2-positive almost order zero maps and decomposition rank
aa r X i v : . [ m a t h . OA ] A ug Yasuhiko Sato
Abstract
We consider 2-positive almost order zero (disjointness preserving) maps on C ∗ -al-gebras. Generalizing the argument of M. Choi for multiplicative domains, we give aninternal characterization of almost order zero for 2-positive maps. It is also shown thatcomplete positivity can be reduced to 2-positivity in the definition of decompositionrank for unital separable C ∗ -algebras. In [33] W. Winter and J. Zacharias gave the structure theorem for completely positiveorder zero maps based on the work of M. Wolff on disjointness preserving linear maps[32]. Recall that a positive linear map ϕ : A → B between two C ∗ -algebras is said to have order zero , if ϕ ( a ) ϕ ( b ) = 0 for any positive elements a , b ∈ A with ab = 0. This notionof order zero maps led to a geometric dimension, known as decomposition rank or nucleardimension, in the context of the classification theorem of nuclear C ∗ -algebras [19, 34].The purpose of this paper is to explore the condition of 2-positivity in connection withorder zero maps.In the first part of the paper we show the following one variable characterization of2-positive almost order zero maps. Theorem 1.1.
For ε > there exists δ > satisfying the following condition: for two C ∗ -algebras A and B , the canonical approximate unit h λ , λ ∈ Λ of A , and a 2-positivecontraction ϕ from A to B , if a positive contraction a ∈ A satisfies lim sup λ (cid:13)(cid:13) ϕ ( a ) − ϕ ( a ) ϕ ( h λ ) (cid:13)(cid:13) < δ, then sup b ∈ A, k b k≤ (cid:18) lim sup λ k ϕ ( a ) ϕ ( b ) − ϕ ( h λ ) ϕ ( ab ) k (cid:19) < ε. Particularly, a 2-positive map ϕ from a unital C ∗ -algebra A to a C ∗ -algebra B has orderzero if ϕ ( a ) = ϕ ( a ) ϕ (1 A ) for any positive element a ∈ A . In the second part of the paper, we study the relation between 2-positivity and de-composition rank. The notion of decomposition rank (Definition 6.1) was introduced byE. Kirchberg and W. Winter in their work [19], in which they showed that finiteness ofdecomposition rank implies quasidiagonality for C ∗ -algebras. In [30] W. Winter showed1hat finiteness of decomposition rank (for separable C ∗ -algebras, see [11] for non-separablecases) also implies the absorpition of the Jiang-Su algebra which plays a central role inthe classification theorem of nuclear C ∗ -algebras [9]. For unital separable simple nuclearmonotracial C ∗ -algebras, we showed the converse, i.e., quasidiagonality and Jiang-Su ab-sorption imply finiteness of decomposition rank [22, 23]. Our second main result charac-terizes finiteness of decomposition rank by 2-positive maps instead of completely positivemaps. Theorem 1.2.
Let A be a unital separable C ∗ -algebra. Then the decomposition rank of A isat most d if and only if for a finite subset F of contractions in A and ε > , there exist finitedimensional C ∗ -algebras F i , i = 0 , , ..., d , a 2-positive contraction ψ : A → d M i =0 F i , and2-positive order zero contractions ϕ i : F i → A , i = 0 , , ..., d such that d X i =0 ϕ i : d M i =0 F i → A is contractive and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d X i =0 ϕ i ! ◦ ψ ( x ) − x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε, for all x ∈ F. Here we simply write d X i =0 ϕ i d M i =0 x i ! = d X i =0 ϕ i ( x i ) for x i ∈ F i . Before closing this section, let us collect some notations and terminologies.For a C ∗ -algebra A , we let A sa and A + denote the set of self-adjoint elements and the coneof positive elements in A . For a subset S ⊂ A , S denotes the set of contractions in S . If A is a unital C ∗ -algebra, 1 A denotes the unit of A .For any two elements a and b in a C ∗ -algebra A we let [ a, b ] denote the commutator ab − ba ∈ A , and by a ≈ ε b for ε > k a − b k < ε .Unless stated otherwise we consider two C ∗ -algebras A and B , and by a “map” ϕ : A → B we mean a “linear map” from A to B . We let id A denote the identity map on A ,i.e., id A ( a ) = a for any a ∈ A . For n ∈ N , M n denotes the C ∗ -algebra of complex n × n matrices. A map ϕ from A to B is called positive if ϕ ( A + ) ⊂ B + . For a natural number k , a map ϕ is called k -positive if ϕ ⊗ id M k : A ⊗ M k → B ⊗ M k is positive. If a map ϕ : A → B is k-positive for any k ∈ N , ϕ is called completely positive . Definition 2.1.
Let A and B be two C ∗ -algebras, and let h λ ∈ A , λ ∈ Λ be an approx-imate unit. For a bounded linear map ϕ from A to B , we define a subspace OD( ϕ ) of A by OD( ϕ ) = { a ∈ A : ϕ ( a ) ϕ ( b ) = lim λ ϕ ( h λ ) ϕ ( ab ) ,ϕ ( b ) ϕ ( a ) = lim λ ϕ ( ba ) ϕ ( h λ ) for any b ∈ A } . It follows from the definition that lim λ k [ ϕ ( h λ ) , ϕ ( a )] k = 0 for any a ∈ OD( ϕ ).2n this section we mainly deal with 2-positive maps for Kadison’s inequality in thefollowing form, which makes OD( ϕ ) into a C ∗ -algebra. For two (not necessarily unital)C ∗ -algebras A and B , if a map ϕ : A → B is contractive and 2-positive, then the originalKadison’s inequality tells us that ϕ ⊗ id M (cid:18)(cid:20) a ∗ a (cid:21)(cid:19) ≤ ϕ ⊗ id M (cid:20) a ∗ a (cid:21) ! for any a ∈ A , see [14, p.770], for example. Then we have ϕ ( a ) ∗ ϕ ( a ) ≤ ϕ ( a ∗ a ) for any a ∈ A , [3, Corollary 2.8]. By using this inequality, we can see that OD( ϕ ) is a C ∗ -algebra. Proposition 2.2.
If a map ϕ : A → B is 2-positive, then the following statements hold. (i) OD( ϕ ) is a C ∗ -algebra which contains the multiplicative domain of ϕ . (ii) OD( ϕ ) is independent of the choice of the approximate unit.Proof. Since OD( ϕ ) = OD( ϕ/ | ϕ k ), we may assume k ϕ k ≤ ϕ is a bounded self-adjoint map, it is straightforward to check that OD( ϕ ) is aself-adjoint Banach space which contains the multiplicative domain of ϕ . It remains toshow that OD( ϕ ) is closed under multiplication. Let a , b be contractions in OD( ϕ ), c acontraction in A , and ε ∈ (0 , k ∈ N we have (1 − t /k ) t < ε / t ∈ [0 , λ ∈ Λ with k h λ / aa ∗ h λ / − aa ∗ k < ε / k (1 − ϕ ( h λ ) /k ) ϕ ( ab ) ϕ ( ab ) ∗ (1 − ϕ ( h λ ) /k ) k ≤ k (1 − ϕ ( h λ ) /k ) ϕ ( aa ∗ )(1 − ϕ ( h λ ) /k ) k < k (1 − ϕ ( h λ ) /k ) ϕ ( h λ ) k + ε / < ε / . Since ϕ ( h λ ), λ ∈ Λ almost commutes with ϕ ( a ), it follows thatlim λ k ϕ ( h λ ) ϕ ( ab ) ϕ ( c ) − ϕ ( h λ ) ϕ ( abc ) k = 0 , which implies lim λ k ϕ ( h λ ) n ( ϕ ( ab ) ϕ ( c ) − ϕ ( h λ ) ϕ ( abc )) k = 0 for any n ∈ N . Then we havelim λ k ϕ ( h λ ) /k ( ϕ ( ab ) ϕ ( c ) − ϕ ( h λ ) ϕ ( abc )) k = 0. Thus, there exists λ ∈ Λ such that for any λ ≥ λ , k ϕ ( ab ) ϕ ( c ) − ϕ ( h λ ) ϕ ( abc ) k < ε. Since ε > ϕ ( ab ) ϕ ( c ) = lim λ ϕ ( h λ ) ϕ ( abc ). By OD( ϕ ) ∗ = OD( ϕ ), wealso have ϕ ( c ) ϕ ( ab ) = lim λ ϕ ( cab ) ϕ ( h λ ) for any a , b ∈ OD( ϕ ) and c ∈ A .(ii) Let k µ ∈ A , µ ∈ I be another approximate unit of A and let OD( ϕ, k ) be the subspacein Definition 2.1 determined by { k µ } µ ∈ I . Since OD( ϕ ) and OD( ϕ, k ) are C ∗ -algebras, itsuffices to show OD( ϕ ) + ⊂ OD( ϕ, k ).Let a ∈ OD( ϕ ) , and let λ ∈ Λ and µ ∈ I be such that k ϕ (( h λ − k µ ) a ) k < ε for any λ ≥ λ and µ ≥ µ . Then it follows that k ϕ ( h λ − k µ ) ϕ ( a ) k = lim ν k ϕ (( h λ − k µ ) a ) ϕ ( h ν ) k < ε.
3y Kadison’s inequality, for any b ∈ A we have k ϕ ( h λ − k µ ) ϕ ( ab ) k ≤ k ϕ ( h λ − k µ ) ϕ ( a ) ϕ ( h λ − k µ ) k < ε, for any λ ≥ λ and µ ≥ µ . Then it follows that lim µ ϕ ( k µ ) ϕ ( ab ) = lim λ ϕ ( h λ ) ϕ ( ab ) = ϕ ( a ) ϕ ( b ). Since a is self-adjoint, we also see that lim µ ϕ ( ba ) ϕ ( k µ ) = ϕ ( b ) ϕ ( a ) for any b ∈ A .To prepare for the Schwartz inequality (Proposition 2.5) and the next section, we needthe following calculation of non-invertible positive elements. This argument is a slightvariation of [25, Lemma 1.4.4]. Lemma 2.3.
Let A be a C ∗ -algebra. For two positive elements a and b in the second dual A ∗∗ with a ≤ b , there exists a unique contraction x in A ∗∗ such that b / x = a / and p ( b ) x = x , where p ( b ) is the strong limit of ( n A ∗∗ + b ) − b ∈ A ∗∗ . If furthermore [ a, b ] = 0 ,then there exists a unique contraction y in A ∗∗ such that by = a and p ( b ) y = y .We write b − / a / = x and b − a = y .Proof. For n ∈ N , we set x n = ( n A ∗∗ + b ) − / a / ∈ A ∗∗ . Then it follows that x n ∗ x n = a / ( n A ∗∗ + b ) − a / ≤ a / ( n A ∗∗ + a ) − a / ≤ A ∗∗ for any n ∈ N . Since the unit ballof A ∗∗ is compact in the σ -weak (ultraweak) topology, there exists a subnet x n λ , λ ∈ Λ of { x n } n ∈ N which converges to a contraction x ∈ A ∗∗ . Thus we have that b / x = σ -weak- lim λ b / (cid:18) n λ A ∗∗ + b (cid:19) − / a / = p ( b ) a / = a / . If x ′ ∈ A ∗∗ satisfies b / x ′ = a / and p ( b ) x ′ = x ′ , then we have x − x ′ = p ( b )( x − x ′ ) =strong- lim n →∞ ( n A ∗∗ + b ) − b ( x − x ′ ) = 0.In the case of [ a, b ] = 0, by a similar argument, we can define a positive contraction y in A ∗∗ as the strong limit of a / ( n A ∗∗ + b ) − a / , n ∈ N . This y also satisfies the desiredconditions. Corollary 2.4.
Let A and B be C ∗ -algebras. (i) Suppose that ϕ : A → B is a 2-positive map and a and b are two elements in A .Then there exists a unique element ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) ∈ B ∗∗ satisfying ϕ ( b ∗ b ) / ( ϕ ( b ∗ b ) − / ϕ ( b ∗ a )) = ϕ ( b ∗ a ) and (1 B ∗∗ − p ( ϕ ( b ∗ b ))) ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) = 0 . (ii) Suppose that ϕ : A → B is a positive map, x is a normal element in A , and y is apositive element in A satisfying xx ∗ ≤ k x k y . Then there exists a unique element ϕ ( y ) − / ϕ ( x ) ∈ B ∗∗ such that ϕ ( y ) / ( ϕ ( y ) − / ϕ ( x )) = ϕ ( x ) and (1 B ∗∗ − p ( ϕ ( y ))) ϕ − / ( y ) ϕ ( x ) = 0 . roof. In both cases we may assume that ϕ is contractive. We may further assume that a and x are contractions in A .(i) By Kadison’s inequality we have ϕ ( a ∗ b ) ∗ ϕ ( a ∗ b ) ≤ ϕ ( b ∗ b ). From Lemma 2.3, we obtainthe contraction ϕ ( b ∗ b ) − / | ϕ ( a ∗ b ) | ∈ B ∗∗ . By the polar decomposition of ϕ ( b ∗ a ) in B ∗∗ ,there exists a contraction ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) ∈ B ∗∗ satisfying the desired conditions. Theuniqueness of ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) ∈ B ∗∗ follows from these conditions automatically.(ii) Since x is normal, Kadison’s inequality implies that ϕ ( x ) ϕ ( x ) ∗ ≤ ϕ ( xx ∗ ) ≤ ϕ ( y ) , see [14, p.770]. By the same argument as in the proof of (i) we obtain a unique element ϕ ( y ) − / ϕ ( x ) ∈ B ∗∗ satisfying the desired conditions.The following Schwartz inequality was given by M. Choi in [4, Proposition 4.1] forstrictly positive maps and invertible elements. Regarding ϕ ( a ∗ b ) ϕ ( b ∗ b ) − ϕ ( b ∗ a ) as ( ϕ ( b ∗ b ) − / ϕ ( b ∗ a )) ∗ ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) obtained in Corollary 2.4, we extend his result to the case ofnon-invertible elements. Proposition 2.5.
Let A and B be C ∗ -algebras. (i) Suppose that ϕ is a 2-positive map from A to B . Then for any a , b ∈ A it followsthat ϕ ( a ∗ b ) ϕ ( b ∗ b ) − ϕ ( b ∗ a ) ≤ ϕ ( a ∗ a ) . (ii) Suppose that ϕ is a positive map from A to B . Then for a self-adjoint element x ∈ A and a positive element y ∈ A with yx = x , it follows that ϕ ( x ) ϕ ( y ) − ϕ ( x ) ≤ ϕ ( x ) . Proof. (i) Since the 2 × (cid:20) ϕ ( a ∗ a ) ϕ ( a ∗ b ) ϕ ( b ∗ a ) ϕ ( b ∗ b ) (cid:21) ∈ B ⊗ M is positive, the matrix (cid:20) B ∗∗
00 ( n B ∗∗ + ϕ ( b ∗ b )) − / (cid:21) (cid:20) ϕ ( a ∗ a ) ϕ ( a ∗ b ) ϕ ( b ∗ a ) ϕ ( b ∗ b ) (cid:21) (cid:20) B ∗∗
00 ( n B ∗∗ + ϕ ( b ∗ b )) − / (cid:21) = (cid:20) ϕ ( a ∗ a ) ϕ ( a ∗ b )( n B ∗∗ + ϕ ( b ∗ b )) − / ( n B ∗∗ + ϕ ( b ∗ b )) − / ϕ ( b ∗ a ) ( n B ∗∗ + ϕ ( b ∗ b )) − ϕ ( b ∗ b ) (cid:21) ∈ B ⊗ M is also positive for any n ∈ N . From k ( n B ∗∗ + ϕ ( b ∗ b )) − / ϕ ( b ∗ a ) k ≤ k ϕ k / k a k for any n ∈ N , we obtain an accumulation point X ∈ B ∗∗ of { ( n B ∗∗ + ϕ ( b ∗ b )) − / ϕ ( b ∗ a ) } n ∈ N in the sense of σ -weak topology. It is straightforward to see that ϕ ( b ∗ b ) / X = ϕ ( b ∗ a )and (1 B ∗∗ − p ( ϕ ( b ∗ b ))) X = 0. By Corollary 2.4, we have X = ϕ ( b ∗ b ) − / ϕ ( b ∗ a ). Then itfollows that the 2 × (cid:20) ϕ ( a ∗ a ) ( ϕ ( b ∗ b ) − / ϕ ( b ∗ a )) ∗ ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) p ( ϕ ( b ∗ b )) (cid:21) ∈ B ∗∗ ⊗ M is alsopositive. Because of0 ≤ (cid:20) B ∗∗ − ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) 1 B ∗∗ (cid:21) ∗ (cid:20) ϕ ( a ∗ a ) ( ϕ ( b ∗ b ) − / ϕ ( b ∗ a )) ∗ ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) p ( ϕ ( b ∗ b )) (cid:21) · (cid:20) B ∗∗ − ϕ ( b ∗ b ) − / ϕ ( b ∗ a ) 1 B ∗∗ (cid:21) = (cid:20) ϕ ( a ∗ a ) − ϕ ( a ∗ b ) ϕ ( b ∗ b ) − ϕ ( b ∗ a ) 00 p ( ϕ ( b ∗ b )) (cid:21) ,
5e conclude that ϕ ( a ∗ a ) ≥ ϕ ( a ∗ b ) ϕ ( b ∗ b ) − ϕ ( b ∗ a ) . (ii) When yx = x , the 2 × (cid:20) x xx y (cid:21) ∈ A ⊗ M is positive. By [4, Corollary 4.4],we can see that (cid:20) ϕ ( x ) ϕ ( x ) ϕ ( x ) ϕ ( y ) (cid:21) ∈ B ⊗ M is also positive, even for a positive map ϕ . Bythe same argument as the proof of (i), we conclude that ϕ ( x ) ≥ ϕ ( x ) ϕ ( y ) − ϕ ( x ). In the following lemma, for a unital C ∗ -algebra A we denote by H A the separable Hilbert A -module A ⊗ ℓ ( N ) and by ( · | · ) H A : H A × H A → A the inner product on H A , whichis defined by( x | y ) H A = ∞ X i =1 x ∗ i y i ∈ A, for x = ( x i ) i ∈ N and y = ( y i ) i ∈ N ∈ H A , (see [15], [20] for detail). Let B ( H A ) denote the set of adjointable operators on H A . Welet { e i } i ∈ N denote the canonical orthonormal basis of ℓ ( N ), and regard a ∈ B ( H A ) as an ∞ -matrix whose ( i, j )-entry is a i,j := (1 A ⊗ e i | a A ⊗ e j ) H A ∈ A for i , j ∈ N . Lemma 3.1.
Let A be a unital C ∗ -algebra. For ε > the following statements hold. (i) If a positive contraction a ∈ B ( H A ) satisfies k a , k < ε , then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 a i, ∗ a i, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε . (ii) If a unitary u ∈ B ( H A ) satisfies k u , ∗ u , − A k < ε , then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =2 u i, ∗ u i, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε .Proof. (i) For a positive contraction a ∈ B ( H A ), we have an element b ∈ B ( H A ) with b ∗ b = a , which implies that a , = ∞ X i =1 b i, ∗ b i, , where the right hand side is in the operatornorm topology on A . Then we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 a i, ∗ a i, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k (1 A ⊗ e | a ∗ a A ⊗ e ) H A k≤ k b k k (1 A ⊗ e | b ∗ b A ⊗ e ) H A k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 b i, ∗ b i, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε. (ii) From u ∗ u = 1 B ( H A ) , it follows that ∞ X i =1 u i, ∗ u i, = 1 A in the operator norm topology.Then we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =2 u i, ∗ u i, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k A − u , ∗ u , k < ε. roof of Theorem 1.1. When a positive contraction a ∈ A satisfieslim sup λ (cid:13)(cid:13) ϕ ( a ) − ϕ ( a ) ϕ ( h λ ) (cid:13)(cid:13) < δ, there exist positive contractions a + , a − ∈ A such that a + a − = a − , k a − a − k < ε , andlim sup λ (cid:13)(cid:13) ϕ ( a − ) − ϕ ( a − ) ϕ ( h λ ) (cid:13)(cid:13) < δ . Once we show thatsup b ∈ A (cid:18) lim sup λ k ϕ ( a − ) ϕ ( b ) − ϕ ( h λ ) ϕ ( a − b ) k (cid:19) < ε, this implies sup b ∈ A (cid:18) lim sup λ k ϕ ( a ) ϕ ( b ) − ϕ ( h λ ) ϕ ( ab ) k (cid:19) < ε . Then we may assume that h λ a = a for a large λ ∈ Λ.By the same argument, we may replace the conditionsup b ∈ A (cid:18) lim sup λ k ϕ ( a ) ϕ ( b ) − ϕ ( h λ ) ϕ ( ab ) k (cid:19) < ε, in Theorem 1.1, by lim sup λ k ϕ ( a ) ϕ ( b ) − ϕ ( h λ ) ϕ ( ab ) k < ε , for any positive contraction b ∈ A with a large λ ∈ Λ such that h λ b = b .For α ∈ (0 ,
1) and t ∈ [0 , f α ( t ) = min { max { , α − t − } , } . Since f α ∈ C ((0 , , there exists g α ∈ C ([0 , + such that g α · id [0 , = f α . Here id [0 , ∈ C ([0 , means the continuous function defined by id [0 , ( t ) = t for t ∈ [0 , ε ∈ (0 ,
1) we let α ∈ (0 ,
1) be such that (cid:13)(cid:13) id [0 , · (1 − f α ) (cid:13)(cid:13) < ε /
16. Set ε =( ε/ (8 k g α k )) >
0. Let α ∈ (0 , /
4) be such that (cid:13)(cid:13) id [0 , · (1 − f α ) (cid:13)(cid:13) < ε /
4, and let δ > δ < ε / (4 k g α k ). By approximating (id [0 , ) / with polynomials, welet δ ∈ (0 , δ ) be such that for any positive contractions x , y in a C ∗ -algebra, the condition k [ x, y ] k < δ implies k [ x / , y ] k < δ .Let A , B , h λ ∈ A , and ϕ be as in the theorem. Suppose that a positive contraction a ∈ A satisfies lim sup λ k ϕ ( a ) − ϕ ( a ) ϕ ( h λ ) k < δ and h λ a = a for a large λ ∈ Λ. Fromlim sup λ k [ ϕ ( a ) , ϕ ( h λ )] k < δ , it follows that lim sup λ k [ ϕ ( a ) , ϕ ( h λ )] k ≤ δ . Let b be apositive contraction in A with h λ b = b for a large λ ∈ Λ. Set self-adjoint elements x = (cid:20) aa b (cid:21) , y = (cid:20) h λ h λ (cid:21) ∈ A ⊗ M . By (ii) of Proposition 2.5, we have ϕ ⊗ id M ( x ) ϕ ⊗ id M ( y λ ) − ϕ ⊗ id M ( x ) ≤ ϕ ⊗ id M ( x ) . This inequality implies that (cid:20) ϕ ( a ) ϕ ( ab ) ϕ ( ba ) ϕ ( a + b ) (cid:21) ≥ (cid:20) ϕ ( h λ ) 00 ϕ ( h λ ) (cid:21) − / (cid:20) ϕ ( a ) ϕ ( a ) ϕ ( b ) (cid:21)! ∗ (cid:20) ϕ ( h λ ) 00 ϕ ( h λ ) (cid:21) − / (cid:20) ϕ ( a ) ϕ ( a ) ϕ ( b ) (cid:21) ≥ (cid:20) ϕ ( a ) ϕ ( a ) ϕ ( b ) (cid:21) (cid:20) g α ( ϕ ( h λ )) 00 g α ( ϕ ( h λ )) (cid:21) (cid:20) ϕ ( a ) ϕ ( a ) ϕ ( b ) (cid:21) = (cid:20) ϕ ( a ) g α ( ϕ ( h λ )) ϕ ( a ) ϕ ( a ) g α ( ϕ ( h λ )) ϕ ( b ) ϕ ( b ) g α ( ϕ ( h λ )) ϕ ( a ) ϕ ( a ) g α ( ϕ ( h λ )) ϕ ( a ) + ϕ ( b ) g α ( ϕ ( h λ )) ϕ ( b ) (cid:21) . y = ϕ ( a + b ) − ϕ ( a ) g α ( ϕ ( h λ )) ϕ ( a ) + ϕ ( b ) g α ( ϕ ( h λ )) ϕ ( b ). Then the following matrix X ∈ B ⊗ M is a positive element, X = (cid:20) ϕ ( h λ )( ϕ ( a ) − ϕ ( a ) g α ( ϕ ( h λ )) ϕ ( a )) ϕ ( h λ ) ϕ ( h λ )( ϕ ( ab ) − ϕ ( a ) g α ( ϕ ( h λ )) ϕ ( b )) ϕ ( h λ ) ϕ ( h λ )( ϕ ( ba ) − ϕ ( b ) g α ( ϕ ( h λ )) ϕ ( a )) ϕ ( h λ ) ϕ ( h λ ) · y · ϕ ( h λ ) (cid:21) . By the choice of g α , f α , δ > δ , and α ∈ (0 , /
4) we have thatlim sup λ (cid:13)(cid:13) ϕ ( h λ )( ϕ ( a ) − ϕ ( a ) g α ( ϕ ( h λ )) ϕ ( a )) ϕ ( h λ ) (cid:13)(cid:13) ≤ lim sup λ (cid:13)(cid:13) ϕ ( h λ ) ϕ ( a ) ϕ ( h λ ) − ϕ ( a ) f α ( ϕ ( h λ )) ϕ ( h λ ) ϕ ( a ) (cid:13)(cid:13) + 2 δ k g α k≤ lim sup λ (cid:13)(cid:13) ϕ ( h λ ) ϕ ( a ) ϕ ( h λ ) − ϕ ( a ) ϕ ( h λ ) ϕ ( a ) (cid:13)(cid:13) + ε ε ≤ lim sup λ (cid:13)(cid:13) ϕ ( a ) ϕ ( h λ ) − ϕ ( a ) (cid:13)(cid:13) + δ + ε ε < δ + ε ε < ε . Applying (i) of Lemma 3.1 to X/ k X k ∈ B ∼ ⊗ M , we have thatlim sup λ k ϕ ( h λ )( ϕ ( ba ) − ϕ ( b ) g α ( ϕ ( h λ )) ϕ ( a )) ϕ ( h λ ) k < k X k ε < ε . Then it follows thatlim sup λ k ϕ ( h λ )( ϕ ( ba ) ϕ ( h λ ) − ϕ ( b ) ϕ ( a )) k < √ ε (cid:18) √ (cid:19) + ε , lim sup λ k f α ( ϕ ( h λ ))( ϕ ( ba ) ϕ ( h λ ) − ϕ ( b ) ϕ ( a )) k < k g α k (cid:18) √ ε (cid:18) √ (cid:19) + ε (cid:19) < ε . By Kadison’s inequality and b ≤ h λ for a large λ ∈ Λ, we see that k f α ( ϕ ( h λ )) ϕ ( ba ) − ϕ ( ba ) k ≤ k (1 − f α ( ϕ ( h λ )) ϕ ( h λ ) k < ε , k f α ( ϕ ( h λ )) ϕ ( b ) ϕ ( a ) − ϕ ( b ) ϕ ( a ) k ≤ k (1 − f α ( ϕ ( h λ )) ϕ ( h λ ) k < ε . Therefore we conclude thatlim sup λ k ϕ ( ba ) ϕ ( h λ ) − ϕ ( b ) ϕ ( a ) k < ε. Theorem 1.1 can be used to give an alternative proof of the structure theorem forcompletely positive order zero maps [32], [33], [10]. Our approach is effective even for2-positive maps.
Corollary 3.2.
Let A , B be two C ∗ -algebras, and h λ ∈ A , λ ∈ Λ be the canonicalapproximate unit of A . Suppose that ϕ is a 2-positive map from A to B such that ϕ ( a ) = lim λ ϕ ( a ) ϕ ( h λ ) , or any positive element a ∈ A . Then there exist a ∗ -homomorphism π from A to B ∗∗ anda positive element h in the multiplier algebra M (C ∗ ( ϕ ( A ))) of C ∗ ( ϕ ( A )) such that π ( a ) ∈ M (C ∗ ( ϕ ( A ))) ∩ { h } ′ and ϕ ( a ) = hπ ( a ) , for any a ∈ A . In particular, ϕ is completely positive.Proof. We may assume that ϕ is contractive.We let h be the strong limit of ϕ ( h λ ), λ ∈ Λ in B ∗∗ . Since hϕ ( a ) = ϕ ( a / ) = ϕ ( a ) h for any a ∈ A , it follows that h ∈ M (C ∗ ( ϕ ( A ))) ∩ (C ∗ ( ϕ ( A ))) ′ . By Lemma 2.3 and by h ≥ ϕ ( a ) for any a ∈ A , we can define a positive element π ( a ) = h − ϕ ( a ) ∈ B ∗∗ for any a ∈ A . Set f n ( h ) = (cid:0) n B ∗∗ + h (cid:1) − ∈ M (C ∗ ( ϕ ( A ))) ⊂ B ∗∗ for n ∈ N . Note that for a , b ∈ A and m , n ∈ N k ϕ ( a ) ( f n ( h ) − f m ( h )) ϕ ( b ) k ≤ (cid:13)(cid:13)(cid:13) h ( f n ( h ) − f m ( h )) (cid:13)(cid:13)(cid:13) , then, by Dini’s theorem, ϕ ( a ) f n ( h ) ϕ ( b ) ∈ C ∗ ( ϕ ( A )) converges to ϕ ( a ) h − ϕ ( b ) in theoperator norm topology. Thus we have h − ϕ ( a ) ∈ M (C ∗ ( ϕ ( A ))) for any a ∈ A .By the uniqueness of h − ϕ (cid:16) a + b k a k + k b k (cid:17) for a , b ∈ A + in Lemma 2.3, it follows that π (cid:16) a + b k a k + k b k (cid:17) = π (cid:16) a k a k + k b k (cid:17) + π (cid:16) b k a k + k b k (cid:17) . Considering the linear span of A , we obtaina self-adjoint linear map π : A → M (C ∗ ( ϕ ( A ))). Applying Theorem 1.1 to ϕ/ k ϕ k , for a , b ∈ A + we have ϕ ( a ) ϕ ( b ) = hϕ ( ab ), which implies that π ( a ) π ( b ) = π ( ab ). Corollary 3.3.
Every 2-positive order zero map is completely positive.More generally, a 2-positive map is completely positive if its restriction to any commutative C ∗ -subalgebra is order zero.Proof. Let ϕ : A → B be a 2-positive map between two C ∗ -algebras, h λ ∈ A , λ ∈ Λ thecanonical approximate unit of A , and a a positive contraction in A . In order to show that ϕ ( a ) = lim λ ϕ ( a ) ϕ ( h λ ), we may assume that ah λ = a for a large λ ∈ Λ. Let C be thecommutative C ∗ -subalgebra of A generated by a and h λ . By the assumption, ϕ | C is anorder zero completely positive map. Then it follows that ϕ ( a ) = ϕ ( a ) ϕ ( h λ ) for λ ≥ λ (see the proof of [32, Lemma 3.1] for a direct argument). By Corollary 3.2, we concludethat ϕ is completely positive on A .Combining the proof above with Corollary 3.2, we see the following structure theorem. Corollary 3.4.
Let A and B be two C ∗ -algebras. For a 2-positive order zero map ϕ : A → B , there exist a representation π of A on B ∗∗ , and a positive contraction h ∈ B ∗∗ satisfying the same condition in Corollary 3.2. The next result is motivated by the question in [13, Section 5] for general C ∗ -algebras. Corollary 3.5.
Let A and B be C ∗ -algebras, and let h λ ∈ A , λ ∈ Λ be the canonicalapproximate unit of A . For a 2-positive linear map ϕ from A to B , the following holds. OD( ϕ ) = span { a ∈ A : ϕ ( a ) = lim λ ϕ ( a ) ϕ ( h λ ) } . Proof.
From Theorem 1.1, the right hand side is contained in OD( ϕ ). Since the orthogo-nality domain OD( ϕ ) is a C ∗ -algebra, it can be decomposed into the span of OD( ϕ ) . Bythe definition of OD( ϕ ), we see that a ∈ OD( ϕ ) implies ϕ ( a ) = lim λ ϕ ( a ) ϕ ( h λ ).9 Examples of k -positive order ε maps In the previous section we have seen that the class of order zero maps is explicitly dividedinto the two cases, positive but not completely positive and completely positive (Corollary3.3). A well-known example of positive order zero map, but not 2-positive, is the transpo-sition on a matrix algebra. This section studies the possibility of constructing k -positivemaps of almost order zero but not k + 1-positive.From now on we denote by { e ( n ) i,j } ni,j =1 the canonical matrix units of M n and tr n thenormalized trace on M n . The following construction of k -positive almost order zero mapsrelies on Tomiyama’s work in [29]. Example 4.1.
Fix a natural number k and ε >
0. Let n be a natural number such that k < n . For λ ∈ (0 , ∞ ), we let ψ λ be the linear map from M n to M n defined by ψ λ ( a ) = λ tr n ( a )1 M n + (1 − λ ) a for a ∈ M n . Because of [29, Theorem2], we can see that ψ λ is k -positive if and only if λ ≤ nk − .We let λ ∈ (0 , ∞ ) be such that n ( k +1) − < λ − ≤ nk − .Let ι : M n → ( e ( m )1 , ⊗ M n ) M m ⊗ M n ( e ( m )1 , ⊗ M n ) be the canonical isomorphism. Wedefine a linear map ϕ ( m ) λ from M m ⊗ M n to M m ⊗ M n by ϕ ( m ) λ ( x ) = (1 − ε ) x + ε M m ⊗ ψ λ ◦ ι − (( e ( m )1 , ⊗ M n ) x ( e ( m )1 , ⊗ M n )) , for x ∈ M m ⊗ M n . Then for any m ∈ N , this map ϕ ( m ) λ is unital and k -positive, satisfying (cid:13)(cid:13)(cid:13) ϕ ( m ) λ ( x ) − ϕ ( m ) λ ( x ) (cid:13)(cid:13)(cid:13) < ε, for any contraction x in M m ⊗ M n . By Theorem 1.1 we can regard ϕ ( m ) λ as an almostorder zero map.For a large m ∈ N , we have that ϕ ( m ) λ is not ( k + 1)-positive. Actually, setting theunital completely positive map Φ n : M m ⊗ M n → M n by Φ n ( a ⊗ b ) = tr m ( a ) b , and e λ = mελ (1 − ε )+ mε >
0, we see thatΦ n ◦ ϕ ( m ) λ ( ι ( a )) = 1 − εm a + ε ( λ tr n ( a )1 M n + (1 − λ ) a )= ελ e λ ( e λ tr n ( a )1 M n + (1 − e λ ) a ) , for a ∈ M n . Since lim m →∞ e λ = λ ∈ (cid:18) n ( k + 1) − , nk − (cid:21) , it follows that e λ > n ( k +1) − fora large m ∈ N . Thus Φ n ◦ ϕ ( m ) λ | ι ( M n ) is not ( k + 1)-positive, so ϕ ( m ) λ is not.In contrast to the above example, by fixing the size of the matrix algebras, the fol-lowing proposition shows how close unital 2-positive almost order zero maps are to beingcompletely positive. 10 roposition 4.2. For ε > , we let δ > be as in Theorem 1.1. Let ϕ is a unital -positive map from M n to a unital C ∗ -algebra B. Suppose that k ϕ ( a ) − ϕ ( a ) k < δ forany positive contraction a ∈ M n . Then the linear map M n ∋ a ϕ ( a ) + nε Tr n ( a )1 B iscompletely positive, where Tr n denotes the non-normalized trace on M n .Proof. We set b = n X i =1 e ( n )1 ,i ⊗ e ( n )1 ,i ∈ M n ⊗ M n and b = b ∗ b ∈ M n ⊗ M n . It is enough toshow that the Choi matrix ( ϕ + εn Tr n ) ⊗ id M n ( b ) is a positive element in B ⊗ M n , (see[2, Proposition 1.5.12] for example). Since (cid:13)(cid:13)(cid:13) ϕ ( e ( n ) i, ) ϕ ( e ( n )1 ,j ) − ϕ ( e ( n ) i,j ) (cid:13)(cid:13)(cid:13) < ε , it follows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ϕ ⊗ id M n ( b ) ∗ ϕ ⊗ id M n ( b ) − n X i,j =1 ϕ ( e ( n ) i,j ) ⊗ e ( n ) i,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < nε. Thus we have that( ϕ + nε Tr n ) ⊗ id M n ( b ) = n X i,j =1 ϕ ( e ( n ) i,j ) ⊗ e ( n ) i,j + nε n X i =1 B ⊗ e ( n ) i,i ≥ . In the rest of this paper, we focus on nuclear C ∗ -algebras and aim to show the second mainresult Theorem 1.2. The following weaker characterization of nuclearity has implicitlyappeared in Ozawa’s survey [24], which was obtained in the context of [16] and [17]. Letus revisit this argument for our self-contained proof.For a C ∗ -algebra B and a net A λ , λ ∈ Λ of C ∗ -subalgebras of B , we denote by Q λ A λ the ℓ ∞ -direct sum of { A λ } λ ∈ Λ (i.e., the set of nets ( a λ ) λ ∈ Λ such that a λ ∈ A λ andsup λ k a λ k < ∞ ), and L λ A λ the c -direct sum (i.e., the set of ( a λ ) λ ∈ Q λ A λ such thatlim λ k a λ k = 0). It is well-known that Q λ A λ is a C ∗ -algebra and L λ A λ is an ideal of Q λ A λ . When A λ = A for any λ ∈ Λ we let ℓ ∞ (Λ , A ) = Y λ A λ and c (Λ , A ) = M λ A λ . We identify a C ∗ -algebra A with the C ∗ -subalgebra of ℓ ∞ (Λ ,A ) c (Λ ,A ) consisting of equivalenceclasses of constant nets. Theorem 5.1. A C ∗ -algebra A is nuclear if and only if there exists a net ϕ λ : M N λ → A , λ ∈ Λ of completely positive contractions such that the canonical completely positivecontraction Φ = ( ϕ λ ) λ : Q λ M N λ L λ M N λ −→ ℓ ∞ (Λ , A ) c (Λ , A ) satisfies Φ (cid:18) Q λ M N λ L λ M N λ (cid:19) ! ⊃ A . ∗ -algebra A , we defineΛ A = { F ⊂ A : a finite subset of unitaries in A } × { ε ∈ R : ε > } , and regard Λ A as the (upward-filtering) ordered set by the inclusion order on 2 A and thestandard order on R . For a C ∗ -algebra A , we let dist( x, F ) denote inf y ∈ F k x − y k for x ∈ A and F ⊂ A . Lemma 5.2.
Let A be a unital C ∗ -algebra and M a unital C ∗ -algebra which is closedunder the polar decomposition by unitaries, i.e., for any x ∈ M there exists a unitary u ∈ M such that x = u | x | . Suppose that for λ = ( F, ε ) ∈ Λ A , a 2-positive contraction ϕ : M → A satisfies dist( x, ϕ ( M )) < ε for all x ∈ F . Then there exist unitaries U x ∈ M , x ∈ F such that k ϕ ( U x ) − x k < √ ε for all x ∈ F. Proof.
Let y x ∈ M be such that k ϕ ( y x ) − x k < ε for x ∈ F . For x ∈ F , by the polardecomposition of y x , there exists a unitary U x ∈ M such that y x = U x | y x | . Since x ∈ F is a unitary, it follows that k ϕ ( y x ) ∗ ϕ ( y x ) − A k < ε . Then Kadison’s inequality impliesthat (1 − ε )1 A ≤ ϕ ( y x ) ∗ ϕ ( y x ) ≤ ϕ ( y x ∗ y x ) ≤ ϕ (1 M ) ≤ A . By ϕ (1 − | y x | ) ≤ ϕ (1 − y x ∗ y x ) ≤ ε A , we have that k ϕ ( U x ) − x k < k ϕ ( U x − y x ) k + ε ≤ (cid:13)(cid:13) ϕ ((1 − | y x | ) ) (cid:13)(cid:13) / + ε ≤ √ ε + ε < √ ε. Lemma 5.3 (Lemma 3.6 of [16], see also Lemma 4.1.4 of [10]) . For N ∈ N and ( F, ε ) ∈ Λ M N , there exist unitaries v i ∈ M N , i = 1 , , ..., M and permuta-tions σ x , x ∈ F of { , , ..., M } such that max i =1 , ,...,M (cid:13)(cid:13) v i · x − v σ x ( i ) (cid:13)(cid:13) < ε for all x ∈ F. Proof of Theorem 5.1.
It is shown in [18, Theorem], [5, Theorem 3.1] that the nuclearityof A implies the completely positive approximation property (CPAP) which is strongerthan the condition in Theorem 5.1. Then it is enough to show the converse direction.It is well-known that A is nuclear if and only if the unitization A ∼ of A is nuclear.Thus we may assume that A is unital. Actually, for e λ = ( F ∼ , ε ) ∈ Λ A ∼ , taking anapproximate unit of A we have a positive contraction e ∈ A and λ x ∈ C for x ∈ F ∼ suchthat (1 A ∼ − e ) x ≈ ε λ x (1 A ∼ − e ) and [ x, e ] ≈ ε x ∈ F ∼ . Let e e ∈ A be such that e / e e / ≈ ε e / . By the assumption of A , we now obtain a completely positive contraction ϕ : M N → A such that dist( y, ϕ ( M N )) < ε for all y ∈ { e e } ∪ { e e / x e e / : x ∈ F ∼ } ⊂ A .Then we have e / ϕ (1 M N ) e / ≈ ε e . Define a completely positive map e ϕ : M N ⊕ C → A by e ϕ ( x ⊕ c ) = e / ϕ ( x ) e / + c (1 A ∼ − e ) for x ∈ M N and c ∈ C . Since e ϕ (1 M N ⊕ ≈ ε A ∼ ,12he canonical extension ϕ e λ : M N +1 → A of ε e ϕ is a completely positive contraction,which satisfies the condition in Theorem 5.1 for A ∼ .Let λ = ( F, ε ) ∈ Λ A be such that ε <
1. By the assumption, we now obtain acompletely positive contraction ϕ : M N → A such that dist( x, ϕ ( M N )) < ( ε/ for all x ∈ F . By Lemma 5.2, there are unitaries U x ∈ M N , x ∈ F such that k ϕ ( U x ) − x k < ε / x ∈ F . By Lemma 5.3, for ( { U x } x ∈ F , ε/ ∈ Λ M N , there exist unitaries v i ∈ M N , i = 1 , , ..., M and permutations σ x , x ∈ F of { , , ..., M } such that (cid:13)(cid:13) v i · U x − v σ x ( i ) (cid:13)(cid:13) < ε/ i = 1 , , ..., M, and x ∈ F. Due to the Kasparov-Stinespring dilation theorem [15], (see also [20, Theorem 6.5]), thereexists a ∗ -homomorphism π : M N → B ( H A ) such that ϕ ( a ) = π ( a ) , ∈ A , where thenotations of H A and a i,j ∈ A for a ∈ B ( H A ) are same as in Lemma 3.1. We set a ( i ) j = π ( v i ) j, ∈ A for i = 1 , , ..., M and j ∈ N .From (ii) of Lemma 3.1 and (cid:13)(cid:13) π ( U x ) ∗ , π ( U x ) , − A (cid:13)(cid:13) = k ϕ ( U x ) ∗ ϕ ( U x ) − A k < ε / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =2 π ( U x ) ∗ j, π ( U x ) j, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε / x ∈ F. Combining this with (cid:13)(cid:13) π ( v i ) · π ( U x ) − π ( v σ x ( i ) ) (cid:13)(cid:13) < ε/
2, we have that for x ∈ F (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =1 (cid:12)(cid:12)(cid:12) a ( i ) j x − a ( σ x ( i )) j (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / = (cid:13)(cid:13)(cid:13) ( a ( i ) j x ) j − ( a ( σ x ( i )) j ) j (cid:13)(cid:13)(cid:13) H A < ε. Since v i , i = 1 , , ..., M are unitaries, we obtain L ∈ N such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L X j =1 a ( i ) j ∗ a ( i ) j − A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε. Let A ∗∗ be the second dual of A faithfully represented on a Hilbert space H i.e., A ⊂ A ∗∗ ⊂ B ( H ). For λ ∈ Λ A , we define a completely positive map Φ λ : B ( H ) → B ( H )by Φ λ ( y ) = 1 M M X i =1 L X j =1 a ( i ) j ∗ y a ( i ) j for y ∈ B ( H ) . Thus we have that for x ∈ F and y ∈ B ( H ) Φ λ ( y ) x ≈ ε M M X i =1 L X j =1 a ( i ) j ∗ y a ( σ x ( i )) j = 1 M M X i =1 L X j =1 a ( σ x − ( i )) j ∗ y a ( i ) j ≈ ε M M X i =1 L X j =1 ( a ( i ) j x ∗ ) ∗ y a ( i ) j = x Φ λ ( y ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L X j =1 a ( i ) j ∗ a ( i ) j − A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε , for y ∈ B ( H ) ∩ A ′ it follows that Φ λ ( y ) ≈ ε y . So, Φ λ isclose to a conditional expectation onto A ′ . Let ω be a (cofinal) ultrafilter on the orderedset Λ A . Then one can define a bounded map Φ : B ( H ) → B ( H ) by the weak ∗ limitΦ( y ) = weak ∗ - lim λ → ω Φ λ ( y ) in B ( H ). By the above conditions of Φ λ , it is straightforwardto check that Φ is a conditional expectation on B ( H ) ∩ A ′ . Hence A ′ is an injective vonNeumann algebra, and so is A ′′ = A ∗∗ . Because of [7], we can see that A ∗∗ is AFD whichimplies the CPAP of A . Remark . In [28] R. Smith showed that the complete positivity of contractive mapsin the CPAP can be replaced by the complete contractivity. However, we cannot expectto replace completely positive contractions ϕ λ in Theorem 5.1 by completely contractivemaps. In fact, there are many non-nuclear C ∗ -algebras with the completely contractiveapproximation property (CCAP), although any C ∗ -algebra A with the CCAP satisfies thefollowing condition : there exists a net of complete contractions ϕ λ : M N λ → A , λ ∈ Λsuch that for a ∈ A there are x a,λ ∈ M N λ , λ ∈ Λ satisfying lim λ ϕ λ ( x a,λ ) = a . Before proving Theorem 1.2, let us recall the definition of decomposition rank.
Definition 6.1 (E. Kirchberg - W. Winter, [19]) . For d ∈ N ∪ { } , a C ∗ -algebra A is saidto have decomposition rank at most d , if for a finite subset F of contractions in A and ε >
0, there exist finite dimensional C ∗ -algebras F i , i = 0 , , ..., d , a completely positivecontraction ψ : A → L di =0 F i , and completely positive order zero contractions ϕ i : F i → A , i = 0 , , ..., d such that P di =0 ϕ i : L di =0 F i → A is contractive and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d X i =0 ϕ i ! ◦ ψ ( x ) − x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε, for all x ∈ F. Theorem 6.2 ( Theorem 1.2 ) . Let A be a unital separable C ∗ -algebra and d ∈ N ∪ { } .Then the following conditions are equivalent. (i) The decomposition rank of A is at most d . (ii) For λ = ( F, ε ) ∈ Λ A , there are finite dimensional C ∗ -algebras F i , i = 0 , , ..., d ,a 2-positive contraction ψ : A → L di =0 F i , and 2-positive order zero contractions ϕ i : F i → A , i = 0 , , ..., d such that P di =0 ϕ i : L di =0 F i → A is contractive and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d X i =0 ϕ i ! ◦ ψ ( x ) − x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε, for all x ∈ F. (iii) There exist finite dimensional C ∗ -algebras F i,λ , i = 0 , , ...d , λ ∈ Λ and nets ϕ i,λ : F i,λ → A , i = 0 , , ..., d , λ ∈ Λ of 2-positive order zero contractions such that X i =0 ϕ i,λ : F λ → A is contractive for any λ ∈ Λ , where F λ = d M i =0 F i,λ , and thecanonical contraction Φ = d X i =0 ϕ i,λ ! λ : Q λ F λ L λ F λ −→ ℓ ∞ (Λ , A ) c (Λ , A ) satisfies Φ (cid:18) Q λ F λ L λ F λ (cid:19) ! ⊃ A . Proof.
The implications (i) = ⇒ (ii) = ⇒ (iii) are trivial. We shall show (iii) = ⇒ (i). ByCorollary 3.3, we see that d X i =0 ϕ i,λ , λ ∈ Λ are completely positive contractions. Taking aconditional expectation from a matrix algebra onto F λ , by Theorem 5.1 we know that A is nuclear.From the assumption of (iii), for µ = ( F, ε ) ∈ Λ A we obtain finite dimensional C ∗ -al-gebras F i,µ , i = 0 , , ..., d , and completely positive order zero contractions ϕ i,µ : F i,µ → A , i = 0 , , ..., d such thatdist x, d X i =0 ϕ i,µ d M i =0 F i,µ ! < ε, for all x ∈ F. Set F µ = d M i =0 F i,µ and ϕ µ = d X i =0 ϕ i,µ : F µ → A for µ ∈ Λ A . By Lemma 5.2 and k ϕ µ k ≤ U x,µ ∈ F µ , x ∈ F , µ = ( F, ε ) ∈ Λ A , such that k ϕ µ ( U x,µ ) − x k < √ ε for all x ∈ F . For any unitary x ∈ A , we set U x,µ = 1 F µ if x F and µ = ( F, ε ), and set U x = ( U x,µ ) µ ∈ Q µ F µ . We let Q : Q µ F µ → Q µ F µ L µ F µ be the quotient map, U x = Q ( U x ),and let C be the C ∗ -subalgebra of Q µ F µ L µ F µ generated by (cid:8) U x : x is a unitary in A (cid:9) .Let ϕ : Q µ F µ L µ F µ → ℓ ∞ (Λ A ,A ) c (Λ A ,A ) be the completely positive contraction defined by ϕ ◦ Q (( x µ ) µ ) = ( ϕ µ ( x µ )) µ in ℓ ∞ (Λ A , A ) /c (Λ A , A ). By regarding A as the C ∗ -subalgebra of ℓ ∞ (Λ A , A ) /c (Λ A , A ), it follows that ϕ (cid:0) U x (cid:1) = x for any unitary x ∈ A , then ϕ ( C ) = A .Because of ϕ (cid:0) U x (cid:1) ∗ ϕ (cid:0) U x (cid:1) = 1 A = ϕ (cid:16) U x ∗ U x (cid:17) and ϕ (cid:0) U x (cid:1) ϕ (cid:0) U x (cid:1) ∗ = 1 A = ϕ (cid:16) U x U x ∗ (cid:17) , we see that ϕ | C : C → A is a unital ∗ -homomorphism. Let e ϕ be the ∗ -isomorphism from C / ker( ϕ | C ) onto A and e ψ = e ϕ − .Applying the Choi-Effros lifting theorem [6] to e ψ , we obtain a unital completely positivemap ψ : A → Q µ F µ such that ϕ ◦ Q ◦ ψ ( a ) = a for any a ∈ A . Note that A is required tobe nuclear and separable in order to apply [6, Theorem 3.10]. Taking unital completelypositive maps ψ µ : A → F µ , µ ∈ Λ A with ( ψ µ ( a )) µ = ψ ( a ) for a ∈ A , we conclude that ψ µ and ϕ i,µ satisfy the conditions in (i). Acknowledgements.
The author would like to thank Professor Marius Dadarlat forhelpful comments on this research, and Professor Narutaka Ozawa for showing him thevaluable survey [24]. He also expresses his gratitude to Professor Huaxin Lin and the15rganizers of Special Week on Operator Algebras 2019 for their kind hospitality duringthe author’s stay in East China Normal University. This work was supported in part bythe Grant-in-Aid for Young Scientists (B) 15K17553, JSPS.
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