J. Math. Anal. Appl. 384 (2011) 375–386

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Journal of Mathematical Analysis andApplications

www.elsevier.com/locate/jmaa

On maximal tensor products and quotient maps of operator systems

Kyung Hoon Han

Department of Mathematical Sciences, Seoul National University, San 56-1 ShinRimDong, KwanAk-Gu, Seoul 151-747, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 March 2011Available online 11 June 2011Submitted by D. Blecher

Keywords:Operator systemTensor productQuotient mapUnitization

We introduce quotient maps in the category of operator systems and show that themaximal tensor product is projective with respect to them. Whereas, the maximal tensorproduct is not injective, which makes the (el,max)-nuclearity distinguish a class in thecategory of operator systems. We generalize Lance’s characterization of C∗-algebras withthe WEP by showing that (el,max)-nuclearity is equivalent to the weak expectationproperty. Applying Werner’s unitization to the dual spaces of operator systems, we considera class of completely positive maps associated with the maximal tensor product andestablish the duality between quotient maps and complete order embeddings.

© 2011 Elsevier Inc. All rights reserved.

1. Introduction

Kadison characterized the unital subspaces of a real continuous function algebra on a compact set [6]. As for its non-commutative counterpart, Choi and Effros gave an abstract characterization of the unital involutive subspaces of B(H) [2].The former is called a real function system or a real ordered vector space with an Archimedean order unit while the latteris termed an operator system. Ever since the work by Choi and Effros, the notion of operator systems has been a useful toolin studying the local structures and the functorial aspects of C∗-algebras.

Recently, the fundamental and systematic developments in the theory of operator systems have been carried out througha series of papers [14,13,8,9]. Perhaps, Ref. [14] is the cornerstone for this program. Under the naive definition of thequotient and that of the tensor product of real function systems, the order unit sometimes fails to be Archimedean. See forexample, [1, p. 67] and [13, Remark 3.12]. The Archimedeanization process introduced in [14] helps remedy this problem. Inorder to apply this idea to the noncommutative situation, the Archimedeanization of a matrix ordered ∗-vector space witha matrix order unit is introduced in [13]. Based on the matricial Archimedeanization process, the tensor products and thequotients of operator systems are defined and studied in [8] and [9] respectively.

Based on these developments, a simple and generalized proof of the celebrated Choi–Effros–Kirchberg approximationtheorem for nuclear C∗-algebras has been given in [5].

In this paper, we continue to study the tensor products in [8] and the quotients in [9]. With the same spirit as in [5],we give a simple and generalized proof of the classical Lance’s theorem [11,12].

In Section 3, we introduce the notion of complete order quotient maps which can be regarded as quotient maps in thecategory of operator systems and show that the maximal tensor product is projective under this definition.

The minimal tensor product is injective functorially, while the maximal tensor product need not be injective. This mis-behavior distinguishes a class which is called (el,max)-nuclear operator systems [9]. In the category of C∗-algebras, Lancecharacterized this tensorial property by the factorization property for the inclusion map into the second dual through B(H).In [9], it is proved that this factorization property for an operator system implies its (el,max)-nuclearity and it is asked

E-mail address: kyunghoon.han@gmail.com.

0022-247X/$ – see front matter © 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2011.05.070

376 K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386

whether the converse holds. In Section 4, we answer this question in the affirmative and deduce Lance’s theorem as acorollary.

The order unit of the dual spaces of operator systems cannot be considered in general. However, the dual spaces of finitedimensional operator systems have a non-canonical Archimedean order unit [2]. This enables the duality between tensorproducts and mapping spaces to work in the proofs of the Choi–Effros–Kirchberg theorem for operator systems [5] andLance’s theorem for operator systems in Section 4. Not only is the finite dimensional assumption restrictive, but also thematrix order unit norm on the dual spaces of finite dimensional operator systems is irrelevant to the matrix norm given bythe standard dual of operator spaces.

To get rid of the finite dimensional assumption and to reflect the operator space dual norm, we apply Werner’s uniti-zation of matrix ordered operator spaces [16] to the dual spaces of operator systems. We consider the completely positivemaps associated with the maximal tensor products and prove their factorization property in Section 5. Finally, we establishthe duality between complete order quotient maps and complete order embeddings in Section 6.

2. Preliminaries

Let S and T be operator systems. As in [8], an operator system structure on S ⊗ T is defined as a family of conesMn(S ⊗τ T )+ satisfying:

(T1) (S ⊗ T , {Mn(S ⊗τ T )+}∞n=1,1S ⊗ 1T ) is an operator system denoted by S ⊗τ T ,(T2) Mn(S)+ ⊗ Mm(T )+ ⊂ Mmn(S ⊗τ T )+ for all n,m ∈ N, and(T3) if ϕ : S → Mn and ψ : T → Mm are unital completely positive maps, then ϕ ⊗ψ : S ⊗τ T → Mmn is a unital completely

positive map.

By an operator system tensor product, we mean a mapping τ : O × O → O, such that for every pair of operator systems Sand T , τ (S, T ) is an operator system structure on S ⊗ T , denoted S ⊗τ T . We call an operator system tensor product τfunctorial, if the following property is satisfied:

(T4) For any operator systems S1, S2, T1, T2 and unital completely positive maps ϕ : S1 → T1, ψ : S2 → T2, the map ϕ ⊗ψ :S1 ⊗ S2 → T1 ⊗ T2 is unital completely positive.

An operator system structure is defined on two fixed operator systems, while the functorial operator system tensor productcan be thought of as the bifunctor on the category consisting of operator systems and unital completely positive maps.

For operator systems S and T , we put

Mn(S ⊗min T )+ = {[pi, j]i, j ∈ Mn(S ⊗ T ): ∀ϕ ∈ Sk(S), ψ ∈ Sm(T ),[(ϕ ⊗ ψ)(pi, j)

]i, j ∈ M+

nkm

}and let ιS : S → B(H) and ιT : T → B(K ) be unital completely order isomorphic embeddings. Then the family {Mn(S ⊗minT )+}∞n=1 is an operator system structure on S ⊗ T rising from the embedding ιS ⊗ ιT : S ⊗ T → B(H ⊗ K ). We call theoperator system (S ⊗ T , {Mn(S ⊗min T )}∞n=1,1S ⊗ 1T ) the minimal tensor product of S and T and denote it by S ⊗min T .

The mapping min : O × O → O sending (S, T ) to S ⊗min T is an injective, associative, symmetric and functorial operatorsystem tensor product. The positive cone of the minimal tensor product is the largest among all possible positive cones ofoperator system tensor products [8, Theorem 4.6]. For C∗-algebras A and B, we have the completely order isomorphicinclusion

A ⊗min B ⊂ A ⊗C∗ min B[8, Corollary 4.10].

For operator systems S and T , we put

Dmaxn (S, T ) = {

α(P ⊗ Q )α∗: P ∈ Mk(S)+, Q ∈ Ml(T )+, α ∈ Mn,kl, k, l ∈ N}.

Then it is a matrix ordering on S ⊗ T with order unit 1S ⊗ 1T . Let {Mn(S ⊗max T )+}∞n=1 be the Archimedeanization of thematrix ordering {Dmax

n (S, T )}∞n=1. Then it can be written as

Mn(S ⊗max T )+ = {X ∈ Mn(S ⊗ T ): ∀ε > 0, X + ε In ∈ Dmax

n (S, T )}.

We call the operator system (S ⊗ T , {Mn(S ⊗max T )+}∞n=1,1S ⊗ 1T ) the maximal operator system tensor product of S andT and denote it by S ⊗max T .

The mapping max : O × O → O sending (S, T ) to S ⊗max T is an associative, symmetric and functorial operator systemtensor product. The positive cone of the maximal tensor product is the smallest among all possible positive cones of operatorsystem tensor products [8, Theorem 5.5]. For C∗-algebras A and B, we have the completely order isomorphic inclusion

A ⊗max B ⊂ A ⊗C∗ max B[8, Theorem 5.12].

K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386 377

The inclusion S ⊗ T ⊂ I(S) ⊗max T for the injective envelope I(S) of S induces the operator system structure onS ⊗ T , which is denoted by S ⊗el T . Here the injective envelope I(S) can be replaced by any injective operator systemcontaining S . The mapping el : O × O → O sending (S, T ) to S ⊗el T is a left injective functorial operator system tensorproduct [8, Theorems 7.3, 7.5].

An operator system S is called (el,max)-nuclear if S ⊗el T = S ⊗max T for any operator system T . An operator systemS is (el,max)-nuclear if and only if S ⊗max T ⊂ S2 ⊗max T for any inclusion S ⊂ S2 and any operator system T [9,Lemma 6.1]. We say that the operator system S has the weak expectation property (in short, WEP) if the inclusion mapι : S ↪→ S ∗∗ can be factorized as Ψ ◦ Φ = ι for unital completely positive maps Φ : S → B(H) and Ψ : B(H) → S ∗∗ . If S hasthe WEP, then it is (el,max)-nuclear [9, Theorem 6.7].

Given an operator system S , we call J ⊂ S a kernel, provided that it is the kernel of a unital completely positive mapfrom S to another operator system. The kernel can be characterized in an intrinsic way: J is a kernel if and only if itis the intersection of a closed two-sided ideal in C∗

u(S) with S [9, Corollary 3.8]. If we define a family of positive conesMn(S/J )+ on Mn(S/J ) by

Mn(S/J )+ = {[xi, j + J ]i, j: ∀ε > 0, ∃ki, j ∈ J , ε In ⊗ 1S + [xi, j + ki, j]i, j ∈ Mn(S)+},

then (S/J , {Mn(S/J )+}∞n=1,1S/J ) satisfies all the conditions of an operator system [9, Proposition 3.4]. We call it thequotient operator system. With this definition, the first isomorphism theorem is proved: If ϕ : S → T is a unital completelypositive map with J ⊂ kerϕ , then the map ϕ : S/J → T given by ϕ(x + J ) = ϕ(x) is a unital completely positive map [9,Proposition 3.6].

Since the kernel J in an operator system S is a closed subspace, the operator space structure of S/J can be interpretedin two ways, one as the operator space quotient and the other as the operator space structure induced by the operatorsystem quotient. The two matrix norms can be different. For a concrete example, see [9, Example 4.4].

3. Projectivity of maximal tensor product

We show that the maximal tensor product is projective functorially in the category of operator systems. To this end, wefirst define the quotient maps in the category of operator systems.

Definition 3.1. For operator systems S and T , we let Φ : S → T be a unital completely positive surjection. We call Φ :S → T a complete order quotient map if for any Q in Mn(T )+ and ε > 0, we can take an element P in Mn(S) so that itsatisfies

P + ε In ⊗ 1S ∈ Mn(S)+ and Φn(P ) = Q .

The key point of the above definition is that the lifting P depends on the choice of ε > 0. A slight modification of [14,Theorem 2.45] implies the following proposition that justifies the above terminology.

Proposition 3.2. For operator systems S and T , we suppose that Φ : S → T is a unital completely positive surjection. Then Φ :S → T is a complete order quotient map if and only if the induced map Φ : S/kerΦ → T is a unital complete order isomorphism.

Proof. Φ : S → T is a complete order quotient map

⇔ ∀Q ∈ Mn(T )+ , ∀ε > 0, ∃P ∈ Mn(S), P + ε In ⊗ 1S ∈ Mn(S)+ and Φn(P ) = Q ,⇔ ∀Q ∈ Mn(T )+ , ∃P ∈ Mn(S), P + kerΦn ∈ Mn(S/kerΦ)+ and Φn(P + kerΦn) = Q ,⇔ the induced map Φ : S/kerΦ → T is a complete order isomorphism. �

Recall that for operator spaces V and W , the linear map Φ : V → W is called a complete quotient map if Φn : Mn(V ) →Mn(W ) is a quotient map for each n ∈ N, that is, Φn maps the open unit ball of Mn(V ) onto the open unit ball of Mn(W ).

Proposition 3.3. For operator systems S and T , we suppose that Φ : S → T is a unital completely positive surjection. If Φ : S → Tis a complete quotient map, then it is a complete order quotient map.

Proof. We choose Q ∈ Mn(T )+ and ε > 0. We then have

−1

2‖Q ‖In ⊗ 1T � Q − 1

2‖Q ‖In ⊗ 1T � 1

2‖Q ‖In ⊗ 1T .

There exists an element P in Mn(S) such that

Φn(P ) = Q − 1‖Q ‖In ⊗ 1T and ‖P‖ � 1‖Q ‖ + ε.

2 2

378 K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386

By considering (P + P∗)/2 instead, we may assume that P is self-adjoint. It follows that

P + 1

2‖Q ‖In ⊗ 1S + ε In ⊗ 1S ∈ Mn(S)+ and Φn

(P + 1

2‖Q ‖In ⊗ 1S

)= Q . �

The following theorem says that the maximal tensor product is projective functorially in the category of operator systems.

Theorem 3.4. For operator systems S1, S2, T and a complete order quotient map Φ : S1 → S2 , the linear map Φ ⊗ idT : S1 ⊗maxT → S2 ⊗max T is a complete order quotient map.

Proof. We choose an element z in Mn(S2 ⊗max T )+ and ε > 0. Then we can write

z + ε In ⊗ 1S2 ⊗ 1T = αP2 ⊗ Q α∗, P2 ∈ Mp(S2)+, Q ∈ Mq(T )+, α ∈ Mn,pq.

We may assume that ‖Q ‖,‖α‖ � 1. There exists an element P1 in M p(S1) such that

Φp(P1) = P2 and P1 + ε I p ⊗ 1S1 ∈ Mp(S1)+.

It follows that

(Φ ⊗ idT )n(αP1 ⊗ Q α∗ − ε In ⊗ 1S1 ⊗ 1T

) = αP2 ⊗ Q α∗ − ε In ⊗ 1S2 ⊗ 1T = z

and (αP1 ⊗ Q α∗ − ε In ⊗ 1S1 ⊗ 1T

) + 2ε In ⊗ 1S1 ⊗ 1T

= α(P1 + ε I p ⊗ 1S1) ⊗ Q α∗ + (ε In ⊗ 1S1 ⊗ 1T − εα

((I p ⊗ 1S1) ⊗ Q

)α∗) ∈ Mn(S1 ⊗max T )+. �

Suppose that we are given an operator system S and a unital C∗-algebra A such that S is an A-bimodule. Moreover, weassume that 1A · s = s for s ∈ S . We call such an S an operator A-system [15, Chapter 15] provided that a · 1S = 1S · a and

[ai, j] · [si, j] · [ai, j]∗ =[

n∑k,l=1

ai,k · sk,l · a∗j,l

]∈ Mn(S)+, [ai, j] ∈ Mn(A), [si, j] ∈ Mn(S)+.

The maximal tensor product A ⊗max S is an operator A-system [8, Theorem 6.7].The converse of Proposition 3.3 does not hold in general since the operator space structure induced by the operator

system quotient by a kernel can be different from the operator space quotient by it [9, Example 4.4]. However, the converseholds in the following special situation. Although the following theorem overlaps with [9, Corollary 5.15], we include it herebecause the proof is so elementary.

Theorem 3.5. Suppose that S is an operator system and A is a unital C∗-algebra with its norm closed ideal I . Then the canonical map

π ⊗ idS : A ⊗max S → A/I ⊗max S

is a complete quotient map.

Proof. By the nuclearity of matrix algebras, it is sufficient to show that the canonical map π ⊗ idS : A ⊗max S → A/I ⊗max Sis a quotient map. We suppose that∥∥(π ⊗ idS )(z)

∥∥A/I⊗max S < 1

for some z ∈ A ⊗ S . Then we have((1 − ε)1A/I ⊗ 1S (π ⊗ idS )(z)

(π ⊗ idS )(z)∗ (1 − ε)1A/I ⊗ 1S

)∈ M2(A/I ⊗max S)+

for ε = 1 − ‖(π ⊗ idS )(z)‖A/I⊗max S . By Theorem 3.4, we can find an element(ω11 ω12ω21 ω22

)in M2(I ⊗ S) such that(

1A ⊗ 1S + ω11 z + ω12z∗ + ω21 1A ⊗ 1S + ω22

)∈ M2(A ⊗max S)+.

Since we have (I ⊗ S)h = Ih ⊗ Sh [2], ω11 can be written as ω11 = ∑ni=1 ai ⊗ si for ai ∈ Ih and si ∈ Sh . Let ai = a+

i − a−i for

1 � i � n and a+,a− ∈ I + . We have

i i

K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386 379

ω11 =n∑

i=1

a+i ⊗ si + a−

i ⊗ (−si) �n∑

i=1

‖si‖(a+

i + a−i

) ⊗ 1S .

Hence, we can find elements a and d in I + such that((1A + a) ⊗ 1S z + ω12

z∗ + ω21 (1A + d) ⊗ 1S

)∈ M2(A ⊗max S)+.

Since A ⊗max S is an operator A-system [8, Theorem 6.7], we have(1A ⊗ 1S (1A + a)− 1

2 · (z + ω12) · (1A + d)− 12

(1A + d)− 12 · (z∗ + ω21) · (1A + a)− 1

2 1A ⊗ 1S

)=

((1A + a)− 1

2 00 (1A + d)− 1

2

)·(

(1A + a) ⊗ 1S z + ω12z∗ + ω21 (1A + d) ⊗ 1S

)·(

(1A + a)− 12 0

0 (1A + d)− 12

)∈ M2(A ⊗max S)+.

It follows that∥∥(1A + a)−12 · (z + ω12) · (1A + d)−

12∥∥

A⊗max S � 1

and

(π ⊗ idS )((1A + a)−

12 · (z + ω12) · (1A + d)−

12) = (π ⊗ idS )(z). �

4. The equivalence of the (el,max)-nuclearity and the WEP

As we have seen in the previous section, the maximal tensor product is projective. However, the maximal tensor productneed not be injective. This misbehavior distinguishes a class which is called (el,max)-nuclear operator systems [9]. In thecategory of C∗-algebras, Lance characterized this tensorial property by the factorization property for the inclusion map intothe second dual through B(H) [11,12]. In [9], it is proved that this factorization property for an operator system implies its(el,max)-nuclearity and it is asked whether the converse holds. In this section, we answer this question in the affirmative,independent of Lance’s original theorem.

Theorem 4.1. Let S be an operator system. The following are equivalent:

(i) we have

S ⊗max T ⊂ S2 ⊗max T

for any inclusion S ⊂ S2 and any operator system T ;(ii) we have

S ⊗max E ⊂ S2 ⊗max E

for any inclusion S ⊂ S2 and any finite dimensional operator system E;(iii) we have

S ⊗max E ⊂ B(H) ⊗max E

for any inclusion S ⊂ B(H) and any finite dimensional operator system E;(iv) there exist unital completely positive maps Φ : S → B(H) and Ψ : B(H) → S ∗∗ such that Ψ ◦ Φ = ι for the canonical inclusion

ι : S ↪→ S ∗∗ .

S ι

Φ

S ∗∗

B(H)

Ψ

Proof. Clearly, (i) implies (ii) and (ii) implies (iii). The direction (iv) ⇒ (i) follows from [9, Theorem 6.7].(iii) ⇒ (iv). Suppose that an operator system S acts on a Hilbert space H . Considering the bidual of the inclusion

S ⊂ B(H) and the universal representation of B(H), we may assume that the inclusions S ⊂ S ∗∗ ⊂ B(H) are given such

380 K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386

that the second inclusion is weak∗-WOT homeomorphic. Let {Pλ} be the family of projections on the finite dimensionalsubspaces of H directed by the inclusions of their ranges. We put

Eλ = PλS Pλ and Φλ := Pλ · Pλ : S → Eλ.

Since each Eλ is a finite dimensional operator system, there exists a non-canonical Archimedean order unit on its dualspace E∗

λ [2, Corollary 4.5]. In other words, the dual space E∗λ is an operator system. By [8, Lemma 5.7], a functional ϕλ on

S ⊗max E∗λ corresponding to the compression Φλ : S → Eλ in the standard way is positive. By assumption, S ⊗max E∗

λ is anoperator subsystem of B(H) ⊗max E∗

λ . By Krein’s theorem, ϕλ extends to a positive functional ψλ on B(H) ⊗max E∗λ .

B(H) ⊗max E∗λ

ψλ

S ⊗max E∗λ ϕλ

C

Let Ψλ : B(H) → Eλ be a completely positive map corresponding to ψλ in the standard way. Since Φλ = Ψλ ◦ ι for theinclusion ι : S ⊂ B(H), Ψλ is a unital completely positive map. We take a state ωλ on Eλ and define a unital completelypositive map θλ : Eλ → B(H) by

θλ(x) = x + ωλ(x)(I − Pλ), x ∈ Eλ.

Let Ψ : B(H) → B(H) be a point-weak∗ cluster point of {θλ ◦ Ψλ}. We may assume that θλ ◦ Ψλ converges to Ψ in thepoint-weak∗ topology. For ξ,η ∈ H and x ∈ S , we have⟨

Ψ (x)ξ,η⟩ = lim

λ

⟨(PλxPλ + ωλ(PλxPλ)(I − Pλ)

)ξ,η

⟩ = 〈xξ,η〉.It follows that Ψ |S = ι. Since Φλ is surjective and Ψλ is an extension of Φλ , for each x ∈ B(H) there exists an element xλ

in S such that Ψλ(x) = Φλ(xλ) = Pλxλ Pλ . For ξ,η ∈ H and x ∈ B(H), we have⟨Ψ (x)ξ,η

⟩ = limλ

⟨(Pλxλ Pλ + ωλ(Pλxλ Pλ)(I − Pλ)

)ξ,η

⟩ = limλ

〈xλξ,η〉.It follows that Ψ (x) belongs to the WOT-closure of S which coincides with S ∗∗ because the inclusion S ∗∗ ⊂ B(H) is weak∗-WOT homeomorphic. �

As a corollary, we deduce the following theorem of Lance. The proof is similar to that of [5, Corollary 3.3].

Corollary 4.2 (Lance). Let A be a unital C∗-algebra. Then we have A ⊗max B ⊂ A2 ⊗max B for any inclusion A ⊂ A2 and any unitalC∗-algebra B if and only if A has the weak expectation property.

Proof. By Theorem 4.1, it will be enough to prove that if A ⊗max B ⊂ B(H) ⊗max B for any unital C∗-algebra B, thenA ⊗max T ⊂ B(H)⊗max T for any operator system T . Due to [8, Theorem 6.4] and [8, Theorem 6.7], we obtain the followingcommutative diagram, which yields the conclusion:

A ⊗max T A ⊗c T A ⊗C∗ max C∗u(T )

B(H) ⊗max T B(H) ⊗c T B(H) ⊗C∗ max C∗u(T ). �

Examples of nuclear operator systems which are not unitally completely order isomorphic to any unital C∗-algebra havebeen constructed in [10,5]. These also provide examples of operator systems with the WEP which are not unitally completelyorder isomorphic to any unital C∗-algebra.

5. Completely positive maps associated with maximal tensor products

The order unit of the dual spaces of operator systems cannot be considered in general. However, the dual spaces offinite dimensional operator systems have a non-canonical Archimedean order unit. This enables the duality between tensorproducts and mapping spaces to work in the proofs of the Choi–Effros–Kirchberg theorem for operator systems [5] and theLance theorem for operator systems in the previous section. Not only is the finite dimensional assumption restrictive, butalso the matrix order unit norm on the dual spaces of finite dimensional operator systems is irrelevant to the matrix normgiven by the standard dual of operator spaces. To get rid of the finite dimensional assumption and to reflect the operatorspace dual norm, we apply Werner’s unitization of matrix ordered operator spaces [16] to the dual spaces of operatorsystems.

K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386 381

Let V be a matrix ordered operator space. We give the involution and the matrix order on V ⊕ C as follows:

(1) (x + a)∗ = x∗ + a, x ∈ V , a ∈ C,(2) X + A ∈ Mn(V ⊕ C)+ iff

A ∈ M+n and ϕ

((A + ε In)

− 12 X(A + ε In)

− 12)� −1

for any ε > 0 and any positive contractive functional ϕ on Mn(V ).

We denote by V the space V ⊕ C with the above involution and matrix order and call it the unitization of V [16, Def-inition 4.7]. The unitization V of a matrix ordered operator space V is an operator system and the canonical inclusionι : V ↪→ V is a completely contractive complete order isomorphism onto its range [16, Lemma 4.8]. However it need notbe completely isomorphic. We apply Werner’s unitization of matrix ordered operator spaces to the dual spaces of operatorsystems. In this case, the canonical inclusion ι : S ∗ ↪→ S ∗ is 2-completely isomorphic [7,4,8].

Lemma 5.1. Suppose that S is an operator system and S ∗ is the unitization of the dual space S ∗ . Let f : S → Mn be a self-adjointlinear map and A ∈ M+

n . Then f + A belongs to Mn(S ∗)+ if and only if we have fm(x) � −Im ⊗ A for all m ∈ N and x ∈ Mm(S)+1 .

Proof. ⇒) The element f + A belongs to Mn(S ∗)+ if and only if

ϕ((A + ε In)

− 12 f (A + ε In)

− 12)� −1

for any ε > 0 and any positive contractive functional ϕ on Mn(S ∗). For x ∈ Mm(S)+1 and ξ ∈ (�2mn)1, the map

ϕx,ξ : f ∈ Mn(

S ∗) = C B(S, Mn) �→ ⟨fm(x)ξ

∣∣ξ ⟩ ∈ C

is a positive contractive functional on Mn(S ∗). It follows that⟨(Im ⊗ (A + ε In)

− 12)

fm(x)(

Im ⊗ (A + ε In)− 1

2)ξ∣∣ξ ⟩

= ϕx,ξ((A + ε In)

− 12 f (A + ε In)

− 12)

� −1.

Hence, we have fm(x) � −Im ⊗ A.⇐) Put Ω = {ϕx,ξ : m ∈ N, x ∈ Mm(S)+1 , ξ ∈ (�2

mn)1} where ϕx,ξ defined as above. Let Γ1 be a weak∗-closed convexhull of Ω and Γ2 a weak∗-closed cone generated by Ω . We want Γ1 = (Mn(S ∗)∗)+1 . Here, (Mn(S ∗)∗)+1 denotes the set ofpositive contractive functionals on Mn(S ∗). If this were not the case, we could choose ϕ0 ∈ (Mn(S ∗)∗)+1 /Γ1. By the Krein–Smulian theorem [3, Theorem 5.12.1], we have R+ ·Γ1 = Γ2, thus ϕ0 does not belong to Γ2. By the Hahn–Banach separationtheorem, there exists f0 ∈ Mn(S ∗)sa which separates Γ2 and ϕ0 strictly. We have Γ2( f0) = {0} or [0,∞) or (−∞,0]. Ifϕx,ξ ( f0) = 0 for all ϕx,ξ ∈ Ω , then we have f0 = 0. We may assume that Γ2( f0) = [0,∞). Then we have f0 ∈ Mn(S ∗)+which is a contradiction since f0 separates Γ2 and ϕ0 strictly. The conclusion follows from Γ1 = (Mn(S ∗)∗)+1 and

ϕx,ξ((A + ε In)

− 12 f (A + ε In)

− 12)� −1. �

Proposition 5.2. Suppose that Φ : S → T is a finite rank map for operator systems S and T . Then Φ is completely positive if andonly if it belongs to (S ∗ ⊗min T )+ .

Proof. ⇒) The finite rank map Φ can be regarded as an element in S ∗ ⊗ T ⊂ S ∗ ⊗ T . For a positive element x in Mn(S),the evaluation map evx : S ∗ → Mn defined by

evx( f ) = fn(x), f ∈ S ∗

is completely positive. For a completely positive map g : T → Mm , we have

(evx ⊗ g)(Φ) = (g ◦ Φ)n(x) ∈ M+mn

(evx ⊗ g : S ∗ ⊗ T → Mmn

).

For a completely positive map f : S ∗ → Mn , its restriction f |S ∗ belongs to C P (S ∗, Mn) = Mn(S ∗∗)+ . It is the point-normlimit of evaluation maps evx for x ∈ Mn(S)+ . Thus, we have

( f ⊗ g)(Φ) = ( f |S ∗ ⊗ g)(Φ) ∈ M+mn.

In other words, Φ belongs to (S ∗ ⊗min T )+ .

382 K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386

⇐) For x ∈ Mn(S)+1 , we define an evaluation map evx : S ∗ → Mn by

evx( f + λ) = fn(x) + λIn.

By Lemma 5.1, the evaluation map evx : S ∗ → Mn is completely positive.From

(g ◦ Φ)n(x) = (evx ⊗ g)(Φ) ∈ M+mn

we see that Φ : S → T is completely positive. �By Proposition 5.2, finite rank completely positive maps from S to T correspond to the elements in S ∗ ⊗ T ∩ (S ∗ ⊗min

T )+ . Our next goal is to study completely positive maps corresponding to the elements in S ∗ ⊗ T ∩ (S ∗ ⊗max T )+ .

Theorem 5.3. Suppose that Φ : S → T is a finite rank map. Then Φ belongs to (S ∗ ⊗max T )+ if and only if for any ε > 0, there exist afactorization Φ = ψ ◦ϕ and a positive semidefinite matrix A ∈ M p such that ϕ : S → M p is a self-adjoint map with ϕm(x) � −Im ⊗ Afor all m ∈ N, x ∈ Mm(S)+1 and ψ : M p → T is a completely positive map with ψ(A) = ε1T .

S Φ

ϕ

T

Mp

ψ

Proof. ⇒) Let Φ ∈ (S ∗ ⊗max T )+ . For any ε > 0, we can write

Φ + ε1S ∗ ⊗ 1T = α((ϕ + A · 1S ∗) ⊗ Q

)α∗

for α ∈ M1,pq,ϕ ∈ M p(S ∗), A ∈ M p, Q ∈ Mq(T )+ and ϕ + A · 1S ∗ ∈ M p( S∗)+ . It follows that

Φ(x) = αϕ(x) ⊗ Q α∗ and ε1T = α(A ⊗ Q )α∗.

By Lemma 5.1, we have

ϕn(x) � −In ⊗ A, n ∈ N, x ∈ Mn(S)+1 .

We define a completely positive map ψ : M p → T by ψ(B) = α(B ⊗ Q )α∗ . Then we have

Φ(x) = αϕ(x) ⊗ Q α∗ = ψ(ϕ(x)

)and ψ(A) = α(A ⊗ Q )α∗ = ε1T .

⇐) We put

Q = [ψ(ei, j)

]1�i, j�p ∈ Mp(T )+ and α = [e1, . . . , ep] ∈ M1,p2 .

Because

ψ([bi, j]

) = ψ

( p∑i, j=1

bi, jei, j

)=

p∑i, j=1

bi, j Q i, j = α([bi, j] ⊗ Q

)α∗,

we can write

Φ(x) = αϕ(x) ⊗ Q α∗ and ε1T = α(A ⊗ Q )α∗.

By Lemma 5.1, we have ϕ + A ∈ Mn(S ∗)+ . It follows that

Φ + ε1S ∗ ⊗ 1T = α((ϕ + A · 1S ∗) ⊗ Q

)α∗ ∈ (

S ∗ ⊗max T)+

. �Let Φ : S → T be a completely positive map factoring through a matrix algebra in a completely positive way. By the

proof of Theorem 5.3, Φ corresponds to an element in the subcone{α(ϕ ⊗ Q )α∗: α ∈ M1,pq, ϕ ∈ Mp

(S ∗)+

, Q ∈ Mq(T )+}

of the cone S ∗ ⊗ T ∩ (S ∗ ⊗max T )+ .

K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386 383

Theorem 5.4. Suppose that Φ : S → T is a completely positive map for operator systems S and T . The map

idR ⊗ Φ : R ⊗min S → R ⊗max T

is completely positive for any operator system R if and only if we have

Φ|E ∈ (E∗ ⊗max T

)+

for any finite dimensional operator subsystem E of S .

Proof. ⇒) By Proposition 5.2, we can regard the inclusion ι : E ⊂ S as an element in (E∗ ⊗min S)+ . By assumption, we seethat Φ|E = (idE∗ ⊗ Φ)(ι) belongs to (E∗ ⊗max T )+ .

⇐) We choose an element

z =n∑

i=1

xi ⊗ yi ∈ (R ⊗min S)+1 .

Let E be a finite dimensional operator subsystem of S which contains {yi: 1 � i � n}. By Theorem 5.3, there exist afactorization Φ|E = ψ ◦ ϕ and a positive semidefinite matrix A ∈ Mn such that ϕ : E → Mn is a self-adjoint map withϕm(x) � −Im ⊗ A for all m ∈ N, x ∈ Mm(S)+1 and ψ : Mn → T is a completely positive map with ψ(A) = ε1T . Let R be aconcrete operator system acting on a Hilbert space H and P the projection onto the finite dimensional subspace of H . Thecompression P R P is the operator subsystem of a matrix algebra M p for p = rank P . From the commutative diagram

Mp ⊗min EidM p ⊗ϕ

Mp ⊗min Mn

P R P ⊗min EidP R P ⊗ϕ

P R P ⊗min Mn

we see that

(idR ⊗ ϕ)(z) � −1R ⊗ A.

From the commutative diagram

R ⊗min S idR⊗Φ R ⊗max T

R ⊗min EidR⊗ϕ R ⊗min Mn = R ⊗max Mn

idR⊗ψ

we also see that

(idR ⊗ Φ)(z) = (idR ⊗ ψ) ◦ (idR ⊗ ϕ)(z)

� −ε(idR ⊗ ψ)(1R ⊗ A)

= −ε1R ⊗ 1T

in R ⊗max T . Since the choice of ε > 0 is arbitrary, we conclude that the map

idR ⊗ Φ : R ⊗min S → R ⊗max T

is positive. �6. Duality

We establish the duality between complete order embeddings and complete order quotient maps.

Theorem 6.1. Suppose that S and T are operator systems with complete norms and Φ : S → T is a unital completely positivesurjection. Then Φ : S → T is a complete order quotient map if and only if its dual map Φ∗ : T ∗ → S ∗ is a complete order embedding.

384 K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386

Proof. ⇒) Let Φ∗n ( f ) ∈ Mn(S ∗)+ = C P (S, Mn) for f ∈ Mn(T ∗). We choose a positive element y in Mm(T ). For any ε > 0,

there exists an element x in Mm(S) such that

Φm(x) = y and x + ε Im ⊗ 1S ∈ Mm(S)+.

We have

( f ◦ Φ)m(x) + ε( f ◦ Φ)m(Im ⊗ 1S ) = (Φ∗

n ( f ))

m(x + ε In ⊗ 1S ) ∈ M+mn.

Since the choice of ε > 0 is arbitrary, we have

fm(y) = ( f ◦ Φ)m(x) ∈ M+mn.

It follows that f : T → Mn is completely positive.⇐) We put

Cn := {y ∈ Mn(T ): ∀ε > 0, ∃x ∈ Mn(S), x + ε In ⊗ 1S ∈ Mn(S)+ and Φn(x) = y

}.

For y ∈ Cn , we have

y + ε In ⊗ 1T = Φn(x + ε In ⊗ 1S ) ∈ Mn(T )+,

thus Cn ⊂ Mn(T )+ . The map Φ : S → T is a complete order quotient map if and only if Cn = Mn(T )+ holds for all n ∈ N.It is easy to check that Cn is a cone. We have the inclusions of the cones

Φn(Mn(S)+

) ⊂ Cn ⊂ Mn(T )+.

Suppose that yk ∈ Cn converges to y ∈ Mn(T ). By the open mapping theorem, there exists M > 0 such that ‖Φ−1n : Mn(T ) →

Mn(S)/Ker Φn‖ � M . We choose yk0 so that ‖y − yk0‖ < ε/M . There exist x, x′ in Mn(S) such that

Φn(x) = yk0 , x + ε In ⊗ 1S ∈ Mn(S)+ and Φn(x′) = y − yk0 , ‖x′‖ < ε.

Replacing x′ by (x′ + x′∗)/2, we may assume that x′ is self-adjoint. It follows that

y = yk0 + (y − yk0) = Φn(x + x′) and x + x′ + 2ε In ⊗ 1S ∈ Mn(S)+.

Hence, the cone Cn is closed. We assume Cn � Mn(T )+ and choose y0 ∈ Mn(T )+/Cn . By the Hahn–Banach separationtheorem, there exists a self-adjoint functional f on Mn(T ) such that f (Cn) = [0,∞) and f (y0) < 0. Even though thefunctional f : Mn(T ) → C is not positive, we have Φ∗

n ( f )(x) = f ◦ Φn(x) � 0 for all x ∈ Mn(S)+ because Φn(Mn(S)+) is asubcone of Cn . Hence, we see that the dual map Φ∗ : T ∗ → S ∗ is not a complete order embedding. �Lemma 6.2. Suppose that f : S → Mn is a self-adjoint linear map for an operator system S . Then we have fm(x) � −Imn for allm ∈ N, x ∈ Mm(S)+1 if and only if the self-adjoint linear map f : S → Mn defined by

f (x) = (f (1S ) + 2In

)− 12 f (x)

(f (1S ) + 2In

)− 12

is completely contractive.

Proof. ⇒) We choose a contractive element x in Mm(S). Then we have

0 �(

Im ⊗ 1S xx∗ Im ⊗ 1S

)� 2

(Im ⊗ 1S 0

0 Im ⊗ 1S

).

By assumption, we have

0 �(

Im ⊗ f (1S ) + 2Imn fm(x)fm(x)∗ Im ⊗ f (1S ) + 2Imn

).

Multiplying both sides by (Im ⊗ f (1S ) + 2Imn)− 12 ⊕ (Im ⊗ f (1S ) + 2Imn)− 1

2 from the left and from the right, we see that‖ fm(x)‖ � 1.

⇐) We choose an element x in Mm(S)+1 . Then we have∥∥∥∥x − 1

2Im ⊗ 1S

∥∥∥∥ � 1

2.

K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386 385

By assumption, we have∥∥∥∥(Im ⊗ f (1S ) + 2Imn

)− 12 fm

(x − 1

2Im ⊗ 1S

)(Im ⊗ f (1S ) + 2Imn

)− 12

∥∥∥∥ =∥∥∥∥ fm

(x − 1

2Im ⊗ 1S

)∥∥∥∥ � 1

2,

thus

−1

2

(Im ⊗ f (1S ) + 2Imn

)� fm

(x − 1

2Im ⊗ 1S

).

It follows that fm(x) � −Imn . �Lemma 6.3. Let S be an operator subsystem of an operator system T and A a positive semidefinite n × n matrix. Suppose thatf : S → Mn is a self-adjoint linear map satisfying fm(x) � −Im ⊗ A for all m ∈ N, x ∈ Mm(S)+1 . Then f : S → Mn extends to aself-adjoint linear map F : T → Mn satisfying Fm(x) � −Im ⊗ A for all m ∈ N, x ∈ Mm(T )+1 .

Proof. We define a self-adjoint linear map g : S → Mn by

g(x) = (A + ε In)− 1

2 f (x)(A + ε In)− 1

2

for 0 < ε < 1. Then we have

gm(x) � −(Im ⊗ A + ε Imn)− 1

2 (Im ⊗ A)(Im ⊗ A + ε Imn)− 1

2 � −Imn

for all m ∈ N, x ∈ Mm(S)+1 . By Lemma 6.2, the self-adjoint linear map g : S → Mn defined by

g(x) = (g(1S ) + 2In

)− 12 g(x)

(g(1S ) + 2In

)− 12

is completely contractive. By the Wittstock extension theorem, g extends to a complete contraction G : T → Mn . By consid-ering 1

2 (G + G∗) instead, we may assume that G is self-adjoint. We put

G(x) = (g(1S ) + 2In

) 12 G(x)

(g(1S ) + 2In

) 12

and

F (x) = (A + ε In)12 G(x)(A + ε In)

12 .

Then F : T → Mn (respectively, G : T → Mn) is a self-adjoint extension of f : S → Mn (respectively, g : S → Mn). Theself-adjoint linear map G can be written as

G(x) = (G(1S ) + 2In

)− 12 G(x)

(G(1S ) + 2In

)− 12 .

By using Lemma 6.2 again, we see that

Fm(x) = (Im ⊗ A + ε Imn)12 Gm(x)(Im ⊗ A + ε Imn)

12 � −Im ⊗ A − ε Imn

for all m ∈ N, x ∈ Mm(T )+1 . The extension F : T → Mn depends on the choice of ε. However the norm of F is uniformlybounded as can be seen from

‖F‖ �∥∥(A + ε In)

12((A + ε In)

− 12 f (1S )(A + ε In)

− 12 + 2In

) 12∥∥2 = ∥∥ f (1S ) + 2(A + ε In)

∥∥ �∥∥ f (1S ) + 2A

∥∥ + 2.

Since the range space is finite dimensional, we can consider the point-norm cluster point of {Fε: 0 < ε < 1}. �Let T : V → W be a completely contractive and completely positive map for matrix ordered operator spaces V and W .

Then its unitization T : V → W defined by

T (x + λ1V ) = T (x) + λ1W , x ∈ V , λ ∈ C

is a unital completely positive map [16, Lemma 4.9].

Theorem 6.4. Suppose that Φ : S → T is a unital completely positive map for operator systems S and T . Then Φ : S → T is acomplete order embedding if and only if the unitization of its dual map Φ∗ : T ∗ → S ∗ is a complete order quotient map.

386 K.H. Han / J. Math. Anal. Appl. 384 (2011) 375–386

Proof. ⇒) Let f + A ∈ Mn(S ∗)+ . By Lemma 5.1, we have fm(x) � −Im ⊗ A for all m ∈ N and x ∈ Mm(S)+1 . We can regard Sas an operator subsystem of T . By Lemma 6.3, there exists a self-adjoint extension F : T → Mn such that Fm(x) � −Im ⊗ Afor all m ∈ N and x ∈ Mm(T )+1 . By Lemma 5.1 again, we have

Φ∗n (F + A) = f + A and F + A ∈ Mn

(T ∗)+

.

⇐) Suppose that Φn(x) belongs to Mn(T )+ for x ∈ Mn(S). Let f : Mn(S) → C be a positive functional. For any ε > 0,there exists a self-adjoint linear functional F : Mn(T ) → C such that

Φ∗n (F ) = f and F + ε In ⊗ 1T ∗ ∈ Mn

(T ∗)+

.

We have

f (x) = F ◦ Φ(x) � −εn2‖x‖.It follows that x ∈ Mn(S)+ . If Φn(x) = 0, then x ∈ Mn(S)+ ∩ −Mn(S)+ = {0}. �References

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