Compactness properties of operator multipliers

Journal of Functional Analysis 256 (2009) 3772–3805 www.elsevier.com/locate/jfa Compactness properties of operator multipliers K. Juschenko a , R.H. Levene b,∗ , I.G. Todorov c , L. Turowska a a Department of Mathematical Sciences, Chalmers & Göteborg University, SE-412 96 Göteborg, Sweden b School of Mathematics, Trinity College, Dublin 2, Ireland c School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland Received 12 September 2008; accepted 10 December 2008 Available online 22 January 2009 Communicated by N. Kalton Abstract We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C ∗ -algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C ∗ -algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C ∗ -Algebra, Diplomarbeit, Universität des Saarlandes, Saarbrücken, 1982].  2008 Published by Elsevier Inc. Keywords: Operator multiplier; Complete compactness; Schur multiplier; Haagerup tensor product 1. Introduction A bounded function ϕ : N×N → C is called a Schur multiplier if (ϕ(i, j )aij ) is the matrix of a bounded linear operator on ℓ2 whenever (aij ) is such. The study of Schur multipliers was initiated by Schur in the early 20th century and since then has attracted considerable attention, much of * Corresponding author. E-mail addresses: jushenko@chalmers.se (K. Juschenko), levene@maths.tcd.ie (R.H. Levene), i.todorov@qub.ac.uk (I.G. Todorov), turowska@chalmers.se (L. Turowska). 0022-1236/$ – see front matter  2008 Published by Elsevier Inc. doi:10.1016/j.jfa.2008.12.018 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3773 which was inspired by A. Grothendieck’s characterisation of these objects in his Résumé [9]. Grothendieck ∞it has the2 form  showed that a function ϕ is a Schur multiplier precisely when a (i)b (j ), where a , b : N → C satisfy the conditions sup ϕ(i, j ) = ∞ k k k i k=1 |ak (i)| < ∞ k=1 k ∞ 2 and supj k=1 |bk (j )| < ∞. In modern terminology, this characterisation can be expressed by saying that ϕ is a Schur multiplier precisely when it belongs to the extended Haagerup tensor product ℓ∞ ⊗eh ℓ∞ of two copies of ℓ∞ . Special classes of Schur multipliers, e.g. Toeplitz and Hankel Schur multipliers, have played an important role in analysis and have been studied extensively (see [19]). Compact Schur multipliers, that is, the functions ϕ for which the mapping (aij ) → (ϕ(i, j )aij ) on B(ℓ2 ) is compact, were characterised by Hladnik [11], who identified them with the elements of the Haagerup tensor product c0 ⊗h c0 . A non-commutative version of Schur multipliers was introduced by Kissin and Shulman [14] as follows. Let A and B be C ∗ -algebras and let π and ρ be representations of A and B on Hilbert spaces H and K, respectively. Identifying H ⊗ K with the Hilbert space C2 (H d , K) of all Hilbert–Schmidt operators from the dual space H d of H into K, we obtain a representation σπ,ρ of the minimal tensor product A ⊗ B acting on C2 (H d , K). An element ϕ ∈ A ⊗ B is called a π ,ρ-multiplier if σπ,ρ (ϕ) is bounded in the operator norm of C2 (H d , K). If ϕ is a π ,ρ-multiplier for any pair of representations (π, ρ) then ϕ is called a universal (operator) multiplier. Multidimensional Schur multipliers and their non-commutative counterparts were introduced and studied in [12], where the authors gave, in particular, a characterisation of universal multipliers as certain weak limits of elements of the algebraic tensor product of the corresponding C ∗ -algebras, generalising the corresponding results of Grothendieck and Peller [9,18] as previously conjectured by Kissin and Shulman in [14]. Let A1 , . . . , An be C ∗ -algebras. Like Schur multipliers, elements of the set M(A1 , . . . , An ) of (multidimensional) universal multipliers give rise to completely bounded (multilinear) maps. Requiring these maps to be compact or completely compact, we define the sets of compact and completely compact operator multipliers denoted by Mc (A1 , . . . , An ) and Mcc (A1 , . . . , An ), respectively. The notion of complete compactness we use is an operator space version of compactness which was introduced by Saar [21] and subsequently studied by Oikhberg [15] and Webster [27]. Our results on operator multipliers rely on the main result of Section 3 where we prove a representation theorem for completely compact completely bounded multilinear maps. In [3] Christensen and Sinclair established a representation result for completely bounded multilinear maps which implies that every such map Φ : K(H2 , H1 ) ⊗h · · · ⊗h K(Hn , Hn−1 ) → K(Hn , H1 ) (where, for Hilbert spaces H ′ and H ′′ , we denote by K(H ′ , H ′′ ) the space of all compact operators from H ′ into H ′′ ) has the form Φ(x1 ⊗ · · · ⊗ xn−1 ) = A1 (x1 ⊗ I )A2 . . . (xn−1 ⊗ I )An , (1) for some index set J and bounded block operator matrices A1 ∈ M1,J (B(H1 )), A2 ∈ MJ (B(H2 )), . . . , An ∈ MJ,1 (B(Hn )). In other words, Φ arises from an element u = A1 ⊙ · · · ⊙ An ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) of the extended Haagerup tensor product of B(H1 ), . . . , B(Hn ). Moreover, if Φ is A′1 , . . . , A′n modular for some von Neumann algebras A1 , . . . , An , then the entries of Ai can be chosen 3774 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 from Ai . We show in Section 3 that a map Φ as above is completely compact precisely when it has a representation of the form (1) where   u = A1 ⊙ · · · ⊙ An ∈ K(H1 ) ⊗h B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 ) ⊗h K(Hn ). This extends a result of Saar [21] in the two-dimensional case. If, additionally, A1 , . . . , An are von Neumann algebras and Φ is A′1 , . . . , A′n -modular then u can be chosen from K′ (A1 ) ⊗h (A2 ⊗eh · · · ⊗eh An−1 ) ⊗h K′ (An ), where K′ (A) denotes the ideal of compact operators contained in A in its identity representation. As a consequence of this and a result of Effros and Kishimoto [4] we point out the completely isometric identifications ∗∗  ∗∗    CC K(H2 , H1 ) ≃ K(H1 ) ⊗h K(H2 ) ≃ CB B(H2 , H1 ) , where CC(X ) and CB(X ) are the spaces of completely compact and completely bounded maps on an operator space X , respectively. In Section 4 we pinpoint the connection between universal operator multipliers and completely bounded maps. This technical result is used in Section 5 to define the symbol uϕ of an operator multiplier ϕ ∈ M(A1 , . . . , An ) which, in the case n is even (resp. odd) is an element of An ⊗eh Aon−1 ⊗eh · · · ⊗eh A2 ⊗eh Ao1 (resp. An ⊗eh Aon−1 ⊗eh · · · ⊗eh Ao2 ⊗eh A1 ). Here Ao is the opposite C ∗ -algebra of a C ∗ -algebra A. This notion extends a similar notion that was given in the case of completely bounded masa-bimodule maps by Katavolos and Paulsen in [13]. We give a symbolic calculus for universal multipliers which is used to establish a universal property of the symbol related to the representation theory of the C ∗ -algebras under consideration. The symbol of a universal multiplier is used in Section 6 to single out the completely compact multipliers within the set of all operator multipliers. In fact, we show that ϕ ∈ Mcc (A1 , . . . , An ) if and only if uϕ ∈  K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh Ao3 ⊗eh A2 ) ⊗h K(Ao1 ) K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh A3 ⊗eh Ao2 ) ⊗h K(A1 ) if n is even, if n is odd, which is equivalent to the approximability of ϕ in the multiplier norm by operator multipliers of finite rank whose range consists of finite rank operators. It follows that a multidimensional Schur multiplier ϕ ∈ ℓ∞ (Nn ) is compact if and only if ϕ ∈ c0 ⊗h (ℓ∞ ⊗eh · · · ⊗eh ℓ∞ ) ⊗h c0 . In Section 7 we use Saar’s construction [21] of a completely bounded compact mapping which is not completely compact to show that the inclusion Mcc (A1 , . . . , An ) ⊆ Mc (A1 , . . . , An ) is proper if both K(A1 ) and K(An ) contain full matrix algebras of arbitrarily large sizes. However, if both K(A1 ) and K(An ) are isomorphic to a c0 -sum of matrix algebras of uniformly bounded sizes then the sets of compact and completely compact multipliers coincide. The case when only one of K(A1 ) and K(An ) contains matrix algebras of arbitrary large size remains, however, unsettled. Finally, for n = 2, we characterise the cases where every universal multiplier is automatically compact: this happens precisely when one of the algebras A1 and A2 is finite dimensional and the other one coincides with its algebra of compact elements. 2. Preliminaries We start by recalling standard notation and notions from operator space theory. We refer the reader to [1,6,16,20] for more details. K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3775 If H and K are Hilbert spaces we let B(H, K) (resp. K(H, K)) denote the set of all bounded linear (resp. compact) operators from H into K. If I is a set we let H I be the direct sum of |I | copies of H and set H ∞ = H N . Writing H ⊗ K for the Hilbertian tensor product of two Hilbert spaces, we observe that H I = H ⊗ ℓ2 (I ) as Hilbert spaces. An operator space E is a closed subspace of B(H, K), for some Hilbert spaces H and K. The opposite operator space E o associated with E is the space E o = {x d : x ∈ E} ⊆ B(K d , H d ). Here, and in the sequel, H d = {ξ d : ξ ∈ H } denotes the dual of the Hilbert space H , where ξ d (η) = (η, ξ ) for η ∈ H . Note that H d is canonically conjugate-linearly isometric to H . We also adopt the notation x d ∈ B(K d , H d ) for the Banach space adjoint of x ∈ B(H, K), so that x d ξ d = (x ∗ ξ )d for ξ ∈ K. As usual, E ∗ will denote the operator space dual of E. If n, m ∈ N, by Mn,m (E) we denote the space of all n by m matrices with entries in E and let Mn (E) = Mn,n (E). The space Mn,m (E) carries a natural norm arising from the embedding Mn,m (E) ⊆ B(H m , K n ). Let I and J be arbitrary index sets. If v is a matrix with entries in E and indexed by I × J , and I0 ⊆ I and J0 ⊆ J are finite sets, we let vI0 ,J0 ∈ MI0 ,J0 (E) be the matrix obtained by restricting v to the indices from I0 × J0 . We define MI,J (E) to be the space of all such v for which   def v = sup vI0 ,J0 : I0 ⊆ I , J0 ⊆ J finite < ∞. Then MI,J (E) is an operator space [6, §10.1]. Note that MI,J (B(H, K)) can be naturally identified with B(H J , K I ) and every v ∈ MI,J (B(H, K)) is the weak limit of {vI0 ,J0 } along the net {(I0 , J0 ): I0 ⊆ I, J0 ⊆ J finite}. We set MI (E) = MI,I (E). For A = (aij ) ∈ MI (E), we write Ad = (aijd ) ∈ MI (E o ). 2.1. Completely bounded maps and Haagerup tensor products If E and F are operator spaces, a linear map Φ : E → F is called completely bounded if the maps Φ (k) : Mk (E) → Mk (F ) given by Φ (k) ((aij )) = (Φ(aij )) are bounded for every k ∈ N and def Φ cb = supk Φ (k) < ∞. Given linear spaces E1 , . . . , En , we denote by E1 ⊙ · · · ⊙ En their algebraic tensor product. If E1 , . . . , En are operator spaces and a k = (aijk ) ∈ Mmk ,mk+1 (Ek ), mk ∈ N, k = 1, . . . , n, we define the multiplicative product a 1 ⊙ · · · ⊙ a n ∈ Mm1 ,mn+1 (E1 ⊙ · · · ⊙ En )  n 1 ⊗ a2 by letting its (i, j )-entry (a 1 ⊙ · · · ⊙ a n )ij be i2 ,...,in ai,i i2 ,i3 ⊗ · · · ⊗ ain ,j . If E is another 2 operator space and Φ : E1 × · · · × En → E is a multilinear map we let Φ (m) : Mm (E1 ) × · · · × Mm (En ) → Mm (E) be the map given by    (m)  1   1 Φ Φ ai,i , ai22 ,i3 , . . . , ainn ,j , a , . . . , a n ij = 2 i2 ,...,in k ) ∈ M (E ), k = 1, . . . , n. The multilinear map Φ is called completely bounded where a k = (as,t m k if there exists a constant C > 0 such that, for all m ∈ N,   Φ (m) a 1 , . . . , a n  C a 1 . . . a n , a k ∈ Mm (Ek ), k = 1, . . . , n. 3776 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 def Set Φ cb = sup{ Φ (m) (a 1 , . . . , a n ) : m ∈ N, a 1 , . . . , a n  1}. It is well known (see [6,17]) that a completely bounded multilinear map Φ gives rise to a completely bounded map on the Haagerup tensor product E1 ⊗h · · · ⊗h En (see [6] and [20] for its definition and basic properties). The set of all completely bounded multilinear maps from E1 × · · · × En into E will be denoted by CB(E1 × · · · × En , E). If E1 , . . . , En and E are dual operator spaces we say that a map Φ ∈ CB(E1 × · · · × En , E) is normal [3] if it is weak∗ continuous in each variable. We write CBσ (E1 × · · · × En , E) for the space of all normal maps in CB(E1 × · · · × En , E). The extended Haagerup tensor product E1 ⊗eh · · · ⊗eh En is defined [5] as the space of all normal completely bounded maps u : E1∗ × · · · × En∗ → C. It was shown in [5] that if u ∈ E1 ⊗eh 1 )∈M · · · ⊗eh En then there exist index sets J1 , J2 , . . . , Jn−1 and matrices a 1 = (a1,s 1,J1 (E1 ), n ∗ n 2 2 a = (as,t ) ∈ MJ1 ,J2 (E2 ), . . . , a = (at,1 ) ∈ MJn−1 ,1 (En ) such that if fi ∈ Ei , i = 1, . . . , n, then def u, f1 ⊗ · · · ⊗ fn = u(f1 , . . . , fn ) = a 1 , f1 . . . a n , fn , (2) k )) and the product of the (possibly infinite) matrices in (2) is defined to where a k , fk = (fk (as,t be the limit of the sums  i1 ∈F1 ,...,in−1 ∈Fn−1  1   2    f1 a1,i f2 ai1 ,i2 . . . fn ainn−1 ,1 1 along the net {(F1 × · · · × Fn−1 ): Fj ⊆ Jj finite, 1  j  n − 1}. We may thus identify u with the matrix product a 1 ⊙ · · · ⊙ a n ; two elements a 1 ⊙ · · · ⊙ a n and ã 1 ⊙ · · · ⊙ ã n coincide if a 1 , f1 . . . a n , fn = ã 1 , f1 . . . ã n , fn for all fi ∈ Ei∗ . Moreover, u eh   = inf a 1 . . . a n : u = a 1 ⊙ · · · ⊙ a n . The space E1 ⊗eh · · · ⊗eh En has a natural operator space structure [5]. If E1 , . . . , En are dual operator spaces then by [5, Theorem 5.3] E1 ⊗eh · · · ⊗eh En coincides with the weak∗ Haagerup tensor product E1 ⊗w∗h · · · ⊗w∗h En of Blecher and Smith [2]. Given operator spaces Fi and completely bounded maps gi : Ei → Fi , i = 1, . . . , n, Effros and Ruan [5] define a completely bounded map g = g1 ⊗eh · · · ⊗eh gn : E1 ⊗eh · · · ⊗eh En → F1 ⊗eh · · · ⊗eh Fn , a 1 ⊙ · · · ⊙ a n → a 1 , g1 ⊙ · · · ⊙ a n , gn where a k , gk = (gk (aijk )). Thus g(u), f1 ⊗ · · · ⊗ fn = u, (f1 ◦ g1 ) ⊗ · · · ⊗ (fn ◦ gn ) (3) for u ∈ E1 ⊗eh · · · ⊗eh En and fi ∈ Fi∗ , i = 1, . . . , n. The following fact is a straightforward consequence of a well-known theorem due to Christensen and Sinclair [3], and it will be used throughout the exposition. K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3777 Theorem 2.1. Let Hi be a Hilbert space and Ri ⊆ B(Hi ) be a von Neumann algebra, i = 1, . . . , n. There exists an isometry γ from R1 ⊗eh · · · ⊗eh Rn onto the space of all R′1 , . . . , R′n -modular maps in CBσ (B(H2 , H1 ) × · · · × B(Hn , Hn−1 ), B(Hn , H1 )), given as follows: if u ∈ R1 ⊗eh · · · ⊗eh Rn has a representation u = A1 ⊙ · · · ⊙ An where Ai ∈ MJ (Ri ) ⊆ B(Hi ⊗ ℓ2 (J )) for some index set J , then γ (u)(T1 , . . . , Tn−1 ) = A1 (T1 ⊗ I )A2 . . . An−1 (Tn−1 ⊗ I )An , for all Ti ∈ B(Hi+1 , Hi ), i = 1, . . . , n − 1, where I is the identity operator on ℓ2 (J ). We now turn to the definition of slice maps which will play an important role in our proofs. Given ω1 ∈ B(H1 )∗ we set Lω1 = ω1 ⊗eh idB(H2 ) . After identifying C ⊗ B(H2 ) with B(H2 ) we obtain a mapping Lω1 : B(H1 ) ⊗eh B(H2 ) → B(H2 ) called a left slice map. Similarly, for  ω2 ∈ B(H2 )∗ we obtain a right slice map Rω2 : B(H1 ) ⊗eh B(H2 ) → B(H1 ). If u = i∈I vi ⊗ wi ∈ B(H1 ) ⊗eh B(H2 ) where v = (vi )i∈I ∈ M1,I (B(H1 )) and w = (wi )i∈I ∈ MI,1 (B(H2 )), then Lω1 (u) =  ω1 (vi )wi and Rω2 (u) = i∈I  ω2 (wi )vi . i∈I Moreover, Rω2 (u), ω1 = u, ω1 ⊗ ω2 = Lω1 (u), ω2 =  ω1 (vi )ω2 (wi ). (4) i∈I It was shown in [24] that if E ⊆ B(H1 ) and F ⊆ B(H2 ) are closed subspaces then, up to a complete isometry, Moreover [23],  E ⊗eh F = u ∈ B(H1 ) ⊗eh B(H2 ): Lω1 (u) ∈ F and Rω2 (u) ∈ E  for all ω1 ∈ B(H1 )∗ and ω2 ∈ B(H2 )∗  = u ∈ B(H1 ) ⊗eh B(H2 ): Lω1 (u) ∈ F and Rω2 (u) ∈ E  for all ω1 ∈ B(H1 )∗ and ω2 ∈ B(H2 )∗ .  E ⊗h F = u ∈ B(H1 ) ⊗h B(H2 ): Lω1 (u) ∈ F and Rω2 (u) ∈ E  for all ω1 ∈ B(H1 )∗ and ω2 ∈ B(H2 )∗ . (5) (6) Thus, E ⊗h F can be canonically identified with a subspace of B(H1 ) ⊗h B(H2 ) which, on the other hand, sits completely isometrically in B(H1 ) ⊗eh B(H2 ). These identifications are made in the statement of the following lemma which will be useful for us later. 3778 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 Lemma 2.2. If H1 , H2 , H3 are Hilbert spaces and E1 , E2 ⊆ B(H1 ), F1 , F2 ⊆ B(H2 ) and G1 , G2 ⊆ B(H3 ) are operator spaces, then (E1 ⊗eh F1 ) ∩ (E2 ⊗h F2 ) = (E1 ∩ E2 ) ⊗h (F1 ∩ F2 ) and (E1 ⊗eh F1 ⊗eh G1 ) ∩ (E2 ⊗h F2 ⊗h G2 ) = (E1 ∩ E2 ) ⊗h (F1 ∩ F2 ) ⊗h (G1 ∩ G2 ). Proof. Since ⊗eh and ⊗h are both associative, the second equation follows from the first. If u ∈ (E1 ⊗eh F1 ) ∩ (E2 ⊗h F2 ) ⊆ B(H1 ) ⊗h B(H2 ) then Lϕ (u) ∈ F1 ∩ F2 and Rψ (u) ∈ E1 ∩ E2 whenever ϕ ∈ B(H1 )∗ and ψ ∈ B(H2 )∗ . By (6), u ∈ (E1 ∩ E2 ) ⊗h (F1 ∩ F2 ). The converse inclusion follows immediately in light of the injectivity of the Haagerup tensor product. ✷ 2.2. Operator multipliers We now recall some definitions and results from [14] and [12] that will be needed later. Let H1 , . . . , Hn be Hilbert spaces and H = H1 ⊗ · · · ⊗ Hn be their Hilbertian tensor product. Set HS(H1 , H2 ) = C2 (H1d , H2 ) and let θH1 ,H2 : H1 ⊗ H2 → HS(H1 , H2 ) be the canonical isometry given by θ (ξ1 ⊗ ξ2 )(ηd ) = (ξ1 , η)ξ2 for ξ1 , η ∈ H1 and ξ2 ∈ H2 . When n is even, we inductively define   def HS(H1 , . . . , Hn ) = C2 HS(H2 , H3 )d , HS(H1 , H4 , . . . , Hn ) , and let θH1 ,...,Hn : H → HS(H1 , . . . , Hn ) be given by   θH1 ,...,Hn (ξ2,3 ⊗ ξ ) = θHS(H2 ,H3 ),HS(H1 ,H4 ,...,Hn ) θH2 ,H3 (ξ2,3 ) ⊗ θH1 ,H4 ,...,Hn (ξ ) , where ξ2,3 ∈ H2 ⊗ H3 and ξ ∈ H1 ⊗ H4 ⊗ · · · ⊗ Hn . When n is odd, we let def HS(H1 , . . . , Hn ) = HS(C, H1 , . . . , Hn ). If K is a Hilbert space, we will identify C2 (Cd , K) with K via the map S → S(1d ). The isomorphism θH1 ,...,Hn in the odd case is given by θH1 ,...,Hn (ξ ) = θC,H1 ,...,Hn (1 ⊗ ξ ). We will omit the subscripts when they are clear from the context and simply write θ . If ξ ∈ H1 ⊗ H2 we let ξ op denote the operator norm of θ (ξ ). By · 2 we will denote the Hilbert–Schmidt norm. Let  (H1 ⊗ H2 ) ⊙ (H2 ⊗ H3 )d ⊙ · · · ⊙ (Hn−1 ⊗ Hn ) if n is even, Γ (H1 , . . . , Hn ) = (H1 ⊗ H2 )d ⊙ (H2 ⊗ H3 ) ⊙ · · · ⊙ (Hn−1 ⊗ Hn ) if n is odd. We equip Γ (H1 , . . . , Hn ) with the Haagerup norm · h where each of the terms of the algebraic tensor product is given the opposite operator space structure to the one arising from the embedding H ⊗K ֒→ (C2 (H d , K), · op ). We denote by · 2,∧ the projective norm on Γ (H1 , . . . , Hn ) where each of the terms is given its Hilbert space norm. K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3779 Suppose n is even. For each ϕ ∈ B(H ) we let Sϕ : Γ (H1 , . . . , Hn ) → B(H1d , Hn ) be the map given by    d   d    d  Sϕ (ζ ) = θ ϕ(ξ1,2 ⊗ ξ3,4 ⊗ · · · ⊗ ξn−1,n ) θ η2,3 θ η4,5 . . . θ ηn−2,n−1 d ⊙ ··· ⊙ ξ where ζ = ξ1,2 ⊙ η2,3 n−1,n ∈ Γ (H1 , . . . , Hn ) is an elementary tensor. In particular, if Ai ∈ B(Hi ), i = 1, . . . , n, and ϕ = A1 ⊗ · · · ⊗ An then  d    Sϕ (ζ ) = An θ (ξn−1,n ) . . . Ad3 θ η2,3 A2 θ ξ1,2 Ad1 . Now suppose that n is odd and let ζ ∈ Γ (H1 , . . . , Hn ) and ξ1 ∈ H1 . Then ξ1 ⊗ ζ ∈ H1 ⊙ Γ (H1 , . . . , Hn ) = Γ (C, H1 , . . . , Hn ). For ϕ ∈ B(H ) we let Sϕ (ζ ) be the operator defined on H1 by Sϕ (ζ )(ξ1 ) = S1 ⊗ ϕ (ξ1 ⊗ ζ ). Note that S1 ⊗ ϕ (ξ1 ⊗ ζ ) ∈ C2 (Cd , Hn ); thus, Sϕ (ζ )(ξ1 ) can be viewed as an element of Hn . It was d ⊗ξ shown in [12] that Sϕ (ζ ) ∈ B(H1 , Hn ). If ζ = η1,2 2,3 ⊗ · · · ⊗ ξn−1,n and ϕ = A1 ⊗ · · · ⊗ An for Ai ∈ B(Hi ), i = 1, . . . , n, then  d  Sϕ (ζ ) = An θ (ξn−1,n ) . . . A3 θ (ξ2,3 )Ad2 θ η1,2 A1 . As observed in [12, Remark 4.3], for any ϕ ∈ B(H ) and ζ ∈ Γ (H1 , . . . , Hn ), Sϕ (ζ ) op  ϕ ζ 2,∧ . (7) On the other hand, an element ϕ ∈ B(H ) is called a concrete operator multiplier if there exists C > 0 such that Sϕ (ζ ) op  C ζ h for each ζ ∈ Γ (H1 , . . . , Hn ). When n = 2, this is equivalent to Sϕ (ζ ) op  C θ (ζ ) op for each ζ ∈ H1 ⊗ H2 . We call the smallest constant C with this property the concrete multiplier norm of ϕ. Now let Ai be a C ∗ -algebra and πi : Ai → B(Hi ) be a representation, i = 1, . . . , n. Set π = π1 ⊗ · · · ⊗ πn : A1 ⊗ · · · ⊗ An → B(H1 ⊗ · · · ⊗ Hn ) (here, and in the sequel, by A ⊗ B we will denote the minimal tensor product of the C ∗ -algebras A and B). An element ϕ ∈ A1 ⊗ · · · ⊗ An is called a π1 , . . . , πn -multiplier if π(ϕ) is a concrete operator multiplier. We denote by ϕ π1 ,...,πn the concrete multiplier norm of π(ϕ). We call ϕ a universal multiplier if it is a π1 , . . . , πn multiplier for all representations πi of Ai , i = 1, . . . , n. We denote the collection of all universal multipliers by M(A1 , . . . , An ); from this definition, it immediately follows that A1 ⊙ · · · ⊙ An ⊆ M(A1 , . . . , An ) ⊆ A1 ⊗ · · · ⊗ An . It was observed in [12] that if ϕ ∈ M(A1 , . . . , An ) then ϕ def m  = sup ϕ π1 ,...,πn :  πi is a representation of Ai , i = 1, . . . , n < ∞. 3780 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 It is obvious that if Ai and Bi are C ∗ -algebras and ρi : Ai → Bi is a ∗-isomorphism, i = 1, . . . , n, then   (ρ1 ⊗ · · · ⊗ ρn ) M(A1 , . . . , An ) = M(B1 , . . . , Bn ). If ϕ is an operator, and {ϕν } a net of operators, acting on H1 ⊗ · · · ⊗ Hn we say that {ϕν } converges semi-weakly to ϕ if (ϕν ξ, η) →ν (ϕξ, η) for all ξ, η ∈ H1 ⊙ · · · ⊙ Hn . The following characterisation of universal multipliers was established in [12] (see Theorem 6.5, the subsequent remark and the proof of Proposition 6.2) and will be used extensively in the sequel. Theorem 2.3. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, i = 1, . . . , n, and ϕ ∈ A1 ⊗ · · · ⊗ An . Suppose that n is even. The following are equivalent: (i) ϕ ∈ M(A1 , . . . , An ); (ii) there exists a net {ϕν } where ϕν = Aν1 ⊙ Aν2 ⊙ · · · ⊙ Aνn and Aνi is a finite block operator man νd trix with entries in Ai such that ϕν → ϕ semi-weakly, ϕν m  ni=1 Aν2i i=1 A2i−1 and the operator norms Aνi for i even and Aνi d for i odd, are bounded by a constant depending only on n. For every net {ϕν } satisfying (ii) we have that Sϕν (ζ ) → Sϕ (ζ ) weakly for all ζ = ξ1,2 ⊗ · · · ⊗ ξn−1,n ∈ Γ (H1 , . . . , Hn ) and that supν ϕν m is finite. Moreover, the net ϕν can be chosen in (ii) so that Aνi → Ai (resp. Aνi d → Adi ) strongly for i even (resp. for i odd) for some bounded block operator matrix Ai with entries in A′′i (resp. (Adi )′′ ) such that     Sid⊗···⊗id(ϕ) (ζ ) = An θ (ξn−1,n ) ⊗ I . . . θ (ξ1,2 ) ⊗ I Ad1 , for all ζ = ξ1,2 ⊗ · · · ⊗ ξn−1,n ∈ Γ (H1 , . . . , Hn ). A similar statement holds if n is odd. Finally, recall that an element a of a C ∗ -algebra A is called compact if the operator x → axa on A is compact. Let K(A) be the collection of all compact elements of A. It is well known [7,29] that a ∈ K(A) if and only if there exists a faithful representation π of A such that π(a) is a compact operator. Moreover, π can be taken to be the reduced atomic representation of A. The notion of a compact element of a C ∗ -algebra will play a central role in Sections 6 and 7 of the paper. 3. Completely compact maps We start by recalling the notion of a completely compact map introduced in [21] and studied further in [27] and [15]. By way of motivation, recall that if X and Y are Banach spaces then a bounded linear map Φ : X → Y is compact if and only if for every ε > 0, there exists a finite dimensional subspace F ⊆ Y such that dist(Φ(x), F ) < ε for every x in the unit ball of X . Now let X and Y be operator spaces. A completely bounded map Φ : X → Y is called completely compact if for each ε > 0 there exists a finite dimensional subspace F ⊆ Y such that   dist Φ (m) (x), Mm (F ) < ε, K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3781 for every x ∈ Mm (X ) with x  1 and every m ∈ N. We extend this definition to multilinear maps: if Y, X1 , . . . , Xn are operator spaces and Φ : X1 × · · · × Xn → Y is a completely bounded multilinear map, we call Φ completely compact if for each ε > 0 there exists a finite dimensional subspace F ⊆ Y such that   dist Φ (m) (x1 , . . . , xn ), Mm (F ) < ε, for all xi ∈ Mm (Xi ), xi  1, i = 1, . . . , n, and all m ∈ N. We denote by CC(X1 × · · · × Xn , Y) the space of all completely bounded completely compact multilinear maps from X1 × · · · × Xn into Y. A straightforward verification shows the following: Remark 3.1. A completely bounded map Φ : X1 × · · · × Xn → Y is completely compact if and only if its linearisation Φ̃ : X1 ⊗h · · · ⊗h Xn → Y is completely compact. In view of this remark, we frequently identify the spaces CC(X1 × · · · × Xn , Y) and CC(X1 ⊗h · · · ⊗h Xn , Y). The next result is essentially due to Saar (see Lemmas 1 and 2 of [21]). Proposition 3.2. (i) CC(X1 × · · · × Xn , Y) is closed in CB(X1 × · · · × Xn , Y). (ii) Let E, F and G be operator spaces. If Φ ∈ CC(E, F ) and Ψ ∈ CB(F , G) then Ψ ◦ Φ ∈ CC(E, G). If Φ ∈ CC(F , G) and Ψ ∈ CB(E, F ) then Φ ◦ Ψ ∈ CC(E, G). Let H1 , . . . , Hn be Hilbert spaces. Recall the isometry   γ : B(H1 ) ⊗eh · · · ⊗eh B(Hn ) → CBσ B(H2 , H1 ) × · · · × B(Hn , Hn−1 ), B(Hn , H1 ) from Theorem 2.1. Let us identify a completely bounded map defined on B(H2 , H1 ) × · · · × B(Hn , Hn−1 ) with the corresponding completely bounded map defined on def Bh = B(H2 , H1 ) ⊗h · · · ⊗h B(Hn , Hn−1 ). For u ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) we let γ0 (u) be the restriction of γ (u) to def Kh = K(H2 , H1 ) ⊗h · · · ⊗h K(Hn , Hn−1 ). Proposition 3.3. The map γ0 is an isometry from B(H1 ) ⊗eh · · · ⊗eh B(Hn ) onto CB(Kh , B(Hn , H1 )). Proof. Let Φ ∈ CB(Kh , B(Hn , H1 )). Since Φ is completely bounded, its second dual Φ ∗∗ : B(H2 , H1 ) ⊗σ h · · · ⊗σ h B(Hn , Hn−1 ) → B(Hn , H1 )∗∗ is completely bounded (here ⊗σ h denotes the normal Haagerup tensor product [5]). Let Q : B(Hn , H1 )∗∗ → B(Hn , H1 ) be the canonical projection. The multilinear map Φ̃ : B(H2 , H1 ) × · · · × B(Hn , Hn−1 ) → B(Hn , H1 ) 3782 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 corresponding to Q ◦ Φ ∗∗ is completely bounded and, by (5.22) of [5], weak∗ continuous in each variable. By Theorem 2.1, there exists an element u ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) such that Φ̃ = γ (u). Hence γ0 (u) = γ (u)|Kh = Φ̃|Kh = Φ. Thus γ0 is surjective. Fix u ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ). From the definition of γ0 we have γ0 (u) cb  γ (u) cb = u eh . On the other hand, the restrictions of the maps Q ◦ γ0 (u)∗∗ and γ (u) to Kh coincide, and since both maps are weak∗ continuous, γ (u) = Q ◦ γ0 (u)∗∗ |Bh . Hence, u eh Thus, γ0 is an isometry.  Q ◦ γ0 (u)∗∗ cb  γ0 (u)∗∗ cb = γ0 (u) cb . ✷ Theorem 3.4. Let H1 , . . . , Hn be Hilbert spaces. The image under γ0 of the operator space def def E = K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ) is F = CC(Kh , K(Hn , H1 )). Proof. We first establish the inclusion γ0 (E) ⊆ F . If Φ = γ0 (u) where u ∈ E then, by Proposition 3.3, Φ is the limit in the cb norm of maps of the form γ0 (v), where   v = a ⊙ B ⊙ b ∈ K(H1 ) ⊙ B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 ) ⊙ K(Hn ), a and b have finite rank and B is a finite matrix with entries in B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 ). But each such map has finite rank and hence is completely compact. Moreover, every operator in the image of γ0 (v) has range contained in the range of a, which is finite dimensional. It follows that Φ takes compact values; it is completely compact by Proposition 3.2. To see that F ⊆ γ0 (E), let Φ ∈ F . We will assume for technical simplicity that H1 , . . . , Hn are separable. Let {pk }k (resp. {qk }k ) be a sequence of projections of finite rank on H1 (resp. Hn ) such that pk → I (resp. qk → I ) in the strong operator topology. Let Ψk : K(Hn , H1 ) → K(Hn , H1 ) be the complete contraction given by Ψk (x) = pk xqk . Let ε > 0. Since Φ is completely compact there exists a subspace F ⊆ K(Hn , H1 ) of dimension ℓ < ∞ such that dist(Φ (m) (x), Mm (F )) < ε whenever x ∈ Mm (Kh ) has norm at most one. Denote the restriction of Ψk to F by Ψk,F and let ι be the inclusion map ι : F ֒→ K(Hn , H1 ). By [6, Corollary 2.2.4], Ψk,F − ι cb  ℓ Ψk,F − ι . Since F ⊆ K(Hn , H1 ), we have that Ψk,F (x) → x in norm for each x ∈ F . It follows easily that there exists k0 such that Ψk,F − ι cb < ε whenever k  k0 . Let x ∈ Mm (Kh ) be of norm at most one. Then there exists y ∈ Mm (F ) such that Φ (m) (x) − y < ε. Note that y  Φ (m) (x) − y + Φ (m) (x)  ε + Φ cb . Let Φk = Ψk ◦ Φ. If k  k0 then  (m)  (m) (m) (m) Φk − Φ (m) (x)  Φk (x) − Ψk (y) + Ψk (y) − y + y − Φ (m) (x)  (m)  = Ψk Φ (m) (x) − y + (Ψk,F − ι)(m) (y) + y − Φ (m) (x)    2ε + ε ε + Φ cb . This shows that Φk − Φ cb → 0. 3783 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 By Proposition 3.2, it only remains to prove that each Φk lies in γ0 (E). By Proposition 3.3, there exists an element u = A1 ⊙ A2 ⊙ · · · ⊙ An−1 ⊙ An ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) where A1 : H1∞ → H1 , Ai : Hi∞ → Hi∞ , i = 2, . . . , n − 1 and An : Hn → Hn∞ are bounded operators, such that Φ = γ0 (u). Observe that Φk = γ0 (uk ) where uk = (pk A1 ) ⊙ A2 ⊙ · · · ⊙ An−1 ⊙ (An qk ). It therefore suffices to show that uk ∈ E for each k. Fix k and let p = pk , q = qk . The operator pA1 : H1∞ → H1 has finite dimensional range and is hence compact. For i = 1, . . . , n, let Qi,r : Hi∞ → Hi∞ be a projection with block matrix whose first r diagonal entries are equal to the identity operator while the rest are zero. Then by compactness, (pA1 )Q1,r → pA1 and Qn,r (An q) → An q in norm as r → ∞. Let B = A2 ⊙ · · · ⊙ An−1 , Cr = (pA1 )Q1,r ⊙ B ⊙ Qn,r (An q), r ∈ N, and C = (pA1 ) ⊙ B ⊙ (An q). Then Cr − C eh  Cr − (pA1 )Q1,r ⊙ B ⊙ (An q)  (pA1 )Q1,r B eh + (pA1 )Q1,r ⊙ B ⊙ (An q) − C Qn,r (An q) − An q + (pA1 )Q1,r − pA1 B eh An q . It follows that Cr − C eh → 0 as r → ∞. Our claim will follow if we show that Cr ∈ E. To this end, it suffices to show that if A1 = [a1 , . . . , ar , 0, . . .] and An = [b1 , . . . , br , 0, . . .]t , where ai , bi are operators of finite rank, then A1 ⊙ B ⊙ An ∈ E. Let A1 and An be as stated and let B ′ = (Q2,r A2 ) ⊙ A3 ⊙ · · · ⊙ An−2 ⊙ (An−1 Qn,r ). Then A1 ⊙ B ⊙ An = A1 ⊙ B ′ ⊙ An+1 belongs to the algebraic tensor product K(H1 ) ⊙ (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊙ K(Hn ) and hence to E = K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ). ✷ Remarks 3.5. (i) It follows from Theorem 3.4 that if Φ : Kh → K(Hn , H1 ) is a mapping of finite rank whose image consists of finite rank operators then there exist finite rank projections p and q on H1 and Hn , respectively, and u ∈ (pK(H1 )) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h (K(Hn )q) such that Φ = γ0 (u). (ii) The identity E1 ⊗h (E2 ⊗eh E3 ) = (E1 ⊗h E2 ) ⊗eh E3 does not hold in general; for an example, take E1 = E3 = B(H ) and E2 = C. J (iii) For every Φ ∈ CC(Kh , K(Hn , H1 )) there exist A1 ∈ K(H1J1 , H1 ), Ai ∈ B(HiJi , Hi i−1 ), i = J 2, . . . , n − 1 and An ∈ K(Hn , Hn n−1 ) such that Φ(x1 ⊗ · · · ⊗ xn−1 ) = A1 (x1 ⊗ I )A2 . . . (xn−1 ⊗ I )An , whenever xi ∈ K(Hi+1 , Hi ), i = 1, . . . , n − 1. Indeed, by Proposition 3.4 we have Φ(x1 ⊗ · · · ⊗ xn−1 ) = A1 (x1 ⊗ I )A2 . . . (xn−1 ⊗ I )An for some A1 ⊙ A2 ⊙ · · · ⊙ An ∈ K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ). Using an idea of Blecher and Smith [2, Theorem 3.1], we can choose A1 = [tj ]j ∈J1 ∈ MJ1 ,1 (K(H1 )) ⊆ B(H1J1 , H1 ) and An =   J [si ]i∈Jn−1 ∈ M1,Jn−1 (K(Hn )) ⊆ B(Hn , Hn n−1 ) such that the sums i si si∗ and j tj∗ tj conF verge uniformly. Then A1 is the norm limit of AF 1 = [tj ]i∈J1 , where F is a finite subset of J1 and tjF = tj if j ∈ F and tjF = 0 otherwise. Therefore A1 ∈ K(H1J1 , H ). Similarly, An ∈ K(Hn , HnJn −1 ). 3784 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 In the case n = 2, Theorem 3.4 reduces to the following result which was established by Saar (Satz 6 of [21]) using the fact that every completely compact completely bounded map on K(H1 , H2 ) is a linear combination of completely compact completely positive maps. Corollary 3.6. A completely bounded map Φ : K(H1 , H2 ) → K(H1 , H2 ) is completely compact if and only if there  exist an index  set J and families {ai }i∈J ⊆ K(H1 ) and {bi }i∈J ⊆ K(H2 ) such that the series i∈J bi bi∗ and i∈J ai∗ ai converge uniformly and Φ(x) =  bi xai , x ∈ K(H1 , H2 ). i∈J We note in passing that Theorem 3.4 together with a result of Effros and Kishimoto [4] yields the following completely isometric identification: Corollary 3.7. CC(K(H2 , H1 ))∗∗ ≃ (K(H1 ) ⊗h K(H2 ))∗∗ ≃ CB(B(H2 , H1 )). Saar [21] constructed an example of a compact map Φ : K(H ) → K(H ) which is not completely compact (see Section 7 where we give a detailed account of this construction). We note that a compact completely positive map Φ : K(H ) → K(H ) is automatically completely compact. Indeed, the Stinespring Theorem implies that  there exist an index set J and a row operator A = [ai ]i∈J ∈ B(H J , H ) such that Φ(x) = i∈J ai xai∗ , x ∈ K(H ). The second dual Φ ∗∗ : B(H ) → B(H ) of Φ is a compact map given by the same formula. A standard Banach space argument shows that Φ ∗∗ takes values in K(H ), and hence Φ ∗∗ (I ) ∈ K(H ). This means that AA∗ ∈ K(H ) and so A ∈ K(H J , H ) which easily implies that Φ is completely compact. The previous paragraph shows that there exists a compact completely bounded map on K(H ) which cannot be written as a linear combination of compact completely positive maps. We finish this section with a modular version of Theorem 3.4. Given von Neumann algebras Ai ⊆ B(Hi ), i = 1, . . . , n, we let CCA′1 ,...,A′n (Kh , K(Hn , H1 )) denote the space of all completely compact multilinear maps from Kh into K(Hn , H1 ) such that the corresponding multilinear map from K(H2 , H1 ) × · · · × K(Hn , Hn−1 ) into K(Hn , H1 ) is A′1 , . . . , A′n -modular. Corollary 3.8. Let Ai ⊆ B(Hi ), i = 1, . . . , n, be von Neumann algebras. Set K′ (Ai ) = K(Hi ) ∩ Ai , for i = 1 and i = n. Then     γ0 K′ (A1 ) ⊗h (A2 ⊗eh · · · ⊗eh An−1 ) ⊗h K′ (An ) = CCA′1 ,...,A′n Kh , K(Hn , H1 ) . Proof. By Theorems 2.1 and 3.4, the image of K′ (A1 ) ⊗h (A2 ⊗eh · · · ⊗eh An−1 ) ⊗h K′ (An ) under γ0 is contained in CCA′1 ,...,A′n (Kh , K(Hn , H1 )). For the converse, fix an element Φ ∈ CCA′1 ,...,A′n (Kh , K(Hn , H1 )). By Theorem 3.4, there exists a unique u ∈ K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ) such that γ0 (u) = Φ. By Theorem 2.1, u ∈ A1 ⊗eh · · · ⊗eh An . Lemma 2.2 now shows that u ∈ K′ (A1 ) ⊗h (A2 ⊗eh · · · ⊗eh An−1 ) ⊗h K′ (An ). ✷ 4. Complete boundedness of multipliers Our aim in this section is to clarify the relationship between universal operator multipliers and completely bounded maps, extending results of [12]. We begin with an observation which will 3785 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 allow us to deal with the cases of even and odd numbers of variables in the same manner. We use the notation established in Section 2. Proposition 4.1. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ M(A1 , . . . , An ). Let πi be a representation of Ai on a Hilbert space Hi , i = 1, . . . , n, and π = π1 ⊗ · · · ⊗ πn . The map Sπ(ϕ) takes values in K(H1 , Hn ) if n is odd, and in K(H1d , Hn ) if n is even. Proof. For even n, this is immediate as observed in [12]. Let n be odd. Assume without loss of generality that Ai = B(Hi ) and πi is the identity representation. We call an element ζ ∈ Γ (H1 , . . . , Hn ) thoroughly elementary if d ζ = η1,2 ⊗ ξ2,3 ⊗ · · · ⊗ ξn−1,n d d d where all ηj,j +1 = ηj ⊗ ηj +1 and ξj −1,j = ξj −1 ⊗ ξj are elementary tensors. The linear span of the thoroughly elementary tensors is dense in the completion of Γ (H1 , . . . , Hn ) in · 2,∧ . Moreover, the linear span of the elementary tensors ϕ = ϕ1 ⊗ · · · ⊗ ϕn is dense in B(H1 ) ⊗ · · · ⊗ B(Hn ). By (7) and since Sϕ (ζ ) is linear in both ϕ and ζ , it suffices to show that Sϕ (ζ ) is compact when ϕ is an elementary tensor and ζ is a thoroughly elementary tensor. However, in this case Sϕ (ζ ) has rank at most 1, since for every ξ1 ∈ H1 , Sϕ (ζ )ξ1 = ϕn θ (ξn−1,n ) . . . ϕ2d θ  d  η1,2 ϕ1 ξ1 = n−1   (ϕj ξj , ηj ) ϕn ξn . j =1 ✷ We now establish some notation. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ A1 ⊗ · · · ⊗ An . Assume that n is even and let π1 , . . . , πn be representations of A1 , . . . , An on H1 , . . . , Hn , respectively. Set π = π1 ⊗ · · · ⊗ πn . Using the natural identifications, we consider the map Sπ(ϕ) : Γ (H1 , . . . , Hn ) → H1 ⊗ Hn as a map (denoted in the same way) We let    d    Sπ(ϕ) : C2 H1d , H2 ⊙ · · · ⊙ C2 Hn−1 , Hn → C2 H1d , Hn .  d      Φπ(ϕ) : C2 Hn−1 , Hn ⊙ · · · ⊙ C2 H1d , H2 → C2 H1d , Hn be the map given on elementary tensors by Φπ(ϕ) (Tn−1 ⊗ · · · ⊗ T1 ) = Sπ(ϕ) (T1 ⊗ · · · ⊗ Tn−1 ). Note that if ϕ ∈ M(A1 , . . . , An ) then Φπ(ϕ) is bounded when the domain is equipped with the Haagerup norm and the range with the operator norm. In this case, Φπ(ϕ) has a unique extension (which will be denoted in the same way)   d    Φπ(ϕ) : K Hn−1 , Hn ⊗h · · · ⊗h K H1d , H2 , · h     → K H1d , Hn , · op  . If n is odd then the map Φπ(ϕ) is defined in a similar way. The map Φπ(ϕ) will be used extensively hereafter. 3786 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 The main result of this section is Theorem 4.3, where we explain how the complete boundedness of the mappings Φπ(ϕ) relates to the property of ϕ being a multiplier. We will need the following lemma. Lemma 4.2. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, i = 1, . . . , n, and let k ∈ N. Let ϕ ∈ A1 ⊗ · · · ⊗ An and write ψ = (id(k) ⊗· · ·⊗id(k) )(ϕ). Suppose that n is even. If Ti ∈ Mk (C2 (Hid , Hi+1 )) for odd i d )) for even i then and Ti ∈ Mk (C2 (Hi , Hi+1 Φϕ(k) (Tn−1 ⊙ · · · ⊙ T1 ) = Φψ (Tn−1 ⊗ · · · ⊗ T1 ), (k) where we identify the operator spaces Mk (C2 (Hid , Hi+1 )) and C2 ((Hid )(k) , Hi+1 ) for odd i, and (k) d )) and C (H , (H d )(k) ) for even i. A similar statement holds for odd n. Mk (C2 (Hi , Hi+1 2 i i+1 Proof. To simplify notation, we give the proof for n = 2; the proof of the general case is similar. If ϕ = a1 ⊗ a2 is an elementary tensor then Φϕ (T ) = a2 T a1d for T ∈ C2 (H1d , H2 ) and it is easily checked that the statement holds. By linearity, it holds for each ϕ ∈ A1 ⊙ A2 . Suppose now that ϕ ∈ A1 ⊗ A2 is arbitrary. Let {ϕm } ⊆ A1 ⊙ A2 be a sequence converging in the operator norm to ϕ and ψm = (id(k) ⊗ id(k) )(ϕm ). By (7), Φϕm (T ) → Φϕ (T ) in the operator norm, for all T ∈ (k) (k) C2 (H1d , H2 ). This implies that if S ∈ Mk (C2 (H1d , H2 )), then Φϕm (S) → Φϕ (S) in the operator norm of Mk (C2 (H1d , H2 )). Since ψm → ψ in the operator norm, we conclude that Φψm (S) → (k) (k) Φψ (S) in the operator norm of C2 ((H1d )(k) , H2 ). It follows that Φψ (S) = Φϕ (S). ✷ Theorem 4.3. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ A1 ⊗ · · · ⊗ An . The following are equivalent: (i) ϕ ∈ M(A1 , . . . , An ); (ii) if πi is a representation of Ai , i = 1, . . . , n, and π = π1 ⊗ · · · ⊗ πn then the map Φπ(ϕ) is completely bounded; (iii) there exist faithful representations πi of Ai , i = 1, . . . , n, such that if π = π1 ⊗ · · · ⊗ πn then the map Φπ(ϕ) is completely bounded. Moreover, if the above conditions hold and π is as in (iii) then ϕ m = Φπ(ϕ) cb . Proof. For technical simplicity we only consider the case n = 3. (i) ⇒ (ii) Let ϕ ∈ M(A1 , A2 , A3 ) and πi : Ai → B(Hi ) be a representation, i = 1, 2, 3. Then π(ϕ) ∈ M(π1 (A1 ), π(A2 ), π3 (A3 )); thus, it suffices to assume that Ai ⊆ B(Hi ) are concrete C ∗ -algebras and that πi is the identity representation, i = 1, 2, 3. Fix k ∈ N and let ψ = (id(k) ⊗ id(k) ⊗ id(k) )(ϕ). Since ϕ ∈ M(A1 , A2 , A3 ), the map       Φψ : K H2d(k) , H3(k) ⊙ K H1(k) , H2d(k) → K H1(k) , H3(k) is bounded with norm not exceeding ϕ m . By Lemma 4.2, Φϕ(k)  ϕ holds for every k ∈ N, the map Φϕ is completely bounded. (ii) ⇒ (iii) is trivial. m . Since this inequality 3787 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 (iii) ⇒ (i) We may assume that Ai ⊆ B(Hi ) and that πi is the identity representation, i = 1, 2, 3. Let λ be a cardinal number, ρi = id(λ) be the ampliation of the identity representation of multiplicity λ, ψ = (ρ1 ⊗ ρ2 ⊗ ρ3 )(ϕ), and H̃i = Hiλ , i = 1, 2, 3. Fix ε > 0 and ζ ∈ Γ (H̃1 , H̃2 , H̃3 ). Let     T̃ = T̃2 ⊙ T̃1 ∈ C2 H̃2d , H̃3 ⊙ C2 H̃1 , H̃2d be the element canonically corresponding to ζ . Then there exist k ∈ N and canonical projections Pi from H̃i onto the direct sum of k copies of Hi such that if T0 = (P3 T̃2 (P2d ⊗ I )) ⊙ (k) (k) (k) ((P2d ⊗ I )T̃1 P1 ) and if ζ0 is the element of Γ (H1 , H2 , H3 ) corresponding to T0 then ζ − ζ0 2,∧  ε. Set ψ0 = (id(k) ⊗ id(k) ⊗ id(k) )(ϕ). Arguing as in Lemma 4.2, we see that Φψ0 (T0 ) op = Φψ (T0 ) op . Using (7) and Lemma 4.2 we obtain Sψ (ζ ) op  Sψ (ζ − ζ0 ) + Sψ (ζ0 ) op 2,∧ + Φψ0 (T0 )  ε ϕ + Φϕ cb T0  ε ϕ + Φϕ cb  ε ϕ + Φϕ cb  ψ ζ − ζ0 op = Sψ (ζ − ζ0 ) op + Φψ (T0 ) op Φϕ(k) (T0 ) op h   P3 T̃2 P2d ⊗ I T̃2 ε ϕ + op op T̃1 op . op  d  P2 ⊗ I T̃1 P1 op It follows that ϕ id(λ) ,id(λ) ,id(λ)  Φϕ cb . Now let ρ1 , ρ2 , ρ3 be arbitrary representations of A1 , A2 , A3 , respectively. Then there exists a cardinal number λ such that each of the representations ρi is approximately subordinate to the representation id(λ) (see [26] and [10, Theorem 5.1]). By Theorem 5.1 of [12], ϕ ρ1 ,ρ2 ,ρ3  ϕ id(λ) ,id(λ) ,id(λ) ; now the previous paragraph implies that ϕ ρ1 ,ρ2 ,ρ3  Φϕ cb . It follows that ϕ ∈ M(A1 , A2 , A3 ) and ϕ m  Φϕ cb . As the reversed inequality was already established, we conclude that ϕ m = Φϕ cb . ✷ 5. The symbol of a universal multiplier Our aim in this section is to generalise the natural correspondence between a function ϕ ∈ ℓ∞ ⊗eh ℓ∞ and the Schur multiplier Sϕ on B(ℓ2 (N)) given by Sϕ ((aij )) = (ϕ(i, j )aij ). To each universal operator multiplier we will associate an element of an extended Haagerup tensor product which we call its symbol. This will be used in the subsequent sections to identify certain classes of operator multipliers. Recall that if A is a C ∗ -algebra, its opposite C ∗ -algebra Ao is defined to be the C ∗ -algebra whose underlying set, norm, involution and linear structure coincide with those of A and whose multiplication · is given by a · b = ba. If a ∈ A we denote by a o the element of Ao corresponding to a. If π : A → B(H ) is a representation of A then the map π d : a o → π(a)d from Ao into B(H d ) is a representation of Ao . Clearly, π is faithful if and only if π d is faithful. If πi : Ai → B(Hi ) are faithful representations, i = 1, . . . , n (n even), then by [5, Lemma 5.4] there exists a d ⊗ · · · ⊗ π d from A ⊗ Ao o complete isometry πn ⊗eh πn−1 eh eh 1 n eh n−1 ⊗eh · · · ⊗eh A1 into B(Hn ) ⊗eh o d ) ⊗ · · · ⊗ B(H d ) which sends a ⊗ a o d B(Hn−1 eh eh n 1 n−1 ⊗ · · · ⊗ a1 to πn (an ) ⊗ πn−1 (an−1 ) ⊗ · · · d ⊗ π1 (a1 ) . 3788 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 Henceforth, we will consistently write π = π1 ⊗ · · · ⊗ πn and  d πn ⊗eh πn−1 ⊗eh · · · ⊗eh π2 ⊗eh π1d if n is even, ′ π = d πn ⊗eh πn−1 ⊗eh · · · ⊗eh π2d ⊗eh π1 if n is odd. Let n ∈ N, A1 , . . . , An be C ∗ -algebras, πi be a representation of Ai , i = 1, . . . , n, and ϕ ∈ M(A1 , . . . , An ). Assume that n is even. By Theorem 4.3, the map  d      , Hn ⊗h · · · ⊗h K H1d , H2 → K H1d , Hn Φπ(ϕ) : K Hn−1 is completely bounded. By Proposition 3.3, there exists a unique element uπϕ ∈ B(Hn ) ⊗eh d ) ⊗ · · · ⊗ B(H d ) such that γ (uπ ) = Φ B(Hn−1 eh eh 0 ϕ π(ϕ) . For example, if each Ai is a concrete 1 ∗ C -algebra and ai ∈ Ai , i = 1, . . . , n, then d d uid a1 ⊗a2 ⊗···⊗an−1 ⊗an = an ⊗ an−1 ⊗ · · · ⊗ a2 ⊗ a1 . If n is odd then we define uπϕ similarly. The main result of this section is the following. Theorem 5.1. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ M(A1 , . . . , An ). There exists a unique element  An ⊗eh Aon−1 ⊗eh · · · ⊗eh A2 ⊗eh Ao1 if n is even, uϕ ∈ An ⊗eh Aon−1 ⊗eh · · · ⊗eh Ao2 ⊗eh A1 if n is odd with the property that if πi is a representation of Ai for i = 1, . . . , n then uπϕ = π ′ (uϕ ). (8) The map ϕ → uϕ is linear and if ai ∈ Ai , i = 1, . . . , n, then ua1 ⊗···⊗an = Moreover, ϕ m = uϕ  o ⊗ · · · ⊗ a2 ⊗ a1o an ⊗ an−1 o an ⊗ an−1 ⊗ · · · ⊗ a2o ⊗ a1 if n is even, if n is odd. eh . Definition 5.2. The element uϕ defined in Theorem 5.1 will be called the symbol of the universal multiplier ϕ. In order to prove Theorem 5.1 we have to establish a number of auxiliary results. If ω ∈ B(H )∗ we let ω̃ ∈ B(H d )∗ be the functional given by ω̃(a d ) = ω(a). Note that if ω = ωξ,η is the vector functional a → (aξ, η) then ω̃ = ωηd ,ξ d . Lemma 5.3. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, ξi , ηi ∈ Hi and ωi = ωξi ,ηi , i = 1, . . . , n. Suppose that ϕ ∈ M(A1 , . . . , An ). Then  uid if n is even,   ϕ , ωn ⊗ ω̃n−1 ⊗ · · · ⊗ ω̃1 ϕ(ξ1 ⊗ · · · ⊗ ξn ), η1 ⊗ · · · ⊗ ηn = (9) id uϕ , ωn ⊗ ω̃n−1 ⊗ · · · ⊗ ω1 if n is odd. K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3789 Proof. We only consider the case n is even; the proof for odd n is similar. Suppose that ϕ is an d d elementary tensor, say ϕ = a1 ⊗ · · · ⊗ an . Then uid ϕ = an ⊗ an−1 ⊗ · · · ⊗ a1 and thus n    ϕ(ξ1 ⊗ · · · ⊗ ξn ), η1 ⊗ · · · ⊗ ηn = (ai ξi , ηi ) = uid ϕ , ωn ⊗ ω̃n−1 ⊗ · · · ⊗ ω̃1 . i=1 By linearity, (9) holds for each ϕ ∈ A1 ⊙ · · · ⊙ An . Now let ϕ be an arbitrary element of M(A1 , . . . , An ). By Theorem 2.3, there exists a net ν ν id {ϕν } ⊆ A1 ⊙ · · · ⊙ An and representations uid ϕ = An ⊙ · · · ⊙ A1 and uϕν = An ⊙ · · · ⊙ A1 , where d ν Ai are finite matrices with entries in Ai if i is even and in Ai if i is odd, such that ϕν → ϕ semi-weakly, Aνi → Ai strongly and all norms Ai , Aνi are bounded by a constant depending only on n. As in (2), we have uid ϕ , ωn ⊗ ω̃n−1 ⊗ · · · ⊗ ω̃1 = An , ωn An−1 , ω̃n−1 . . . A1 , ω̃1 . (10) Moreover, all norms Aνi , ωi (for even i) and Aνi , ω̃i (for odd i) are bounded by a constant depending only on n, and the strong convergence of Aνi to Ai implies that Aνi , ωi converges strongly to Ai , ωi . Indeed, it is easy to check that if ξ, η ∈ H , A ∈ MI (B(H )) = B(H ⊗ ℓ2 (I )) and ζ ∈ ℓ2 (I ) for some index set I then A, ωξ,η ζ 2   = A(ξ ⊗ ζ ), η ⊗ A, ωξ,η ζ . This implies that Ai − Aνi , ωi η  C (Ai − Aνi )(ξi ⊗ η) for some constant C > 0, and the strong convergence follows. Since operator multiplication is jointly strongly continuous on bounded sets, it now follows from (10) that id uid ϕν , ωn ⊗ ω̃n−1 ⊗ · · · ⊗ ω̃1 → uϕ , ωn ⊗ ω̃n−1 ⊗ · · · ⊗ ω̃1 . On the other hand, since ϕν → ϕ semi-weakly,     ϕν (ξ1 ⊗ · · · ⊗ ξn ), η1 ⊗ · · · ⊗ ηn → ϕ(ξ1 ⊗ · · · ⊗ ξn ), η1 ⊗ · · · ⊗ ηn . The proof is complete. ✷ Lemma 5.4. Let Hi be a Hilbert space and Ei ⊆ B(Hi ) be an operator space, i = 1, . . . , n. Suppose that X and Y are closed subspaces of E1 and En , respectively and let u, v ∈ E1 ⊗eh · · · ⊗eh En . If Rω (u) ∈ X and Lω′ (v) ∈ Y ′ whenever ω = ω2 ⊗ · · · ⊗ ωn and ω′ = ω1′ ⊗ · · · ⊗ ωn−1 where every ωi , ωi′ ∈ B(Hi )∗ is a vector functional, then u ∈ X ⊗eh E2 ⊗eh · · · ⊗eh En and v ∈ E1 ⊗eh · · · ⊗eh En−1 ⊗eh Y. 3790 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 Proof. Let Fi be the span of the vector functionals on B(Hi ). By linearity, Rω (u) ∈ X for each ω ∈ F2 ⊙ · · · ⊙ Fn . Now suppose that   ω ∈ B(H2 ) ⊗eh · · · ⊗eh B(Hn ) ∗ = C1 (H2 ) ⊗h · · · ⊗h C1 (Hn ). There exists a sequence (ωm ) ⊆ F2 ⊙ · · · ⊙ Fn such that ωm → ω in norm. Hence Rω (u) − Rωm (u) B(H )  ω − ωm 1 u eh → 0, whence Rω (u) = limm Rωm (u) ∈ X . Spronk’s formula (5) now implies that u ∈ X ⊗eh E2 ⊗eh · · · ⊗eh En . The assertion concerning v has a similar proof. ✷ We will use slice maps defined on the minimal tensor product of several C ∗ -algebras as follows. Assume that Ai ⊆ B(Hi ) and ωi ∈ B(Hi )∗ , i = 1, . . . , n, and let ϕ ∈ A1 ⊗ · · · ⊗ An . If 1  i1 < · · · < ik  n and {ℓ1 < ℓ2 < · · · < ℓn−k } is the complement of {i1 , . . . , ik } in {1, . . . , n}, let Λωi1 ,...,ωik : A1 ⊗ · · · ⊗ An → Aℓ1 ⊗ · · · ⊗ Aℓn−k be the unique norm continuous linear mapping given on elementary tensors by Λωi1 ,...,ωik (a1 ⊗ · · · ⊗ an ) = ωi1 (ai1 ) . . . ωik (aik ) aℓ1 ⊗ · · · ⊗ aℓn−k . Proposition 5.5. Let Ai ⊆ B(Hi ), i = 1, . . . , n, be C ∗ -algebras and let ϕ ∈ M(A1 , . . . , An ). Then  An ⊗eh Adn−1 ⊗eh · · · ⊗eh A2 ⊗eh Ad1 if n is even, id uϕ ∈ An ⊗eh Adn−1 ⊗eh · · · ⊗eh Ad2 ⊗eh A1 if n is odd. d Proof. We only consider the case n = 3. Let u = uid ϕ ; by definition, u ∈ B(H3 ) ⊗eh B(H2 ) ⊗eh B(H1 ). Let ξi , ηi ∈ Hi and ωi = ωξi ,ηi , i = 1, 2, 3. Then by (4) and Lemma 5.3, Thus   Rω̃2 ⊗ω1 (u)ξ3 , η3 = Rω̃2 ⊗ω1 (u), ω3 = u, ω3 ⊗ ω̃2 ⊗ ω1     = ϕ(ξ1 ⊗ ξ2 ⊗ ξ3 ), η1 ⊗ η2 ⊗ η3 = Λω1 ,ω2 (ϕ)ξ3 , η3 . Rω̃2 ⊗ω1 (u) = Λω1 ,ω2 (ϕ) ∈ A3 . Lemma 5.4 now implies that u ∈ A3 ⊗eh B(H2d ) ⊗eh B(H1 ). Let w = Rω1 (u). By the previous paragraph, w ∈ A3 ⊗eh B(H2d ). By (4) and Lemma 5.3,   Lω3 (w)η2d , ξ2d = Lω3 (w), ω̃2 = Rω1 (u), ω3 ⊗ ω̃2     = u, ω3 ⊗ ω̃2 ⊗ ω1 = Λω1 ,ω3 (ϕ)ξ2 , η2 = Λω1 ,ω3 (ϕ)d η2d , ξ2d . K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3791 Hence Lω3 (w) = Λω1 ,ω3 (ϕ)d ∈ Ad2 and, by Lemma 5.4, w ∈ A3 ⊗eh Ad2 . Applying this lemma again shows that u ∈ A3 ⊗eh Ad2 ⊗eh B(H1 ). Continuing in this fashion we see that u ∈ A3 ⊗eh Ad2 ⊗eh A1 . ✷ Lemma 5.6. Let A1 , . . . , An be C ∗ -algebras and let ρi : Ai → B(Ki ), θi : ρi (Ai ) → B(Hi ) be representations, i = 1, . . . , n. Suppose that (κ) (i) for any cardinal number κ, the representations θi : ρi (Ai ) → B(Hiκ ) are strongly continuous, and (ii) whenever ϕ ∈ M(A1 , . . . , An ) and {ϕν } is a net in A1 ⊙ · · · ⊙ An such that ρ(ϕν ) → ρ(ϕ) semi-weakly and supν ϕν m < ∞ then Φθ◦ρ(ϕν ) → Φθ◦ρ(ϕ) pointwise weakly. θ◦ρ Then uϕ ρ = θ ′ (uϕ ), for each ϕ ∈ M(A1 , . . . , An ). Proof. We suppose that n is even, the proof for odd n being similar. If ϕ = a1 ⊗ · · · ⊗ an is an ρ o ⊗ · · · ⊗ a1o ), so elementary tensor, then uϕ = ρ ′ (an ⊗ an−1     o uϕθ◦ρ = (θ ◦ ρ)′ an ⊗ an−1 ⊗ · · · ⊗ a1o = θ ′ uρϕ . By linearity, the claim also holds for ϕ ∈ A1 ⊙ · · · ⊙ An . If ϕ ∈ M(A1 , . . . , An ) is arbitrary then ρ(ϕ) ∈ M(ρ1 (A1 ), . . . , ρn (An )) and by Theorem 2.3 and Proposition 5.5, there exist a net {ϕν } ⊆ A1 ⊙ · · ·⊙ An such that ρ(ϕν ) → ρ(ϕ) semi-weakly, ρ a representation uϕ = An ⊙ · · · ⊙ A1 , where Ai ∈ Mκ (ρi (Ai )) ⊆ B(Kiκ ) if i is even and Ai ∈ Mκ (ρid (Aoi )) ⊆ B(Kiκ )d if i is odd (κ being a suitable index set), whose operator matrix entries ρ belong to ρi (Ai ) if i is even and to ρid (Aoi ) if i is odd, and representations uϕν = Aνn ⊙ · · · ⊙ Aν1 ν where the Ai are finite matrices with operator entries in ρi (Ai ) if i is even and ρid (Aoi ) if i is odd such that Aνi → Ai strongly and all norms Aνi , Ai are bounded. ρ ρ Now θ ′ (uϕ ) = Ãn ⊙ · · · ⊙ Ã1 and θ ′ (uϕν ) = Ãνn ⊙ · · · ⊙ Ãν1 where Ãi and Ãνi are the images (κ) of Ai and Aνi under θi or (θid )(κ) according to whether i is even or odd. By assumption (i),       (11) γ0 θ ′ uρϕν (Tn−1 ⊗ · · · ⊗ T1 ) → γ0 θ ′ uρϕ (Tn−1 ⊗ · · · ⊗ T1 ) d , H ),. . .,T ∈ C (H d , H ). On the other hand, assumption (ii) and weakly for all Tn−1 ∈ C2 (Hn−1 n 1 2 2 1 the first paragraph of the proof show that        = Φθ◦ρ(ϕν ) → Φθ◦ρ(ϕ) = γ0 uϕθ◦ρ γ0 θ ′ uρϕν = γ0 uϕθ◦ρ ν θ◦ρ ρ pointwise weakly. Using (11) we conclude that γ0 (uϕ ) = γ0 (θ ′ (uϕ )); since γ0 is injective we θ◦ρ ρ have that uϕ = θ ′ (uϕ ). ✷ Proof of Theorem 5.1. We will only consider the case n is even. Let ρi : Ai → B(Ki ) be the d ⊗ · · · ⊗ ρd. universal representation of Ai , i = 1, . . . , n. Set ρ = ρ1 ⊗ · · · ⊗ ρn and ρ ′ = ρn ⊗ ρn−1 1 ρ ρ ′ ′ −1 By Proposition 5.5, uϕ lies in the image of ρ ; we define uϕ = (ρ ) (uϕ ). 3792 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 (κ) (κ) Let κ be a nonzero cardinal number and let σi = ρi . If θi = idρi (Ai ) = σi ◦ ρi−1 then it follows from the proof of Proposition 6.2 of [12] that the hypotheses of Lemma 5.6 are satisfied, so writing θ = θ1 ⊗ · · · ⊗ θn , we have   uσϕ = uϕθ◦ρ = θ ′ uρϕ = (θ ′ ◦ ρ ′ )(uϕ ) = σ ′ (uϕ ). Now let πi be an arbitrary representation of Ai . It is well known (see e.g. [25]) that πi is unitarily (κ) equivalent to a subrepresentation of σi = ρi for some κ. Hence there exist unitary operators vi , i = 1, . . . , n (acting between appropriate Hilbert spaces) and subspaces Hi of Kiκ , such that if τi (x) = vi xvi∗ |Hi then πi = τi ◦ σi . Examining the proof of Proposition 6.2 of [12], we see that τ = τ1 ⊗ · · · ⊗ τn satisfies the hypotheses of Lemma 5.6, so   uπϕ = uϕτ ◦σ = τ ′ uσϕ = (τ ◦ σ )′ (uϕ ) = π ′ (uϕ ). The uniqueness of uϕ follows from the injectivity of γ0 . The linearity of the map ϕ → uϕ and its values on elementary tensors are straightforward. The fact that ϕ m = uϕ eh follows from Proposition 3.3 and Theorem 4.3. ✷ Remarks. (i) Let Ai ⊆ B(Hi ), i = 1, . . . , n be concrete C ∗ -algebras of operators. Taking πi to be the identity representation for i = 1, . . . , n and writing id = π1 ⊗ · · · ⊗ πn gives uϕ = uid ϕ if we identify Aoi with Adi . (ii) Theorem 5.1 implies that if Ai , i = 1, . . . , n, are concrete C ∗ -algebras then the entries of the block operator matrices Ai appearing in the representation of ϕ in Theorem 2.3 can be chosen from Ai , i = 1, . . . , n. 6. Completely compact multipliers In this section we introduce the class of completely compact multipliers and characterise them within the class of all universal multipliers using the notion of the symbol introduced in Section 5. We will need the following lemma. Lemma 6.1. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, i = 1, . . . , n, a ∈ A1 , b ∈ An and ϕ ∈ M(A1 , . . . , An ). Let ψ ∈ A1 ⊗ · · · ⊗ An be given by  (a ⊗ I ⊗ · · · ⊗ I ⊗ b)ϕ if n is even, ψ= (I ⊗ · · · ⊗ I ⊗ b)ϕ(a ⊗ I ⊗ · · · ⊗ I ) if n is odd. Then ψ ∈ M(A1 , . . . , An ) and Φψ (x) =  bΦϕ (x)a d bΦϕ (x)a if n is even, if n is odd. (12) Proof. For technical simplicity, we will only consider the case n = 2. Let ai ∈ Ai , i = 1, 2, and ϕ = a1 ⊗ a2 . In this case ψ = (aa1 ) ⊗ (ba2 ) so Φψ (T ) = ba2 T (aa1 )d = ba2 T a1d a d = bΦϕ (T )a d . By linearity, (12) holds whenever ϕ ∈ A1 ⊙ A2 . K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3793 Assume that ϕ ∈ M(A1 , A2 ) is arbitrary. Fix an operator T ∈ C2 (H1d , H2 ). By Theorem 2.3, there exists a net {ϕν } ⊆ A1 ⊙ A2 such that ϕν → ϕ semi-weakly, supν ϕν m < ∞ and Φϕν (T ) → Φϕ (T ) weakly. Let ψν = (a ⊗ b)ϕν ; then ψν → ψ semi-weakly. Clearly, ψν ∈ A1 ⊙ A2 ; in particular ψν ∈ M(A1 , A2 ). By the previous paragraph, Φψν (·) = bΦϕν (·)a d and hence Φψν (T ) → bΦϕ (T )a d weakly. If ϕν = B1ν ⊙ B2ν then ψν = (aB1ν ) ⊙ ((b ⊗ I )B2ν ). It follows from Theorem 2.3 that ψ ∈ M(A1 , A2 ) and that Φψν (T ) → Φψ (T ) weakly. Thus Φψ (T ) = bΦϕ (T )a d . ✷ Given faithful representations π1 , . . . , πn of the C ∗ -algebras A1 , . . . , An , respectively, we define   π (A1 , . . . , An ) = ϕ ∈ M(A1 , . . . , An ): Φπ(ϕ) is completely compact Mcc  Mffπ (A1 , . . . , An ) = ϕ ∈ M(A1 , . . . , An ): the range of Φπ(ϕ)  is a finite dimensional space of finite-rank operators . Theorem 6.2. Let Ai ⊆ B(Hi ) be a C ∗ -algebra, i = 1, . . . , n, and ϕ ∈ M(A1 , . . . , An ). The following are equivalent: id (A , . . . , A ); (i) ϕ ∈ Mcc 1 n  (K(Hn ) ∩ An ) ⊗h (Adn−1 ⊗eh · · · ⊗eh A2 ) ⊗h (K(H1d ) ∩ Ad1 ) (ii) uid ∈ ϕ (K(Hn ) ∩ An ) ⊗h (Adn−1 ⊗eh · · · ⊗eh Ad2 ) ⊗h (K(H1 ) ∩ A1 ) (iii) there exists a net {ϕα } ⊆ Mffid (A1 , . . . , An ) such that ϕα − ϕ m if n is even, if n is odd; → 0. Proof. We will only consider the case n is even. (i) ⇒ (ii) Theorem 3.4 implies that   d    d uid ϕ ∈ K(Hn ) ⊗h B Hn−1 ⊗eh · · · ⊗eh B(H2 ) ⊗h K H1 while, by Proposition 5.5, d d uid ϕ ∈ An ⊗eh An−1 ⊗eh · · · ⊗eh A2 ⊗eh A1 . The conclusion now follows from Lemma 2.2. (ii) ⇒ (i) By Theorem 3.4, Φϕ = γ0 (uid ϕ ) is completely compact. (ii) ⇒ (iii) Let p ∈ B(H1 ) (resp. q ∈ B(Hn )) be the projection onto the span of all ranges of operators in K(H1 ) ∩ A1 (resp. K(Hn ) ∩ An ), and let {pα } ⊆ K(H1 ) ∩ A1 (resp. {qα } ⊆ K(Hn ) ∩ An ) be a net of finite rank projections which tends strongly to p (resp. q). It is easy to see that Φϕ (Tn−1 ⊗ · · · ⊗ T1 ) = qΦϕ (Tn−1 ⊗ · · · ⊗ T1 )p d , for all T1 ∈ K(H1d , H2 ), . . . , Tn−1 ∈ d , H ). Let ϕ = (p ⊗ I ⊗ · · · ⊗ I ⊗ q )ϕ. By Lemma 6.1, ϕ ∈ M(A , . . . , A ) and K(Hn−1 n α α α α 1 n Φϕα (·) = qα Φϕ (·)pαd ; hence ϕα ∈ Mffid (A1 , . . . , An ). We have already seen that Φϕ is completely compact, and it follows from the proof of Theorem 3.4 that Φϕα → Φϕ in the cb norm. By Theorem 4.3, ϕ − ϕα m → 0. (iii) ⇒ (i) is immediate from Proposition 3.2 and Theorem 4.3 and the fact that finite rank maps are completely compact. ✷ 3794 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 Now consider the sets Mcc (A1 , . . . , An ) =  π Mcc (A1 , . . . , An ), π Mff (A1 , . . . , An ) =  Mffπ (A1 , . . . , An ) π where the unions are taken over all π = π1 ⊗ · · · ⊗ πn , each πi being a faithful representation of Ai . We refer to the first of these as the set of completely compact multipliers. Lemma 6.3. If ρi is the reduced atomic representation of Ai , i = 1, . . . , n, and ρ = ρ1 ⊗ · · · ⊗ ρn ρ then Mff (A1 , . . . , An ) = Mff (A1 , . . . , An ). Proof. Again, we give the proof for the even case only. We must show that Mffπ (A1 , . . . , An ) ⊆ ρ Mff (A1 , . . . , An ) whenever π = π1 ⊗ · · · ⊗ πn where each πi is a faithful representation of Ai . Without loss of generality, we may assume that each πi is the identity representation of Ai ⊆ B(Hi ). Let ϕ ∈ Mffπ (A1 , . . . , An ) so that the range of Φϕ is finite dimensional and consists of finite rank operators. By Remark 3.5 (i) there exist finite rank projections p and q on H1d and Hn , respectively, such that uid ϕ lies in the intersection of          d  ⊗eh · · · ⊗eh B(H2 ) ⊗h K H1d p qK(Hn ) ⊗h B Hn−1 and An ⊗eh · · · ⊗eh Ad1 . By Lemma 2.2, uid ϕ lies in          d  ⊗eh · · · ⊗eh B(H2 ) ⊗h K H1d p ∩ Ad1 . qK(Hn ) ∩ An ⊗h B Hn−1 id Hence there exists a representation uid ϕ = An ⊙ · · · ⊙ A1 of uϕ such that An = qAn and A1 = A1 p. Suppose that An = [b1 , b2 , . . .], where bj∈ An for each j , and let qj be the orthogsee that {Qm } is an increasing onal projection onto the range of bj . Setting Qm = m j =1 qj we sequence of projections in An dominated by q. It follows that ∞ m=1 Qm ∈ An . We may thus assume that q ∈ An . Similarly, we may assume that p ∈ Ad1 . Now     ρ ′ (uϕ ) = ρn (q)ρn (An ) ⊙ · · · ⊙ ρ1 (A1 )ρ1 (p) . ρ By [29], ρn (q) and ρ1 (p) have finite rank. By Lemma 6.1, ϕ ∈ Mff (A1 , . . . , An ). ✷ We are now ready to prove the main result of this section. Theorem 6.4. Let A1 , . . . , An be C ∗ -algebras and ϕ ∈ M(A1 , . . . , An ). The following are equivalent: (i) ϕ ∈ Mcc (A1 , . . . , An );  K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh A2 ) ⊗h K(Ao1 ) (ii) uϕ ∈ K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh Ao2 ) ⊗h K(A1 ) if n is even, if n is odd; (iii) there exists a net {ϕα } ⊆ Mff (A1 , . . . , An ) such that ϕα − ϕ m → 0. 3795 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 Proof. We will only consider the case n is even. π (A , . . . , A ); after identifying A (i) ⇒ (ii) Choose π = π1 ⊗ · · · ⊗ πn such that ϕ ∈ Mcc 1 n i with its image under πi , we may assume that each πi is the identity representation of a concrete C ∗ -algebra Ai ⊆ B(Hi ). By Theorem 6.2, uid ϕ lies in         K(Hn ) ∩ An ⊗h Aon−1 ⊗eh · · · ⊗eh A2 ⊗h K H1d ∩ Ao1 . The conclusion follows from the fact that K(Hi ) ∩ Ai ⊆ K(Ai ) for i = 1, n. (ii) ⇒ (i) Let ρi be the reduced atomic representation Ai → B(Hi ) for i = 1, . . . , n. Since ρ ′ ρ is an isometry, uϕ = ρ ′ (uϕ ) lies in        d  o  ρn K(An ) ⊗h ρn−1 An−1 ⊗eh · · · ⊗eh ρ2 (A2 ) ⊗h ρ1d K Ao1 . ρ By Theorem 7.5 of [28], K(Hi ) ∩ ρi (Ai ) = ρi (K(Ai )). By Theorem 6.2, ϕ ∈ Mcc (A1 , . . . , An ). (i) ⇒ (iii) is immediate from Theorem 6.2. (iii) ⇒ (i) Suppose that {ϕα } ⊆ Mff (A1 , . . . , An ) is a net such that ϕα − ϕ m → 0. By ρ Lemma 6.3, {ϕα } ⊆ Mff (A1 , . . . , An ), where ρ is the tensor product of the reduced atomic repρ resentations of A1 , . . . , An . By Theorem 6.2, ϕ ∈ Mcc (A1 , . . . , An ) ⊆ Mcc (A1 , . . . , An ). ✷ In the next theorem we show that in the case n = 2 one more equivalent condition can be added to those of Theorem 6.4. Theorem 6.5. Let A and B be C ∗ -algebras and ϕ ∈ M(A, B). The following are equivalent: (i) ϕ ∈ Mcc (A, B); (ii) there exists a sequence {ϕk }∞ k=1 ⊆ K(A) ⊙ K(B) such that ϕk − ϕ m → 0 as k → ∞.  o ⊗h K(Ao ); thus uϕ = ∞ Proof. (i) ⇒ (ii) By Theorem 6.4, uϕ ∈ K(B) i=1 bi ⊗ ai where   ∞ ∞ o o∗ o ∗ o ai ∈ K(A ), bi ∈ K(B), i ∈ N, and the series i=1 bi bi and i=1 ai ai converge in norm.   Let ϕk = ki=1 ai ⊗ bi ∈ A ⊙ B. By Theorem 5.1, uϕk = ki=1 bi ⊗ aio and ϕ − ϕk m = uϕ − uϕk eh → 0 as k → ∞. (ii) ⇒ (i) Assume that A and B are represented concretely. It is clear that ϕk ∈ Mcc (A, B). By Theorem 4.3, Φid(ϕ) − Φid(ϕk ) cb = ϕ − ϕk m . Proposition 3.2 now implies that Φid(ϕ) is completely compact, in other words, ϕ ∈ Mcc (A, B). ✷ 7. Compact multipliers In this section we compare the set of completely compact multipliers with that of compact multipliers. We exhibit sufficient conditions for these two sets of multipliers to coincide, and show that in general they are distinct. Finally, we address the question of when any universal multiplier in the minimal tensor product of two C ∗ -algebras is automatically compact. We show that this happens precisely when one of the C ∗ -algebras is finite dimensional while the other coincides with the set of its compact elements. 7.1. Automatic complete compactness We will need the following result complementing Theorem 3.4. Notation is as in Section 3. 3796 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 Proposition 7.1. If Φ : Kh → K(Hn , H1 ) is a compact completely bounded map then γ0−1 (Φ) ∈ K(H1 ) ⊗eh B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 ) ⊗eh K(Hn ). Proof. Fix ε > 0. By compactness, there exist y1 , . . . , yℓ ∈ K(Hn , H1 ) such that min1iℓ Φ(x) − yi < ε for each x ∈ Kh with x  1. Let {pα } (resp. {qα }) be a net of finite rank projections in K(H1 ) (resp. K(Hn )) such that pα → I (resp. qα → I ) strongly and let Φα : Kh → K(Hn , H1 ) be the map given by Φα (x) = pα Φ(x)qα . Let u = γ0−1 (Φ) and uα = γ0−1 (Φα ). Since each yi is compact there exists α0 such that pα yi qα − yi < ε for i = 1, . . . , ℓ and α  α0 . Moreover, for any x ∈ Kh , x  1 and α  α0 , we have Φα (x) − Φ(x)  min  1iℓ Φα (x) − pα yi qα + pα yi qα − yi + yi − Φ(x)    min 2 Φ(x) − yi + pα yi qα − yi  3ε,  1iℓ so Φα − Φ → 0. Remark 3.5 (i) shows that uα ∈ K(H1 ) ⊗h (B(H2 ) ⊗eh · · · ⊗eh B(Hn−1 )) ⊗h K(Hn ); it follows that for every ω ∈ (B(H2 )⊗eh · · ·⊗eh B(Hn−1 )⊗eh B(Hn ))∗ we have Rω (uα ) ∈ K(H1 ). Suppose that ξi , ηi ∈ Hi and let ωi = ωξi ,ηi be the corresponding vector functional. Lemma 5.3 and a straightforward verification shows that if v ∈ B(H1 ) ⊗eh · · · ⊗eh B(Hn ) has a representation of the form v = A1 ⊙ · · · ⊙ An and ω = ω2 ⊗ · · · ⊗ ωn then     Rω (v)ξ1 , η1 = v, ω1 ⊗ · · · ⊗ ωn = γ0 (v)(ζ )ξn , η1 , where ζ= (13)     ∗     ∗ ⊗ ξn−2 ⊗ ηn∗ ⊗ ξn−1 ∈ Kh η2 ⊗ ξ1 ⊗ η3∗ ⊗ ξ2 ⊗ · · · ⊗ ηn−1 is an elementary tensor whose components are rank one operators. Since γ0 (uα ) → γ0 (u) in norm, (13) implies that Rω (uα ) → Rω (u) in the operator norm of K(H1 ). Since Rω (uα ) ∈ K(H1 ), we obtain Rω (u) ∈ K(H1 ). By Lemma 5.4, u ∈ K(H1 ) ⊗eh B(H2 ) ⊗eh · · · ⊗eh B(Hn ). Similarly we see that u ∈ B(H1 ) ⊗eh B(H2 ) ⊗eh · · · ⊗eh K(Hn ); the conclusion now follows. ✷ Remark. The converse of Proposition 7.1 does not hold, even for n = 2. Indeed, let {pi }∞ i=1 be a family of pairwise orthogonal rank one projections on a Hilbert space H and let u = ∞ p ⊗ p . Then u ∈ K(H ) ⊗ K(H ) and the range of γ (u) consists of compact operai eh 0 i=1 i tors, but γ0 (u)(pi ) = pi for each i, so γ0 (u) is not compact. Given C ∗ -algebras A1 , . . . , An , we let Mc (A1 , . . . , An ) be the collection of all ϕ ∈ M(A1 , . . . , An ) for which there exist faithful representations π1 , . . . , πn of A1 , . . . , An , respectively, such that if π = π1 ⊗ · · · ⊗ πn then the map Φπ(ϕ) is compact. We call the elements of Mc (A1 , . . . , An ) compact multipliers. As a consequence of the previous result we obtain the following fact. 3797 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 Proposition 7.2. Let A1 , . . . , An be C ∗ -algebras and let ϕ ∈ Mc (A1 , . . . , An ). Then uϕ ∈  K(An ) ⊗eh Aon−1 ⊗eh · · · ⊗eh A2 ⊗eh K(Ao1 ) K(An ) ⊗eh Aon−1 ⊗eh · · · ⊗eh Ao2 ⊗eh K(A1 ) if n is even, if n is odd. Proof. We only consider the case n is even. We may assume that Ai ⊆ B(Hi ) is a concrete nondegenerate C ∗ -algebra, i = 1, . . . , n, and that Φϕ is compact. By Propositions 5.5 and 7.1, uid ϕ belongs to       d  ⊗eh · · · ⊗eh B(H2 ) ⊗eh K H1d ∩ An ⊗eh · · · ⊗eh Ad1 . K(Hn ) ⊗eh B Hn−1 Since K(Hn ) ∩ An ⊆ K(An ) and K(H1d ) ∩ Ad1 ⊆ K(Ad1 ), an application of (5) shows that uid ϕ ∈ d d K(An ) ⊗eh An−1 ⊗eh · · · ⊗eh A2 ⊗eh K(A1 ). ✷ If {Aj }j ∈J is a family of C ∗ -algebras, we will denote by and ℓ∞ -direct sums, respectively.  c0 j ∈J Aj and ℓ∞ j ∈J Aj their c0 - and suppose that K(A1 ) is isomorphic to Theorem 7.3. Let A1 , . . . , An be C ∗ -algebras, c0  c0 and K(A ) is isomorphic to M M m n n j j where J is some index set and supj ∈J mj j ∈J j ∈J and supj ∈J nj are finite. Then Mc (A1 , . . . , An ) = Mcc (A1 , . . . , An ). Proof. We give the proof for n = 3; the case of a general n is similar. Let m = sup{mj , nj : def j ∈ J }. By hypothesis, K(A1 ) and K(A3 ) may both be embedded in the C ∗ -algebra C =  c0 j ∈J Mm for some m ∈ N; without loss of generality, we may assume that this embedding is an inclusion and that Ai is represented faithfully on some Hilbert space Hi such that H1 and H3 both contain the Hilbert space H = j ∈J Cm . Given ϕ ∈ Mc (A1 , A2 , A3 ), Proposition 7.2 implies that the symbol uϕ of ϕ can be written in the form uϕ = A3 ⊙ A2 ⊙ A1 , where the entries of A3 and A1 belong to C. Let us write {eij : i, j = 1, . . . , m} for the canonical matrix unit ∞  system of Mm and let Pk = j ∈J ekk ∈ ℓj ∈J Mm , k = 1, . . . , m. For k, ℓ, s, t = 1, . . . , m, we s,t set Ak,ℓ 3 = Pk A3 (Pℓ ⊗ I ) and A1 = (Ps ⊗ I )A1 Pt and define s,t uk,ℓ,s,t = Ak,ℓ 3 ⊙ A2 ⊙ A1 and Φk,ℓ,s,t = γ0 (uk,ℓ,s,t ).  Then γ0 (uϕ ) = Φ = k,ℓ,s,t Φk,ℓ,s,t so it suffices to show that each of the maps Φk,ℓ,s,t is completely compact. Now     s,t Φk,ℓ,s,t (T2 ⊗ T1 ) = Pk Φ(Pℓ T2 ⊗ T1 Ps )Pt = Ak,ℓ 3 (Pℓ T2 ) ⊗ I A2 (T1 Ps ) ⊗ I A1 . Thus, Φk,ℓ,s,t can be considered as a completely bounded multilinear map from K(H2d , Pℓ H ) × K(Ps H, H2d ) into K(Pt H, Pk H ). Since Φ is compact, it follows that Φk,ℓ,s,t is compact.  j Take a basis {ei : i = 1, . . . , m, j ∈ J } of H = j ∈J Cm , where for each j ∈ J , the standard j basis of the j th copy of Cm is {ei : i = 1, . . . , m}. Let Uk : Pk H → P1 H be the unitary operator 3798 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 j j defined by Uk ek = e1 . Consider the mapping Ψ : K(H2d , P1 H )×K(P1 H, H2d ) → K(P1 H, P1 H ) given by Ψ (T2 ⊗ T1 ) = Uk Φk,ℓ,s,t (Uℓ T2 ⊗ T1 Us )Ut . To show that Φk,ℓ,s,t is completely compact it suffices to show that Ψ is. Let C0 = P1 CP1 ; then C0 is isomorphic to c0 and its commutant C0′ has a cyclic vector. Moreover, Ψ is a C0′ modular multilinear map. Let {pα } be a net of finite rank projections belonging to C0 , such that s-lim pα = IP1 H . Consider the completely bounded multilinear maps Ψα (x) = pα Ψ (x)pα . Since the range of each pα is finite dimensional, Ψα has finite rank, so is completely compact. Since Ψ is compact, we may argue as in the proof of Proposition 7.1 to show that Ψα − Ψ → 0. Now the maps Ψ and Ψα are C0′ -modular and C0′ has a cyclic vector, so by the generalisation [12, Lemma 3.3] of a result of Smith [23, Theorem 2.1], Ψα − Ψ cb = Ψα − Ψ → 0. Proposition 3.2 now implies that Ψ is completely compact. ✷ The following corollary extends Proposition 5 of [11] to the case of multidimensional Schur multipliers. Let n  2 be an integer. We recall from [12] that with every ϕ ∈ ℓ∞ (Nn ) we associate a mapping Sϕ : ℓ2 (N2 ) ⊙ · · · ⊙ ℓ2 (N2 ) → ℓ2 (N2 ) which extends the usual Schur multiplication in the case n = 2. We equip the domain of Sϕ with the Haagerup norm where each of the terms is given its operator space structure arising from its embedding into the corresponding space of Hilbert–Schmidt operators endowed with the operator norm. Corollary 7.4. Let n > 2 and ϕ ∈ ℓ∞ (Nn ). The following are equivalent: (i) Sϕ is compact; (ii) ϕ ∈ c0 ⊗h (ℓ∞ ⊗eh · · · ⊗eh ℓ∞ ) ⊗h c0 .    n−2 Proof. Assume first that Sϕ is compact. It follows from [12, Section 3] that the map Sϕ induces a completely bounded compact map Ŝϕ : C2 × · · · × C2 → C2 defined by Ŝϕ (Tf1 , . . . , Tfn ) = TSϕ (f1 ,...,fn ) , where Tf is the Hilbert–Schmidt operator with kernel f . By Proposition 7.1, ϕ = γ0−1 (Ŝϕ ) ∈ K(ℓ2 ) ⊗eh B(ℓ2 ) ⊗eh . . . ⊗eh B(ℓ2 ) ⊗eh K(ℓ2 ). Since Sϕ is bounded, ϕ is a Schur multiplier and by [12, Theorem 3.4], ϕ ∈ ℓ∞ ⊗eh . . . ⊗eh ℓ∞ . Hence ϕ ∈ c0 ⊗eh ℓ∞ ⊗eh . . . ⊗eh ℓ∞ ⊗eh c0 . We may now argue as in the last paragraph of the preceding proof to show that ϕ ∈ c0 ⊗h (ℓ∞ ⊗eh · · · ⊗eh ℓ∞ ) ⊗h c0 . ✷ Our next aim is to show that if both K(A1 ) and K(An ) contain full matrix algebras of arbitrarily large sizes then the completely compact multipliers form a proper subset of the compact multipliers. Saar [21] has provided an example of a compact completely bounded map on K(H ) (where H is a separable Hilbert space) which is not completely compact. It turns out that Saar’s K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3799 example also shows that the sets of compact and completely compact multipliers are distinct, in the case under consideration. We will need some preliminary results. Let A and B be C ∗ -algebras. Recall that a linear map Φ : A → B is called symmetric (or hermitian) if Φ = Φ ∗ where Φ ∗ : A → B is the map given h = {a ∈ S : a = a ∗ }. The by Φ ∗ (a) = (Φ(a ∗ ))∗ . By SA we denote the unit ball of A and set SA A following lemma is a special case of Satz 6 of [21]. We include a direct proof for the convenience of the reader. Lemma 7.5. Let H be a Hilbert space. If Φ : A → K(H ) is a symmetric, completely compact linear map with Φ cb  1, then there exists a positive operator c ∈ K(H ) such that Φ (n) (a)  h c ⊗ 1n for all a ∈ SM and all n ∈ N. Moreover, c can be chosen to have norm arbitrarily n ( A) close to one. Proof. We first show that for a given ε > 0 there exists a finite rank projection p on H such that Φ (n) (a) − (p ⊗ 1n )Φ (n) (a)(p ⊗ 1n )  ε for any a ∈ SMn (A) . (14) Since Φ is completely compact, there exists a finite dimensional subspace F ⊂ K(H ) such that dist(Φ (n) (a), Mn (F ))  ε/3 for any a ∈ Mn (A), a  1 and any n ∈ N. Let SF,1+ε = {x ∈ F : x  1 + ε} and let k = dim F . Choose a finite rank projection p ∈ K(H ) such that x − pxp < ε k(3 + ε) for all x ∈ SF,1+ε and let Ψ : F → K(H ) be defined by Ψ (x) = x − pxp. By [6, Corollary 2.2.4], Ψ is completely bounded and Ψ cb  k Ψ . This implies that Ψ (n) (y)  k Ψ y  ε ε y  3+ε 3 for all y ∈ Mn (F ) with y  1 + ε/3. h Now for a ∈ SM let y ∈ Mn (F ) be such that n ( A) (n) Φ (a) + ε/3  1 + ε/3. Hence Φ (n) (a) − y  ε/3. Then y  Φ (n) (a) − (p ⊗ 1n )Φ (n) (a)(p ⊗ 1n )    Φ (n) (a) − y + Ψ (n) (y) + (p ⊗ 1n ) y − Φ (n) (a) (p ⊗ 1n )  ε/3 + ε/3 + ε/3 = ε, proving (14). Next we fix ε > 0 and choose a finite rank projection q1 on H such that ε Φ (n) (a) − (q1 ⊗ 1n )Φ (n) (a)(q1 ⊗ 1n )  , 2 a ∈ Mn (A), a  1, n ∈ N. Let r1 : A → K(H ) be the mapping given by r1 (a) = Φ(a) − q1 Φ(a)q1 , a ∈ A. Then r1 = Ψ ◦ Φ, where Ψ : K(H ) → K(H ) is the completely bounded map given by Ψ (x) = x − q1 xq1 . By Proposition 3.2, r1 is completely compact. Moreover, r1 cb  ε/2 and Φ(a) = 3800 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 q1 Φ(a)q1 + r1 (a), a ∈ A. Proceeding by induction, we can find sequences of finite rank projections qi and completely compact symmetric mappings ri such that ri cb  ε/2i and Φ(a) = q1 Φ(a)q1 + ∞  qi+1 ri (a)qi+1 , a ∈ A. i=1 Let c = q1 + ∞ ε i=1 2i qi+1 . (n) We have that Φ (n) and ri Φ (n) (a) = (q1 ⊗ 1n )Φ (n) (a)(q1 ⊗ 1n ) + are symmetric and ∞  (n) (qi+1 ⊗ 1n )ri (a)(qi+1 ⊗ 1n ), i=1 for each a ∈ A. Now Φ (n) (a)  (q1 ⊗ 1n ) Φ cb + ∞  i=1  ∞  ε qi+1 ⊗ 1n = c ⊗ 1n (qi+1 ⊗ 1n ) ri cb  q1 + 2i i=1 h for all a ∈ SM . By construction, c is compact and c  1 + ε. n ( A) ✷ Let H be an infinite dimensional separable Hilbert space  and {qk }k∈N be a family n of pairwise orthogonal projections in B(H ) with rank qk = k and ∞ q = I . Set p = k n k=1 k=1 qk , n ∈ N. Let Φk : B(qk H ) → B(qk H ), k ∈ N, be symmetric linear maps such that Φk cb = 1, Φk → 0 as k → ∞, and ∞  Φk 2 2 < ∞, (15) k=1 where Φk 2 denotes the norm of the mapping Φk when B(qk H ) ≃ C2 (qk H ) is equipped with the Hilbert–Schmidt norm. Identifying B(qk H ) with qk B(H )qk , let Φ : K(H ) → B(H ) be the map given by the norm-convergent sum Φ(x) = ∞  ⊕ Φk (qk xqk ), x ∈ K(H ). (16) k=1 An example of such a map is obtained by taking Φk = k −1 τk where τk is the transposition map B(qk H ) ≃ Mk → Mk ≃ B(qk H ), which is symmetric and an isometry for both the operator and the Hilbert–Schmidt norm. It is well known [20, p. 419] that τk cb = k and hence conditions (15) are satisfied. The next lemma is a straightforward extension of [21, pp. 32–34]. Lemma 7.6. If Φ is a map satisfying (15) and (16) then the range of Φ consists of compact operators. Moreover, Φ is completely contractive and compact but not completely compact. Proof. Fix x ∈ K(H ). Since Φk → 0 as k → ∞, we have pn Φ(x)pn → Φ(x) in norm, so Φ(x) ∈ K(H ). Each of the maps x → Φk (qk xqk ) is completely contractive, so Φ is completely contractive. K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3801 def Next, note that Φ maps the unit ball of K(H ) into U = U1 ⊕ U2 ⊕ · · ·, where Uk is the ball of radius Φk in qk B(H )qk . Since U is compact, the map Φ is compact. If Φ were completely compact then by Lemma 7.5, there would exist a positive compact operator c on H such that Φ (k) (x)  c ⊗ 1k h for all x ∈ SM and all k ∈ N. k (K(H )) h Hence for every k ∈ N and x ∈ SM , k (K(H )) (k)  Φk (k) However, Φk  (qk ⊗ 1k )x(qk ⊗ 1k ) = (qk ⊗ 1k )Φ (k) (x)(qk ⊗ 1k )  qk cqk ⊗ 1k . = Φk cb = 1 by [22], so qk cqk = qk cqk ⊗ 1k  sup which is impossible since c is compact.   1 (k) h Φk (x) : x ∈ SM  , k (qk K(H )qk ) 2 ✷ Lemma 7.7. Given a map Φ be as above, let C = sal multiplier ϕ ∈ M(C d , C) with Φ = Φid(ϕ) . c0 k∈N B(qk H ) ⊆ K(H ). There exists a univer- Proof. Let ϕk ∈ B(qk H )d ⊗ B(qk H ) be such that Φid(ϕk ) = Φk , k ∈ N, where the family {Φk }∞ k=1 n satisfies (15). Then ϕ = Φ . Let ψ = ϕ . If n < m then ψ − ψ = k min k 2 n k m n min k=1 m Φ so k 2 k=n+1 ψm − ψn min  m  Φk 22 k=n+1 1/2 . By (15), the sequence {ψn } converges to an element ϕ ∈ C d ⊗ C. Moreover, for every x ∈ C2 (H ) we have Φid(ϕ) (x) = lim pn Φid(ϕ) (x)pn = lim Φid(ψn ) (x) = Φ(x), n→∞ n→∞ where the limits are in the operator norm. So Φid(ϕ) = Φ which is completely contractive by Lemma 7.6, so ϕ ∈ M(C d , C) by Theorem 4.3. ✷ Given C ∗ -algebras Ai ⊆ B(Hi ), i = 1, . . . , n, and ψ = c2 ⊗ · · · ⊗ cn−1 ∈ A2 ⊙ · · · ⊙ An−1 , we may define a bounded linear map A1 ⊗ An → B1 ⊗ A2 ⊗ · · · ⊗ An , where B1 = A1 if n is even and B1 = Ad1 if n is odd, by a ⊗ b →  a⊗ψ ⊗b ad ⊗ ψ ⊗ b if n is even, if n is odd. We write ιψ for the restriction of this map to M(A1 , An ). 3802 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 Lemma 7.8. (i) The range of ιψ is contained in M(B1 , A2 , . . . , An ). (ii) ιψ (Mcid (A1 , An )) ⊆ Mcid (B1 , A2 , . . . , An ). d ) ⊗ · · · ⊗ B(H )) . Writing (iii) Suppose that n is even and ω ∈ (B(Hn−1 eh eh 2 ∗      d  ⊗eh · · · ⊗eh B(H2 ) ⊗eh B H1d → B(Hn ) ⊗eh B H1d Mω : B(Hn ) ⊗eh B Hn−1 for the “middle slice map” Mω = Rω ⊗eh idB(H d ) , we have 1 Mω (uιψ (ϕ) ) = ω(ψ̃)uϕ d where ψ̃ = cn−1 ⊗ · · · ⊗ c2 . The same is true, mutatis mutandis, if n is odd. Proof. Let ϕ ∈ M(A1 , An ). By Theorem 2.3, there exist a net {ϕν } ⊆ A1 ⊙ An and representad ν ν ν id tions uid ϕν = A2 ⊙ A1 and uϕ = A2 ⊙ A1 , where Ai are finite matrices with entries in A1 if i = 1 ν ν and in An if i = 2, such that ϕν → ϕ semi-weakly, Ai → Ai strongly and supi,ν Ai < ∞. (i) It is easy to see that ιψ (ϕν ) satisfies the boundedness conditions of Theorem 2.3 and converges semi-weakly to ιψ (ϕ), which is therefore a universal multiplier. (ii) Suppose that n is even and let ι = ιψ . It is immediate to check that if ϕ ∈ A1 ⊙ An and d , H ) then T1 ∈ K(H1d , H2 ), . . ., Tn−1 ∈ K(Hn−1 n   d . . . c 2 T1 . Φι(ϕ) (Tn−1 ⊗ · · · ⊗ T1 ) = Φϕ Tn−1 cn−1 Note that this equation holds for any ϕ ∈ M(A1 , An ) since Φϕν (T ) → Φϕ (T ) and Φι(ϕν ) (Tn−1 ⊗ · · · ⊗ T1 ) → Φι(ϕ) (Tn−1 ⊗ · · · ⊗ T1 ) weakly for any T , T1 , . . . , Tn−1 . Since Φι(ϕ) is the compod sition of the bounded mapping Xn−1 ⊗ · · · ⊗ X1 → Xn−1 cn−1 . . . c2 X1 with Φϕ , it follows that if ϕ is a compact operator multiplier then so is ι(ϕ). (iii) We have that  d  Φι(ϕν ) (Tn−1 ⊗ · · · ⊗ T1 ) = Aν2 (Tn−1 ⊗ I ) cn−1 ⊗ I . . . (c2 ⊗ I )(T1 ⊗ I )Aν1  d  → A2 (Tn−1 ⊗ I ) cn−1 ⊗ I . . . (c2 ⊗ I )(T1 ⊗ I )A1 weakly. On the other hand, Φι(ϕν ) (Tn−1 ⊗ · · · ⊗ T1 ) → Φι(ϕ) (Tn−1 ⊗ · · · ⊗ T1 ) which implies that d uι(ϕ) = A2 ⊙ (cn−1 ⊗ I ) ⊙ · · · ⊙ (c2 ⊗ I ) ⊙ A1 . It follows that Mω (uι(ϕ) ) = ω(ψ̃)uϕ . ✷ Theorem 7.9. Let A1 , . . . , An be C ∗ -algebras with the property that both K(A1 ) and K(An ) contain full matrix algebras of arbitrarily large sizes. Then the inclusion Mcc (A1 , . . . , An ) ⊆ Mc (A1 , . . . , An ) is proper. Proof. We may assume that Ai ⊆ B(Hi ), i = 1, . . . , n for some Hilbert spaces H1 , . . . , Hn . First suppose that n = 2. By hypothesis, we may assume that there is an infinite separable cdimensional 0 Hilbert space H with H d ⊆ H1 and H ⊆ H2 , and a C ∗ -algebra C = k∈N Mk ⊆ K(H ) as in Lemma 7.7 with C d ⊆ A1 and C ⊆ A2 . By the injectivity of the minimal tensor product of C ∗ algebras, C d ⊗ C ⊆ A1 ⊗ A2 . K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3803 Let ϕ ∈ C d ⊗ C be given by Lemma 7.7. It follows from Lemma 7.6 that ϕ ∈ Mc (A1 , A2 ) \ id (A , A ). Since faithful representations of A and A restrict to representations of C conMcc 1 2 1 2 taining the identity subrepresentation up to unitary equivalence, we have that ϕ ∈ Mc (A1 , A2 ) \ Mcc (A1 , A2 ). Suppose now that n is even. Let ϕ ∈ Mc (A1 , An ) \ Mcc (A1 , An ), fix any non-zero ψ = c2 ⊗ · · · ⊗ cn−1 ∈ A2 ⊙ · · · ⊙ An−1 and let us write ι = ιψ . Suppose that ι(ϕ) is a completely compact multiplier. By Theorem 6.4, uι(ϕ) ∈ K(An ) ⊗h (Aon−1 ⊗eh · · · ⊗eh A2 ) ⊗h K(Ao1 ). d ) ⊗ · · · ⊗ B(H )) d ⊗ · · · ⊗ c2 ∈ Adn−1 ⊗eh · · · ⊗eh A2 and fix ω ∈ (B(Hn−1 Let ψ̃ = cn−1 eh eh 2 ∗ such that ω(ψ̃) = 0. By Lemma 7.8 (iii), Mω (uι(ϕ) ) = ω(ψ̃)uϕ and hence uϕ ∈ K(An ) ⊗h K(Ao1 ) which by Theorem 6.4 contradicts the assumption that ϕ is not a completely compact multiplier. If n is odd then the same proof works with minor modifications. ✷ Remark 7.10. We do not know whether the sets Mcc (A, B) and Mc (A, B) are distinct if K(A) contains matrix algebras of arbitrarily large sizes, while K(B) does not (and vice versa). Let C be the C ∗ -algebra defined in Lemma 7.7. To show that the inclusion Mcc (C, c0 ) ⊆ Mc (C, c0 ) is proper it would suffice to exhibit mappings Φk : Mk → Mk which satisfy (15) and are left Dk modular (where Dk is the subalgebra of all diagonal matrices of Mk ). This modularity condition would enable us to find ϕk ∈ Mkd ⊗ Dk such that Φk = Φid(ϕk ) using the method of Lemma 7.7 and we could then conclude from Lemma 7.6 that Mcc (C, c0 )  Mc (C, c0 ). However, we do not know if such mappings Φk exist. This prompts the following question: if D is a masa in B(H ), does there exist a constant C such that whenever Φ : K(H ) → K(H ) is a bounded and left D-modular map then Φ cb  C Φ ? If such a version of Smith’s automatic complete boundedness result holds then it would follow that Mcc (C, c0 ) = Mc (C, c0 ). 7.2. Automatic compactness We now turn to the question of when every universal multiplier is automatically compact. We will restrict to the case n = 2 for the rest of the paper. We will first establish an auxiliary result in a different but related setting. Suppose that A and B are commutative C ∗ -algebras and assume that A = C0 (X) and B = C0 (Y ) for some locally compact Hausdorff spaces X and Y . The C ∗ -algebra C0 (X) ⊗ C0 (Y ) will be identified with C0 (X × Y ) and M(A, B) with a subset of C0 (X × Y ). Elements of the Haagerup tensor product C0 (X) ⊗h C0 (Y ), as well as of ˆ 0 (Y ), will be identified with functions in C0 (X × Y ) in the projective tensor product C0 (X)⊗C ˆ 0 (Y ) the natural way. Note that, by Grothendieck’s inequality, C0 (X) ⊗h C0 (Y ) and C0 (X)⊗C coincide as sets of functions. Proposition 7.11. Let X and Y be infinite, locally compact Hausdorff spaces. Then C0 (X) ⊗h C0 (Y ) ⊆ M(C0 (X), C0 (Y )) and this inclusion is proper. Proof. The inclusion C0 (X) ⊗h C0 (Y ) ⊆ M(C0 (X), C0 (Y )) follows from Corollary 6.7 of [14]. To show that this inclusion is proper, suppose first that X and Y are compact. By Theorem 11.9.1 of [8], there exists a sequence (fi )∞ i=1 ⊆ C(X) ⊗h C(Y ) such that supi∈N fi h < ∞, converging uniformly to a function f ∈ C(X × Y ) \ C(X) ⊗h C(Y ). By Corollary 6.7 of [14], f ∈ M(C(X), C(Y )). The conclusion now follows. Now assume that both X and Y are locally compact but not compact (the case where one of the spaces is compact while the other is not is similar). Let X̃ = X ∪ {∞} and Ỹ = Y ∪ {∞} be the 3804 K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 one point compactifications of X and Y . Then C(X̃) = C0 (X) + C1 and C(Ỹ ) = C0 (Y ) + C1, where 1 denotes the constant function taking the value one. Moreover, it is easy to see that C(X̃) ⊗ C(Ỹ ) = C0 (X × Y ) + C0 (X) + C0 (Y ) + C1 and ˆ Ỹ ) = C0 (X)⊗C ˆ 0 (Y ) + C0 (X) + C0 (Y ) + C1. C(X̃)⊗C( (17) By the first part of the proof, there exists ϕ ∈ M(C(X̃), C(Ỹ )) \ C(X̃) ⊗h C(Ỹ ). Write ϕ = ϕ1 + ϕ2 + ϕ3 + ϕ4 where ϕ1 ∈ C0 (X × Y ), ϕ2 ∈ C0 (X), ϕ3 ∈ C0 (Y ) and ϕ4 ∈ C1. Suppose that ˆ Ỹ ), a contradiction. ✷ ϕ1 ∈ C0 (X) ⊗h C0 (Y ). By (17), ϕ ∈ C(X̃)⊗C( Theorem 7.12. Let A and B be C ∗ -algebras. The following are equivalent: (i) either A is finite dimensional and K(B) = B, or B is finite dimensional and K(A) = A; (ii) Mc (A, B) = M(A, B); (iii) Mcc (A, B) = M(A, B). Proof. (i) ⇒ (iii) Suppose that A is finite dimensional and K(B) = B, and that A ⊆ B(H1 ) and B ⊆ B(H2 ) for some Hilbert spaces H1 and H2 where H1 is finite dimensional. Fix ϕ ∈ M(A, B). Then ϕ is the sum of finitely many elements of the form a ⊗ b where a has finite rank and b ∈ K(H2 ); such elements are completely compact multipliers by Theorem 6.4. (iii) ⇒ (ii) is trivial. (ii) ⇒ (i) Assume that both A and B are infinite dimensional and are identified with their image under the reduced atomic representation. If either K(A) or K(B) is finite dimensional then there exists an elementary tensor a ⊗ b ∈ (A ⊙ B) \ (K(A) ⊙ K(B)). By Proposition 7.2, a ⊗ b ∈ Mc (A, B). We can therefore assume that both K(A) and K(B) are infinite dimensional. Then, up to a ∗-isomorphism, c0 is contained in both K(A) and K(B). By Proposition 7.11, there exists ϕ ∈ M(c0 , c0 ) \ (c0 ⊗h c0 ). Then ϕ ∈ M(A, B) and Φid(ϕ) is not compact by Hladnik’s characterisation [11]. Since the restrictions to c0 of any faithful representations of A, B contain representations unitarily equivalent to the identity representations, we see that ϕ is not a compact multiplier. Thus at least one of the C ∗ -algebras A and B is finite dimensional; assume without loss of generality that this is A. Suppose that B = K(B) and fix an element b ∈ B \ K(B). Let a ∈ A be a non-zero element. By Proposition 7.2, the elementary tensor a ⊗ b is not a compact multiplier. ✷ Acknowledgments We are grateful to V.S. Shulman for stimulating results, questions and discussions. We would like to thank M. Neufang for pointing out to us Corollary 3.7 and R. Smith for a discussion concerning Remark 7.10. The first named author is grateful to G. Pisier for the support of the one semester visit to the University of Paris 6 and the warm atmosphere at the department, where one of the last drafts of the paper was finished. The first named author was supported by The Royal Swedish Academy of Sciences, Knut och Alice Wallenbergs Stiftelse and Jubileumsfonden of the University of Gothenburg’s Research K. Juschenko et al. / Journal of Functional Analysis 256 (2009) 3772–3805 3805 Foundation. The second and the third named authors were supported by Engineering and Physical Sciences Research Council grant EP/D050677/1. The last named author was supported by the Swedish Research Council. References [1] D.P. Blecher, C. Le Merdy, Operator Algebras and Their Modules—An Operator Apace Approach, Oxford University Press, 2004. [2] D.P. Blecher, R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. (2) 45 (1992) 126–144. [3] E. Christensen, A.M. Sinclair, Representations of completely bounded multilinear operators, J. 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