) for any k ∈ N because A ∩ J is generated by { a l | l ∈ N } . Repeating this process, we obtain a sequence { A n } n ∈ N of separable C ∗ -subalgebras of B ω such that A n ⊆ A n +1 and A n ∩ J ⊆ CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 17 ( A n +1 ∩ J ) b k ( A n +1 ∩ J ) b k ( A n +1 ∩ J ) for any k, n ∈ N . Put A := S ∞ n =1 A n . Sincewe have A ∩ J = S ∞ n =1 ( A n ∩ J ), we see that A is a desired separable C ∗ -algebra. (cid:3) By Lemma 5.8 and Lemma 5.9, [1, II.8.5.3] implies the following lemma.
Lemma 5.10.
With notation as above, let { b k | k ∈ N } be a countable subset of B ω \ J and S a separable subset of B ω . Then there exists a separable C ∗ -algebra A such that { b k | k ∈ N } ∪ S ⊆ A ⊂ B ω , A ⊆ GL( A ∼ ) and b k ( A ∩ J ) b k is full in A ∩ J for any k ∈ N .We shall construct a separable extension η .Since ̺ is surjective and D is separable, there exists a separable subset S of B ω such that ̺ ( S ) = Π( D ). Applying Lemma 5.8 to S , we obtain a separableC ∗ -algebra B such that S ⊆ B ⊂ B ω and B ⊆ GL( B ∼ ). Since B is separable,there exist a countable subset { a ,m | m ∈ N } of ( B ∩ J ) + and a countable subset { b ,k | k ∈ N } of B such that { a ,m | m ∈ N } = ( B ∩ J ) + and { b ,k | k ∈ N } = B . Put T := { ( k, l ) ∈ N × N | ( b ,k − /l ) + / ∈ J } . By Proposition 5.4, for any ( k, l ) ∈ T and m ∈ N , there exist a positive element c , , ( k,l ) ,m and an element z , , ( k,l ) ,m in( b ,k − /l ) + J ( b ,k − /l ) + such that( b ,k − /l ) + a ,m ( b ,k − /l ) + c , , ( k,l ) ,m = 0 ,z ∗ , , ( k,l ) ,m z , , ( k,l ) ,m = ( b ,k − /l ) + a ,m ( b ,k − /l ) + and z , , ( k,l ) ,m z ∗ , , ( k,l ) ,m = c , , ( k,l ) ,m . Let S := B ∪ { c , , ( k,l ) ,m , z , , ( k,l ) ,m | ( k, l ) ∈ T , m ∈ N } . Applying Lemma 5.10to { ( b ,k − /l ) + | ( k, l ) ∈ T } and S , we obtain a separable C ∗ -algebra B suchthat B ∪ { c , , ( k,l ) ,m , z , , ( k,l ) ,m | ( k, l ) ∈ T , m ∈ N } ⊆ B ⊂ B ω ,B ⊆ GL( B ∼ ) and ( b ,k − /l ) + ( B ∩ J )( b ,k − /l ) + is full in B ∩ J for any( k, l ) ∈ T . By the same way as above, there exist a countable subset { a ,m | m ∈ N } of ( B ∩ J ) + and a countable subset { b ,k | k ∈ N } of B such that { a ,m | m ∈ N } = ( B ∩ J ) + and { b ,k | k ∈ N } = B , and put T := { ( k, l ) ∈ N × N | ( b ,k − /l ) + / ∈ J } . By Proposition 5.4, for any1 ≤ i ≤
2, ( k, l ) ∈ T i and m ∈ N , there exist a positive element c ,i, ( k,l ) ,m and anelement z ,i, ( k,l ) ,m in ( b i,k − /l ) + J ( b i,k − /l ) + such that( b i,k − /l ) + a ,m ( b i,k − /l ) + c ,i, ( k,l ) ,m = 0 ,z ∗ ,i, ( k,l ) ,m z ,i, ( k,l ) ,m = ( b i,k − /l ) + a ,m ( b i,k − /l ) + and z ,i, ( k,l ) ,m z ∗ ,i, ( k,l ) ,m = c ,i, ( k,l ) ,m . Let S := B ∪ { c ,i, ( k,l ) ,m , z ,i, ( k,l ) ,m | ≤ i ≤ , ( k, l ) ∈ T i , m ∈ N } . ApplyingLemma 5.10 to { ( b i,k − /l ) + | ≤ i ≤ , ( k, l ) ∈ T i } and S , we obtain a separableC ∗ -algebra B such that B ∪ { c ,i, ( k,l ) ,m , z ,i, ( k,l ) ,m | ≤ i ≤ , ( k, l ) ∈ T i , m ∈ N } ⊆ B ⊂ B ω ,B ⊆ GL( B ∼ ) and ( b i,k − /l ) + ( B ∩ J )( b i,k − /l ) + is full in B ∩ J for any 1 ≤ i ≤ k, l ) ∈ T i . Repeating this process, for any n ∈ N , we obtain B n ⊂ B ω , { a n,m | m ∈ N } ⊂ ( B n ∩ J ) + , { b n,k | k ∈ N } ⊂ B n + ,T n ⊂ N × N , { c n,i, ( k,l ) ,m , z n,i, ( k,l ) ,m | ≤ i ≤ n, ( k, l ) ∈ T i , m ∈ N } such that B n is separable, B n ⊆ B n +1 , B n ⊆ GL( B ∼ n ) , { a n,m | m ∈ N } = ( B n ∩ J ) + , { b n,k | k ∈ N } = B n + , T n = { ( k, l ) ∈ N × N | ( b n,k − /l ) + / ∈ J } ,c n,i, ( k,l ) ,m , z n,i, ( k,l ) ,m ∈ ( b i,k − /l ) + ( B n +1 ∩ J )( b i,k − /l ) + , ( b i,k − /l ) + a n,m ( b i,k − /l ) + c n,i, ( k,l ) ,m = 0 ,z ∗ n,i, ( k,l ) ,m z n,i, ( k,l ) ,m = ( b i,k − /l ) + a n,m ( b i,k − /l ) + ,z n,i, ( k,l ) ,m z ∗ n,i, ( k,l ) ,m = c n,i, ( k,l ) ,m and ( b i,k − /l ) + ( B n +1 ∩ J )( b i,k − /l ) + is full in B n +1 ∩ J for any 1 ≤ i ≤ n and( k, l ) ∈ T i . Define B := ∞ [ n =1 B n , J := B ∩ J and M := ̺ ( B ) . Then η : 0 / / J / / B ̺ / / M / / D ) ⊆ M . Corollary 2.3 implies B ⊆ GL( B ∼ ) sincewe have B n ⊆ GL( B ∼ n ) for any n ∈ N . Furthermore, for any i ∈ N and ( k, l ) ∈ T i ,( b i,k − /l ) + J ( b i,k − /l ) + is full in J by a similar argument as in the proof ofLemma 5.9. Note that for any n ∈ N , J = ∞ [ n = n { a n,m | m ∈ N } and B = ∞ [ n = n { b n,k | k ∈ N } . We shall show that J is stable and η is purely large. Proof of stability of J . Let a ∈ J \ { } and ε >
0. Set ε ′ := min (cid:26) ε k a k , r ε , ε (cid:27) . Since B is separable, there exists an approximate unit { h n } n ∈ N for B . Note that h n / ∈ J for sufficiently large n because of M = { } . Hence there exists N ∈ N suchthat h N / ∈ J and k h N ah N − a k < ε ′ /
2. Since B = S ∞ n =1 { b n,k | k ∈ N } , for any l ∈ N , there exist n ( l ) and k ( l ) in N such that k h N − b n ( l ) ,k ( l ) k < l Note that ( b n ( l ) ,k ( l ) − /l ) + → h N as l → ∞ because we have k h N − ( b n ( l ) ,k ( l ) − /l ) + k ≤ k h N − b n ( l ) ,k ( l ) k + k b n ( l ) ,k ( l ) − ( b n ( l ) ,k ( l ) − /l ) + k < l . Hence there exists l ∈ N such that ( b n ( l ) ,k ( l ) − /l ) + / ∈ J , that is, ( k ( l ) , l ) ∈ T n ( l ) and k a − ( b n ( l ) ,k ( l ) − /l ) + a ( b n ( l ) ,k ( l ) − /l ) + k < ε ′ . Since J = S ∞ n = n ( l ) { a n,m | m ∈ N } , there exist n ≥ n ( l ) and m ∈ N such that k a − a n ,m k < ε ′ k b n ( l ) ,k ( l ) k . Put a ′ := ( b n ( l ) ,k ( l ) − /l ) + a n ,m ( b n ( l ) ,k ( l ) − /l ) + , then k a − a ′ k < ε ′ ≤ ε. CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 19
By construction of B and J , there exist z = z n ,n ( l ) , ( k ( l ) ,l ) ,m , c = c n ,n ( l ) , ( k ( l ) ,l ) ,m ∈ J such that a ′ c = 0, z ∗ z = a ′ and zz ∗ = c . Hence a ′ ∼ c and k ac k = k ac − a ′ c k ≤ k a − a ′ kk c k = k a − a ′ kk a ′ k < ε ′ ( k a k + ε ′ ) ≤ ε. Therefore J is stable by Hjelmborg and Rørdam’s characterization (Theorem 5.3). (cid:3) Proof of purely largeness of η . Let x ∈ B \ J . Note that we have xx ∗ / ∈ J . Since B = S ∞ n =1 { b n,k | k ∈ N } , for any l ∈ N , there exist n ( l ) and k ( l ) in N such that k xx ∗ − b n ( l ) ,k ( l ) k < l . By a similar argument as in the proof of stability of J , there exists l ∈ N suchthat ( b n ( l ) ,k ( l ) − /l ) + / ∈ J , that is, ( k ( l ) , l ) ∈ T n ( l ) . On the other hand, [24,Lemma 2.2] implies that ( b n ( l ) ,k ( l ) − / l ) + is Cuntz smaller than xx ∗ . Since wehave B ⊆ GL( B ∼ ), there exists a unitary element u in B ∼ such that u ( b n ( l ) ,k ( l ) − /l ) + u ∗ = u (( b n ( l ) ,k ( l ) − / l ) + − / l ) + u ∗ ∈ xx ∗ B xx ∗ = xB x ∗ by Lemma 5.5. Put C := u ( b n ( l ) ,k ( l ) − /l ) + J ( b n ( l ) ,k ( l ) − /l ) + u ∗ ⊆ xJ x ∗ , then C is full in J because ( b n ( l ) ,k ( l ) − /l ) + J ( b n ( l ) ,k ( l ) − /l ) + is full in J .We shall show that C is stable. Let a ∈ C + \ { } and ε >
0. Set ε ′ := min (cid:26) ε k a k , r ε , ε (cid:27) . By the definition of C and J = S ∞ n = n ( l ) { a n,m | m ∈ N } , there exist n ≥ n ( l )and m ∈ N such that k a − u ( b n ( l ) ,k ( l ) − /l ) + a n ,m ( b n ( l ) ,k ( l ) − /l ) + u ∗ k < ε ′ ≤ ε. Put a ′ = u ( b n ( l ) ,k ( l ) − /l ) + a n ,m ( b n ( l ) ,k ( l ) − /l ) + u ∗ ∈ C , then k a − a ′ k <ε ′ ≤ ε . By construction of B and J , there exist elements z n ,n ( l ) , ( k ( l ) ,l ) ,m , c n ,n ( l ) , ( k ( l ) ,l ) ,m in ( b n ( l ) ,k ( l ) − /l ) + J ( b n ( l ) ,k ( l ) − /l ) + such that u ∗ a ′ uc n ,n ( l ) , ( k ( l ) ,l ) ,m = 0 , z ∗ n ,n ( l ) , ( k ( l ) ,l ) ,m z n ,n ( l ) , ( k ( l ) ,l ) ,m = u ∗ a ′ u and z n ,n ( l ) , ( k ( l ) ,l ) ,m z ∗ n ,n ( l ) , ( k ( l ) ,l ) ,m = c n ,n ( l ) , ( k ( l ) ,l ) ,m . Put c := uc n ,n ( l ) , ( k ( l ) ,l ) ,m u ∗ . It is easy to see that c ∈ C , a ′ c = 0 and c ∼ c n ,n ( l ) , ( k ( l ) ,l ) ,m ∼ u ∗ a ′ u ∼ a ′ in B . Since C is a hereditary C ∗ -subalgebra of B and a ′ , c ∈ C , we see that a ′ is Murray-von Neumann equivalent to c in C . Therefore, the same argument as in the proofof stability of J shows k ac k < ε , and C is stable. Consequently, η is a purelylarge extension. (cid:3) Therefore we obtain the following lemma.
Lemma 5.11.
With notation as above, there exist separable C ∗ -subalgebras J ⊂ J , B ⊂ B ω and M ⊂ M such that J is stable, η : 0 / / J / / B ̺ | B / / M / / D ) ⊂ M .Consequently, we obtain the following theorem by Lemma 5.1, Lemma 5.2 andthe lemma above. Theorem 5.12.
Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } and B a simple separable exact monotracial Z -stable C ∗ -algebra. Then there exists a trace preserving homomorphism from D to B . Remark 5.13.
Actually, we need not assume that D is M ∞ -stable in the theoremabove. Indeed, define a homomorphism ϕ from D to D ⊗ M ∞ by ϕ ( a ) = a ⊗ ϕ is a trace preserving homomorphism from D to D ⊗ M ∞ . By the thoremabove, there exists a trace preserving homomorphism ψ from D ⊗ M ∞ to B . Then ψ ◦ ϕ is is a trace preserving homomorphism from D to B .The following corollary is an immediate consequence of the theorem above. Corollary 5.14.
Let B a simple separable exact monotracial Z -stable C ∗ -algebra.Then there exists a trace preserving homomorphism from W to B .The injective II factor can embed unitally into every II factor. Hence thefollowing question is natural and interesting. Question 5.15. (1) Let B be a simple monotracial infinite-dimensional C ∗ -algebra.Does there exist a trace preserving homomorphism from W to B ?(2) Let B be a simple non-type I C ∗ -algebra. Does there exists a (non-zero) homo-morphism from W to B ?Note that Dadarlat, Hirshberg, Toms and Winter [10] showed that there exists aunital simple separable nuclear infinite-dimensional C ∗ -algebra B such that Z doesnot embed unitally into B .6. Characterization of W In this section we shall show that if D is a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } , then D is isomorphic to W .Also, we shall characterize W by using properties of F ( W ). Theorem 6.1.
Let D be a simple separable nuclear monotracial M ∞ -stable C ∗ -algebra which is KK -equivalent to { } . Then D is isomorphic to W . Proof.
By Theorem 5.12 and Corollary 5.14, there exist trace preserving homo-morphisms ϕ and ψ from D to W and from W and D , respectively. Since D and W have property W by Corollary 3.11, Theorem 4.3 implies that ψ ◦ ϕ and ϕ ◦ ψ are approximately inner. Therefore D is isomorphic to W by Elliott’s approximateintertwining argument [11] (see also [44, Corollary 2.3.4]). (cid:3) The following corollary is an immediate consequence of the theorem above.
Corollary 6.2. (i) If A is a simple separable nuclear monotracial C ∗ -algebra, then A ⊗ W is isomorphic to W . In particular, W ⊗ W is isomorphic to W .(ii) For any non-zero positive element h in W , h W h is isomorphic to W .Following the definition in [29], we say that a C ∗ -algebra A is W -embeddable ifthere exists an injective homomorphism from A to W . CHARACTERIZATION OF THE RAZAK-JACELON ALGEBRA 21
Lemma 6.3.
Let A be a monotracial W -embeddable C ∗ -algebra. Then there existsa trace preserving homomorphism from A to W . Proof.
By the assumption, there exists an injective homomorphism ϕ from A to W .Let s be a strictly positive element in A . (Note that A is separable because A is W -embeddable.) Since ϕ is injective, ϕ ( s ) is a non-zero positive element. Corollary6.2 implies that there exists a isomorphism Φ from ϕ ( s ) W ϕ ( s ) onto W . Note that ϕ can be ragarded as a homomorphism from A to ϕ ( s ) W ϕ ( s ). Define ψ := Φ ◦ ϕ ,then ψ is a trace preserving homomorphism from A to W . (cid:3) The following theorem is a characterization of W . Theorem 6.4.
Let D be a simple separable nuclear monotracial C ∗ -algebra. Then D is isomorphic to W if and only if D has property W and is W -embeddable, thatis, D satisfies the following properties:(i) for any θ ∈ [0 , p in F ( D ) such that τ D,ω ( p ) = θ ,(ii) if p and q are projections in F ( D ) such that 0 < τ D,ω ( p ) = τ D,ω ( q ), then p isMurray-von Neumann equivalent to q ,(iii) there exists a homomorphism from D to W . Proof.
The only if part is obvious by Corollary 3.11. We shall show the if part.Since D is W -embeddable, there exists a trace preserving homomorphism ϕ from D to W by Lemma 6.3. Lemma 4.1 implies that D is Z -stable because D hasproperty W. Hence there exists a trace preserving homomorphism ψ from W to D by Corollary 5.14. The rest of proof is same as the proof of Theorem 6.1. (cid:3) We think that every simple separable nuclear monotracia C ∗ -algebra with prop-erty W ought to be W -embeddable. Note that every simple separable nuclearmonotracial C ∗ -algebra with property W is stably projectionless by [22, Remark2.13] and a similar argument as in the proof of [34, Corollary 5.9]. Hence an affir-mative answer to the following question, which can be regarded as an analogous ofKrichberg’s embedding theorem [23], would imply this. Question 6.5.
Let A be a simple separable exact stably projectionless monotracialC ∗ -algebra. Assume that τ A is amenable. Is A W -embeddable?Note that we need to assume that τ A is amenable because π τ W ( W ) ′′ is theinjective II factor. References [1] B. Blackadar,
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