Quasi-multipliers of operator spaces

arXiv:math/0312501v1 [math.OA] 30 Dec 2003 QUASIMULTIPLIERS OF OPERATOR SPACES MASAYOSHI KANEDA AND VERN I. PAULSEN Abstract. We use the injective envelope to study quasimultipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasimultipliers. We obtain generalizations of the Banach-Stone theorem. 1. Introduction We begin with some general algebraic comments, inspired by [10], that make clear the role that quasimultipliers can play. Let A be an algebra and let X ⊆ A be a subspace. We shall call an element z ∈ A a quasimultiplier of X (relative to A) provided that XzX ⊆ X, i.e., x1 zx2 ∈ X for every x1 , x2 ∈ X. Clearly, the set of quasimultipliers of X is a linear subspace of A. Moreover, each quasimultiplier z induces a bilinear map mz : X × X → X defined by mz (x1 , x2 ) := x1 zx2 . The associativity of the product on A, implies that each mz is an associative bilinear map and hence can be regarded as a product on X. This product gives X the structure of an algebra, which we denote by (X, mz ). There are two homomorphisms, πl and πr from this algebra into A, defined by πl (x) := xz and πr (x) := zx. The range of πl is contained in the subalgebra of left multipliers of X (relative to A), while the range of πr is contained in the subalgebra of right multipliers of X (relative to A). Recall that an element a ∈ A is called a left (respectively, right) multiplier of X provided that aX ⊆ X (respectively, Xa ⊆ X). Finally, note that since the quasimultipliers are a linear subspace of A, the set of “products” on X that one obtains in this manner is a linear subspace of the vector space of bilinear maps from X × X into X. In general, linear (or, even convex) combinations of associative bilinear maps need not be associative. For an example of this phenomena, consider X = C2 and the associative bilinear maps, m1 ((a, b), (c, d)) := (ac, bd) and m2 ((a, b), (c, d)) := (ac, bc). Their convex combination, m := (m1 + m2 )/2, is not associative. One shortcoming of the above representation of quasimultipliers is that it is extrinsic. The quasimultipliers that one obtains and their induced bilinear maps, could easily depend on the algebra A and on the particular 1991 Mathematics Subject Classification. Primary 46L05; Secondary 46A22, 46H25, 46M10, 47A20. Key words and phrases. injective, multipliers, operator space, Banach-Stone. Research supported in part by a grant from the National Science Foundation. 1 2 MASAYOSHI KANEDA AND VERN I. PAULSEN embedding of X into A and not on intrinsic properties of X. Thus, the totality of bilinear maps that one could obtain in this manner would be a union of linear spaces, taken over all embeddings of X into an algebra, which would no longer need to be a linear space. In this paper, we develop a theory of quasimultipliers of operator spaces and then use the injective envelope to give an intrinsic characterization of quasimultipliers and of their associated bilinear maps. Among the results that we obtain are that an operator space endowed with a completely contractive product can be represented completely isometrically as an algebra of operators on some Hilbert space if and only if the product is a bilinear map that belongs to this space of “bilinear quasimultipliers”. As a corollary, we find that the set of “representable” completely contractive products is a convex set. In fact, it is affinely isomorphic with the unit ball of the space of quasimultipliers. We then turn our attention to generalizations of the Banach-Stone theorem. Our basic result is that a linear complete isometry between any two operator algebras induces a quasimultiplier and that by using the quasimultiplier, one recovers earlier generalizations of the Banach-Stone theorem. 2. Quasimultipliers In this section we introduce various spaces of quasimultipliers of an operator space and develop some of their key properties. Let X be an operator space, H be a Hilbert space, B(H) denote the algebra of bounded, linear operators on H, and let φ : X → B(H) be a complete isometry. We set QMφ (X) := {z ∈ B(H) : φ(X)zφ(X) ⊆ φ(X)}, and we call QMφ (X) the space of quasimultipliers of X relative to φ. Note that QMφ (X) is a norm closed subspace of B(H). Each z ∈ QMφ (X) induces a bilinear map, mz : X × X → X defined by mz (x1 , x2 ) := φ−1 (φ(x1 )zφ(x2 )). The bilinear map mz is completely bounded in the sense of Christensen-Sinclair, that is, its linear extension is completely bounded as a map from X ⊗h X to X with kmz kcb ≤ kzk. We say that mz is the bilinear map induced by z. Definition 2.1. Let QM B(X) denote the set of bilinear maps from X×X to X, that are of the form mz for some quasimultiplier z and some completely isometric map φ from X into the bounded linear operators on some Hilbert space. For m ∈ QM B(X) we set kmkqm := inf{kzk : m = mz }, where the infimum is taken over all possible completely isometric maps φ and z as above. Note that by the above remarks, every m ∈ QM B(X) is completely bounded as a bilinear map and kmkcb ≤ kmkqm . We shall show in Example 2.11 that this inequality can be sharp. For a fixed map φ, the set of bilinear maps, {mz : z ∈ QMφ (X)} is a linear subspace of the set QM B(X). However, since QM B(X) is the union of these subspaces, it is not clear that QUASIMULTIPLIERS OF OPERATOR SPACES 3 it is a vector subspace of the vector space of bilinear maps from X × X to X. We shall prove that it is a vector space later. The above definitions are extrinsic, in the sense that they could depend on the particular embedding. We now seek intrinsic characterizations of these maps by using the injective envelope as in [12] and [7] (see also [18]). We begin by recalling a construction used in [7], but we prefer the notation from [18]. Recall that if X ⊆ B(K, H) is a (concrete) operator space, then we may form the (concrete) operator system, in B(H ⊕ K),    x λIH : λ, µ ∈ C, x, y ∈ X . SX := y ∗ µIK Given a complete isometry ϕ : X → B(K1 , H1 ), the operator system    λIH1 ϕ(x) Sϕ(X) := : λ, µ ∈ C, x, y ∈ X ϕ(y)∗ µIK1 is completely order isomorphic to SX via the map, Φ : SX → Sϕ(X) defined by     λIH x λIH1 ϕ(x) Φ := . y ∗ µIK ϕ(y)∗ µIK1 Thus, the operator system SX only depends on the operator space structure of X and not on any of X. n particular  representation o λIH 0 ∗ ∼ Since C ⊕ C = 0 µIK : λ, µ ∈ C is a C -subalgebra of SX , C ⊕ C

will still be a C ∗ -subalgebra of the C ∗ -algebra, I(SX ) with I0H 00 and 00 I0K corresponding to orthogonal projections e1 and e2 , respectively, in the C ∗ algebra I(SX ). We have that e1 + e2 is equal to the identity and e1 · e2 = 0. A few words on such a situation are in order. Let A be any unital C ∗ algebra with orthogonal projections e1 and e2 satisfying e1 +e2 = 1, e1 ·e2 = 0 and let π : A → B(H) be a one-to-one unital ∗-homomorphism. Setting H1 = π(e1 )H, H2 = π(e2 )H we have that H = H1 ⊕ H2 and relative to this decomposition every T ∈ B(H) has the form T = (Tij ) where Tij ∈ B(Hj , Hi ). In particular, identifying A with π(A) we have that    a11 a12 A= : aij ∈ Aij a21 a22 where Aij = ei Aej , with Aii ⊆ B(Hi ) unital C ∗ -subalgebras and A21 = A∗12 . The operator space A12 ⊆ B(H2 , H1 ) will be referred to as a corner of A. Note that A11 · A12 · A22 ⊆ A12 so that A12 is an A11 − A22 -bimodule. Returning to I(SX ), relative to e1 and e2 , we wish to identify each of these 4 subspaces. Note that X ⊆ e1 I(SX )e2 = I(SX )12 . As shown in Chapter 16 of [18], we may identify I(SX )12 = I(X). 4 MASAYOSHI KANEDA AND VERN I. PAULSEN We define, I11 (X) := I(SX )11 and I22 (X) := I(SX )22 . Thus we have the following picture of the C ∗ -algebra I(SX ), namely,    a z I(SX ) = : a ∈ I11 (X), b ∈ I22 (X), z, w ∈ I(X) w∗ b where I11 (X) and I22 (X) are injective C ∗ -algebras and I(X) is an operator I11 (X) − I22 (X)-bimodule. Moreover, the fact that I(SX ) is a C ∗ -algebra means that for z, w ∈ I(X),     ∗  0 z 0 z zw 0 = w∗ 0 w∗ 0 0 w∗ z and consequently there are natural products, z·w∗ ∈ I11 (X), w∗ ·z ∈ I22 (X). It is interesting to note that setting hz, wi = zw∗ defines an I11 (X)-valued inner product that makes I(X) a Hilbert C ∗ -module over I11 (X), but we shall not use this additional structure. Definition 2.2. We set QM (X) := {z ∈ I(X)∗ : XzX ⊆ X}, where the products are all taken in the C ∗ -algebra I(SX ). In [10] a theory was developed of quasimultipliers of Hilbert C ∗ -bimodules. If X is a Hilbert C ∗ -bimodule, then they also use the notation, QM (X), for their quasimultipliers. We warn the reader, and apologize, that although we are using the same notation, our quasimultipliers and theirs are not the same objects. In the first place their quasimultipliers are defined using the second dual of the linking algebra and their quasimultipliers are a subset of the second dual of X. Briefly, if X is a Hilbert A − B-bimodule, then an element t in the second dual of X is a quasimultiplier in the sense of [10] provided that AtB ⊆ X where the products are defined in the second dual of the linking algebra. Note that if A and B are both unital C ∗ -algebras, then this forces t ∈ X. To make our quasimultiplier theory fit with theirs a bit better, we should have perhaps taken adjoints of elements so that our QM (X) is a subset of I(X). However, this alternate definition would have made several natural maps, that we define later, conjugate linear and we believe that it would have led to the “opposite”, i.e., transposed operator space structure. Theorem 2.3. Let X be an operator space, H be a Hilbert space and let φ : X → B(H), be a complete isometry. Then there exists a unique, completely contractive map γ : QMφ (X) → QM (X) such that φ(x1 )zφ(x2 ) = φ(x1 γ(z)x2 ) for all x1 , x2 ∈ X and every z ∈ QMφ (X). Proof. The proof is similar to that of Theorem 1.7 in [7]. Let Sφ(X) ⊆ B(H ⊕ H) be the concrete operator system defined above and let C ∗ (Sφ(X) ) be the C ∗ -subalgebra of B(H ⊕ H) that it generates. The C ∗ -subalgebra of I(SX ) generated by SX is known to be the C ∗ -envelope of SX , Ce∗ (SX ). Consequently, by [15] Corollary 4.2 the identity map on SX extends to be a surjective ∗-homomorphism π : C ∗ (Sφ(X) ) → Ce∗ (SX ). QUASIMULTIPLIERS OF OPERATOR SPACES 5 Let Γ : B(H ⊕ H) → I(SX ) be a completely positive map that extends this ∗-homomorphism. Since Γ extends π, it will be a π-bimodule map, that is, for A, B ∈ C ∗ (Sφ(X) ) we will have that Γ(AT B) = π(A)Γ(T )π(B). This   T11 T12 forces Γ to be a matrix of maps, that is, for T = ∈ B(H ⊕ H) T21 T22   γ11 (T11 ) γ12 (T12 ) . we will have that Γ(T ) = γ21 (T21 ) γ22 (T22 )   0 φ(x) In particular, for x ∈ X and z ∈ QMφ (X) we will have that Γ( )= z 0   0 x . γ21 (z) 0 Hence by the π-bimodule property, for x1 , x2 ∈ X and z ∈ QMφ (X) we have       0 φ(x1 )zφ(x2 ) 0 φ(x1 ) 0 0 0 φ(x2 ) Γ( ) = Γ( ) 0 0 0 0 z 0 0 0       0 x1 0 0 0 x2 0 x1 γ21 (z)x2 = = . 0 0 γ21 (z) 0 0 0 0 0 Thus, it follows that γ21 (z) ∈ QM (X) and φ(x1 γ21 (z)x2 ) = φ(x1 )zφ(x2 ). Because Γ is a unital, completely positive map, γ21 is completely contractive. Finally, the uniqueness of the map comes from the following observation. Suppose q1 , q2 ∈ QM (X) have the property that x1 q1 x2 = x1 q2 x2 for every x1 , x2 ∈ X. This implies that (x1 q1 − x1 q2 )X = 0 and so by Corollary 1.3 of [7], we have that x1 (q1 − q2 ) = 0 for every x1 ∈ X. Applying the corollary again, we see that X(q1 − q2 )(q1 − q2 )∗ = 0 and so q1 = q2 .  Remark 2.4. Let A be a C ∗ -algebra and let πu : A → B(Hu ) be its universal representation. The classical definition of the quasimultiplier space of A as given in [19] is the set QMπu (A). In [17], the first author proves that the map γ : QMπu (A) → QM (A) is an onto complete isometry, and preserve quasimultiplication. Thus, at least in the case of a C ∗ -algebra our definition and the classical definition agree. Corollary 2.5. Let X be an operator space. The map z 7→ mz from QM (X) to QM B(X) equipped with k · kqm is an onto isometry, where mz : X × X → X is defined by mz (x1 , x2 ) := x1 zx2 . Consequently, QM B(X) is a linear subspace of the vector space of bilinear maps from X × X to X. Let X be an operator space. Given an associative, bilinear map m : X × X → X, we let (X, m) denote the resulting algebra. We let CCP (X) denote the set of associative bilinear maps on X that are completely contractive in the sense of Christensen-Sinclair, that is, CCP (X) denotes the set of completely contractive products on X. We let OAP (X) ⊆ CCP (X) denote those maps such that the algebra (X, m) has a completely isometric homomorphism into B(H) for some Hilbert space H. That is, OAP (X) denotes the set of operator algebra products on X. 6 MASAYOSHI KANEDA AND VERN I. PAULSEN We let CBP (X) denote those associative, bilinear maps from X × X to X (i.e., products on X), that are completely bounded in the sense of Christensen-Sinclair, that is the set of completely bounded algebra products. By a result of [1], m ∈ CBP (X) if and only if there exists a Hilbert space H and a completely bounded homomorphism, π : (X, m) → B(H) with completely bounded inverse, π −1 : π(X) → (X, m). In fact, by [1] one may choose π such that kπkcb kπ −1 kcb ≤ 1 + ǫ for any ǫ ≥ 0. Finally, we let SOAP (X) ⊆ CBP (X) denote those associative, bilinear maps for which one can choose, π satisfying, kπkcb kπ −1 kcb = 1, i.e., such that π is a scalar multiple of a complete isometry. These are the scaled operator algebra products. The following theorem illustrates the importance of quasimultipliers. Theorem 2.6. Let X be an operator space. Then QM B(X) = SOAP (X) and OAP (X) = {m ∈ QM B(X) : kmkqm ≤ 1}. Proof. We prove the second equality first. Let m ∈ OAP (X) and let π : (X, m) → B(H) be a completely isometric homomorphism. Then IH is a quasimultiplier of π(X) that induces the bilinear map m. Thus, m ∈ QM B(X) and kmkqm ≤ 1. Conversely, let m ∈ QM B(X) with kmkqm ≤ 1. Then there exists z ∈ QM (X) with kzk ≤ 1 such that m = mz . As [8] Remark 2, define π : (X, m) → I(SX ) by   √ xz x 1 − zz ∗ . π(x) := 0 0 It is easily seen that, π(x1 )π(x2 ) = π(x1 zx2 ) = π(m(x1 , x2 )) and that kπ(x)k2 = kπ(x)π(x)∗ k = kxx∗ k = kxk2 , where all products take place in I(SX ). Thus, π is an isometric homomorphism. The proof that π is completely isometric is similar and thus m ∈ OAP (X). If m ∈ SOAP (X) and π : (X, m) → B(H) is a completely bounded homomorphism such that φ = π/kπkcb , is a complete isometry, then IH is a quasimultiplier of φ(X). We have that −1 −2 φ(x1 )IH φ(x2 ) = kπk−2 cb π(x1 )π(x2 ) = kπkcb π(m(x1 , x2 )) = kπkcb φ(m(x1 , x2 )). Hence, m ∈ QM B(X) and kmkqm ≤ kπkcb . Conversely, if m(x1 , x2 ) = x1 zx2 for z ∈ QM (X) with kzk = r, then consider   √ xz x r 2 − zz ∗ π(x) = 0 0 and argue as above to prove that π is a homomorphism and is r times a complete isometry.  Corollary 2.7. Let X be an operator space, then OAP (X) is a convex set, and SOAP (X) is a vector space. The following example shows that, in general, CCP (X) and CBP (X) are not convex sets. QUASIMULTIPLIERS OF OPERATOR SPACES 7 Example 2.8. The following example shows that, in general, OAP (X) 6= CCP (X), SOAP (X) 6= CBP (X), and that the sets CCP (X) and CBP (X) need not be convex. Let X = C2 , that is, the subspace of the two by two matrices, M2 consisting of those matrices  thatare   whose   firstcolumn  columnand  0in thesecond ac c a ac c a be ) := , and m2 ( ) := , is arbitrary. Let m1 ( bc d b bd d b as in the introduction. Since m2 is the product on C2 induced by the inclusion of C2 into M2 , we have that m2 ∈ OAP (C2 ) ⊆ CCP (C2 ). We claim that  m1 ∈ CCP (C2 ). To see this claim,   note  that if we identify  A A C AC (n) Mn (C2 ) = { : A, B ∈ Mn }, then m1 ( , ) = . If B   D BD     B A C AC k k ≤ 1 and k k ≤ 1, then k k2 = kC ∗ A∗ AC + D ∗ B ∗ BDk. B D BD However, 0 ≤ C ∗ A∗ AC + D ∗ B ∗ BD ≤ C ∗ C + D ∗ D ≤ I and so km1 kcb ≤ 1. Thus, m1 ∈ CCP (X) as claimed. Since (m1 + m2 )/2 is not even associative, we see that CCP (C2 ) is not convex and hence cannot be equal to OAP (C2 ). Since m1 and m2 are both also in CBP (C2 ), we have that this set is also not convex and hence it is not equal to SOAP (C2 ). Also, since SOAP (C2 ) is convex and m2 ∈ SOAP (C2 ), it must be the case that m1 is not in SOAP (C2 ). This last fact can also be seen by using the injective envelope and Theorem 2.6. Since C2 is already an injective operator space, we have that I(C2 )∗ = C2∗ = R2 , where R2 denotes the corresponding row space. Since C2 R2 C2 ⊆ C2 , we have that QM (C2 ) = R2 . Now it is easily checked that there is no z = (e, f ) ∈ R2 such that m1 (x1 , x2 ) = x1 zx2 and so m1 is not in QM B(C2 ) = SOAP (C2 ).     a a 0 Finally, note that π : (C2 , m1 ) → M2 defined by π( ) := is b 0 b a completely contractive homomorphism with completely bounded inverse. Which gives a direct way, independent of Blecher’s theorem [1], to see that (C2 , m1 ) is completely boundedly representable. It is also interesting to note that for the natural inclusion φ : C2 → M2 , we have that QMφ (C2 ) is all of M2 , while QM (C2 ) = R2 is 2-dimensional. Thus, we see that the map γ of Theorem 2.3 need not be one-to-one. Remark 2.9. Although, OAP (X) is a convex set and CCP (X) is not in general, we know little else about the structure of CCP (X) or about the subset of CCP (X) consisting of those products that can be induced by a completely contractive, but not completely isometric representation. The product m1 from Example 2.8 is one such product. For a finite dimensional vector space the set of associative bilinear maps is an algebraic set. Thus, when X is a finite-dimensional operator space, CCP (X) is the intersection of this algebraic set with the set of completely 8 MASAYOSHI KANEDA AND VERN I. PAULSEN contractive bilinear maps. Generally, the set of completely contractive bilinear maps need not even be a semialgebraic set. But it is still possible that the intersection, CCP (X), is a semialgebraic set. The following result illustrates how some of the results of [5] on operator algebras with one-sided identities can be deduced from the theory of quasimultipliers. Proposition 2.10. Let X be an operator space and let m ∈ OAP (X). Then (X, m) has a right contractive identity e if and only if m = mz where z ∈ QM (X) satisfies z ∗ = e, and zz ∗ is the identity of I22 (X). In this case the map x → xz defines a completely isometric homomorphism of (X, m) into Ml (X). Proof. Since m ∈ OAP (X) we have that m = mz for some z ∈ QM (X) with kzk ≤ 1. Assume that (X, m) has a contractive right identity e. Then we have that for every x ∈ X, x = m(x, e) = xze. Hence, x(122 − ze) = 0 for every x ∈ X. By [7] Corollary 1.3, this implies that 122 − ze = 0. But since both z and e are contractions, e = z ∗ must hold. Conversely, if z ∗ = e and zz ∗ = 122 , then clearly, m(x, e) = xze = x and so e is a contractive, right identity. Finally, since kxk = kxzz ∗ k ≤ kxzk, we see that the completely contractive homomorphism of (X, m) into Ml (X) given by x → xz is a complete isometry.  Note that by the above result, the relationship between the product m and the product in I(SX ) is that m(x1 , x2 ) = x1 e∗ x2 . There is an analogous result for left identities. It is possible for a concrete algebra of operators to have a two-sided identity e of norm greater than one. For an example see [18], page 279. In this case the multiplication will still be given by a contractive quasimultiplier z and one has ze = 122 , ez = 111 but one no longer has that e = z ∗ . We close this section with a number of examples of spaces of quasimultipliers that illustrate the limits of some of the above results. Example 2.11. This example shows that it is possible to have kzk > kmz kcb for a quasimultiplier. It is based on Example 4.4 of [2]. Let A ⊆ M3 denote the subalgebra that is the span of {E12 , E13 , I3 }, where I3 denotes the identity matrix. Let Q := I3 + J where J is the matrix whose entries are all 1’s, and set P := Q1/2 . A little calculation shows that P = I3 + 31 J and that P −1 = I3 − 16 J. We let X = AP . Since the C ∗ -subalgebra of M3 generated by X is all of M3 , which is irreducible, one finds that I11 (X ) = I(X ) = M3 , with the usual product. From this we see quite easily that Ml (X ) = A and QM (X ) = P −1 A. QUASIMULTIPLIERS OF OPERATOR SPACES 9 p Let Z := P −1 (E12 − E13 ), so that kZk = 3/2. Writing Xi , Yi ∈ X as Xi = (ai I3 + Ni )P, Yi = (bi I3 + Mi )P where Ni and Mi are in the span of E12 and E13 , we have that, m√Z (Xi , Yi ) := Xi ZYi = ai bi (E12√− E13 )P . Since P k(E12 − E13 )P k = 2, we have that kmZ k = κ 2, where κ = sup{| ai bi | : k(X1 , . . . , Xn )k ≤ 1, k(Y1∗ , . . . , Yn∗ )k ≤ 1} and mZ isPthe linear map mZ : A ⊗h A → A. The first inequality implies that I3 ≥ Xi Xi∗ . Examining the (3,3)-entry of these matrices leads P P to the conclusion that 2 |ai |2 ≤ 1. The second inequality implies that (bi I3 +Mi )∗ (bi I3 +Mi ) ≤ P −2 . Examining the (1,1)-entry pof these matrices leads to the conclusion that P 2 |bi | ≤ 3/4 and hence, κ ≤ 3/8. p p Thus, we are led to conclude that kmZ k ≤ 3/4 < 3/2. The same calculation, p using matrix coefficients for the entries of Xi and Yi shows that kmZ kcb ≤ 3/4 too, and so the result follows. Indeed, if we write X, Y ∈ Mn (A) as X = A ⊗ E11 + C ⊗ E12 + D ⊗ E13 + A ⊗ E22 + A ⊗ E33 and Y = B ⊗ E11 + G⊗ E12 + H ⊗ E13 + B ⊗ E22 + B ⊗ E33 where A, B, C, D, G, H (n) are in Mn , then it is easily seen that mZ (X, Y ) = AB ⊗ (E12 P− E13∗)P Now taking Xi ∈ Mn (A) with k(X1 , . . . , Xm )k ≤ 1 implies that Xi Xi ≤ IP ⊗ I and examining in the (1,1)-block of this matrix inequality 3 n P ∗ P yields, ∗ Ai Ai ≤ In . Similarly one gets that Bi Bi ≤ 3/4In Hence, Ai Bi = (A1 , . . . , Am ) · (B1 , . . . , Bm )t and these estimates yield that the norm p of the row of A’s is less than one and norm of the column of B’s is less than 3/4. p In fact, it is not too hard to show that kmZ k = kmZ kcb = 3/4. Example 2.12. Let {Eij } denote the canonical matrix units and let X = span{E11 , E12 , E21 , E32 } ⊆ M3,2 . We compute QM (X) for this space and illustrate some of its properties.   M3 M3,2 It is not difficult to show that I(SX ) = = M5 , with the M2,3 M2 obvious identifications. To see this one first shows that since X is a D3 −D2 bimodule, where Dn denotes the n×n diagonal matrices, then any completely contractive map Φ from M3,2 into itself that fixes X must be a bimodule map. From this it follows that Φ must be given as a Shur product map, but then the fact that Φ is completely contractive forces Φ to be the identity map. Now a direct calculation shows that QM (X) = span{E12 , E23 } ⊆ M2,3 , that Ml (X) = span{E11 , E12 , E13 , E22 , E33 } ⊆ M3 and that Mr (X) = span{E11 , E22 } ⊆ M2 . Note that the span of the products X · QM (X) is not dense in Ml (X) but that the span of the products QM (X) · X is all of Mr (X). For the contractive quasimultiplier z = E12 + E23 , we see that the induced homomorphism πl (x) = xz into Ml (X) is a complete isometry, but that πr (x) = zx is not even one-to-one. For the quasimultipliers E12 and E23 neither πl nor πr is one-to-one. Example 2.13. Let X = span{E11 + E32 , E21 + E33 } ⊆ M3 . This space can be identified as a concrete representation of the maximum of C2 and R2 , that 10 MASAYOSHI KANEDA AND VERN I. PAULSEN is, as the least operator space structure on C2 that is greater than both C2 and R2 . We will show that QM (X) = (0) and consequently, there are no nontrivial operator algebra products on this operator space, i.e., OAP (X) = (0). However, since the natural maps from X to the concrete operator algebras C2 ⊆ M2 and R2 ⊆ M2 are both complete contractions, we see that there are at least 4 different products (up to scaling) in CCP (X) that have completely contractive representations whose inverses are completely bounded. To see these claims, one first shows that if one regards SX ⊆ M6 , then C ∗ (SX ) = I(SX ). From this it follows that I11 (X) = M2 ⊕ C, I22 (X) = C ⊕ M2 , I(X) = span{E11 , E32 , E21 , E33 } and that Ml (X) and Mr (X) are both just the scalar multiples of the identity. Once these things are seen, it is straightforward to check that QM (X) = (0). To prove that C ∗ (SX ) = I(SX ), first note that there is a *-homomorphism of C ∗ (SX ) onto the C*-subalgebra of I(SX ) generated by the copy of SX , i.e., onto the boundary C*-algebra. But the original C*-algebra has only 2 ideals that could be the kernel of this map. Now argue that if you mod out by either ideal then you will not have a 2-isometry on SX . Hence this homomorphism must be 1-1. But C ∗ (SX ) is injective so we are done. 3. A Non-commutative Banach-Stone Theorem In this section we use quasimultipliers to obtain a characterization of linear complete isometries from one operator algebra onto another. Our theorem needs no assumptions concerning the existence of units or approximate units. To understand the statement of the theorem, it is perhaps instructive ∗ to keep the following example in   mind. Let A ⊆ B(H) be a unital C 0 a subalgebra and let B := { : a ∈ A} ⊆ B(H ⊕ H). These are both 0 0 algebras of operators, although the product of any two elements of B is 0. The identification of A with B is a complete isometry, onto, but clearly the only possible homomorphism between these two algebras is the 0 map. However, in this example one sees that the left multipliers of B can be identified with the C ∗ -algebra A. Theorem 3.1. Let A and B be algebras of operators and let ψ : A → B be a linear complete isometry that is onto. Then we have the following: (1) there exists a unique z ∈ QM (A), with kzk ≤ 1 such that ψ(a1 )ψ(a2 ) = ψ(a1 za2 ) for every a1 , a2 ∈ A; (2) there exists a unique w ∈ QM (B), with kwk ≤ 1 such that ψ(a1 a2 ) = ψ(a1 )wψ(a2 ) for every a1 , a2 ∈ A; (3) setting πl (a) = ψ(a)w and πr (a) = wψ(a) defines completely contractive homomorphisms of A into Ml (B) and Mr (B), respectively; (4) if A has a contractive right (respectively, left) approximate identity, then πl (respectively, πr ) is a completely isometric homomorphism; QUASIMULTIPLIERS OF OPERATOR SPACES 11 (5) if A has a contractive right (respectively, left) identity, e, then w = ψ(e)∗ and ww∗ is the identity of I22 (B) (respectively, w∗ w is the identity of I11 (B)). Proof. Set γ := ψ −1 and define m : A×A → A by m(a1 , a2 ) := γ(ψ(a1 )ψ(a2 )). It is easily checked that m(a1 , m(a2 , a3 )) = γ(ψ(a1 )ψ(a2 )ψ(a3 )) = m(m(a1 , a2 ), a3 ), so that m is an associative bilinear map on A and defines a new product on A. Moreover, because the product on B is completely contractive this new product on A is completely contractive and the map ψ : (A, m) → B is a completely isometric algebra isomorphism. Thus, since B is an algebra of operators, we see that m ∈ OAP (A) and hence by Theorem 2.6 there exists a unique z ∈ QM (A) such that m(a1 , a2 ) = a1 za2 . Hence, ψ(a1 za2 ) = ψ(m(a1 , a2 )) = ψ(a1 )ψ(a2 ) and (1) follows. Applying (1) to γ yields w ∈ QM (B) such that γ(ψ(a1 )wψ(a2 )) = γ(ψ(a1 ))γ(ψ(a2 )) = a1 a2 . Thus, ψ(a1 )wψ(a2 ) = ψ(a1 a2 ) and so (2) follows. To see (3), note that πl (a1 )πl (a2 ) = ψ(a1 )wψ(a2 )w = ψ(a1 a2 )w = πl (a1 a2 ) with a similar calculation for πr . Since πl (a)b = ψ(a)wψ(γ(b)) = ψ(aγ(b)) ∈ B for every b ∈ B, we see that πl (a) ∈ Ml (B) for every a ∈ A, with a similar calculation for πr . Now let {eα } be a contractive, approximate right identity for A. We then have that kψ(a)k = lim kψ(aeα )k = lim kψ(a)wψ(eα )k = lim kπl (a)ψ(eα )k ≤ kπl (a)k. Thus, πl is an isometry. The proof that πl is a complete isometry and the case for πr are similar. Finally, if A has a right identity e, then ψ(a) = ψ(ae) = ψ(a)wψ(e). This shows that bwψ(e) = b for every b ∈ B and hence wψ(e) is a right identity for Mr (B). By [7] Corollary 1.3, we have that wψ(e) is the identity of I22 (B). Since kwk ≤ 1 and kψ(e)k ≤ 1, we have that ψ(e) = w∗ and (5) follows.  Remark 3.2. (1) When B has a contractive right identity, then one may identify B ⊆ Ml (B), but it is not clear if the image of πl maps onto this copy of B. However, in this case it is clear how to define a homomorphism into B. Let ψ(a0 ) = eB and define ρ : A → B by setting ρ(a) = eB wψ(a) = ψ(a0 a). Letting the product in B be denoted by ⊙ to avoid confusion, we have that b1 ⊙ b2 = b1 e∗B b2 , where the latter product is taken in I(SB ). Since e∗B eB = 122 , by Proposition 2.10, we have that ρ(a1 ) ⊙ ρ(a2 ) = eB wψ(a1 )e∗B eB wψ(a2 ) = eB wψ(a1 a2 ) = ρ(a1 a2 ), and so ρ is a homomorphism. Note that ρ is onto B if and only if a0 A = A. (2) If one considers A = B = C2 ⊆ M2 and lets ψ be the identity map, then we are in the situation of the last remark. Thus, πl is a complete isometry, but since Mr (C2 ) = C, we have that πr is not a complete isometry. In fact, it is the compression to the (1, 1)-entry. 12 MASAYOSHI KANEDA AND VERN I. PAULSEN 4. Further Results on QMB(X) In [2] and [4] various characterizations are given of the linear maps of an operator space X into itself that are given as left multiplication by an element from the left multiplier algebra of X, Ml (X). In this section we present characterizations of the bilinear maps of an operator space into itself that are in QM B(X). Among the results that we obtain is a characterization of when a linear map from X into Ml (X) is given as right multiplication by a quasimultiplier. We also identify a subspace of QM (X), related to ternary structures on X, that we denote by T ER(X)∗ for which we have kzk = kmz kqm = kmz kcb . In [4] it was shown that one could determine whether or not a linear map from X into X was given by a contractive left multiplier by determining whether or not an associated linear map was completely contractive. The following is an analogous result for determining when a map is given as multiplication by a quasimultiplier. Recall that given any operator space X, R2 (X) denotes the operator subspace of M2 (X) consisting of 1 × 2 matrices. Theorem 4.1. Let X be an operator space and let γ : X → I11 (X) be a linear map. There exists y ∈ I(X)∗ with kyk ≤ 1 such that γ(x) = xy for every x ∈ X if and only  if the map β : R2 (X) → I(SX ) defined by γ(x1 ) x2 β((x1 , x2 )) := is completely contractive. 0 0 Proof. Note that if such an element y exists,  then  β is given, at least formally, y 0 as right multiplication by the matrix and since this matrix has 0 122 norm 1, β should be a complete contraction. To complete this argument, we create a C ∗ -algebra where these products occur. To this end consider the following C ∗ -algebra,   I11 (X) I(X) I(X) B :=  I(X)∗ I22 (X) I22 (X) , I(X)∗ I22 (X) I22 (X) where the products are allinduced 0 X R2 (X) with the subspace 0 0 0 0 from  the products in I(SX ). Identifying X 0  ⊆ B, we see that β is given as right 0   0 0 0 multiplication in the C ∗ -algebra B by the matrix y 0 0. 0 122 0 For the converse, we must assume that β is a complete contraction and produce the element y. To this end we create a second C ∗ -algebra, C and an operator system S. QUASIMULTIPLIERS OF OPERATOR SPACES Let 13   I11 (X) I11 (X) I(X) C := I11 (X) I11 (X) I(X)  , I(X)∗ I(X)∗ I22 (X) where the products are all induced from the products in I(SX ) and let   C111 X X 0  ⊆ B. S =  X ∗ C122 ∗ X 0 C122 We define Φ : S → C by     λ111 x1 x2 λ111 γ(x1 ) x2 0 ) := γ(x3 )∗ µ111 0 . Φ( x∗3 µ122 ∗ ∗ 0 ν122 x4 0 ν122 x4 Since β is completely contractive, Φ is completely positive and since C is clearly an injective C ∗ -algebra, we may extend Φ to a completely positive map on all of B, which we still denote by Φ. Because Φ fixes the diagonal, it will be a bimodule map over the diagonal. Also note that the compression of S to the span of the first and third entries is a copy of SX and that Φ fixes this operator system. By the rigidity properties of the injective envelope, we see that necessarily     a 0 b a 0 b Φ( 0 µ122 0 =  0 µ111 0 , c∗ 0 d c∗ 0 d for every a ∈ I11 (X), b, c ∈ I(X), d ∈ I22 (X) and µ ∈ C. These matrices that are fixed by Φ form a common C ∗ -subalgebra of B and C and hence Φ will necessarily be a bimodule map over this C ∗ -subalgebra. Thus, we will have for any x ∈ X that         0 γ(x) 0 0 x 0 0 0 x 0 0 0 0 0 0 = Φ(0 0 0) = Φ(0 0 0  · 0 0 0) 0 122 0 0 0 0 0 0 0 0 0 0        0 0 x 0 0 0 0 0 x 0 0 0 = 0 0 0 Φ(0 0 0) = 0 0 0 0 0 0 , 0 0 0 0 122 0 0 0 0 0 y 0 where y is the image of 122 under the restriction of Φ to the subspaces corresponding to the (3,2)-entries. Equating entries of the matrices occurring in the first and last expressions, we see that γ(x) = xy, and the proof is complete.  We state the corresponding result for maps into the right multiplier algebra without proof. Theorem 4.2. Let X be an operator space and let ψ : X → I22 (X) be a linear map. There exists y ∈ I(X)∗ such that ψ(x) = yx for every x ∈ X if 14 MASAYOSHI KANEDA AND VERN I. PAULSEN     x 0 x1 and only if the map α : C2 (X) → I(SX ) defined by α( 1 ) := x2 0 ψ(x2 ) is completely contractive. The above theorem yields a method for determining whether or not a bilinear map m : X × X → X is a contractive quasimultiplier, i.e., whether or not m ∈ OAP (X). Corollary 4.3. Let X be an operator space and let m : X × X → X be a bilinear map. Then the following are equivalent: (1) m ∈ OAP (X); (2) there exists a linear map γ : X → Ml (X) such that m(x, y) = γ(x)y and the   map β : R2 (X) → I(SX ) defined by β((x1 , x2 )) = γ(x1 ) x2 is completely contractive; 0 0 (3) there exists a linear map ψ : X → Mr (X) such that m(x 1 , x2 ) = x x1 ψ(x2 ) and the map α : C2 (X) → I(SX ) defined by α( 1 ) = x2   0 x1 is completely contractive. 0 ψ(x2 ) Recall that by the results of [4] (see also [18]) to determine whether or not x → m(x1 , x) is a contractive left multiplier it is  necessary and   sufficient  x2 m(x1 , x2 ) that the map τx1 : C2 (X) → C2 (X) defined by τx1 ( ) := x3 x3 be a complete contraction. There is a similar result involving R2 (X) for determining when a map is a contractive right multiplier. Thus, by combining Corollary 4.3 with the characterizations of multipliers, one obtains a bootstrap method for determining whether or not a bilinear map m : X × X → X is a contractive quasimultiplier, i.e., whether or not m ∈ OAP (X). Remark 4.4. Since we have obtained necessary and sufficient conditions for a bilinear map to be in OAP (X), we have in some sense given a full generalization of the Blecher-Ruan-Sinclair characterization of unital operator algebras [8] to arbitrary operator algebras. However, to prove the original BRS theorem by applying Corollary 4.3, one still needs to use the theory of multipliers and the proof that one obtains in this fashion is not really different from the proof given in [18]. In the thesis of the first author [17], a new direct characterization of the bilinear maps in OAP (X) is given that is independent of the theory of multipliers and is sufficiently simple that the BRS theorem can be deduced directly from this characterization. The space QM B(X) currently is endowed with two generally different norms, k · kcb and k · kqm . The first norm comes from its natural inclusion into the space of completely bounded bilinear maps from X into X and QUASIMULTIPLIERS OF OPERATOR SPACES 15 the second from its identification with the space QM (X). The next results allow us to prove that for a subspace of QM (X), that is related to ternary structures on X, these two norms are the same.   X Theorem 4.5. Let X be an operator space and let Y = where Mr (X) we give Y the operator space structure that comes from its identification as a subspace of I(SX ). Then I(SY ) can be identified with the injective I11 (X) I(X) I(X) C ∗ -algebra B :=  I(X)∗ I22 (X) I22 (X) in such a way that I11 (Y ) = I(X)∗ I22 (X) I22 (X)   I(X) and I22 (Y ) = I22 (X). I(SX ), I(Y ) = I22 (X)   C Y Proof. We identify SY = with an operator system in B via the Y∗ C       α111 0 x1 α y1 xi α122 t1 . The map that sends where yi = to  0 y2∗ β ti β122 t∗2 x∗2 proof of the theorem will be complete if we can show that any completely positive map Φ : B → B that is the identity on SY must be the identity on B.     a 0 b a b To this end let γ : I(SX ) → B be defined by γ( ) := 0 0 0 c d c 0 d   a11 a13 and let δ : B → I(SX ) be defined by δ((ai,j )) := . Since δ ◦ Φ ◦ γ a31 a33 is the identity on SX by rigidity it must be the identity on  I(SX ).  111 0 0  0 0 0 , E23 = To simplify notation, we define elements of B by E11 := 0 0 0   0 0 0 0 0 122  and give similar definitions to E22 , E32 , E33 . Note that be0 0 0 cause I(X) need not contain an identity we do not attempt to define E12 , E13 , E21 and E31 . We first prove that Φ fixes the five “matrix units” defined above. Note that since Φ fixes SY we already have that Φ(E11 +E22 ) = E11 +E22 , Φ(E33 ) = E33 and Φ(E23 ) = E23 .

Since δ ◦ Φ ◦ γ( 1011 00 ) = 1011 00 , it follows that Φ(E11 ) =: P = (Pij ) with P11 = 111 . Since Φ is contractive and positive, it follows that Pij = 0 when (i, j) is (1, 2), (1, 3), (2, 1), or (3, 1) and that P33 ≥ 0. But since Φ(E33 ) = E33 and kΦ(E11 + E33 )k ≤ 1, we have that P33 = 0. Now the positivity of P implies that P23 = P32 = 0. ∗ ∗ E Since E23 23 = E33 = Φ(E23 E23 ) and Φ(E23 ) = E23 , we have that E23 is in the right multiplicative domain of Φ, that is Φ(BE23 ) = Φ(B)E23 , ∀B ∈ 16 MASAYOSHI KANEDA AND VERN I. PAULSEN B. If we let Φ(E22 ) =: Q = (Qij ), then E23 = Φ(E22 E23 ) = QE23 and it follows that Q22 = 122 . But since Φ(E11 + E22 ) = E11 + E22 , it follows that P22 = 0 and Q = E22 . Thus, we have shown that these five matrix units are fixed by Φ as was claimed. Since the span of these matrix units are a C ∗ -subalgebra of B, we have that Φ must be a bimodule map over this C ∗ -subalgebra. Since this subalgebra contains the diagonal matrices we see that there exist maps φij such that Φ((Bij )) = (φij (Bij )). To prove that Φ is the identity map, it will be enough to show that each φij is the identity map on its respective domain. Using the fact that δ ◦ Φ ◦ γ is the identity on I(SX ) yields that φ11 is the identity map on I11 (X) and similarly φ13 , φ31 and φ33 are the identities on their respective domains. To see that φ12 is the identity on its domain, note that for any u ∈ I(X) we have that       0 u 0 0 0 u 0 0 u Φ(0 0 0) = Φ(0 0 0  E32 ) = Φ(0 0 0)E32 = 0 0 0 0 0 0 0 0 0     0 0 u 0 u 0 0 0 0  E32 = 0 0 0 . 0 0 0 0 0 0 Note that what was used in this argument was the bimodularity property of the matrix units and the fact that certain maps were the identity maps. A similar argument shows that φ21 , φ23 and φ32 are all the identity maps on their respective domains. Finally, that φ22 is the identity follows from the rigidity of the upper left corner I(SX ) of B. This completes the proof of the theorem.  Given any operator space X the sets Ml (X) ∩ Ml (X)∗ and Mr (X) ∩ Mr (X)∗ are C ∗ -subalgebras of I11 (Y ) and I22 (X), respectively. In [7] these sets were denoted IMl∗ (X) and IMr∗ (X), respectively. They were shown to be equal to the sets Al (X) and Ar (X) of adjointable left and right multipliers, respectively, introduced in [2]. We shall use the latter notation for these sets. Definition 4.6. Given an operator space X, we set T ER(X) := X ∩ QM (X)∗ and we call this the ternary subspace of X. Note that in the multiplication inherited from I(SX ) we have that T ER(X)· T ER(X)∗ · T ER(X) ⊆ T ER(X). The following results give further properties of this subspace.   X where we Corollary 4.7. Let X be an operator space, let Y := Mr (X) give Y the operator space structure that comes from its identification as a QUASIMULTIPLIERS OF OPERATOR SPACES 17 ∗ subspace of I(SX ) and let I(SY ) be identified with the  C -algebra B as above.  Ml (X) X Al (X) T ER(X) Then Ml (Y ) = and Al (Y ) = . QM (X) Mr (X) T ER(X)∗ Ar (X) Proof. One simply checks that Ml (X) is exactly the matrix in I11 (Y ) = I(SX ) that leaves Y invariant under left multiplication.  Corollary 4.8. Let X be an operator space, let x ∈ T ER(X), set z = x∗ ∈ QM (X) and let mz : X × X → X be the associated bilinear map, then kzk = kmz kqm = kmz kcb . Proof. We have that kzk  = km  z kqm by definition and clearly, kmz kcb ≤ 0 0 kzk. Clearly, kzk = k k and this latter matrix is in Al (Y ). By z 0 Theorem 1.9 (i) of [7] the norm of this latter matrix is equal to the norm of the map it induces acting by left multiplication on Y , regarded as an element of CBl (Y ). Hence given any ǫ > 0, there exists a matrix over X, k(xij )k ≤ 1 such that k(zxij )k ≥ kzk − ǫ. The matrix (zxij ) is a matrix of right multipliers, and so applying [7] again, we can find another matrix over X, k(yij )k ≤ 1 such that X X k( mz (yik , xkj ))k = k( yik zxkj )k ≥ k(zxij )k − ǫ ≥ kzk − 2ǫ. k k Thus, we have that kmz kcb ≥ kzk−2ǫ and since ǫ was arbitrary, the result follows.  5. Quasicentralizers and Quasihomomorphisms We introduce a new family of bilinear maps that we call the quasicentralizers of an operator space and a set of maps that we call the quasihomomorphisms and explore the relationships between these maps and the space QM B(X). Definition 5.1. Let X be an operator space and let m : X × X → X be a bilinear map. We call m a quasicentralizer1 provided that there exists completely bounded maps, γ : X → Ml (X) and ψ : X → Mr (X) such that m(x, y) = γ(x)y = xψ(y) for every x, y ∈ X. We let QC(X) denote the set of quasicentralizers. We call a linear map γ : X → Ml (X) (respectively, ψ : X → Mr (X)) a left (right) quasihomomorphism2 provided that γ(x)γ(y) = γ(γ(x)y) (respectively, ψ(x)ψ(y) = ψ(xψ(y)) for every x, y ∈ X. These definitions are motivated by the following observations. If m = mz ∈ QM B(X) for some z ∈ QM (X), then m(x, y) = πl (x)y = xπr (y) 1This is a generalization of a quasi-centralizer defined for an operator algebra with a two-sided contractive approximate identity in [17] Definition 3.2.1. 2This is different from a quasi-homomorphism which the first author defined in [17] Definition 3.1.1 (2). 18 MASAYOSHI KANEDA AND VERN I. PAULSEN where πl (x) = xz and πr (y) = zy. Thus, QM B(X) ⊆ QC(X). Moreover, the maps πl and πr are left and right quasihomomorphisms, respectively. Note that QC(X) is a linear subspace of the space of bilinear maps from X to X. We begin with a few elementary observations about the relationships between these concepts. Proposition 5.2. Let X be an operator space. A linear map γ : X → Ml (X) (respectively, ψ : X → Mr (X)) is a left (respectively, right) quasihomomorphism if and only if the bilinear map m(x, y) = γ(x)y (respectively, m(x, y) = xψ(y)) is associative. In this case, γ (respectively, ψ) is a homomorphism of the algebra (X, m) into Ml (X) (respectively, Mr (X)). Proof. The proof is straightforward.  We shall refer to m as the product associated with the quasihomomorphism. Proposition 5.3. Let X be an operator space, let γ : X → Ml (X) (respectively, ψ : X → Mr (X)) be a linear map and let m(x, y) := γ(x)y (respectively, m(x, y) := xψ(y)), then kmkcb = kγkcb (respectively, kmkcb = kψkcb ). Proof. By [7], we have that the norm of a left (respectively, right) multiplier is given by the cb-norm of its action as a left (respectively, right) multiplication. Thus, given k(xij )k ≤ 1 and k(yij )k ≤ 1, we have that X X k( m(xik , ykj )k = k( γ(xik )ykj )k ≤ k(γ(xij )k · k(yij )k ≤ kγkcb k k and it follows that kmkcb ≤ kγkcb . The other inequalities follow similarly.  By the above results, the product associated with a completely contractive quasihomomorphism is completely contractive, i.e., is in CCP (X). Example 5.4. This is an example of a product associated with a completely contractive quasihomomorphism that is in CCP (X) but not in OAP (X). Recall the product m1 on C2 of Example 2.8 that is not in  OAP  (C2 ). We  a a 0 have that Ml (C2 ) = M2 and that γ : C2 → M2 defined by γ( ) := b 0 b is a completely contractive quasihomomorphism with m1 the associated product. Thus, γ is a completely contractive homomorphism of (C2 , m1 ) into an operator algebra, but since m1 is not an operator algebra product on C2 , there can be no completely isometric homomorphism of (C2 , m1 ) into an operator algebra. It is interesting to note that m1 is also not a quasicentralizer. In fact, it is not hard to show by a direct calculation that QC(C2 ) = QM B(C2 ). Remark 5.5. 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E-mail address: [email protected], URL: http://www.math.uci.edu/∼mkaneda/ Vern I. Paulsen: Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204-3008 U.S.A. E-mail address: [email protected], URL: http://www.math.uh.edu/∼vern/