An abstract characterization of unital operator spaces
aa r X i v : . [ m a t h . OA ] M a y An abstract characterization of unital operator spaces ∗ Xu-Jian Huang and Chi-Keung NgNovember 21, 2018
Abstract
In this article, we give an abstract characterization of the “identity” of an operator space V bylooking at a quantity n cb ( V , u ) which is defined in analogue to a well-known quantity in Banachspace theory. More precisely, we show that there exists a complete isometry from V to some L ( H )sending u to id H if and only if n cb ( V , u ) = 1. We will use it to give an abstract characterization ofoperator systems. Moreover, we will show that if V is a unital operator space and W is a propercomplete M -ideal, then V / W is also a unital operator space. As a consequece, the quotient of anoperator system by a proper complete M -ideal is again an operator system. In the appendix, wewill also give an abstract characterisation of “non-unital operator systems” using an idea arose fromthe definition of n cb ( V , u ). Operator spaces are subspaces of L ( H ) (where H is a Hilbert spaces) together with the induced “matrixnorm structures”. The theory of operator spaces is a very important tool in the study of operatoralgebras. Since the discovery of an abstract characterization of operator spaces by Ruan (see e.g. [19] or[10]), there have been many more applications of operator spaces to other branches in functional analysis(see e.g. [2], [11], [12], [13], [15], [16], [20], [21] [22] and [23]).In the theory of operator spaces, sometimes the starting point is not general subspaces of L ( H ) butunital subspaces of L ( H ) (i.e. subspaces that contain the identity; see e.g. [3]). It is natural to askwhether there is an abstract characterization of unital operator spaces.In fact, there is a Banach space characterization for the identity of a unital C ∗ -algebra (see e.g. [1,Theorem 2], [6, 4.1] and [18, 9.5.16]) which gives rise to the concept of “geometric unitary” (see e.g. [14]).More precisely, a norm one element u in a Banach space X is a geometric unitary if certain quantity n ( X ; u ) is non-zero. This quantity is also related to numerical indexes of Banach spaces (see e.g. [7]).It is natural to think that certain operator space analogue of such quantity may give a characterizationof the “identity” of an operator space.In this paper, we will define and study such a quantity n cb ( V ; u ) for an operator space V and anorm one element u ∈ V . We will give some properties about it and show that there exists a completeisometry (respective, complete isomorphism) from V to some L ( H ) that sends u to id H if and only if n cb ( V ; u ) = 1 (respectively, n cb ( V ; u ) > n cb ( V ; u ) and n ( V ; u ). As an application, we give an abstract characterization for operator systems. Furthermore, wewill show that if V is a unital operator space (respectively, operator system), then any proper complete ∗ This work is jointly supported by Hong Kong RGC Research Grant (2160255), the National Natural Science Foundationof China (10771106) and NCET-05-0219 -ideal W of V will not contain the identity and the quotient V / W is again a unital operator space(respectively, operator system).In the appendix, we give an abstract characterisation for “non-unital operator systems” using thesimilar idea as that for ordinary operator systems as in the main body of the article.In the following, let us first recall the notion of “geometric unitary” for normed spaces. Suppose that X is a normed space and S ( X ) is the unit sphere of X . For any u ∈ S ( X ), we set S ( X ; u ) := { f ∈ X ∗ : k f k = 1 = f ( u ) } ,γ u ( x ) := sup {k f ( x ) k : f ∈ S ( X ; u ) } ( x ∈ X ) , as well as n ( X ; u ) := inf { γ u ( x ) : x ∈ S ( V ) } . A norm one element u in a normed space X is called a geometric unitary (respectively, strict geometricunitary ) if n ( V ; u ) > n ( V ; u ) = 1).The following result is probably well-known. Part (a) of which is one of the motivations behind thiswork. Since the arguments for it is straight forward (and also follows from a similar arguments as thosefor Theorem 2.9 and Proposition 2.7(b)), we leave it to the readers to check them. Proposition 1.1
Let X be a normed space and u ∈ S ( X ) .(a) u is a strict geometric unitary (respectively, geometric unitary) if and only if there exist a compactHausdorff space Ω and an isometry (respectively, a contractive topological injection) ϕ : X → C (Ω) suchthat ϕ ( u ) = 1 Ω .(b) Suppose that Y is another normed space and Ψ : X → Y is a contractive topological injection. If Ψ( u ) ∈ S ( Y ) , then n ( Y ; Ψ( u )) ≤ k Ψ − k · n ( X ; u ) . In analogue to the notion of geometric unitaries, one can define complete geometric unitaries for operatorspaces. In fact, we will go a step further and start with matricially normed spaces (as defined in [19,p.217]) instead of operator spaces.
Notation 2.1
Throughout this article, unless specified, V is a matricially normed space (i.e. a matrixnormed space that satisfies only condition (M2) in [10, p.20]) while V is the underlying normed space(or underlying vector space) of V . We will denote by S ( V ) the unit sphere of V . Remark 2.2 If M ∞ ( V ) denote the set of all infinite matrices on V with only finitely many non-zero en-tries, then by [19, 2.1], one can turn M ∞ ( V ) into an essential normed M ∞ ( C ) -bimodule (where M ∞ ( C ) is regarded as a normed subalgebra of L ( ℓ ) ). Therefore, one can construct the so-called regularization of M ∞ ( V ) (as in [17, 1.2]) which produces an operator space V (by [17, 2.1]). In fact, V is theimage of V in V ∗∗ ( V ∗∗ is as defined in [10, p.41] rather than [19]) together with the induced norm.Now, by a similar argument as that of [17, 2.5], if W is an operator space, then the canonical map from CB( V , W ) to CB( V , W ) given by the canonical map κ V : V → V is a complete isometry. .1 Definition and main results Let u ∈ S ( V ). For any n ∈ N , we define S n ( V ; u ) := { ϕ ∈ CB( V , M n ) : k ϕ k cb ≤ ϕ ( u ) = I n } ,γ uk ( x ) := sup {k ϕ k ( x ) k : ϕ ∈ S n ( V ; u ); n ∈ N } ( k ∈ N ; x ∈ M k ( V ))as well as n cb ( V ; u ) := inf { γ uk ( x ) : x ∈ S ( M k ( V )); k ∈ N } . In the case when V is an operator algebra and u being the identity of V , the quantity γ uk ( x ) wasdefined in [5, p.192] and is called the k th numerical radius of x ∈ V . Definition 2.3
A norm one element u in V is called a complete geometric unitary (respectively, com-plete strict geometric unitary ) if n cb ( V ; u ) > (respectively, n cb ( V ; u ) = 1 ). Remark 2.4 (a) S ( V ; u ) = S ( V ; u ) .(b) If V is the operator space as given in Remark 2.2, then S n ( V , u ) = S n ( V , κ V ( u )) .(c) S n ( V ; u ) is a compact convex set under the point-norm topology. In fact, it is not hard to seethat S n ( V ; u ) is point-norm closed in the closed unit ball of CB( V , M n ) which is compact under thepoint-norm topology (by part (b) and the corresponding fact for operator spaces).(d) Suppose that V is a matrix normed subspace of a matricially normed space W and γ u,Wk is definedby S n ( W , u ) in a similar way as γ uk . By the Arveson extension theorem and part (b), the canonical mapgives a surjection from S ( W ; u ) to S ( V ; u ) . Therefore, γ uk = γ u,Wk for all k ∈ N (so we can use γ uk instead of γ u,Vk ). Consequently, n cb ( W , u ) ≤ n cb ( V , u ) .(e) Suppose that X is a vector space and σ k is a seminorm on M k ( X ) ( k ∈ N ) satisfying the followingtwo conditions: for any m, n ∈ N , x ∈ M m ( X ) , y ∈ M n ( Y ) , α ∈ M n,m and β ∈ M m,n , I. σ m + n ( x ⊕ y ) ≤ max { σ m ( x ) , σ n ( y ) } ; II. σ n ( α · x · β ) ≤ k α k σ m ( x ) k β k .Let N = { v ∈ V : σ u ( v ) = 0 } . Then σ k induces a semi-norm ˜ σ k on M k ( X/N ) (since σ k (( x i,j ) ki,j =1 ) ≤ P ni,j =1 σ ( x i,j ) ) which also satisfies the two conditions as the above. Therefore, by [10, 2.3.6], { ˜ σ k } givesan operator space structure on X/N . Lemma 2.5 γ uk is a semi-norm on M k ( V ) ( k ∈ N ) satisfying the two properties in Remark 2.4(e) andthey induce an operator space structure on V u := V /N u where N u := { v ∈ V : γ u ( v ) = 0 } . In the following, when we talk about the operator space V u , we consider the operator space structureas given in the above lemma (even when N u = (0)). Moreover, we denote by Q u the canonical completecontraction from V to V u . Lemma 2.6
Let V be a matricially normed space and u ∈ S ( V ) .(a) n cb ( V ; u ) > if and only if Q u is a complete isomorphism. In this case, n cb ( V ; u ) = k Q − u k − .(b) n cb ( V u , Q u ( u )) = 1 . roof: (a) Since the norm on M n ( V u ) is given by ˜ γ un (( Q u ) n ( x )) := γ un ( x ) ( n ∈ N , x ∈ M n ( V )) and n cb ( V ; u ) = sup { λ ∈ R + : λ k x k ≤ γ un ( x ); n ∈ N ; z ∈ M n ( V ) } , (1)we obtain the first statement. If n cb ( V ; u ) >
0, then n cb ( V ; u ) − = inf { µ ∈ R + : k ( Q − u ) n ( y ) k ≤ µ ˜ γ un ( y ); n ∈ N ; y ∈ M n ( V u ) } which gives the second statement.(b) Consider the map ( ˜ Q u ) n : CB( V u , M n ) → CB( V , M n ) given by composition with Q u . It is clearthat ( ˜ Q u ) n ( S n ( V u ; Q u ( u ))) ⊆ S n ( V ; u ) ( n ∈ N ). On the other hand, for any ϕ ∈ S n ( V ; u ), we have k ϕ k ( x ) k ≤ γ uk ( x ) = ˜ γ uk (( Q u ) k ( x )) ( x ∈ M k ( V )) . Hence there exists ψ ∈ CB( V u , M n ) with ϕ = ψ ◦ Q u and k ψ k cb ≤
1. This shows that ( ˜ Q u ) n is asurjection (and hence a bijection) from S n ( V u ; Q u ( u )) to S n ( V ; u ). Consequently, N Q u ( u ) = (0) and˜ γ uk = γ Q u ( u ) k on M k ( V u ) = M k ( V Q u ( u ) ). Hence Q Q u ( u ) is a completely isometric isomorphism and n cb ( V u ; Q u ( u )) = 1 by part (a). (cid:3) Proposition 2.7
Let V and W be matricially normed spaces. Suppose that Ψ : V → W is a completecontraction and u ∈ S ( V ) such that Ψ( u ) ∈ S ( W ) .(a) There exists a complete contraction Ψ u : V u → W Ψ( u ) with Q Ψ( u ) ◦ Ψ = Ψ u ◦ Q u .(b) If Ψ is a complete topological injection, then n cb ( W ; Ψ( u )) ≤ k Ψ − k cb · n cb ( V ; u ) . Proof: (a) The composition with Ψ gives a map ˜Ψ n : S n ( W ; Ψ( u )) → S n ( V ; u ) and we have γ Ψ( u ) n ◦ Ψ ≤ γ un ( n ∈ N ). This induces the required map Ψ u .(b) By Remark 2.4(d), Equality (1) and the inequality in the argument for part (a), n cb ( W ; Ψ( u )) ≤ n cb (Ψ( V ); Ψ( u ))= sup { λ ∈ R + : λ k Ψ n ( x ) k ≤ γ Ψ( u ) n (Ψ n ( x )); n ∈ N ; x ∈ M n ( V ) }≤ sup { λ ∈ R + : λ k Ψ − k − k x k ≤ γ un ( x ); n ∈ N ; x ∈ M n ( V ) } = k Ψ − k cb · n cb ( V ; u ) . (cid:3) Consequently, for any complete contraction Ψ : V → L ( H ) with Ψ( u ) = id H (where u ∈ S ( V )),there exists a complete contraction Ψ u : V u → L ( H ) with Ψ = Ψ u ◦ Q u .The following is our first theorem which tells us that one can use strict complete geometric unitaryto describe the “identity” of an operator space. Note that one direction of this theorem appeared inthe disguised form in [5, Proposition 1.5] in the case of operator algebras and its proof can be carriedover directly to the case of operator spaces. However, we will give its easy proof here for completeness.To do this, we need the following well-known fact (an argument for it can be found in the proof of [5,Proposition 1.5]). Lemma 2.8
For any Hilbert space H and any T ∈ L ( H n ) , k T k = sup {k ( P ⊗ T ( P ⊗ k : P ∈ L ( H ) is a finite dimensional projection } . heorem 2.9 Let V be a matricially normed space and u ∈ S ( V ) . Then u is a complete strictgeometric unitary of V if and only if there exists a Hilbert space H and a complete isometry Θ : V → L ( H ) such that Θ( u ) = id H . Proof: ⇐ ). For any finite dimensional projection P on H , if the rank of P is n , then x P Θ( x ) P can be regarded as an element of S n ( V ; u ). Therefore, k · k k ≤ γ uk (and hence equal) on M k ( V ) ( k ∈ N )by Lemma 2.8. Thus, Q u is a complete isometry and n cb ( V , u ) = 1 (see e.g. Lemma 2.6(a)). ⇒ ). If Ω n is the compact Hausdorff space S n ( V ; u ) (under the point-norm topology; see Remark 2.4(c)),then A = L ∞ n =1 C (Ω n , M n ) is a unital C ∗ -algebra. We define Θ : V → A by Θ( x ) = (Θ ( n ) ( x )) n ∈ N whereΘ ( n ) : V → C (Ω n , M n ) given byΘ ( n ) ( x )( ϕ ) = ϕ ( x ) ( x ∈ V ; ϕ ∈ Ω n ) . For any k ∈ N and z ∈ M k ( V ), we have Θ k ( z ) ∈ M k ( A ) ∼ = L ∞ n =1 C (Ω n , M nk ) and Θ ( n ) k ( z )( ϕ ) = ϕ k ( z )( ϕ ∈ Ω n ). Thus, k Θ k ( z ) k = sup {k ϕ k ( z ) k : n ∈ N ; ϕ ∈ S n ( V ; u ) } = γ uk ( z ) = k z k (as n cb ( V , u ) = 1) and so, Θ is a complete isometry. It is not hard to see that Θ( u ) = id H . (cid:3) One can also prove the necessity of the above theorem by taking H = L ∞ n =1 L ϕ ∈ S n ( V ; u ) C n andadapting the argument of [10, 2.3.5]. Corollary 2.10 (a) Suppose that V is a matricially normed space. Then n cb ( V ; u ) > if and only ifthere exists a Hilbert space H and a completely contractive complete topological injection Ψ : V → L ( H ) such that Ψ( u ) = id H .(b) If V is an operator space, then n cb ( V ; u ) = n cb ( V ∗∗ ; u ) . Proof: (a) If n cb ( V ; u ) >
0, then by Lemma 2.6(a), Q u : V → V u is a completely contractive completeisomorphism, and one can apply Theorem 2.9 to V u (because of Lemma 2.6(b)). The converse followsfrom Proposition 2.7(b) and Theorem 2.9.(b) Note that if n cb ( V , u ) = 0, then n cb ( V ∗∗ , u ) = 0 (by Remark 2.4(d)). Moreover, if n cb ( V , u ) = 1,then there is a Hilbert space H and a complete isometry Θ : V → L ( H ) such that Θ( u ) = id H (Theorem 2.9), and so, Θ ∗∗ : V ∗∗ → L ( H ) ∗∗ is a complete isometry with Θ ∗∗ ( u ) = id H which impliesthat n cb ( V ∗∗ , u ) = 1 (note that there exists a faithful unital representation for L ( H ) ∗∗ and we canapply Theorem 2.9). We now suppose that n cb ( V , u ) >
0. Then by Lemma 2.6(a), Q u is a completeisomorphism and so is Q ∗∗ u : V ∗∗ → ( V u ) ∗∗ . By Lemma 2.6(b), we have n cb ( V u , Q u ( u )) = 1 and theabove implies that n cb (( V u ) ∗∗ , Q u ( u )) = 1. Consequently, Lemma 2.6(a), Proposition 2.7(b) as well asRemark 2.4(d) tell us that n cb ( V ; u ) = k Q − u k − = k ( Q ∗∗ u ) − k − ≤ n cb ( V ∗∗ ; u ) ≤ n cb ( V ; u ) . (cid:3) Remark 2.11 (a) It is clear that if V is a matricially normed space having a complete strict geometricunitary, then V is an operator space (by Theorem 2.9). More generally, if V has a complete geometricunitary, then V is pseduo L ∞ in the sense that there exists κ ≥ such that k u ⊕ v k ≤ κ max {k u k , k v k} for any u ∈ M m ( V ) , v ∈ M n ( V ) and m, n ∈ N (by Corollary 2.10(a)). Consequently, if p ∈ [1 , ∞ ) , thenany L p -matricially normed space will not have a complete geometric unitary. b) Let V be a matricially normed space and u ∈ S ( V ) . Suppose that H is a Hilbert space and Θ : V u → L ( H ) is a complete isometry such that Θ( Q u ( u )) = id H (by Lemma 2.6(b) and Theorem2.9). Then Θ ◦ Q u : V → L ( H ) satisfies certain universal property in the following sense: if K is anyHilbert space and Ψ : V → L ( K ) is a complete contraction with Ψ( u ) = id K , then there exists a completecontraction (not necessarily unique) Λ : L ( H ) → L ( K ) such that Ψ = Λ ◦ Θ ◦ Q u . Indeed, Ψ = Ψ u ◦ Q u (where Ψ u : V u → L ( K ) is as in Proposition 2.7(a)) and there exists, by the Averson extension theorem,a complete contraction Λ : L ( H ) → L ( K ) such that Ψ u = Λ ◦ Θ . In this subsection, we will give some comparison between n cb ( V , u ) and n ( V, u ). Corollary 2.12
Let V be a matricially normed space. If n cb ( V ; u ) > (respectively, n cb ( V ; u ) = 1 ),then n ( V ; u ) > (respectively, n ( V ; u ) ≥ ). Proof:
Let Ψ be the completely contractive complete topological injection (respectively, completeisometry) given by Corollary 2.10(a) (respectively, Theorem 2.9). Then Ψ is a contractive topologicalinjection (respectively, isometry) and Proposition 1.1(b) and [8, Theorem 3] shows that n ( V ; u ) ≥ n ( L ( H );1) k Ψ − k ≥ k Ψ − k . (cid:3) Next, we compare n and n cb for the minimal quantization. Proposition 2.13
Let X be a normed space and u ∈ S ( X ) .(a) n ( X ; u ) ≤ n cb (min X ; u ) .(b) n ( X ; u ) > if and only if n cb (min X ; u ) > . Proof: (a) Since S (min X ; u ) = S ( X ; u ), it suffices to prove that for any k ∈ N and x = ( x ij ) ∈ S ( M k (min X )), we have n ( X ; u ) ≤ sup {k f k ( x ) k : f ∈ S ( X ; u ) } . (2)Suppose that Ω is a compact Hausdorff space such that min X ⊆ C (Ω) as operator subspace (hence, M k (min X ) ⊆ C (Ω; M k )). There exist ω ∈ Ω and ( c i ) , ( d i ) ∈ S ( C n ) with1 = k x k = k x ( ω ) k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ij c i x ij ( ω ) d j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) which implies that k P ij c i x ij d j k ≥
1. On the other hand, it is easy to see that k P ij c i x ij d j k ≤ k x k = 1.Now, for any f ∈ S ( X ; u ), k f k ( x ) k ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* ( f ( x ij )) d ... d k , c ... c k +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f X ij c i x ij d j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence, γ u ( P ij c i x ij d j ) ≤ sup {k f k ( x ) k : f ∈ S ( X ; u ) } and Equality (2) is verified.(b) This part follows from part (a) as well as Corollary 2.12. (cid:3) emark 2.14 (a) The arguement of the above actually shows that for any ( x ij ) ∈ M k (min X ) , thereexists ( c i ) , ( d i ) ∈ S ( C n ) such that k ( x ij ) k = k P ij c i x ij d j k . This could be a known fact although we donot find it in the literatures.(b) Let X be a finite dimensional normed space and X be any quantization of X . Then the identity mapis a completely contractive complete isomorphism from X to min X (see e.g. [10, 2.2.4]). Consequently,by Propositions 2.7(b) and 2.13(b), if u ∈ S ( X ) , then n cb ( X , u ) > if and only if n ( X, u ) > .(c) If V is an operator space and u ∈ S ( V ) such that n cb ( V ; u ) > , then n cb (min V ; u ) > (byCorollary 2.12 and Proposition 2.13). This fact is not easy to obtain directly from the definition. Let us start this subsection with the following result concerning n cb for C ∗ -algebras. Note that (i) ⇒ (ii)follows from Theorem 2.9 while (iii) ⇒ (i) follows from Corollary 2.12 as well as [6, 4.1]. Corollary 2.15
For a C ∗ -algebra A and u ∈ S ( A ) , the following statements are equivalent.(i) u is a unitary.(ii) n cb ( A ; u ) = 1 .(iii) n cb ( A ; u ) > . Note that from [8, Theorem 3], if A is a commutative unital C ∗ -algebra, then n ( A ; 1) = 1 but if A is noncommutative, then n ( A ; 1) = . By considering n cb , the above corresponding result looks cleanerbut one losses the ability to detect whether A is commutative.On the other hand, for a general operator spaces V and any u, v ∈ S ( V ), it is possible that both n cb ( V , u ) and n cb ( V , u ) are non-zero but n cb ( V ; u ) = n cb ( V ; v ). In order to give such an example, weneed the following lemma which is probably known. However, since we cannot find it in the literatures,we give a proof here for completeness. Lemma 2.16
Let X and Y be two normed spaces and u ∈ S ( X ) . Then n ( X ⊕ Y ; ( u, n ( X ; u ) . Proof:
Let E = X ⊕ Y . For any ( f, g ) ∈ X ∗ ⊕ ∞ Y ∗ , we have ( f, g ) ∈ S ( E ; ( u, {k f k , k g k} = 1 = f ( u ). Hence, S ( E ; ( u, S ( X ; u ) × Ball( Y ∗ ) (where Ball( Y ∗ ) is the closed unitball of Y ∗ ). Thus, γ ( u, ( x, y ) ≤ γ u ( x ) + k y k (( x, y ) ∈ E ) . On the other hand, for any ǫ >
0, there exists f ∈ S ( X ; u ) and g ∈ Ball( Y ∗ ) such that γ u ( x ) < | f ( x ) | + ǫ and g ( y ) = k y k . If f ( x ) = | f ( x ) | e iθ , then (cid:12)(cid:12) f ( x ) + ( e iθ g )( y ) (cid:12)(cid:12) = | f ( x ) | + k g k ≥ γ u ( x ) + k y k − ǫ . Thisshows that γ ( u, ( x, y ) = γ u ( x ) + k y k . Consequently, n ( E ; ( u, { γ u ( x ) + k y k : ( x, y ) ∈ E ; k x k + k y k = 1 } ≤ n ( X ; u )(since k y k = 0 is possible). On the other hand, for any n ∈ N , there exists ( x n , y n ) ∈ E with k x n k + k y n k = 1 and γ u ( x n ) + k y n k < n ( E ; ( u, n . If there are infinitely many n with x n = 0, then there exists a subsequence such that 1 = k y n k k Proposition 3.1 Suppose that V is a matricially normed space and u is a complete strict geometricunitary of V . For any n ∈ N , we set K nu := { x ∈ M n ( V ) : ϕ n ( x ) ≥ ϕ ∈ S k ( V ; u ); k ∈ N } . If Ψ : V → L ( H ) is any complete isometry with Ψ( u ) = id H , then Ψ n ( K nu ) = Ψ n ( M n ( V )) ∩ L ( H ( n ) ) + .Conseqently, V s = span K u is an operator system with matrix order structure given by K nu ∩ M n ( V s ) . Proof: If A and Θ are as in the argument for Theorem 2.9, then clearly, Θ n ( K nu ) = Θ n ( M n ( V )) ∩ M n ( A ) + . Regard A as a unital C ∗ -subalgebra of some L ( K ). If Ψ : V → L ( H ) is as in the statement,then there exist (by Remark 2.11(c)) complete contractions Γ : L ( H ) → L ( K ) and Γ : L ( K ) → L ( H )such that Θ = Γ ◦ Ψ and Ψ = Γ ◦ Θ. Hence, both Γ and Γ are completely positive, and so,Ψ n ( K nu ) = Ψ n ( M n ( V )) ∩ L ( H ( n ) ) + . (cid:3) Theorem 3.2 Let V be a matrically normed space and u ∈ S ( V ) . Then there exists a matrix orderstructure on V turning it into an operator system with order unit u if and only if n cb ( V , u ) = 1 and K u spans V . Corollary 3.3 Let V be matricially normed space and u ∈ S ( V ) .(a) V u,s is an operator system whose cone is Q u ( K u ) .(b) If n cb ( V ; u ) > , there exists an equivalent matrix norm on V under which V s is an operator systemwith order unit u .(c) If ϕ : V → L ( H ) is a complete contraction such that ϕ ( u ) = id H , then ϕ factor through a “completelypositive map” from V u to L ( H ) . One can also use the above ideal to give an abstract characterisation for “non-unital operator sys-tems”. However, since this is not directly related to complete geometric unitary, we will do this in theAppendix. 8 .2 Quotient by a complete M -ideal In this subsection, we will show that the quotient of a unital operator space with a proper complete M -ideal is also unital. As an application, the quotient of any operator system by a proper complete M -ideal is again an operator system. Let us begin with the following lemma which may also be known. Lemma 3.4 Let X and Y be two normed spaces. If ( u, v ) ∈ S ( X ⊕ ∞ Y ) with n ( X ⊕ ∞ Y ; ( u, v )) > ,then k u k = 1 = k v k . Proof: Suppose that k u k = 1 but k v k < 1. For any ( f, g ) ∈ S ( X ⊕ ∞ Y ; ( u, v )), we have k f k + k g k =1 = f ( u ) + g ( v ). Hence g ( v ) = k g k and so, either k g k = 0 or k v k ≥ S ( X ⊕ ∞ Y ; ( u, v )) = S ( X ; u ) × { } . Now, for any y ∈ S ( Y ) and ( f, ∈ S ( X ⊕ ∞ Y ; ( u, v )), we have( f, , y ) = 0 and so, n ( X ⊕ ∞ Y ; ( u, v )) = 0 which is a contradiction. A similar contradiction occur if k v k = 1 but k u k < (cid:3) If A is a unital operator algebra with identity 1 A and I is a proper closed ideal of A , then it isobvious that 1 A / ∈ I and the image of 1 A in A/I is the identity. Interesting, this fact can be regarded asa geometric statement and can be generalized to the following result. Theorem 3.5 (a) Let V be an operator space and W ( V be a complete M -ideal. If v ∈ S ( V ) with n cb ( V ; v ) > , then k Q ( v ) k = 1 and n cb ( V ; v ) ≤ n cb ( V / W ; Q ( v )) (where Q : V → V / W is thecanonical quotient map).(b) Let V be an operator system and W is a proper complete M -ideal of V . Then V / W is also anoperator system. Proof: (a) By Corollary 2.10(b), one can assume that W is a complete M -summand of V (since W ⊥⊥ is a complete M -summand of V ∗∗ ). Let P : V → V be the complete M -projection such that P ( V ) = W and let Z = ( I − P )( V ). Then V ∼ = Z ⊕ ∞ W as operator spaces and Q ( x ) ( I − P )( x ) is a completeisometry from V / W to Z . Suppose that v = ( u, w ) with u ∈ Z and w ∈ W . Then by Corollary 2.12and Lemma 3.4, we see that k u k = 1 = k w k . Pick any f ∈ S ( Z ∗ ) with f ( u ) = 1 and define Ψ : Z → V by Ψ( x ) = ( x, f ( x ) w ). It is not hard to check that Ψ is a complete isometry (as k f n ( x ) w k ≤ k x k for any x ∈ M n ( Z )) such that Ψ( u ) = ( u, w ). Now, n cb ( V ; ( u, w )) ≤ n cb ( Z ; u ) (by Proposition 2.7(b)).(b) Let v be the order unit of V . Since the composition with the canonical quotient map Q : V → V / W is a map from S ( V / W ; Q ( v )) to S ( V , v ), we know that Q ( K v ) ⊆ K Q ( v ) and so, K Q ( v ) spans V / W . Onthe other hand, part (a) tells us that n cb ( V / W ; Q ( v )) = 1. Now, the result follows from Theorem 3.2. (cid:3) Remark 3.6 One can also show that the image of a (strict) geometric unitary of a Banach space in thequotient by a proper M -ideal is also a (strict) geometric unitary. A Appendix: non-unital operator systems We begin this appendix with the following probably well-known result.9 emma A.1 Let H be a Hilbert space and T ∈ L ( H n ) . Then T ≥ if and only if ( P ⊗ T ( P ⊗ ≥ for any finite dimensional projection P ∈ L ( H ) . Proof: We only need to show the sufficiency. Let B be an orthonormal basis for H and H := span B .Suppose that F ( B ) is the collection of all finite subsets of B and P F is the projection onto span F ( F ∈ F ( B )). Then ( P F ⊗ T ( P F ⊗ ≥ F ∈ F ( B ) will imply that h η, T η i ≥ η ∈ H n ,and so, T ≥ (cid:3) Let V be a matrically normed space and M n ( V ) + be a cone in M n ( V ) ( n ∈ N ). For every n ∈ N ,we set S + n ( V ) := { ϕ ∈ CB( V , M n ) : k ϕ k cb ≤ ϕ m ( M m ( V ) + ) ⊆ ( M mn ) + ; m ∈ N } ,γ + k ( x ) := sup (cid:8) k ϕ k ( x ) k : ϕ ∈ S + n ( V ); n ∈ N (cid:9) ( k ∈ N ; x ∈ M k ( V )) ,n +cb ( V ) := inf (cid:8) γ + k ( x ) : x ∈ S ( M k ( V )); k ∈ N (cid:9) as well as K n := { v ∈ M n ( V ) : ϕ n ( v ) ∈ ( M nk ) + ; k ∈ N ; ϕ ∈ S + k ( V ) } . It is easy to check that S + n ( V ) is compact under the point-norm topology and M n ( V ) + ⊆ K n ( n ∈ N ). Theorem A.2 Suppose that V and M n ( V ) + be as in the above. Then there exist a Hilbert space H and a complete isometry Φ : V → L ( H ) with Φ( M n ( V ) + ) = Φ( M n ( V )) ∩ L ( H n ) + ( n ∈ N ) if and onlyif n +cb ( V ) = 1 and M n ( V ) + = K n for any n ∈ N . Proof: ⇒ ). n +cb ( V ) = 1 because of Lemma 2.8 and M n ( V ) + = K n follows from Lemma A.1. ⇐ ). Let A := L ∞ k =1 C ( S + k ( V ) , M k ) and define Θ : V → A by Θ( v ) = (Θ ( k ) ( v )) where Θ ( k ) ( v )( ϕ ) = ϕ ( v )( ϕ ∈ V ). As in the argument for Theorem 2.9, Θ is a complete isometry (because n +cb ( V ) = 1). Moreover,it is easy to see that Θ n ( K n ) = Θ n ( M n ( V )) ∩ M n ( A ) + ( n ∈ N ). (cid:3) This gives an abstract characterization of “possibly non-self-adjoint and non-unital operator system”as follows:Suppose that V is a matrically normed space and M n ( V ) + is a cone in M n ( V ) for any n ∈ N . Then ( V , M n ( V ) + ) is called an abstract (not necessarily unital) operator system if n +cb ( V ) = 1 and M n ( V ) + = K n for any n ∈ N . Acknowledgement 1. 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Vol. 2. ∗ -algebras , Encyclopediaof Mathematics and its Applications 79, Cambridge University Press, Cambridge (2001).[19] Z.-J. Ruan, Subspaces of C ∗ -algebras, J. Funct. Anal. 76 (1988), 217-230.[20] Z.-J. Ruan, The operator amenability of A ( G ), Amer. J. Math. 117 (1995), 1449-1474.1121] Z.-J. Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139(1996), 466-499.[22] V. Runde, Applications of operator spaces to abstract harmonic analysis, Expo. Math. 22 (2004),317-363.[23] N. Spronk, Operator weak amenability of the Fourier algebra, Proc. Amer. Math. Soc. 130 (2002),3609-3617.Xu-Jian Huang, Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China. Email address: [email protected] Ng, Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China.