Minimal and Maximal Operator Space Structures on Banach Spaces
aa r X i v : . [ m a t h . OA ] N ov MINIMAL AND MAXIMAL OPERATOR SPACESTRUCTURES ON BANACH SPACES
VINOD KUMAR P. AND M. S. BALASUBRAMANI
Abstract.
Given a Banach space X , there are many operator space struc-tures possible on X , which all have X as their first matrix level. Blecherand Paulsen [4] identified two extreme operator space structures on X ,namely M in ( X ) and M ax ( X ) which represents respectively, the smallestand the largest operator space structures admissible on X . In this note,we consider the subspace and the quotient space structure of minimal andmaximal operator spaces.AMS Mathematics Subject Classification(2000) No: 46L07,47L25Key Words: operator spaces, maximal operator spaces, minimal operatorspaces, submaximal spaces, Q -spaces. Introduction
Operator spaces form a natural quantization of Banach spaces and theirstudy took a rigorous form with the representation theorem obtained by Z. J.Ruan [17] in 1988 and after that it has seen considerable development withapplications to the theory of operator algebras and various aspects of operatorspaces are being studied extensively.A concrete operator space X is a closed linear subspace of B ( H ), for someHilbert space H . Here, in each matrix level M n ( X ), we have a norm k . k n ,induced by the inclusion M n ( X ) ⊂ M n ( B ( H )), where the norm in M n ( B ( H ))is given by the natural identification M n ( B ( H )) ∼ = B ( H n ). More precisely, for[ x ij ] ∈ M n ( X ), we have k [ x ij ] k n = sup { (Σ ni =1 (cid:13)(cid:13) Σ nj =1 x ij h j (cid:13)(cid:13) H ) / | h j ∈ H , (Σ k h j k H ) / ≤ } . Thus, a concrete operator space carries not just an inherited norm, but theseadditional sequence of matrix norms.An abstract operator space , or simply an operator space is a pair ( X, {k . k n } n ∈ N )consisting of a linear space X and a complete norm k . k n on M n ( X ) for every n ∈ N , such that there exists a linear complete isometry ϕ : X → B ( H ) for VINOD KUMAR P. AND M. S. BALASUBRAMANI some Hilbert space H . The sequence of matrix norms {k . k n } n ∈ N is called an operator space structure on the linear space X . An operator space structure {k . k n } n ∈ N on a Banach space ( X, k . k ) is said to be an admissible operatorspace structure on X , if k . k = k . k . An important type of operator spaces arethose X ⊂ B ( H ) which are isometric (as a Banach space) to a Hilbert space.Such spaces are called Hilbertian operator spaces [13].If X is a Banach space, then any linear isometry from X to B ( H ), forsome Hilbert space H , endows an operator space structure on X . Generally,for a Banach space X , the matrix norms so obtained are not unique. Inother words, a given Banach space has, in general, many realizations as anoperator space. Blecher and Paulsen observed that the set of all operatorspace structures admissible on a given Banach space X admits a minimal anda maximal element. The minimal and the maximal operator space structureson a Banach space were introduced and their dual relations were explored in [4]and further structural properties were investigated in [10] and [11]. In whatfollows, we focus on the subspace and quotient space structure of minimal andmaximal operator spaces.It is known that any subspace of a minimal operator space is again mini-mal, but quotient of a minimal space need not be minimal. A subspace of amaximal operator space need not be maximal. But quotient spaces inheritsthe maximality. We address the following question: If every proper, nontrivialsubspace of an operator space X is minimal (maximal), is X minimal? (max-imal?). We give an example to show that the answer is negative in the case offinite dimensional operator spaces, and show that the answer is affirmative inthe case of infinite dimensional operator spaces.Regarding quotient operator spaces, we prove that, if X is an infinite dimen-sional operator space, and if every quotient of X by a proper closed nontrivialsubspace of X is minimal (maximal), then X is minimal (maximal). We alsogive an example to show that the result is invalid in the case of finite dimen-sional operator spaces.2. Minimal and Maximal Operator Spaces
Let X be a Banach space and X ∗ be its dual space. Let K = Ball ( X ∗ ) bethe closed unit ball of the dual space of X with its weak* topology. Then thecanonical embedding J : X → C ( K ), defined by J ( x )( f ) = f ( x ) , x ∈ X and INIMAL AND MAXIMAL OPERATOR SPACE STRUCTURES ON BANACH SPACES 3 f ∈ K is a linear isometry. Since, subspaces of C ∗ -algebras are operator spaces(by Gelfand-Naimark Theorem), this identification of X induces matrix normson M n ( X ) that makes X an operator space and the matrix norms on X aregiven by k [ x ij ] k n = sup {k [ f ( x ij )] k | f ∈ K } for all [ x ij ] ∈ M n ( X ) and for all n ∈ N .Here k [ f ( x ij )] k indicates the norm of the scalar n × n matrix [ f ( x ij )] viewedas a linear map from C n → C n .The above defined operator space structure on X is called the minimaloperator space structure on X , and we denote this operator space as M in ( X ).Thus, M in ( X ) can be regarded as a space of continuous functions defined onthe closed unit ball of X ∗ . An operator space X is said to be minimal if M in ( X ) = X . The minimal operator space structure of a Banach space ischaracterized by the universal property that for any arbitrary operator space Y , any bounded linear map ϕ : Y → M in ( X ) is completely bounded andsatisfies k ϕ : Y → M in ( X ) k cb = k ϕ : Y → X k . The above described universalproperty implies that
M in ( X ) is indeed the smallest admissible operator spacestructure on a Banach space X . For, if {k . k ′ n } n ∈ N is any other admissibleoperator space structure on X , and if e X denotes the space X with these matrixnorms, then id : e X → M in ( X ) is a linear isometry and k id k cb = k id k = 1.This shows that k . k ′ n dominates the corresponding matrix norms in M in ( X ).It is known that, an operator space is minimal if and only if it is completelyisometric to a subspace of a commutative C ∗ -algebra [5].If X is a Banach space, there is a maximal way to consider it as an operatorspace. For [ x ij ] ∈ M n ( X ), the matrix norms given by k [ x ij ] k = sup k [ ϕ ( x ij )] k where the supremum is taken over all operator spaces Y and all linear maps ϕ ∈ Ball ( B ( X, Y )), define an admissible operator space structure on X . Wedenote this operator space as M ax ( X ) and is called the maximal operator spacestructure on X . An operator space X is said to be maximal if M ax ( X ) = X .By the definition of M ax ( X ), any operator space structure that we can puton X , must be smaller than M ax ( X ).The maximal operator space structure of a Banach space is characterized bythe universal property that for any arbitrary operator space Y , any boundedlinear map ϕ : M ax ( X ) → Y is completely bounded and satisfies VINOD KUMAR P. AND M. S. BALASUBRAMANI k ϕ : M ax ( X ) → Y k cb = k ϕ : X → Y k . Thus, if X and Y are Banach spaces and ϕ ∈ B ( X, Y ), then ϕ is completelybounded and k ϕ k cb = k ϕ k , when considered as a map from M ax ( X ) → Y. To see that the space
M ax ( X ) satisfies the above mentioned property, let ϕ : M ax ( X ) → Y be a bounded linear map. Then u = ϕ k ϕ k ∈ Ball ( X, Y ),and by definition of the matrix norms of
M ax ( X ), k [ u ( x ij )] k is dominatedby k [ x ij ] k M n ( Max ( X )) , for all [ x ij ] ∈ M n ( X ) and for all n ∈ N , showing that k u k cb ≤
1. Thus k ϕ k cb ≤ k ϕ k . Therefore, ϕ : M ax ( X ) → Y is completelybounded and k ϕ k cb = k ϕ k .The above described universal property implies that M ax ( X ) is indeed thelargest admissible operator space structure on a Banach space X . For, if {k . k ′ n } n ∈ N is any other admissible operator space structure on X , and if e X denotes the space X with these matrix norms, then id : M ax ( X ) → e X is alinear isometry and k id k cb = k id k = 1. This shows that k . k ′ n is dominated bythe corresponding matrix norms in M ax ( X ).The following proposition gives characterizations of minimal and maximaloperator spaces up to complete isomorphisms . These characterizations identifylarger classes of operator space structures which are completely isomorphic(need not be completely isometric) to minimal and to maximal operator spaces. Proposition . (i). An operator space X is completely isomorphic to a minimal operator spaceif and only if for any arbitrary operator space Y , any completely bounded linearbijection ϕ : X → Y is a complete isomorphism.(ii). An operator space X is completely isomorphic to a maximal operatorspace if and only if for any arbitrary operator space Y , any completely boundedlinear bijection ϕ : Y → X is complete isomorphism. Proof.
We prove only (i) and (ii) will follow in a similar way. Assume that ϕ : X → Y is a completely bounded linear bijection. Let ψ : X → M in ( Z )be a complete isomorphism. Then by the universal property of minimal op-erator spaces, ψ ◦ ϕ − : Y → M in ( Z ) is completely bounded. Therefore, k ϕ − k cb = k ψ − ◦ ψ ◦ ϕ − k cb ≤ k ψ − k cb k ψ ◦ ϕ − k cb < ∞ . This shows that ϕ : X → Y is a complete isomorphism. For the converse, take Y = M in ( X ) INIMAL AND MAXIMAL OPERATOR SPACE STRUCTURES ON BANACH SPACES 5 and consider the formal identity mapping id : X → M in ( X ). By assumption, id − : M in ( X ) → X is completely bounded, showing that X is completelyisomorphic to M in ( X ). (cid:3) Remark . The above theorem describes the complete isomorphism class of minimal andmaximal operator spaces. Recently, T. Oikhberg [9] proved that the completeisomorphism class of any infinite dimensional operator space has infinite diam-eter with respect to the completely bounded Banach-Mazur distance. i.e., for n ∈ N , ∀ C >
0, and for any infinite dimensional operator space X , there existsan operator space structure ˜ X on X such that the identity map id : X → ˜ X isa complete isomorphism, id ( n ) is an isometry, and d cb ( X, ˜ X ) > C. Measuringthe diameter of the complete isomorphism class of an operator space is stillopen in the case of finite dimensional operator spaces.Let X be an infinite dimensional Banach space. Then operator space struc-tures on X , which are completely isomorphic to M in ( X ) can be constructed asfollows: Choose an operator space Y which is isometric to X and completelyisomorphic to M in ( X ), say v : M in ( X ) → Y be a complete isomorphism(From the above remark, such a choice is always possible). On X , define anew operator space structure, say e X , by setting k [ x ij ] k M n ( e X ) = k [ v ( x ij )] k M n ( Y ) , ∀ [ x ij ] ∈ M n ( X ) and ∀ n ∈ N . Then, e X and Y are completely isometricallyisomorphic, and so M in ( X ) and e X are completely isomorphic. In a similarway, we can construct operator space structures on X which are completelyisomorphic to M ax ( X ).The following theorem describes the dual nature of minimal and maximaloperator space structures on a Banach space. Theorem 2.3 ( [1]) . For any Banach space X , we have M in ( X ) ∗ ∼ = M ax ( X ∗ ) and M ax ( X ) ∗ ∼ = M in ( X ∗ ) completely isometrically. Submaximal Spaces and Q -Spaces From the definition of minimal operator spaces, it is clear that any subspaceof a minimal operator space is again minimal, but a quotient of a minimalspace need not be minimal. An operator space that is a quotient of a minimaloperator space (up to complete isometric isomorphism) is called a Q -space [14].Since an operator space is minimal if and only if it is completely isometric to a VINOD KUMAR P. AND M. S. BALASUBRAMANI subspace of a commutative C ∗ -algebra [5], Q -spaces are precisely the quotientsof subspaces of commutative C ∗ -algebras. Also, the category of Q -spaces isstable under taking quotients and subspaces. Q -spaces were investigated byM. Junge [6] and by Blecher and Le Merdy [2]. Q -spaces need not be minimal, for instance, the space R ∩ C is a Q -space, asit can be identified with the quotient space L ∞ [0 , /S , where S is the subspaceorthogonal to the Rademacher functions [12]. But, R ∩ C is not minimal, andmoreover d cb ( R n ∩ C n , M in ( ℓ n )) = √ n [13], so that R ∩ C is not completelyisomorphic to M in ( ℓ ).Another example for a Q -space, which is not minimal is furnished by thespace of Hankel matrices that can be identified with L ∞ /H ∞ . It can be shownthat it has a subspace which is completely isometric to R ∩ C , so that thespace L ∞ /H ∞ is not minimal [12].Subspace structure of various maximal operator spaces were studied in [8].Subspaces of maximal operator spaces are called submaximal spaces and ingeneral they need not be maximal, i.e., if Y is a subspace of X and if x ij ∈ Y for i, j = 1 , , ..., n , then the norm of [ x ij ] in M n ( M ax ( Y )) can be larger thanthe norm of [ x ij ] as an element of M n ( M ax ( X )). For example, the space R + C is submaximal, as it can be identified as a closed subspace of M ax ( L ) spannedby the Rademacher functions [7]. But R + C is not maximal and moreover d cb ( R n + C n , M ax ( ℓ n )) = √ n [13], so that R + C is not completely isomorphicto M ax ( ℓ ). However, Paulsen obtained the following result. Theorem 3.1 ( [11]) . Let X be an infinite dimensional operator space and x ij ∈ X , for i, j = 1 , , ..., n , then k [ x ij ] k M n ( Max ( X )) = inf { k [ x ij ] k M n ( Max ( Y )) | x ij ∈ Y, Y ⊂ X, f inite dimensional } But quotient spaces inherits the maximality as illustrated in the followingtheorem [13].
Theorem 3.2 ( [13]) . If X is a maximal operator space and Y is a closedsubspace of X , then M ax ( X/Y ) ∼ = M ax ( X ) /Y completely isometrically. Also, if every subspace of
M ax ( X ) is maximal, then any two Banach isomor-phic subspaces of X will be completely isomorphic as subspaces of M ax ( X ).For, if E and F are Banach isomorphic subspaces of X , then E and F arecompletely isomorphic as subspaces of M ax ( X ). In [15], the notion of heredi-tarily maximal spaces is introduced. Hereditarily maximal spaces determine a INIMAL AND MAXIMAL OPERATOR SPACE STRUCTURES ON BANACH SPACES 7 subclass of maximal operator spaces with the property that the operator spacestructure induced on any subspace coincides with the maximal operator spacestructure on that subspace. Also, it is proved that the class of hereditarilymaximal spaces includes all Hilbertian maximal operator spaces. Since ℓ hasa unique operator space structure, ℓ is a maximal operator space and all ofits subspaces are maximal. So, ℓ is an example for a hereditarily maximalspace which is not Hilbertian. The smallest submaximal space structure µ ( X ),admissible on an operator space X is studied in [8] and [16].The following result reveals the natural duality between subspaces and quo-tient spaces. Theorem 3.3 ( [5]) . If Y is a closed subspace of an operator space X , then, ( X/Y ) ∗ ∼ = Y ⊥ and Y ∗ ∼ = X ∗ /Y ⊥ completely isometrically, where Y ⊥ = { f ∈ X ∗ | f ( y ) = 0 , ∀ y ∈ Y } . Let Y be a submaximal space, say Y ⊂ M ax ( X ). Then by Theorem 3.3, Y ∗ ∼ = ( M ax ( X )) ∗ /Y ⊥ . But by using Theorem 2.3, we get ( M ax ( X )) ∗ /Y ⊥ ∼ = M in ( X ∗ ) /Y ⊥ showing that the dual of a submaximal space is a Q -space.Conversely, if Z = M in ( X ) /Y is a Q -space, then by Theorem 3.3, Z ∗ =( M in ( X ) /Y ) ∗ ∼ = Y ⊥ . But here, Y ⊥ = { f ∈ ( M in ( X )) ∗ | f ( y ) = 0 , ∀ y ∈ Y } = { f ∈ M ax ( X ∗ ) | f ( y ) = 0 , ∀ y ∈ Y } so that, Z ∗ is a submaximal space. Thus, the dual of a submaximal space is a Q -space and vice versa.We have noted that Q -spaces need not be minimal. But, using Theorem3.3, we observe that if X is a Hilbertian operator space, then any Q -space in X (i.e., any quotient of M in ( X )) is minimal.Let X be a Hilbertian operator space and Y be a closed subspace of X . Let id : M in ( Y ⊥ ) → M in ( X ) be the formal identity mapping, and π : M in ( X ) → M in ( X ) /Y be the quotient map.Then π ◦ id : M in ( Y ⊥ ) → M in ( X ) /Y is a complete contraction. Since, X/Y is isometric to Y ⊥ , M in ( X/Y ) is completely isometric to
M in ( Y ⊥ ), sothat k [ x ij + Y ] k M n ( Min ( X ) /Y ) ≤ k [ x ij + Y ] k M n ( Min ( X/Y )) . But, by definition of minimal operator spaces,
VINOD KUMAR P. AND M. S. BALASUBRAMANI k [ x ij + Y ] k M n ( Min ( X/Y )) ≤ k [ x ij + Y ] k M n ( Min ( X ) /Y ) . Thus,
M in ( X ) /Y is completely isometrically isomorphic to M in ( X/Y ). Remark . There are non-Hilbertian operator spaces X for which all Q -spaces in X areminimal. For instance, ℓ is a minimal operator space, which is not Hilbertianand all Q -spaces in ℓ are minimal. Identification of a subclass of minimaloperator spaces with the property that the operator space structure inducedon any quotient space coincides with the minimal operator space structure onthat quotient space is still open.4. Main Results
We have noted that a subspace of a minimal operator space is minimal,whereas a subspace of a maximal space need not be maximal. On the otherhand, let us consider the following question: If every proper, nontrivial sub-space of an operator space X is minimal (maximal), is X minimal? (maxi-mal?) The answer to this question is No. For instance, if X is of dimension2, and if e X is any operator space structure on X such that e X = M in ( X )( resp. e X = M ax ( X )) (such a space exists since any Banach space of di-mension greater than 2 has more than one quantization, i.e., the space hasmore than one admissible operator space structure [13].) Then any proper,nontrivial subspace of e X will be of dimension 1, and so is minimal (resp. max-imal). (Since there is only one operator space of dimension 1, up to completeisometric isomorphism.)But in the case of infinite dimensional operator spaces, we have an affirma-tive answer. Theorem 4.1.
Let X be an infinite dimensional operator space. If everyfinite dimensional subspace of X is minimal (maximal), then X is minimal(maximal).Proof. First, assume that every finite dimensional subspace of X is minimal. Let[ x ij ] ∈ M n ( X ). Then, [ x ij ] ∈ M n ( E ), where E = span { x ij } and the dimensionof E ≤ n . By assumption, E is minimal and by using the Hahn-Banach INIMAL AND MAXIMAL OPERATOR SPACE STRUCTURES ON BANACH SPACES 9 theorem, we have k [ x ij ] k M n ( X ) = k [ x ij ] k M n ( E ) = k [ x ij ] k M n ( Min ( E )) = sup {k [ f ( x ij )] k | f ∈ Ball ( E ∗ ) }≤ sup {k [ f ( x ij )] k | f ∈ Ball ( X ∗ ) } = k [ x ij ] k M n ( Min ( X )) Since
M in ( X ) is the smallest operator space structure on X , this shows that k [ x ij ] k M n ( X ) = k [ x ij ] k M n ( Min ( X )) and hence X is minimal.Now to prove the maximal case, by Theorem 3.1, for any [ x ij ] ∈ M n ( X ), wehave k [ x ij ] k M n ( Max ( X )) = inf {k [ x ij ] k M n ( Max ( Y )) | x ij ∈ Y, Y ⊂ X, f inite dimensional } = inf {k [ x ij ] k M n ( Y ) | x ij ∈ Y, Y ⊂ X, f inite dimensional }≤ k [ x ij ] k M n ( X ) Since
M ax ( X ) is the largest operator space structure on X , this shows that k [ x ij ] k M n ( Max ( X )) = k [ x ij ] k M n ( X ) and hence X is maximal. (cid:3) We have noted that quotients of minimal operator spaces need not be mini-mal, whereas quotients of maximal operator spaces are maximal. In the case ofquotient spaces of an infinite dimensional operator space, we have the followingresult.
Theorem 4.2.
Let X be an infinite dimensional operator space.(i). If every quotient of X by a proper closed nontrivial subspace of X isminimal, then X is minimal.(ii). If every quotient of X by a proper closed nontrivial subspace of X ismaximal, then X is maximal. For proving this, we make use of the following theorems.
Theorem 4.3 ( [5]) . If X is any operator space and [ x ij ] ∈ M n ( X ) , thereexists a complete contraction ϕ : X → M n such that k ϕ ( n ) ([ x ij ]) k = k [ x ij ] k . Theorem 4.4 ( [3]) . Let X and Z be operator spaces. If φ : X → Z iscompletely bounded, and if Y is a closed subspace of X contained in ker ( φ ) ,then the canonical map ˜ ϕ : X/Y → Z induced by ϕ is also completely bounded, with k ˜ ϕ k cb = k ϕ k cb . If Y = ker ( ϕ ) , then ϕ is a complete quotient map if andonly if ˜ ϕ is a completely isometric isomorphism.Proof of Theorem 4.2. We first prove the second part. Let [ x ij ] ∈ M n ( X ). By Theorem 4.3, wehave k [ x ij ] k M n ( X ) = sup {k [ ϕ ( x ij )] k | ϕ : X → M n , k ϕ k cb ≤ } (1)Since X is infinite dimensional, and the range of ϕ is finite dimensional, ϕ hasa nontrivial kernel, ker ( ϕ ). Let e ϕ : X/ker ( ϕ ) → M n be the canonical mapdefined by e ϕ ( x + ker ( ϕ )) = ϕ ( x ). Then by Theorem 4.4, k ϕ k cb = k e ϕ k cb = k e ϕ k = k ϕ k , where the last but one equality follows from the assumption that X/ker ( ϕ ) is maximal. Thus in equation (1), we can replace k ϕ k cb ≤ k ϕ k ≤
1, so that we have k [ x ij ] k M n ( X ) = sup {k [ ϕ ( x ij )] k | ϕ : X → M n , k ϕ k ≤ } (2)Applying equation (1) to the operator space M ax ( X ), and using the universalproperty of M ax ( X ) and the equation (2), we obtain k [ x ij ] k M n ( Max ( X )) = sup {k [ ϕ ( x ij )] k | ϕ : M ax ( X ) → M n , k ϕ k cb ≤ } = sup {k [ ϕ ( x ij )] k | ϕ : M ax ( X ) → M n , k ϕ k ≤ } = sup {k [ ϕ ( x ij )] k | ϕ : X → M n , k ϕ k ≤ } = k [ x ij ] k M n ( X ) This shows that X = M ax ( X ) and so X is maximal.For proving the first part, we note that for any [ x ij ] ∈ M n ( X ), k [ x ij ] k M n ( X ) = sup {k [ x ij + Y ] k M n ( X/Y ) | Y ⊂ X } (3)Now, we assume that every quotient of X by a proper closed nontrivial sub-space of X is minimal.Let [ x ij ] ∈ M n ( X ). We claim that k [ x ij ] k M n ( X ) = sup {k [ x ij + Y ] k M n ( X/Y ) | Y ⊂ X, dim ( X/Y ) < ∞} From equation (3), we have k [ x ij ] k M n ( X ) ≥ sup {k [ x ij + Y ] k M n ( X/Y ) | Y ⊂ X, dim ( X/Y ) < ∞} INIMAL AND MAXIMAL OPERATOR SPACE STRUCTURES ON BANACH SPACES11
By Theorem 4.3, there exists a complete contraction ϕ : X → M n such that k ϕ ( n ) ([ x ij ]) k = k [ x ij ] k M n ( X ) . Let Y = ker ( ϕ ), then X/Y is finite dimensionaland let ˜ ϕ be the canonical map. Then, k [ x ij ] k M n ( X ) = k ϕ ( n ) ([ x ij ]) k = k ˜ ϕ ( n ) ([ x ij + Y ]) k ≤ k [ x ij + Y ] k M n ( X/Y ) . This implies, k [ x ij ] k M n ( X ) ≤ sup {k [ x ij + Y ] k M n ( X/Y ) | Y ⊂ X, dim ( X/Y ) < ∞} This proves the claim.Now let id : M in ( X ) → X be the formal identity map. Let Y be a subspaceof X such that dim ( X/Y ) < ∞ and π : X → X/Y be the quotient mapping.Then, e id = π ◦ id : M in ( X ) → X/Y is given by e id ( x ) = x + Y . Since, byassumption, X/Y is minimal and by using the universal property of minimalspaces, e id is completely bounded and k e id k cb = k e id k ≤ k π kk id k ≤
1. Thisshows that e id is completely contractive. Therefore, k [ x ij ] k M n ( X ) = sup {k [ x ij + Y ] k M n ( X/Y ) | Y ⊂ X, dim ( X/Y ) < ∞} = sup {k [ e id ( x ij )] k M n ( X/Y ) | Y ⊂ X, dim ( X/Y ) < ∞}≤ k [ x ij ] k M n ( Min ( X )) . This shows that X = M in ( X ), and so X is minimal. (cid:3) Remark . If X is finite dimensional, the above result need not be true. For instance,if X is an operator space of dimension 2 which is not minimal (maximal),then every quotient of X by a proper, nontrivial closed subspace will be ofdimension 1 and hence is minimal (maximal). Acknowledgements . The first author acknowledge the financial supportby University Grants Commission of India, under Faculty Development Pro-gramme. The first author is grateful to Prof. Gilles Pisier, Prof. Vern Paulsenand Prof. Eric Ricard for having many discussions on this subject.
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Corresponding Author : Vinod Kumar. P
Department of Mathematics,, Thunchan Memorial Govt. College, Tirur,,Kerala , India.
E-mail: [email protected]