Introduction. In C -algebra theory, the minimal and maximal tensor products (denoted by A 1 min A 2 and A 1 max A 2 ) of two C -algebras A 1 ; A 2, pl

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1 The \maximal" tensor product of operator spaces by Timur Oikhberg Texas A&M University College Station, TX 77843, U. S. A. and Gilles Pisier* Texas A&M University and Universite Paris 6 Abstract. In analogy with the maximal tensor product of C -algebras, we dene the \maximal" tensor product E 1 E 2 of two operator spaces E 1 and E 2 and we show that it can be identied completely isometrically with the sum of the two Haagerup tensor products: E 1 h E 2 + E 2 h E 1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E E = E min E holds isometrically i E = R n or E = C n (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F min E = F E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space AMS Classication Numbers: 47 D 15, 47 D 25, 46 M 05. * Partially supported by the N.S.F. 1

2 Introduction. In C -algebra theory, the minimal and maximal tensor products (denoted by A 1 min A 2 and A 1 max A 2 ) of two C -algebras A 1 ; A 2, play an important r^ole, in connection with \nuclearity" (a C -algebra A 1 is nuclear if A 1 min A 2 = A 1 max A 2 for any A 2 ). See [T] and [Pa] for more information and references on this. In the recently developed theory of operator spaces [ER1-6, Ru1-2, BP1, B1-3], some specic new versions of the injective and projective tensor products (going back to Grothendieck for Banach spaces) have been introduced. The \injective" tensor product of two operator spaces E 1 ; E 2 coincides with the minimal (or spatial) tensor product and is denoted by E 1 min E 2. Another tensor product of paramount importance for operator spaces is the Haagerup tensor product, denoted by E 1 h E 2 (cf. [CS1-2, PaS, BRS, BS]). Assume given two completely isometric embeddings E 1 A 1, E 2 A 2. Then E 1 min E 2 (resp. E 1 h E 2 ) can be identied with the closure of the algebraic tensor product E 1 E 2 in A 1 min A 2 (resp. in the \full" free product C -algebra A 1 A 2, see [CES]). (The \projective" case apparently cannot be described in this fashion and will not be considered here.) It is therefore tempting to study the norm induced on E 1 E 2 by A 1 max A 2. When A i = B(H i ) (i = 1; 2) the resulting tensor product is studied in [JP] and denoted by E 1 M E 2. See also [Ki] for other tensor products. In the present paper, we follow a dierent route: we work in the category of (a priori nonself-adjoint) unital operator algebras, and we use the maximal tensor product in the latter category (already considered in [PaP]), which extends the C -case. The resulting tensor product, denoted by E 1 E 2 is the subject of this paper. A brief description of it is as follows: we rst introduce the canonical embedding of any operator space E into an associated \universal" unital operator algebra, denoted by OA(E), then we can dene the tensor product E 1 E 2 as the closure of E 1 E 2 in OA(E 1 ) max OA(E 2 ). Our main result is Theorem 1 which shows that E 1 E 2 coincides with a certain \symmetrization" of the Haagerup tensor product. We apply this (see Corollary 10 and Theorem 16) to nd which spaces E 1 have the property that E 1 E 2 = E 1 min E 2 for all operator spaces E 2. We give two proofs of the main result, in the bilinear case. The second one (in x2) is shorter, but we feel the rst proof is more instructive, and easier to generalize to more general situations (cf. Remark 20). Besides, each proof seems to yield 1

3 a dierent n-linear extension for n > 2, not obtainable (as far as we can see) by the other argument (see the extension of Theorem 1, stated after its proof, and Theorem 19). We refer the reader to the book [Pa] for the precise denitions of all the undened terminology related to operator spaces and complete boundedness, and to [KaR, T] for operator algebras in general. We recall only that an \operator space" is a closed subspace E B(H) of the C -algebra of all bounded operators on a Hilbert space H. We will use freely the notion of a completely bounded (in short c:b:) map u: E 1! E 2 between two operator spaces, as dened e.g. in [Pa]. We denote by kuk cb the corresponding norm and by cb(e 1 ; E 2 ) the Banach space of all c.b. maps from E 1 to E 2. We will denote by A 0 the commutant of a subset A B(H). Let E 1 ; : : : ; E n be a family of operator spaces. Let i : E i! B(H) be complete contractions (i = 1; 2; : : : ; n). We denote by 1 : : : n : E 1 E n! B(H) the linear map taking x 1 x n to the operator 1 (x 1 ) 2 (x 2 ) : : : n (x n ). We dene the norm k k on E 1 E n as follows: (1) 8x 2 E 1 E n kxk = sup k 1 : : : n (x)k B(H) where the supremum runs over all possible H and all n-tuples ( i ) of complete contractions as above, with the restriction that we assume that for any i 6= j, the range of i commutes with the range of j. We will denote by (E 1 E 2 E n ) the completion of E 1 E n for this norm. P In the particular case n = 2, we denote this simply by E 1 E 2, so we have for any x = x 1 i x2 i 2 E 1 E 2 kxk = supfk X 1 (x 1 i ) 2 (x 2 i )k B(H) g where the supremum runs over all possible pairs ( 1 ; 2 ) of complete contractions (into some common B(H)) with commuting ranges, i.e. such that 1 (x 1 ) 2 (x 2 ) = 2 (x 2 ) 1 (x 1 ) for all x 1 2 E 1, x 2 2 E 2. The space (E 1 E n ) can obviously be equipped with an operator space structure associated to the embedding (2) J: (E 1 E n )! M B(H ) B( M H ) 2

4 where the direct sum runs over all n-tuples = ( 1 ; : : : ; n ) with i : E i! B(H ) such that k i k cb 1 and the i 's have commuting ranges. Here as well as throughout this paper, we observe that we can always restrict ourselves in the above direct sum to the case when the cardinal of H is majorized by a suitably xed cardinal, thus eliminating set theoretic objections. We note in passing that if we dene ^ i : E i! L B(H ) B( L H ) by ^ i (x) = i (x), then we have J(x) = ^ 1 :::^ n and the maps ^ i have commuting ranges. Thus we can now unambiguously refer to (E 1 E n ), and in particular to E 1 E 2 as operator spaces. We will give more background on operator spaces and c.b. maps below. For the moment, we merely dene a \complete metric surjection": by this we mean a surjective mapping Q: E 1! E 2 between two operator spaces, which induces a complete isometry from E 1 = ker(q) onto E 2. To state our main result, we also need the notion of `1-direct sum of two operator spaces E 1 ; E 2 : this is an operator space denoted by E 1 1 E 2. The norm on the latter space is as in the usual `1-direct sum, i.e. we have k(e 1 ; e 2 )k E1 1 E 2 = ke 1 k E1 + ke 2 k E2 but the operator space structure is such that for any pair u 1 : E 1! B(H), u 2 : E 2! B(H) of complete contractions, the mapping (e 1 ; e 2 )! u 1 (e 1 ) + u 2 (e 2 ) is a complete contraction from E 1 1 E 2 to B(H). The simplest way to realize this operator space E 1 1 E 2 as a subspace of B(H) for some H is to consider the collection I of all pairs p = (u 1 ; u 2 ) as above with H = H p (say) and to dene the embedding dened by J(e 1 ; e 2 ) = L (u 1 ;u 2 )2I J: E 1 1 E 2! B M 1 H p A p2i [u 1 (e 1 ) + u 2 (e 2 )]. Then, we may as well dene the operator space structure of E 1 1 E 2 as the one induced by the isometric embedding J. In other words, E 1 1 E 2 can be viewed as the \maximal" direct sum for operator spaces, in accordance with the general theme of this paper. 3

5 We will denote by E 1 E 2 the linear tensor product of two vector spaces and by v! t v the transposition map, i.e. for any v = P x i y i in E 1 E 2, we set t v = P y i x i. The identity map on a space E will be denoted by Id E. We will denote by E 1 h E 2 the Haagerup tensor product of two operator spaces for which we refer to [CS1-2, PaS]. Convention: We reserve the term \morphism" for a unital completely contractive homomorphism u: A! B between two unital operator algebras. x1. Main results. Our main result is the following one. Theorem 1 (bilinear case). Let E 1, E 2 be two operator spaces. Consider the mapping Q: (E 1 h E 2 ) 1 (E 2 h E 1 )! E 1 E 2 dened on the direct sum of the linear tensor products by Q(u v) = u + t v. Then Q extends to a complete metric surjection from (E 1 h E 2 ) 1 (E 2 h E 1 ) onto E 1 E 2. In particular, for any u in E 1 E 2, we have: kuk < 1 i there are v; w in E 1 E 2 such that u = v + w and kvk E1 h E 2 + k t wk E2 h E 1 < 1: In the terminology of [P2], the preceding statement means that E 1 E 2 is completely isometric to the \sum" (in the style of interpolation theory, see [P2]) E 1 h E 2 + E 2 h E 1 (in analogy with R + C). Remark 2. We rst recall a simple consequence of the Cauchy-Schwarz inequality: for any a 1 ; : : : ; a n, b 1 ; : : : ; b n in a C -algebra A, we have P a i b i P a i a P i 1=2 b i b i 1=2 : Hence if a i b i = b i a i we also have P a i b i P a i a P i 1=2 b i b i 1=2 : >From these it is easy to deduce that the above map Q is completely contractive. The main idea of the proof of Theorem 1 is to use the universal unital operator algebras of operator spaces as initiated in [P4], to relate their free product with their \maximal" tensor product, and to use the appearance of the Haagerup tensor product inside the free product. Let E be an operator space. Let P T (E) = C E (E E) be its tensor algebra, so that any x in T (E) is a sum x = x n with x n 2 E E (n times) with x n = 0 for all n suciently large. For each linear : E! B(H) we denote by T (): T (E)! B(H) the unique unital homomorphism extending. 4

6 Let C be the collection of all : E! B(H ) with kk cb 1. We dene an embedding M! J: T (E)! B H 2C by setting J(x) = M 2C T ()(x): Then J is a unital homomorphism. We denote by OA(E) the unital operator algebra obtained by completing J(T (E)). We will always view T (E) as a subset of OA(E), so we identify x and J(x) when x 2 T (E). Observe that the natural inclusion E! OA(E) is obviously a complete isometry. More generally, the natural inclusion of E E (n times) into OA(E) denes a completely isometric embedding of E h h E into OA(E). (This follows from a trick due to Varopoulos, and used by Blecher in [B1], see [P4] for details). The algebra OA(E) is characterized by the following universal property: for any : E! B(H) with kk cb 1, there is a unique morphism ^: OA(E)! B(H) extending (here we view E as embedded into OA(E) in the natural way). See [Pes] for the self-adjoint analogue. We now turn to the maximal tensor product in the category of unital operator algebras. This is dened in [PaP], so we only briey recall the denition: Let A 1 ; A 2 be two unital operator algebras. For any pair = ( 1 ; 2 ) of morphisms i : A i! B(H ) with commuting ranges, we denote by 1 2 the morphism from A 1 A 2 to B(H ) which takes a 1 a 2 to 1 (a 1 ) 2 (a 2 ). Then we consider the embedding M! J: A 1 A 2! B H dened by J(x) = L 1 2 (x). We dene kxk max = sup k 1 2 (x)k and we denote by A 1 max A 2 the completion of A 1 A 2 for this norm. We will consider A 1 max A 2 as a unital operator algebra, using the isometric embedding J just dened. 5

7 Lemma 3. The natural inclusion of E 1 E 2 into OA(E 1 ) max OA(E 2 ) is a completely isometric embedding. We now turn to the free product in the category of unital operator algebras. Let A 1 ; A 2 be two such algebras and let F be their algebraic free product as unital algebras (i.e. we identify the units and \amalgamate over C"). For any pair u = (u 1 ; u 2 ) of morphisms, as follows u i : A i! B(H u ) (i = 1; 2), we denote by u 1 u 2 : F! B(H u ) the unital homomorphism extending u 1 ; u 2 to the free product. Then we consider the embedding J: F! M u B(H u ) L dened by J(x) = [u 1 u 2 (x)], for all x in F. Note that J is a unital homomorphism. u We dene the free product A 1 A 2 (in the category of unital operator algebras) as the closure of J(F). Actually, we will identify F with J(F) and consider that A 1 A 2 is the completion of F relative to the norm induced by J. Moreover, we will consider A 1 A 2 as a unital operator algebra equipped with the operator space structure induced by J. It is easy to see that A 1 A 2 is characterized by the (universal) property that for any pair of morphisms u i : A i! B(H) (i = 1; 2) there is a unique morphism from A 1 A 2 to B(H) which extends both u 1 and u 2. We will use several elementary facts which essentially all follow from the universal properties of the objects we have introduced. Lemma 4. OA(E 1 ) OA(E 2 ) ' OA(E 1 1 E 2 ) completely isometrically. Remark. The \functor" E! OA(E) is both injective and projective: indeed, if E 2 E 1 is a closed subspace then the associated maps j: OA(E 2 )! OA(E 1 ) and q: OA(E 1 )! OA(E 1 =E 2 ) are respectively a complete isometry and a complete metric surjection.(to check the projectivity, let B = OA(E 1 )= ker(q). Note that we have a complete contraction E 1! OA(E 1 )! B which vanishes on E 2, whence a complete contraction E 1 =E 2! B, which extends to a completely contractive morphism OA(E 1 =E 2 )! B.) We will use the following fact which was observed by Blecher and Paulsen ([BP2, 4.4]). 6

8 Lemma 5. Let A 1 ; A 2 be two unital operator algebras. Then the natural morphism Q: A 1 A 2! A 1 max A 2 is a complete metric surjection. More precisely, the restriction of Q to the algebraic free product F denes a complete isometry between F= ker(q jf ) and A 1 A 2 A 1 max A 2. The next lemma (already used in [B1]) is elementary. Lemma 6. Consider an element x = x 0 +x 1 + +x n + in T (E), with x n 2 E E (n times). Then the mapping x! x n denes a completely contractive projection on OA(E). Proof. Let m denote the normalized Haar measure on the unidimensional torus T. For z in T, let x(z) = P n0 z n x n. By denition of OA(E), we clearly have kx(z)k = kxk; hence kx n k = Z z?n x(z)m(dz) kxk: This shows that x! x n is a contractive linear projection. The argument for complete contractivity is analogous and left to the reader. The next result, which plays an important r^ole in the sequel, might be of independent interest. Lemma 7. Let E 1 ; E 2 ; F 1 ; F 2 be two operator spaces. Let X = (E 1 1 E 2 ) h (F 1 1 F 2 ). With the obvious identications, we may view E 1 F 2 + E 2 F 1 as a linear subspace of X. Let S be its closure in X. Then we have S ' (E 1 h F 2 ) 1 (E 2 h F 1 ) completely isometrically. Moreover, the natural (coordinatewise) projection P : X! S is completely contractive. Proof. Obviously we have completely contractive natural inclusions E 1 h F 2! X and E 2 h F 1! X, whence a natural inclusion (E 1 h F 2 ) 1 (E 2 h F 1 )! X. To show that this is completely isometric it clearly suces to show that S has the \universal" property characteristic of the 1 -direct sum. Equivalently, it suces to show that every completely 7

9 contractive mapping : (E 1 h F 2 ) 1 (E 2 h F 1 )! B(H) denes a completely contractive mapping from S to B(H) (then we may apply this when is a completely isometric embedding). So let be such a map. Clearly, we can assume that (xy) = u(x)+v(y) with u: E 1 h F 2! B(H), v: E 2 h F 1! B(H) such that kuk cb 1, kvk cb 1. By the factorization of c:b:-bilinear maps ([CS, PaS]) we can further write u(x 1 x 2 ) = u 1 (x 1 )v 2 (x 2 ) and v(y 2 y 1 ) = u 2 (y 2 )v 1 (y 1 ) where u i : E i! B(H) and v i : F i! B(H) are all completely contractive. Let us then dene : E 1 1 E 2! B(H) and : F 1 1 F 2! B(H) by (x 1 x 2 ) = u 1 (x 1 ) + u 2 (x 2 ) and (x 1 x 2 ) = v 1 (x 1 ) + v 2 (x 2 ). By denition of 1, these maps are still complete contractions. Moreover, we have for any z in S, say z = x + y with x 2 E 1 F 2, y 2 E 2 F 1 (z) = u(x) + v(y) = (z): hence we conclude that admits an extension ~ (namely ~ = ) dened on the whole of X with k~k cb(x;b(h)) kk cb kk cb 1; a fortiori kk cb(s;b(h)) 1. This establishes the rst part. To check that P is completely contractive, just observe that if we denote by Q i (resp. R i ) the i-th canonical projection on E 1 1 E 2 (resp. F 1 1 F 2 ), then P is equal to the average of (" 1 Q 1 +" 2 Q 2 )(" 2 R 1 +" 1 R 2 ) over all the choices of signs (" i ). Lemma 8. Let A 1 ; : : : ; A n be unital operator algebras. Then the natural mapping from A 1 A 2 A n to A 1 A 2 A n denes a completely isometric embedding of A 1 h A 2 h A n into A 1 A 2 A n. Proof. In essence, this is proved in [CES], but only for the non-unital free product. The unital case is done in detail in [P3] so we skip it. Proof of Theorem 1. Consider u in E 1 E 2 with kuk < 1. Let F be, as before, the algebraic free product of OA(E 1 ) and OA(E 2 ). By Lemma 5, we have kuk OA(E1 ) max OA(E 2 ) < 1, hence by Lemma 4, there is an element ^u in F with k^uk < 1 such that Q(^u) = u. By 8

10 Lemma 3 we may write as well k^uk OA(E1 1 E 2 ) < 1: Let us write ^u = ^u 0 + ^u 1 + ^u 2 + where ^u d 2 (E 1 1 E 2 ) h h (E 1 1 E 2 ) (d times). By Lemma 6 we have k^u d k < 1. Let z 1 ; z 2 be complex numbers with jz i j 1. There is a unique morphism zi : OA(E i )! OA(E i ) extending z i Id Ei. We will use the morphisms z1 z2 acting on OA(E 1 ) max OA(E 2 ) and z1 z2 acting on OA(E 1 ) OA(E 2 ): Note that we trivially have the following relation: [ z1 z2 ] Q = Q [ z1 z2 ]: It follows that z 1 z 2 u = Q[ z1 z2 (^u)]: Hence identifying the coecient of z 1 z 2 on the right hand side we obtain u = Q[~u] where ~u is in the subspace S = span[e 1 E 2 + E 2 E 1 ] (E 1 1 E 2 ) h (E 1 1 E 2 ) considered in Lemma 7, and where we view (E 1 1 E 2 ) h (E 1 1 E 2 ) as the subspace of OA(E 1 1 E 2 ) formed of all terms of degree 2, according to Lemma 8. Hence by Lemma 7, we conclude that ~u can be written as v + w with v 2 E 1 E 2 and w 2 E 2 E 1 such that kvk E1 h E 2 + kwk E2 h E 1 < 1: This shows that 8u 2 E 1 E 2 with kuk < 1 there are v; w as above with u = v + t w. Thus the natural mapping is a metric surjection from E 1 h E 2 1 E 2 h E 1 onto E 1 E 2. To show that this is a complete surjection, one simply repeats the argument with M n (E 1 E 2 ) instead of E 1 E 2. We leave the easy details to the reader. Using the same techniques, one can prove the following isomorphic generalization of Theorem 1 for more than two spaces: Theorem 1 (general case). Let E 1 ; : : : ; E n be an n-tuple of operator spaces. Let L be the `1-direct sum of the family (E (1) h : : : h E (n) ) indexed by all permutations of f1; 2; :::; ng. Let Q be the natural completely contractive mapping from L to (E 1 E n ), and let : L= ker(q)! (E 1 E n ) be the canonically associated map. Then is a complete isomorphism satisfying kk cb 1 and k?1 k cb (n? 1)!. 9

11 Proof. We will use the following fact. Let E ij be a family of operator spaces. Let X = (E 11 1 : : : 1 E 1n ) h : : : h (E k1 1 : : : 1 E kn ); Y = (E 11 h : : : h E k1 ) 1 : : : 1 (E 1n h : : : h E kn ); S = (E 11 h : : : h E k1 ) : : : (E 1n h : : : h E kn ),! X; then Y ' S completely isometrically (here S is equipped with the operator space structure induced by X). Moreover, the natural (coordinatewise) projection P : X! S is completely contractive. This follows from Lemma 7 by iteration. We skip the details. Let us now prove the preceding statement. Let E = E 1 1 : : : 1 E n. Let G n be the set of all permutations of f1; 2; :::; P ng. Let X = E h : : : h E (n times) and let X be the subspace P dened by = 2G n E (1) : : : E (n). For any subset A G n, we set (A) = 2A E (1) : : :E (n). We equip and (A) with the operator space structure induced by X. Let us say that A is admissible if for each i the set f(i) j 2 Ag is the whole of f1; 2; :::; ng. By the same argument as above for n = 2, the natural product map Q:! (E 1 E n ) is a complete metric surjection. We clearly have a natural completely contractive map : L!. To conclude, it suces to prove that is a complete isomorphism with k?1 k cb (n? 1)!. To prove this we use a partition of G n into (n? 1)! admissible subsets, each with n elements (for instance the left cosets associated with the subgroup formed of all the n cyclic permutations). Indeed, by the preceding fact, for any admissible A, the restriction of?1 to (A) is a complete isometry, and the natural projection from to (A) is completely contractive. This implies k?1 k cb (n? 1)!. Remark. We do not believe that the isometric analog of Theorem 1 holds true for n > 2. While we do not have an explicit example, at least we have checked that the map appearing above is not completely isometric in general. Remark. If X; Y are Banach spaces, and if v 2 Y X, let us denote by 2 (v) the norm of factorization through Hilbert space of the linear map ~v: Y! X associated to v. This is a classical notion in Banach space theory (cf. e.g. [P1, p. 21]). Note that Theorem 1 10

12 obviously implies that for any v in E 1 E 2 (E 1 ; E 2 being arbitrary operator spaces), we have (3) 2 (v) kvk : Remark. Note that (E 1 E 2 ) E 3 = E 1 h E 2 h E 3 + E 3 h E 1 h E 2 + E 2 h E 1 h E 3 + E 3 h E 2 h E 1 ; but the above expression does not necessarily coincide with (E 1 E 2 E 3 ), and moreover the -tensor product is not associative, in sharp contrast with the Haagerup one (or with the maximal tensor product for unital operator algebras). In particular, in general the natural mapping from E 1 (E 2 E 3 ) into (E 1 E 2 ) E 3 is unbounded (and actually only makes sense on the linear tensor products). All this follows from the counterexample below, kindly communicated to us by C. Le Merdy. Let X be a Banach space and let K be the algebra of all compact operators on `2. Take E 1 = C, E 2 = R, and E 3 = min(x) in the sense of [BP1]. Assume that we have a bounded map (E 1 E 2 E 3 )! (E 1 E 2 ) E 3 : Then a fortori we have a bounded map E 1 (E 2 E 3 )! (E 1 E 2 ) E 3, and consequently a bounded map E 1 h E 3 h E 2! (E 1 E 2 ) E 3 : But then C h E 3 h R = K min E 3 completely isometrically (see [BP1, ER4]), hence it is isometric to the (Banach space theoretic) injective tensor product K X. Moreover, since R h C is isometric to K, by Theorem 1 we have C R = K isometrically. Thus, we would have a bounded map from K X to (C R) E 3, and this would imply by (3), that for some constant C, for all v in K X, we would have 2 (v) Ckvk _. However, it is well known that this fails at least for some Banach space X (take for example X = `1 and v = P n 1 e ii e i, so that v represents an isometric embedding of `1n into K, then kvk _ = 1 and 2 (v) = p n, cf. [P1, p. 48] for more on this question). We now give several consequences and reinterpret Theorem 1, in terms of factorization. The following notation will be convenient. Let X be an operator space. We will say that a linear map u: E 1! E 2 between operator spaces factors through X if there are 11

13 maps w: E 1! X and v: X! E 2 such that u = vw. We will denote by? X (E 1 ; E 2 ) the class of all such mappings and moreover we let X (u) = inffkvk cb kwk cb g where the inmum runs over all possible such factorizations. Let us denote by K the C -algebra of all compact operators on `2, with its natural \basis" (e ij ). The preceding notation applies in particular when X = K and gives us the space? K (E 1 ; E 2 ). In the case X = K, it is easy to check that K is a norm with which? K (E 1 ; E 2 ) becomes a Banach space. We wish to relate the possible factorizations of a map through K with its possible factorizations through two specic subspaces of K, namely the row and column Hilbert spaces dened by R = span(e 1j j j = 1; 2; : : :) C = span(e i1 j i = 1; 2; : : :): Clearly these subspaces of K admit a natural completely contractive projection onto them (namely x! e 11 x is a projection onto R, and x! xe 11 one onto C). Therefore we have K (Id R ) = 1 and K (Id C ) = 1. A fortiori any linear map u: E 1! E 2 which factors either through R or through C factors through K and we have K (u) R (u) and K (u) C (u): A fortiori, if u = v + w for some v: E 1! E 2 and w: E 1! E 2, we have K (u) R (v) + C (w): Note that if v and w are of nite rank, then with the obvious identications, we have R (v) = kvk E1 h E 2 and C (w) = k t wk E2 h E 1 : Thus, from Theorem 1 we deduce: Corollary 9. Let E 1 ; E 2 be operator spaces. Consider u in E 1 E 2 and let ~u: E 1! E 2 be the associated nite rank operator. Then we have (4) K (~u) kuk 12

14 Corollary 10. Let E be an n-dimensional operator space. Let i E 2 E E be associated to the identity of E and let (E) = ki E k : Then (5) maxf K (Id E ); K (Id E )g (E): Moreover (E) = 1 i either E = R n or E = C n (completely isometrically). Proof. Note that (5) clearly follows from (4). Assume that (E) = 1. Then by Theorem 1 (and an obvious compactness argument) we have a decomposition Id E = u 1 + u 2 with (6) R (u 1 ) + C (u 2 ) = 1: In particular, this implies that 2 (Id E ) = 1, (where 2 (:) denotes the norm of factorization through Hilbert space, see e.g. [P1, chapter 2] for more background) whence that E is isometric to `n2 (n = dim E). Moreover, for any e in the unit sphere of E we have 1 = kek ku 1 (e)k + ku 2 (e)k ku 1 k + ku 2 k R (u 1 ) + C (u 2 ) 1: Therefore we must have (7) ku 1 (e)k = ku 1 k = R (u 1 ) and ku 2 (e)k = ku 2 k = C (u 2 ): Let i = ku i k so that (by (6)) = 1. Assume that both 1 > 0 and 2 > 0. We will show that this is impossible if n > 1. Indeed, then U i = ( i )?1 u i (i = 1; 2) is an isometry on `n2, such that, for any e in the unit sphere of E, we have e = 1 U 1 (e) + 2 U 2 (e): By the strict convexity of `n2, this implies that U 1 (e) = U 2 (e) = e for all e. Moreover, by (7) we have R (U 1 ) = 1 and C (U 2 ) = 1. This implies that E = R n and E = C n completely isometrically, which is absurd when n > 1. Hence, if n > 1, we conclude that either 1 = 0 or 2 = 0, which implies either C (Id E ) = 1 or R (Id E ) = 1, equivalently either E = C n or E = R n completely isometrically. The remaining case n = 1 is trivial. Remark. Alternate proof: if (E) = 1, then, using (5) and the relexivity of E, we see that both for E and E the identity factors through K, therefore E is an injective operator 13

15 space as well as its dual. Now, in [Ru2], Ruan gives the complete list of the injective operator subspaces of nite dimensional C -algebras (see also [Ro] for more on this theme). Running down this list, and using an unpublished result of R. Smith saying that a nite dimensional injective operator space is completely contractively complemented in a nite dimensional C -algebra (see [B2]), we nd that R n and C n are the only possibilities. Remark. We suspected that there did not exist an operator space X such that (with the notation of Corollary 9) we had for any E 1 ; E 2 and any u 2 E 1 E 2 (8) X (~u) = kuk ; and indeed C. Le Merdy has kindly provided us with an argument, as follows. Let X be such a space. Let E be an arbitrary nite dimensional subspace of X and let v E 2 E X denote the tensor representing the inclusion map ~v E : E! X. Then, by (3) and (8), 2 (~v E ) kv E k = X (~v E ) = 1. By a well known ultraproduct argument (cf. e.g. [P1, p. 22]), this implies that X is isometric to a Hilbert space. But then, a variant of the proof of Corollary 10 shows that we must have either X = R or X = C completely isometrically, and this is absurd. However, (8) is true up to equivalence if we take for X the direct sum of R and C, in any reasonable way. For instance, it is easy to check that for any u 2 E 1 E 2 (~u: E 1! E 2 being the associated nite rank operator) we have (9) 1 2 kuk R1 C(~u) kuk Remark. It follows from Theorem 1 and the projectivity of Haagerup tensor product that is also projective, i.e, if q i : F i! E i (i = 1; 2) are quotient maps, so is q 1 q 2 : F 1 F 2! E 1 E 2. On the other hand, is not injective. To show this, consider the identity operator i n : R n \ C n! R n \ C n and the natural (completely isometric) embedding j n : R n \ C n,! R n 1 C n. The preceding remark (applied with ~u = (j n i n ) ) implies that jjj n i n jj (Rn +C n ) (R n 1C n ) 2: However, by [HP, p. 912] we have K (i n ) (1 + p n)=2; hence by Corollary 9, we have ki n k (1 + p n)=2: 14

16 This proves that the tensor product is not injective. Remark. The examples in [P1, chapter 10] imply that there are (innite dimensional) operator spaces E such that E min E = E E with equivalent norms, but E is not completely isomorphic to R or C, and actually (as a Banach space) E is not isomorphic to any Hilbert space. Thus (in the isomorphic case) the second part of Corollary 10 does not seem to extend to the innite dimensional setting without assuming some kind of approximation property. We recall that any Hilbert space H (resp. K) can be equipped with a column (resp. row) operator space, by identifying H (resp. K) with H c = B(C; H) (resp. with K r = B(K ; C)). Any operator space of this form will be called a \column space" (resp. a \row space"). We will use the following result from [O]: Theorem 11. ([O]) Let E be an operator space such that Id E can be factorized completely boundedly through the direct sum X = H c 1 K r of a column space and a row space, (i.e. there are c:b: maps u: E! X and v: X! E such that Id E = vu), then there are subspaces E 1 H c and E 2 K r such that E is completely isomorphic to E 1 1 E 2. More precisely, if we have kuk cb kvk cb c for some number c, then we can nd a complete isomorphism T : E! E 1 1 E 2 such that kt k cb kt?1 k cb f(c) where f: R +! R + is a certain function f. Theorem 12. The following properties of an an operator space E are equivalent: (i) For any operator space F, we have F min E = F E isomorphically. (ii) E is completely isomorphic to the direct sum of a row space and a column space. Proof. The implication (i) ) (ii) is easy and left to the reader. Conversely, assume (i). Then, a routine argument shows that there is a constant K such that for all F and all u in F E we have kuk Kkuk min. Let S E be an arbitrary nite dimensional subspace let j S : S! E be the inclusion map, and let ^js 2 S E be the associated tensor. Then we have by (9) sup R1 C(j S ) = sup k^j S k K. By a routine ultraproduct argument, S S this implies that the identity of E can be written as in Theorem 11 with c = K, thus we conclude that E ' E 1 1 E 2 where E 1 is a row space and E 2 a column space. Note that we 15

17 obtain an isomorphism T : E! E 1 1 E 2 such that kt k cb kt?1 k cb f(k). In particular, if E is nite dimensional, we nd T such that kt k cb kt?1 k cb f(ki E k ): We now turn to a result at the root of the present investigation. Let E be an operator space and let A be a unital operator algebra. For any x in E A, we dene (x) = sup k (x)k where the supremum runs over all pairs (; ) where : E! B(H) is a complete contraction, : A! B(H) a morphism and moreover and have commuting ranges. Let E A be the completion of E A for this L norm. We may clearly also view E A as an operator space using the embedding x! ( )(x) where the direct sum runs over all pairs as above. Note that the natural inclusion E A OA(E) max A is a complete isometry. In particular, if F is another operator space, we have a natural completely isometric embedding of E F into E OA(F ), which explains the connection of the delta tensor product to the present paper. Then we may state. Theorem 13. Consider the linear mapping q: A E A! E A dened by (;) q(a e b) = e (ab): This mapping q denes a complete metric surjection from A h E h A onto E A. More precisely, for any n and any x in M n (E A) with kxk Mn (E A) < 1, there is ~x in M n (A E A) with k~xk Mn (A h E h A) < 1 such that I Mn q(~x) = x. Remark. This statement is due to the second author [P4] (who is indebted to C. Le Merdy for observing this useful reformulation). A proof (somewhat dierent from the original one in [P4]) can be given following the lines of the above proof of Theorem 1 (here, one considers OA(E) max A as a quotient of the free product OA(E) A, and one uses the fact that A:E:A spans inside OA(E) A a subspace completely isometric to A h E h A), so we 16

18 skip it. This result yields simpler proofs and extensions of several statements concerning nuclear C - algebras. See the nal version of [P4] for more details on this topic. x2. An alternate approach. Using [CS1-2, PaS], one can see that the following statement is a dual reformulation of Theorem 1. Theorem 18. Let E 1 ; E 2 be two operator spaces, and let ': E 1 E 2! B(H) be a linear mapping. The following are equivalent: (i) k'k cb(e1 E 2 ;B(H)) 1. (ii) For some Hilbert space H, there are complete contractions 1 : E 1! B(H; H), 2 : E 2! B(H; H), and 1 : E 1! B(H; H), 2 : E 2! B(H; H), such that 8(x 1 ; x 2 ) 2 E 1 E 2 '(x 1 x 2 ) = 1 (x 1 ) 2 (x 2 ) = 2 (x 2 ) 1 (x 1 ): (iii) For some Hilbert space H, there are complete contractions i : E i! B(H) (i = 1; 2), with commuting ranges, and contractions V : H! H and W : H! H, such that 8(x 1 ; x 2 ) 2 E 1 E 2 '(x 1 x 2 ) = W 1 (x 1 ) 2 (x 2 )V: Proof. Assume (i). By Remark 2, ' denes a complete contraction into B(H) both from E 1 h E 2 and from E 2 h E 1. Then (ii) follows from the Christensen-Sinclair factorization theorem for bilinear maps, extended to general operator spaces by Paulsen and Smith in [PaS]. Now assume (ii). Let H 1 = H, H 2 = H and H 3 = H. We dene maps 1 : E 1 7?! B(H 1 H 2 H 3 ) and 2 : E 2?! B(H 1 H 2 H 3 ) using matrix notation, as follows Then, by (ii) we have 0 1 (x 1 ) 0 1(x 1 ) (x 1 ) A 1 2 (x 2 ) 0 2(x 2 ) (x 2 ) A : (x 1 ) 2 (x 2 ) = 2 (x 2 ) 1 (x 1 ) = '(x 1 x 2 ) A

19 hence 1 ; 2 have commuting ranges, and are complete contractions. Therefore if we let W : H 1 H 2 H 3! H be the projection onto the rst coordinate and V : H! H 1 H 2 H 3 be the isometric inclusion into the third coordinate, then we obtain (iii). Finally, the implication (iii) implies (i) is obvious by the very denition of E 1 E 2. Alternate proof of Theorem 1 (n = 2). By duality, it clearly suces to show that for any linear map ': E 1 E 2! B(H) the norms k'k cb(e1 E 2!B(H)) and k'qk cb are equal. But this is precisely the meaning of the equivalence between (i) and (ii) in Theorem 18. Thus we conclude that Theorem 18 implies Theorem 1 for n = 2. In the n-linear case with n > 2, the preceding argument yields the following statement, which seems rather dierent from the n-linear version of Theorem 1. To formulate this we need to introduce a variant of the denitions appearing in (1) and (2) where we restrict ourselves to the n cyclic permutations of f1; : : : ; ng. We will say that an n-tuple of operators (T 1 ; : : : ; T n ) on H cyclically commute if we have T 1 : : : T n = T (1) : : : T (n) for any cyclic permutation. (When n = 2, this is the same as ordinary commutation.) Let E 1 ; : : : ; E n be operator spaces. We will dene the cyclic analog of the -tensor product. We rst dene the norm as follows. 8x 2 E 1 E n kxk c = sup k 1 : : : n (x)k B(H) where the supremum runs over all possible H and all n-tuples ( i ) of complete contractions i : E i! B(H) which cyclically commute (i.e. such that (T 1 ; : : : ; T n ) cyclically commute for any choice of T i in the range of i ). We will denote by (E 1 E 2 E n ) c the completion of E 1 E n for this norm, and will consider it as an operator space in the same way as in (2). Theorem 19. Let E 1 ; : : : ; E n be an n-tuple of operator spaces. Let Y 1 = E 1 h E 2 : : : h E n and, for k > 2, Y k = E k h E k+1 : : : h E n h E 1 h : : : h E k?1. Let L c = Y 1 1 : : : 1 Y n. Let Q c be the natural completely contractive mapping from L c to (E 1 E n ) c, and let c : L c = ker(q c )! (E 1 E n ) c be the canonically associated map. Then c is a complete isometry. 18

20 We leave the proof, which is an easy modication of the argument for Theorem 18, to the reader. Note that we do not see how to prove Theorem 19 using the ideas of x1 (nor do we see how to prove Theorem 1 when n > 2 using the ideas of x2). Remark 20. Following [Di], we say that a collection of Banach algebras which is stable by arbitrary direct sums, subalgebras and quotients is a variety. Let V be a variety formed of unital operator algebras. Of course, we are interested in their operator space (and not only their Banach space) structure and we use unital completely contractive homomorphisms as morphisms. One of the advantages of the rst proof over the second one is that its \categorical principle" can be easily adapted to compute the analogue of the -tensor product obtained when one restricts all maps to take their values into an algebra belonging to some xed given variety V. Acknowledgement. We are very grateful to C. Le Merdy for letting us include several useful remarks and his example showing the non-associativity of the -tensor product. References [B1] D. Blecher. Tensor products of operator spaces II. Canadian J. Math. 44 (1992) [B2] D. Blecher. The standard dual of an operator space. Pacic J. Math. 153 (1992) [B3] D. Blecher. A completely bounded characterization of operator algebras. Math. Ann. 303 (1995) [BP1] D. Blecher and V. Paulsen. Tensor products of operator spaces J. Funct. Anal. 99 (1991) [BP2] D. Blecher and V. Paulsen. Explicit constructions of universal operator algebras and applications to polynomial factorization. Proc. Amer. Math. Soc. 112 (1991) [BRS] D. Blecher, Z. J. Ruan and A. Sinclair. A characterization of operator algebras. J. Funct. Anal. 89 (1990) [BS] D. Blecher and R. Smith. The dual of the Haagerup tensor product. Journal London Math. Soc. 45 (1992)

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