The Effros-Ruan conjecture for bilinear forms on C^*-algebras
aa r X i v : . [ m a t h . OA ] D ec . THE EFFROS-RUAN CONJECTURE FOR BILINEAR FORMS ON C ∗ -ALGEBRAS UFFE HAAGERUP (1)
AND MAGDALENA MUSAT (2)
Abstract.
In 1991 Effros and Ruan conjectured that a certain Grothendieck-type inequality for a bilin-ear form on C ∗ -algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002Pisier and Shlyakhtenko proved that this inequality holds in the more general setting of operator spaces,provided that the operator spaces in question are exact. Moreover, they proved that the conjecture ofEffros and Ruan holds for pairs of C ∗ -algebras, of which at least one is exact. In this paper we prove thatthe Effros-Ruan conjecture holds for general C ∗ -algebras, with constant one. More precisely, we showthat for every jointly completely bounded (for short, j.c.b.) bilinear form on a pair of C ∗ -algebras A and B , there exist states f , f on A and g , g on B such that for all a ∈ A and b ∈ B , | u ( a, b ) | ≤ k u k jcb ( f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / ) . While the approach by Pisier and Shlyakhtenko relies on free probability techniques, our proof uses moreclassical operator algebra theory, namely, Tomita-Takesaki theory and special properties of the Powersfactors of type III λ , 0 < λ < Introduction
In 1956 Grothendieck published the celebrated ”R´esum´e de la th´eorie m´etrique des produits tensorielstopologiques”, containing a general theory of tensor norms on tensor products of Banach spaces, describingseveral operations to generate new norms from known ones, and studying the duality theory between thesenorms. Since 1968 it has had considerable influence on the development of Banach space theory (see e.g.,[11]) . The highlight of the paper [8], now referred to as the ”R´esum´e” is a result that Grothendieck called”The fundamental theorem on the metric theory of tensor products”. Grothendieck’s theorem asserts thatgiven compact spaces K and K and a bounded bilinear form u : C ( K ) × C ( K ) → K (where K = R or C ) , then there exist probability measures µ and µ on K and K , respectively, such that | u ( f, g ) | ≤ K K G k u k (cid:18)Z K | f ( t ) | dµ ( t ) (cid:19) / (cid:18)Z K | g ( t ) | dµ ( t ) (cid:19) / , for all f ∈ C ( K ) and g ∈ C ( K ) , where K K G is a universal constant.The non-commutative version of Grothendieck’s inequality (conjectured in the ”R´esum´e”) was firstproved by Pisier under some approximability assumption (cf. [12]) , and obtained in full generality in [9].The theorem asserts that given C ∗ -algebras A and B and a bounded bilinear form u : A × B → C , thenthere exist states f , f on A and states g , g on B such that for all a ∈ A and b ∈ B , | u ( a, b ) | ≤ k u k ( f ( a ∗ a ) + f ( aa ∗ )) / ( g ( b ∗ b ) + g ( bb ∗ )) / . † (1) Partially supported by the Danish Natural Science Research Council. (2)
Partially supported by the National Science Foundation, DMS-0703869.2000
Mathematics Subject Classification.
Primary: 46L10; 47L25.
Key words and phrases.
Grothendieck inequality for bilinear forms on C ∗ -algebras ; jointly completely bounded bilinearforms ; Powers factors ; Tomita-Takesaki theory. s a corollary, it was shown in [9] that given C ∗ -algebras A and B , then any bounded linear operator T : A → B ∗ admits a factorization T = SR through a Hilbert space H , where A R −→ H S −→ B ∗ , and k R kk S k ≤ k T k . Let E ⊆ A and F ⊆ B be operator spaces sitting in C ∗ -algebras A and B , and let u : E × F → C be abounded bilinear form. Then, there exists a unique bounded linear operator e u : E → F ∗ such that(1.1) u ( a , b ) := h e u ( a ) , b i , a ∈ E , b ∈ F , where h · , · i denotes the duality bracket between F and F ∗ . The map u is called jointly completelybounded (for short, j.c.b.) if the associated map e u : E → F ∗ is completely bounded, in which case we set(1.2) k u k jcb := k e u k cb . (Otherwise, we set k u k jcb = ∞ .) It is easily checked that(1.3) k u k jcb = sup n ∈ N k u n k , where for every n ≥ u n : M n ( E ) ⊗ M n ( F ) → M n ( C ) ⊗ M n ( C ) is given by u n k X i =1 a i ⊗ c i , l X j =1 b j ⊗ d j = k X i =1 l X j =1 u ( a i , b j ) c i ⊗ d j , for all finite sequences { a i } ≤ i ≤ k in E , { b j } ≤ j ≤ l in F , { c i } ≤ i ≤ k and { d j } ≤ j ≤ l in M n ( C ) , k , l ∈ N . Moreover, k u k jcb is the smallest constant κ for which, given arbitrary C ∗ -algebras C and D and finitesequences { a i } ≤ i ≤ k in E , { b j } ≤ j ≤ l in F , { c i } ≤ i ≤ k in C and { d j } ≤ j ≤ l in D , where k , l ∈ N , thefollowing inequality holds(1.4) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =1 l X j =1 u ( a i , b j ) c i ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ⊗ min D ≤ κ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ⊗ min C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) l X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F ⊗ min D . For a reference, see the discussion following Definition 1.1 in [15]It was conjectured by Effros and Ruan in 1991 (cf. [5] and [15], Conjecture 0.1) that if A and B areC ∗ -algebras and u : A × B → C is a jointly completely bounded bilinear form, then there exist states f , f on A and states g , g on B such that for all a ∈ A and b ∈ B ,(1.5) | u ( a, b ) | ≤ K k u k jcb ( f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / ) , where K is a universal constant.In [15] Pisier and Shlyakhtenko proved an operator space version of (1.5), namely, if E ⊆ A and F ⊆ B are exact operator spaces with exactness constants ex( E ) and ex( F ) , respectively, and u : E × F → C isa j.c.b. bilinear form, then there exist states f , f on A and states g , g on B such that for all a ∈ E and b ∈ F , | u ( a, b ) | ≤ / ex( E )ex( F ) k u k jcb ( f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / ) . Moreover, by the same methods they were able to prove the Effros-Ruan conjecture for C ∗ -algebras withconstant K = 2 / , provided that at least one of the C ∗ -algebras A , B is exact (cf. [15], Theorem 0.5).The main result of this paper is that the Effros-Ruan conjecture is true. Moreover, it holds with constant K = 1 , that is, heorem 1.1. Let A and B be C ∗ -algebras and u : A × B → C a jointly completely bounded bilinearform. Then there exist states f , f on A and states g , g on B such that for all a ∈ A and b ∈ B , | u ( a, b ) | ≤ k u k jcb ( f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / ) . It follows from Theorem 1.1 that every completely bounded linear map T : A → B ∗ from a C ∗ -algebra A to the dual B ∗ of a C ∗ -algebra B has a factorization T = vw through H r ⊕ K c (the direct sum of a rowHilbert space and a column Hilbert space), such that k v k cb k w k cb ≤ k T k cb . (See Proposition 3.5 of this paper.) Theorem 1.1 also settles in the affirmative a related conjecture byBlecher (cf. [1]; see also [15], Conjecture 0.2). For details, see Remark 3.2 of this paper.Furthermore, thanks to Theorem 1.1 we can strengthen a number of results from [15] , cf. Corollaries 3.7through 3.10 in this paper. For instance, it follows that if an operator space E and its dual E ∗ both embedin noncommutative L -spaces, then E is completely isomorphic to a quotient of a subspace of H r ⊕ K c ,for some Hilbert spaces H and K .It also follows from Theorem 1.1 that if u : A × B → C is a j.c.b. bilinear form on C ∗ -algebras A and B , then the inequality (1.4) holds, as well, when the C ⊗ min D -norm on the left-hand side is replaced bythe C ⊗ max D -norm (with constant 2 k u k jcb instead of k u k jcb ) , cf. Proposition 3.11. Moreover, we showthat for bilinear forms u on operator spaces E ⊆ A and F ⊆ B sitting in C ∗ -algebras A and B , the abovementioned variant of (1.4) , namely the inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 n X j =1 u ( a i , b j ) c i ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ⊗ max D ≤ κ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ⊗ min C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F ⊗ min D (where C and D are arbitrary C ∗ -algebras) characterizes those j.c.b. bilinear forms that satisfy an Effros-Ruan type inequality . That is, there exists a constant κ ≥ f , f on A and states g , g on B such that, for all a ∈ E and b ∈ F , | u ( a, b ) | ≤ κ ( f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / ) . For details on operator spaces and completely bounded maps we refer to the monographs [7] and [14].2.
Proof of the Effros-Ruan Conjecture
Let 0 < λ < M , φ ) be the Powers factor of type III λ with product state φ , that is,( M , φ ) = ∞ O n =1 ( M ( C ) , ω λ ) , where φ = N ∞ n =1 ω λ , ω λ ( · ) = Tr( h λ · ) and h λ = λ λ λ ! (cf. [4]) . The modular automorphismgroup ( σ φt ) t ∈ R of φ is given by σ φt = ∞ O n =1 σ ω λ t , where for any matrix x = [ x ij ] ≤ i,j ≤ ∈ M ( C ) , σ ω λ t ( x ) = h itλ xh − itλ = x λ it x λ − it x x ! , t ∈ R . herefore σ ω λ t and σ φt are periodic in t ∈ R with minimal period t := − π log λ . Let M φ denote the centralizer of φ , that is, M φ := { x ∈ M : σ φt ( x ) = x , ∀ t ∈ R } . It was proved by Connes (cf. [3] , Theorem 4.26) that the relative commutant of M φ in M is trivial, i.e., M ′ φ ∩ M = C M , where 1 M denotes the identity of M . In particular, φ is homogeneous in the sense of Takesaki (cf. [16]).Furthermore, it is shown in [10] , (see Theorem 3.1 therein) that the following strong Dixmier propertyholds for the Powers factor M . Namely, for all x ∈ M , φ ( x ) · M ∈ conv { vxv ∗ : v ∈ U ( M φ ) } k·k , where the closure is taken in norm topology and U ( M φ ) denotes the unitary group on M φ . Moreover, byCorollary 3.4 in [10] , this can be extended to finite sets in M , i.e., for every finite set { x , . . . x n } ∈ M and every ε > α of elements from { ad( v ) : v ∈ U ( M φ ) } such that k α ( x i ) − φ ( x i ) · M k < ε , ≤ i ≤ n . By standard arguments, it follows that there exists a net { α i } i ∈ I ⊆ conv { ad( v ) : v ∈ U ( M φ ) } such that(2.1) lim i ∈ I k α i ( x ) − φ ( x ) · M k = 0 , x ∈ M . In the following, we will identify M with π φ ( M ) , where ( π φ , H φ , ξ φ ) is the GNS representation of M associated to the state φ . Then H φ := M ξ φ = L ( M , φ ) . By Tomita-Takesaki theory (cf. [17]), the operator S defined by S ( xξ φ ) = x ∗ ξ φ , x ∈ M is closable. Its closure S := S has a unique polar decomposition(2.2) S = J ∆ / , where ∆ is a positive self-adjoint unbounded operator on L ( M , φ ) and J is a conjugate-linear involution.Moreover, for all t ∈ R , σ φt ( x ) = ∆ it x ∆ − it , x ∈ M and J M J = M ′ , where M ′ denotes the commutant of M .Following Takesaki’s construction from [16] , define for all n ∈ Z M n := { x ∈ M : σ φt ( x ) = λ int x , ∀ t ∈ R } . Then, by Lemma 1.16 in [16] , M n = { x ∈ M : φ ( xy ) = λ n φ ( yx ) , ∀ y ∈ M} . n particular, M φ = M . It was proved in [16] (cf. Lemma 1.10) that M n = { } , for all n ∈ Z .Furthermore, by a combination of Lemma 1.4 and Corollary 1.16 in [16] , it follows that for all n ∈ Z ,∆( η ) = λ n η , η ∈ M n ξ φ and that L ( M , φ ) = ∞ M n = −∞ M n ξ φ . As a consequence, one has the following
Lemma 2.1.
For every n ∈ Z , there exists c n ∈ M such that (2.3) φ ( c ∗ n c n ) = λ − n/ , φ ( c n c ∗ n ) = λ n/ and, moreover, (2.4) h c n Jc n Jξ φ , ξ φ i H φ = 1 . Proof.
Let n ∈ Z . Take z ∈ M n \ { } . Then φ ( zz ∗ ) = λ n φ ( z ∗ z ) . Moreover, JzJξ φ = S ∆ − / zξ φ = S ( λ − n/ z ) ξ φ = λ − n/ z ∗ ξ φ . Therefore, h zJzJξ φ , ξ φ i H φ = λ − n/ h zz ∗ ξ φ , ξ φ i H φ = λ − n/ φ ( zz ∗ ) = λ n/ φ ( z ∗ z ) . Hence c n := ( λ n/ φ ( z ∗ z )) − / z satisfies relations (2.3) and (2.4) . (cid:3) Since M is an injective factor, it is known (cf. [4]) that for all finite sequences x , . . . , x n ∈ M and y , . . . , y n ∈ M ′ , where n is a positive integer, the following holds(2.5) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 c i d i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ( L ( M ,φ )) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 c i ⊗ d i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M⊗ min M ′ . That is, the map defined by c ⊗ d cd , where c ∈ M and d ∈ M ′ extends uniquely to a C ∗ -algebraisomorphism of M ⊗ min M ′ onto C ∗ ( M , M ′ ) .Now let A and B be C ∗ -algebras and let u : A × B → C be a jointly completely bounded bilinear form. Proposition 2.2.
There exists a bounded bilinear form b u : ( A ⊗ min M ) × ( B ⊗ min M ′ ) → C such that (2.6) b u ( a ⊗ c , b ⊗ d ) = u ( a , b ) h cdξ φ , ξ φ i H φ , a ∈ A , b ∈ B , c ∈ M , d ∈ M ′ , and, moreover, (2.7) k b u k ≤ k u k jcb . roof. Let a , . . . , a m ∈ A , b , . . . , b n ∈ B , c , . . . , c m ∈ M , d , . . . , d n ∈ M ′ , where m and n arepositive integers. Then, by (2.5) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 n X j =1 u ( a i , b j ) h c i d j ξ φ , ξ φ i H φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 n X j =1 u ( a i , b j ) c i d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ( L ( M ,φ )) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 n X j =1 u ( a i , b j ) c i ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) M⊗ min M ′ ≤ k u k jcb (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ⊗ min M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ⊗ min M ′ , which yields the conclusion. (cid:3) Lemma 2.3.
Let v ∈ U ( M φ ) and set v ′ = JvJ ∈ M ′ . Then, for all x ∈ A ⊗ M and y ∈ B ⊗ M ′ , (2.8) b u (( Id A ⊗ ad ( v ))( x ) , ( Id B ⊗ ad ( v ′ ))( y )) = b u ( x , y ) . Proof.
It suffices to prove that formula (2.8) holds for elementary tensors x = a ⊗ c and y = b ⊗ d , where a ∈ A , b ∈ B , c ∈ M and d ∈ M ′ . By (2.6), it is enough to show that for all c ∈ M and d ∈ M ′ ,(2.9) h vcv ∗ v ′ d ( v ′ ) ∗ ) ξ φ , ξ φ i H φ = h cdξ φ , ξ φ i H φ . Since { v , c , v ∗ } commutes with { v ′ , d , ( v ′ ) ∗ } , we have h vcv ∗ v ′ d ( v ′ ) ∗ ξ φ , ξ φ i H φ = h v ′ vcdv ∗ ( v ′ ) ∗ ξ φ , ξ φ i H φ (2.10) = h cdv ∗ ( v ′ ) ∗ ξ φ , v ∗ ( v ′ ) ∗ ξ φ i H φ . But since Jξ φ = ξ φ , we deduce that v ∗ ( v ′ ) ∗ ξ φ = v ∗ ( JvJ ) ∗ ξ φ = v ∗ Jv ∗ Jξ φ = v ∗ Jv ∗ ξ φ . Furthermore, since v ∗ ∈ M φ and ∆ it ξ φ = ξ φ , for all t ∈ R , we have∆ it ( v ∗ ξ φ ) = σ φt ( v ∗ )∆ it ξ φ = v ∗ ξ φ , t ∈ R . Hence v ∗ ξ φ is an eigenvector for ∆ with corresponding eigenvalue equal to 1. Using the polar decomposition(2.2) of S , we infer that v ∗ Jv ∗ ξ φ = v ∗ S ∆ − / u ∗ ξ φ = v ∗ Sv ∗ ξ φ = v ∗ vξ φ = ξ φ , i.e., v ∗ ( v ′ ) ∗ ξ φ = ξ φ . Therefore h cdv ∗ ( v ′ ) ∗ ξ φ , v ∗ ( v ′ ) ∗ ξ φ i H φ = h cdξ φ , ξ φ i H φ . This gives (2.10) , which completes the proof of the lemma. (cid:3)
Lemma 2.4.
Let { α i } i ∈ I ⊆ conv { ad ( v ) : v ∈ U ( M φ ) } be a net satisfying (2.1) . For every i ∈ I , considerthe corresponding map α ′ i on M ′ = J M J given by α ′ i ( JxJ ) = Jα i ( x ) J , x ∈ M . oreover, let φ ′ be the state on M ′ defined by (2.11) φ ′ ( JxJ ) := φ ( x ) , x ∈ M . Furthermore, let ˆ f be a state on A ⊗ min M and ˆ g be a state on B ⊗ min M ′ , arbitrarily chosen, and definestates f on A , respectively, g on B by f ( a ) = ˆ f ( a ⊗ M ) , a ∈ A (2.12) g ( b ) = ˆ g ( b ⊗ M ′ ) , b ∈ B , (2.13) where M ′ denotes the identity of M ′ . Then, (2.14) lim i ∈ I ˆ f (( Id A ⊗ α i )( z )) = ( f ⊗ φ )( z ) , z ∈ A ⊗ min M , and, respectively, (2.15) lim i ∈ I ˆ g (( Id B ⊗ α ′ i )( w )) = ( g ⊗ φ ′ )( w ) , w ∈ B ⊗ min M ′ . Proof.
Note that for i ∈ I , k α i k cb ≤ k α ′ i k cb ≤ A ⊗ α i and Id B ⊗ α ′ i are well-definedcontractions on A ⊗ min M and B ⊗ min M ′ , respectively. Hence, in order to prove (2.14) and (2.15) , itsuffices to consider elementary tensors z = a ⊗ c and w = b ⊗ d , where a ∈ A , b ∈ B , c ∈ M and d ∈ M ′ .Let a ∈ A and c ∈ M . By (2.1) we deduce that the following holds in norm topologylim i ∈ I (Id A ⊗ α i )( a ⊗ c ) = lim i ∈ I a ⊗ α i ( c ) = φ ( c )( a ⊗ M ) . It follows that lim i ∈ I ˆ f ((Id A ⊗ α i )( a ⊗ c )) = φ ( c ) ˆ f ( a ⊗ M ) = φ ( c ) f ( a ) = ( f ⊗ φ )( a ⊗ c ) , which proves (2.14) . Further, for all x ∈ M ,lim i ∈ I α ′ i ( JxJ ) = lim i ∈ I Jα i ( x ) J = J ( φ ( x ) · M ) J = φ ( x ) J · J = φ ( x ) · M = φ ′ ( JxJ ) · M , where the limit is taken in norm topology. Then (2.15) can be proved in the same way as (2.14) . (cid:3) Proposition 2.5.
Let u , b u and φ ′ be as above. Then there exist states f , f on A and states g , g on B such that for all x ∈ A ⊗ min M and y ∈ B ⊗ min M ′ , (2.16) | b u ( x, y ) | ≤ k u k jcb (( f ⊗ φ )( xx ∗ ) + ( f ⊗ φ )( x ∗ x )) / (( g ⊗ φ ′ )( y ∗ y ) + ( g ⊗ φ ′ )( yy ∗ )) / . Proof.
By the Grothendieck inequality for C ∗ -algebras (cf. [9]) applied to the bilinear form b u , thereexist states ˆ f , ˆ f on A ⊗ min M and states ˆ g , ˆ g on B ⊗ min M ′ such that for all x ∈ A ⊗ min M and y ∈ B ⊗ min M ′ , | b u ( x, y ) | ≤ k b u k ( ˆ f ( xx ∗ ) + ˆ f ( x ∗ x )) / ( ˆ g ( y ∗ y ) + ˆ g ( yy ∗ )) / (2.17) ≤ k u k jcb ( ˆ f ( xx ∗ ) + ˆ f ( x ∗ x )) / ( ˆ g ( y ∗ y ) + ˆ g ( yy ∗ )) / , wherein we have used inequality (2.7) .Since √ αβ ≤ ( α + β ) / α , β ≥ | b u ( x, y ) | ≤ k u k jcb (cid:16) ˆ f ( xx ∗ ) + ˆ f ( x ∗ x ) + ˆ g ( y ∗ y ) + ˆ g ( yy ∗ ) (cid:17) . or i = 1 , f i be the state on A constructed from ˆ f i by formula (2.12), and, respectively, let g i bethe state on B constructed from ˆ g i by formula (2.13). We show in the following that these are the stateswe are looking for.By Lemma 2.3 , we deduce that for all v ∈ U ( M φ ) (and v ′ := JvJ , as defined therein) , | b u ( x, y ) | ≤ k u k jcb h ˆ f ((Id A ⊗ ad( v ))( xx ∗ )) + ˆ f ((Id A ⊗ ad( v ))( x ∗ x )) +(2.18) + ˆ g ((Id B ⊗ ad( v ′ ))( y ∗ y )) + ˆ g ((Id B ⊗ ad( v ′ ))( yy ∗ )) i . Next choose nets { α i } i ∈ I and { α ′ i } i ∈ I as in Lemma 2.4 . For all i ∈ I , it follows that | b u ( x, y ) | ≤ k u k jcb h ˆ f ((Id A ⊗ α i )( xx ∗ )) + ˆ f ((Id A ⊗ α i )( x ∗ x )) +(2.19) + ˆ g ((Id B ⊗ α ′ i )( y ∗ y )) + ˆ g ((Id B ⊗ α ′ i )( yy ∗ )) i , since the right-hand side of (2.19) is a convex combination of the possible right-hand sides of (2.18) . Then,by Lemma 2.4 we obtain in the limit that(2.20) | b u ( x, y ) | ≤ k u k jcb (( f ⊗ φ )( xx ∗ ) + ( f ⊗ φ )( x ∗ x ) + ( g ⊗ φ ′ )( y ∗ y ) + ( g ⊗ φ ′ )( yy ∗ )) . Recall that x and y were arbitrarily chosen in A ⊗ min M and B ⊗ min M ′ , respectively. Hence, replacing x by t / x and y by t − / y , where t > x ∈ A ⊗ min M , y ∈ B ⊗ min M ′ and t > | b u ( x, y ) | ≤ k u k jcb (cid:18) t ( f ⊗ φ )( xx ∗ ) + t ( f ⊗ φ )( x ∗ x ) + 1 t ( g ⊗ φ ′ )( y ∗ y ) + 1 t ( g ⊗ φ ′ )( yy ∗ ) (cid:19) . Since for all α , β ≥ t> ( tα + t − β ) = 2 p αβ , the assertion then follows by taking infimum over all t > (cid:3) Lemma 2.6.
Let α , β ≥ . Then (2.23) inf n ∈ Z ( λ n α + λ − n β ) ≤ ( λ / + λ − / ) p αβ . Proof.
The statement is obvious if α = 0 or β = 0 . Assume that α , β > < λ < , ∞ ) = S n ∈ Z [ λ n +1 , λ n − ] . Hence, we can choose n ∈ Z such that λ n +1 ≤ β/α ≤ λ n − . Set α := λ n α and β := λ − n β . Then λ ≤ β /α ≤ /λ . Since the function t t / + t − / is decreasingon [ λ ,
1] and increasing on [1 , /λ ] , it follows thatmax { t / + t − / : t ∈ [ λ , /λ ] } = λ / + λ − / . Hence, we deduce that λ n α + λ − n β = α + β = (cid:16)p α /β + p β /α (cid:17) p α β ≤ ( λ / + λ − / ) p α β = ( λ / + λ − / ) p αβ , which proves (2.23) . (cid:3) roposition 2.7. Set C ( λ ) := q(cid:0) λ / + λ − / (cid:1) / . Let u be as above and let f , f be states on A , respectively, g , g be states on B as in Proposition 2.5 .Then, for all a ∈ A and b ∈ B , (2.24) | u ( a, b ) | ≤ C ( λ ) k u k jcb (cid:16) f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / (cid:17) . that is, the Effros-Ruan conjecture holds with constant C ( λ ) .Proof. Let n ∈ Z and choose c n ∈ M as in Lemma 2.1 . Then, for all a ∈ A and all b ∈ B , it follows by(2.6) and (2.4) that b u ( a ⊗ c n , b ⊗ Jc n J ) = u ( a, b ) h c n Jc n Jξ φ , ξ φ i H φ = u ( a, b ) . By Proposition 2.5, together with (2.11) and (2.3) , it follows that | u ( a, b ) | = | b u ( a ⊗ c n , b ⊗ Jc n J ) | (2.25) ≤ k u k ( f ( aa ∗ ) φ ( c n c ∗ n ) + f ( a ∗ a ) φ ( c ∗ n c n )) ( g ( b ∗ b ) φ ( c n ∗ c n ) + g ( bb ∗ ) φ ( c n c n ∗ ))= k u k (cid:16) λ n/ f ( aa ∗ ) + λ − n/ f ( a ∗ a ) (cid:17) (cid:16) λ − n/ g ( b ∗ b ) + λ n/ g ( bb ∗ ) (cid:17) = k u k (cid:0) f ( aa ∗ ) g ( b ∗ b ) + f ( a ∗ a ) g ( bb ∗ ) + λ n f ( aa ∗ ) g ( bb ∗ ) + λ − n f ( a ∗ a ) g ( b ∗ b ) (cid:1) . Note that λ / + λ − / = 2 C ( λ ) . By taking infimum in (2.25) over all n ∈ Z , we deduce from Lemma2.6 that | u ( a, b ) | ≤ k u k (cid:16) f ( aa ∗ ) g ( b ∗ b ) + f ( a ∗ a ) g ( bb ∗ ) + 2 C ( λ ) f ( a ∗ a ) g ( b ∗ b ) f ( aa ∗ ) g ( bb ∗ ) (cid:17) ≤ C ( λ ) k u k (cid:16) f ( aa ∗ ) g ( b ∗ b ) + f ( a ∗ a ) g ( bb ∗ ) (cid:17) , wherein we have used the fact that C ( λ ) > (cid:3) Proof of Theorem 1.1 : Thus far we have proved that given C ∗ -algebras A and B and a j.c.b. bilinear form u : A × B → C , then the Effros-Ruan conjecture holds with constant C ( λ ) = q(cid:0) λ / + λ − / (cid:1) / < λ < Q ( A ) := { f ∈ A ∗ + : k f k ≤ } , Q ( B ) := { g ∈ B ∗ + : k g k ≤ } are compact in the weak ∗ -topology , where A ∗ + and B ∗ + denote the sets of positive functionals on A and B , respectively. Since C ( λ ) → λ → f , f ∈ Q ( A ) and g , g ∈ Q ( B ) such that for all a ∈ A and b ∈ B , | u ( a, b ) | ≤ k u k jcb (cid:16) f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / (cid:17) . But f i ≤ f i , respectively, g i ≤ g i , i = 1 , f , f are states on A and g , g are states on B .Therefore the Effros-Ruan conjecture holds with constant one. (cid:3) . Applications
Let E ⊆ A and F ⊆ B be operator spaces sitting in C ∗ -algebras A and B . Let u : E × F → C be abounded bilinear form. Define k u k ER to be the smallest constant 0 ≤ κ ≤ ∞ for which there exist states f , f on A and states g , g on B such that for all a ∈ E and b ∈ F ,(3.1) | u ( a, b ) | ≤ κ ( f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / ) . In the case when E = A and F = B , we have from Theorem 1.1 that k u k ER ≤ k u k jcb . Moreover, if E and F are exact operator spaces and u : E × F → C is a j.c.b. bilinear form, then by [15] (cf. Theorem0.3 and 0.4) , k u k ER ≤ / ex( E )ex( F ) k u k jcb . However, for bilinear forms on general operator spaces E and F it can happen that k u k jcb < ∞ , while k u k ER = ∞ (see Example 3.6 below) . Therefore Theorem 1.1 cannot be generalized to arbitrary operatorspaces.Recall that a bilinear map u : E × F → C is called completely bounded (in the sense of Christensen andSinclair) (see [2] , [15] and the references given therein) if the bilinear forms u n : M n ( E ) × M n ( F ) → M n ( C )defined by u n ( a ⊗ x , b ⊗ y ) := u ( a , b ) xy , a ∈ E , b ∈ F , x , y ∈ M n ( C )are uniformly bounded, in which case we set(3.2) k u k cb := sup n ∈ N k u n k . Moreover, u is completely bounded if and only if there exists a constant κ ≥ f on A and g on B such that for all a ∈ E and b ∈ F ,(3.3) | u ( a, b ) | ≤ κ f ( aa ∗ ) / g ( b ∗ b ) / and k u k cb is the smallest constant κ for which (3.3) holds (see also the Introduction to [15]).It was shown by Effros and Ruan (cf. [6]) that if u : E × F → C is completely bounded, then theassociated map e u : E → F ∗ defined by (1.1) admits a factorization of the form e u = vw through a rowHilbert space H r , where E v −→ H r w −→ F ∗ and k v k cb k w k cb = k u k cb . In particular, it follows that(3.4) k u k jcb := k e u k cb ≤ k u k cb . Lemma 3.1. ( cf. [15] and [18]) Let u : E × F → C be a bounded bilinear form on operator spaces E ⊆ A and F ⊆ B sitting in C ∗ -algebras A and B . Let f , f be states on A and g , g be states on B such thatfor all a ∈ E and b ∈ F , | u ( a, b ) | ≤ k u k ER ( f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / ) . Then u can be decomposed as u = u + u , where u and u are bilinear forms satisfying the followinginequalities, for all a ∈ A and b ∈ B : | u ( a, b ) | ≤ k u k ER f ( aa ∗ ) / g ( b ∗ b ) / (3.5) | u ( a, b ) | ≤ k u k ER f ( a ∗ a ) / g ( bb ∗ ) / . (3.6) In particular, k u k cb ≤ k u k ER , k u t k cb ≤ k u k ER , where u t ( b, a ) := u ( a, b ) , for all a ∈ E and b ∈ F . roof. Such a decomposition was obtained in [15] (cf. last statement in Theorem 0.4 in [15]), except thatthe states f , f , g , g satisfying (3.5) and (3.6) were possibly different from the original ones. Later,following a suggestion of Pisier, Xu proved the above decomposition without change of states. (See [18],Proposition 5.1 and the Remark following the proof of this proposition.) (cid:3) Remark 3.2.
Note that our main result combined with the above splitting lemma solves conjecture (0 . ′ )in [15] (with constant K = 2) , and hence it solves Blecher’s conjecture (cf. [1] and Conjecture (0 .
2) in[15]).
Proposition 3.3. ( i ) Let u : A × B → C be a bounded bilinear form on C ∗ -algebras A and B . Then (3.7) k u k ER ≤ k u k jcb ≤ k u k ER . ( ii ) Let c , c denote the best constants in the inequalities (3.8) c k u k ER ≤ k u k jcb ≤ c k u k ER , where u : A × B → C is any bounded bilinear form on arbitrary C ∗ -algebras A and B . Then c = 1 and c = 2 .Proof. ( i ) . The left-hand side inequality follows from our main theorem, while the right-hand side in-equality follows from the splitting lemma above. Indeed, we can assume that k u k ER < ∞ . Then with u , u : A × B → C as in Lemma 3.1 , k u k jcb ≤ k u k jcb + k u k jcb = k u k jcb + k u t k jcb ≤ k u k cb + k u t k cb ≤ k u k ER . ( ii ) . By ( i ) we know that c ≥ c ≤ c = 2 . Let τ be a tracial state ona C ∗ -algebra A and define a bilinear form u : A × A → C by u ( a, b ) := τ ( ab ) , for all a , b ∈ A . Then k u k jcb ≥ k u k = 1 , and for all a , b ∈ A , | u ( a, b ) | ≤ τ ( aa ∗ ) / τ ( b ∗ b ) / = 12 (cid:16) τ ( aa ∗ ) / τ ( b ∗ b ) / + τ ( a ∗ a ) / τ ( bb ∗ ) / (cid:17) , which implies that k u k ER ≤ . By (3.7) , k u k ER ≥ k u k jcb . Hence k u k ER = and k u k jcb = 1 , and theassertion follows. To prove that c = 1 , let φ be any state on a unital, properly infinite C ∗ -algebra A .Let u : A ⊗ A → C be defined by u ( a, b ) := φ ( ab ) , for all a , b ∈ A . Note that k u k ER ≤ k u k jcb ≤ k u k cb ≤ , where the last inequality follows immediately from (0 . ′ ) in [15] (by taking f = g = φ therein). We claimthat k u k ER = 1 . For this, let f , f , g , g be states on A and let { s n } n ≥ be a sequence of isometriesin A with orthogonal ranges. Then f k ( s n s ∗ n ) → n → ∞ , respectively g k ( s n s ∗ n ) → n → ∞ , for k = 1 , u ( s n , s ∗ n ) = 1 , for all n ≥ n →∞ f ( s n s ∗ n ) / g ( s n s ∗ n ) / + f ( s ∗ n s n ) / g ( s ∗ n s n ) / = 1 . This shows that k u k ER ≥ (cid:3) emma 3.4. Let E ⊆ A and F ⊆ B be operator spaces sitting in C ∗ -algebras A and B , and let u : E × F → C be a bounded bilinear form. If k u k ER < ∞ , then the associated map e u : E → F ∗ admits acb-factorization e u = vw through H r ⊕ K c for some Hilbert spaces H and K , where E v −→ H r ⊕ K c w −→ F ∗ ,satisfying k v k cb k w k cb ≤ k e u k ER . Proof.
Choose states f , f on A and states g , g on B such that (3.1) holds. Then, by Lemma 3.1 , u can be decomposed as u = u + u , where u and u are bounded bilinear forms satisfying (3.5) and (3.6).The rest of the proof follows from the proof of Corollary 0.7 on p. 206 in [15]. (cid:3) Proposition 3.5.
Let A and B be C ∗ -algebras. Then every completely bounded linear map T : A → B ∗ admits a cb-factorization T = vw through H r ⊕ K c for some Hilbert spaces H and K , such that k u k cb k w k cb ≤ k T k cb . Proof.
Let T : A → B ∗ be a completely bounded linear map. Then T is of the form T = e u , for a j.c.b.bilinear form u : A × B → C with k u k jcb = k T k cb . The assertion follows now from Lemma 3.4, by usingthe fact that k u k ER ≤ k u k jcb . (cid:3) The following example is implicit in the proof of Corollary 3.2 in [15]:
Example 3.6.
Let E be an operator space which is not Banach space isomorphic to a Hilbert space, andlet E ⊆ A and E ∗ ⊆ B be completely isometric embeddings of E and E ∗ , respectively, into C ∗ -algebras A and B . Define u : E × E ∗ → C by u ( a, b ) := b ( a ) , a ∈ E , b ∈ E ∗ . Then e u : E → E ∗∗ is the standard inclusion of E into its second dual. Therefore k u k jcb = k e u k cb = 1 .We will show that k u k ER = ∞ . If k u k ER < ∞ , then it follows from Lemma 3.4 that e u admits a cb-factorization through H r ⊕ K c , for some Hilbert spaces H and K . In particular, e u : E → E ∗∗ hasa Banach space factorization through a Hilbert space. This contradicts the assumption on E . Hence k u k ER = ∞ .The following result was proved in [15] with constant 2 / instead of √ Corollary 3.7.
Let T be a completely bounded linear map from a C ∗ -algebra A to the operator Hilbertspace OH ( I ) , I being an arbitrary index set. Then there exist states f and f on A such that k T ( a ) k ≤ √ f ( aa ∗ ) / f ( a ∗ a ) / , a ∈ A .
Proof.
Given a vector space E , we let ¯ E denote the conjugate vector space. Let J : OH ( I ) → OH ( I ) ∗ be the canonical cb-isomorphism of OH ( I ) with the conjugate of its dual space (cf. [13]), and set V := T ∗ JT , where T ∗ : OH ( I ) ∗ → A ∗ is the adjoint of T . Then V is a completely bounded linear map from A to A ∗ = ( ¯ A ) ∗ . Therefore V = e v for a j.c.b. bilinear form v : A × ¯ A → C . Moreover, k v k jcb = k V k cb ≤ k T k . ctually, equality holds above (cf. [15], proof of Corollary 3.4), but we shall not need this. By our maintheorem, there exist states f , f on A and states g , g on ¯ A such that for all a ∈ A and b ∈ ¯ A , | v ( a, b ) | ≤ k T k (cid:16) f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / (cid:17) . The canonical isomorphism J of OH ( I ) onto OH ( I ) ∗ satisfies J ( x )( x ) = k x k = J ( x )(¯ x ) , x ∈ OH ( I ) . For all a ∈ A we then have v ( a , ¯ a ) = ( V a )(¯ a ) = ( T ∗ JT a )(¯ a ) = ( JT a )( T a ) = k T a k , and therefore k T a k = | v ( a , ¯ a ) | ≤ k T k (cid:16) f ( aa ∗ ) / g ( a ∗ a ) / + f ( a ∗ a ) / g ( aa ∗ ) / (cid:17) ≤ k T k (cid:0) f ( aa ∗ ) + g ( aa ∗ ) (cid:1) / (cid:0) f ( a ∗ a ) + g ( a ∗ a ) (cid:1) / ≤ k T k f ( aa ∗ ) / f ( a ∗ a ) / , where f and f are states on A given by f ( a ) := 12 (cid:16) f ( a ) + g (¯ a ) (cid:17) , f ( a ) := 12 (cid:16) f ( a ) + g (¯ a ) (cid:17) , a ∈ A .
This completes the proof. (cid:3)
As a consequence of Proposition 3.3 we also obtain (by adjusting the corresponding proofs in [15]) thefollowing strengthening of Corollaries 3.1 and 3.3 in [15]:
Corollary 3.8.
Let E be an operator space such that E and its dual E ∗ embed completely isomorphicallyinto preduals M ∗ and N ∗ , respectively, of von Neumann algebras M and N . Then E is cb-isomorphic toa quotient of a subspace of H r ⊕ K c , for some Hilbert spaces H and K . Corollary 3.9.
Let E be an operator space and let E ⊆ A and E ∗ ⊆ B be completely isometric embeddingsinto C ∗ -algebras A and B such that both subspaces are completely complemented. Then E is cb-isomorphicto H r ⊕ K c for some Hilbert spaces H and K . Note that as another consequence of our main theorem we obtain (with essentially the same proof asthe corresponding Corollary 0.6 in [15]) the following result:
Corollary 3.10.
Let A , A , B and B be C ∗ -algebras such that A ⊆ A and B ⊆ B . Then any j.c.b.bilinear form u : A × B → C extends to a bilinear form u : A × B → C such that k u k jcb ≤ k u k jcb . Let u : A × B → C be a j.c.b. bilinear form on C ∗ -algebras A and B . Recall that k u k jcb is the smallestconstant κ for which inequality (1.4) holds, for arbitrary C ∗ -algebras C and D . The following resultshows that if the inequality (1.4) holds for the given bilinear form u with constant κ , then the sameinequality (with κ replaced by 2 κ ) holds for u , when the ( C ⊗ min D )-norm on the left-hand side isreplaced by the ( C ⊗ max D )-norm. roposition 3.11. Let A and B be C ∗ -algebras, and let u : A × B → C be a j.c.b. bilinear form. Then,for all C ∗ -algebras C and D , all m, n ∈ N and all finite sequences a , . . . , a m ∈ A , b , . . . , b n ∈ B , c , . . . , c m ∈ C , d , . . . , d n ∈ D , (3.9) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 n X j =1 u ( a i , b j ) c i ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ⊗ max D ≤ k u k jcb (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ⊗ min C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ⊗ min D . Proof.
There exist states f , f on A and g , g on B such that inequality (3.1) holds. Then, as explainedin the proof of Lemma 3.4 , u can be decomposed as u = u + u , where u and u are bounded bilinearforms satisfying (3.5) and (3.6) .By the definition of k · k max , in order to prove (3.9) we have to show that for all pairs of commutingrepresentations π : A → B ( H ) , ρ : B → B ( H ) , where H is an arbitrary Hilbert space, and all finitesequences a , . . . , a m ∈ A , b , . . . , b n ∈ B , c , . . . , c m ∈ C , d , . . . , d n ∈ D , where m, n ∈ N , we have(3.10) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 n X j =1 u ( a i , b j ) π ( c i ) ρ ( d j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k u k jcb (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ⊗ min C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ⊗ min D . By our main theorem, k u k ER ≤ k u k jcb < ∞ . Let ξ , η be unit vectors in H . Let u = u + u be thedecomposition of u satisfying (3.5) and (3.6) as above. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h m X i =1 n X j =1 u ( a i , b j ) π ( c i ) ρ ( d j ) ξ, η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.11) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h m X i =1 n X j =1 u ( a i , b j ) π ( c i ) ρ ( d j ) ξ, η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 n X j =1 h u ( a i , b j ) π ( c i ) ρ ( d j ) ξ, η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where h· , ·i denotes the inner product on H . By using the GNS construction for the states f on A and g on B and inequality (3.5), we obtain for any a ∈ A and b ∈ B that | u ( a, b ) | ≤ k u k ER f ( aa ∗ ) / g ( b ∗ b ) / = k u k ER k π f ( a ∗ ) ξ f k · k π g ( b ) ξ g k , where ( H f , π f , ξ f ) is the GNS triple associated to ( A, f ) , respectively, ( H g , π g , ξ g ) is the GNS tripleassociated to ( B, g ) . Hence, there exists V ∈ B ( H g , H f ) such that k V k ≤ k u k ER , satisfying u ( a, b ) = h V π g ( b ) ξ g , π f ( a ∗ ) ξ f i , a ∈ A , b ∈ B .
Therefore, for any a ∈ A and b ∈ B , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h m X i =1 n X j =1 u ( a i , b j ) π ( c i ) ρ ( d j ) ξ, η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h m X i =1 n X j =1 h V π g ( b ) ξ g , π f ( a ∗ ) ξ f i ρ ( d j ) π ( c i ) ξ , η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( V ⊗ H )( π g ⊗ ρ )( n X j =1 b j ⊗ d j )( ξ g ⊗ ξ ) , ( π f ⊗ π )( m X i =1 a ∗ i ⊗ c ∗ i )( ξ f ⊗ η ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k u k ER (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ⊗ min C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ⊗ min D , wherein we used the fact that the representations π and ρ do commute, and that P i a ∗ i ⊗ c ∗ i = ( P i a i ⊗ c i ) ∗ . imilarly, by using the GNS construction for the states f on A and g on B and inequality (3.6), weobtain for any a ∈ A and b ∈ B that | u ( a, b ) | ≤ k u k ER f ( a ∗ a ) / g ( bb ∗ ) / = k u k ER k π f ( a ) ξ f k · k π g ( b ∗ ) ξ g k , where ( H f , π f , ξ f ) is the GNS triple associated to ( A, f ) , respectively, ( H g , π g , ξ g ) is the GNS tripleassociated to ( B, g ) . Hence, there exists V ∈ B ( H f , H g ) such that k V k ≤ k u k ER , satisfying u ( a, b ) = h V π f ( a ) ξ f , π g ( b ∗ ) ξ g i , a ∈ A , b ∈ B .
Therefore, for any a ∈ A and b ∈ B , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h m X i =1 n X j =1 u ( a i , b j ) π ( c i ) ρ ( d j ) ξ, η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h m X i =1 n X j =1 h V π f ( a ) ξ f , π g ( b ∗ ) ξ g i π ( c i ) ρ ( d j ) ξ , η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.13) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( V ⊗ H )( π f ⊗ π )( m X i =1 a i ⊗ c i )( ξ f ⊗ ξ ) , ( π g ⊗ ρ )( n X j =1 b ∗ j ⊗ d ∗ j )( ξ g ⊗ η ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k u k ER (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ⊗ min C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ⊗ min D . The inequality (3.10) follows now by (3.11) , (3.12) and (3.13) , since k u k ER ≤ k u k jcb . The proof iscomplete. (cid:3) Our next proposition gives a complete characterization of those bilinear forms u : E × F → C on operatorspaces E ⊆ A and F ⊆ B sitting in C ∗ -algebras A and B , for which k u k ER < ∞ . Proposition 3.12.
Let E ⊆ A and F ⊆ B be operator spaces sitting in C ∗ -algebras A and B , and let u : E × F → C be a bounded bilinear map. The following two conditions are equivalent: ( i ) k u k ER < ∞ . ( ii ) There exists a constant κ ≥ such that for all C ∗ -algebras C and D , all m, n ∈ N and all a , . . . , a m ∈ E , b , . . . , b n ∈ F , c , . . . , c m ∈ C , d , . . . , d n ∈ D , we have (3.14) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 n X j =1 u ( a i , b j ) c i ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ⊗ max D ≤ κ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ⊗ min C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F ⊗ min D . Moreover, if κ ( u ) denotes the best constant in ( ii ) , then k u k ER ≤ κ ( u ) ≤ k u k ER . Proof.
The implication ( i ) ⇒ ( ii ) can be obtained from the proof of Proposition 3.11 with minor modifi-cations. In the case when E = A and F = B we have by (3.12) and (3.13) that(3.15) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 n X j =1 u ( a i , b j ) c i ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ⊗ max D ≤ k u k ER (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 a i ⊗ c i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ⊗ min C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 b j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ⊗ min D . o extend the proof of (3.15) to the general case of operator spaces E ⊆ A and F ⊆ B , the operators V ∈ B ( H g , H f ) and V ∈ B ( H f , H g ) will instead be operators in B ( H g , H f ) and B ( H f , H g ) ,respectively, where H f := π f ( E ) ∗ ξ f , H g := π g ( F ) ξ g , H f := π f ( E ) ξ f , H g := π g ( F ) ∗ ξ g . The rest of the proof of the implication ( i ) ⇒ ( ii ) can then be completed as in the Proof of Proposition3.11 . It also follows that κ ( u ) ≤ k u k ER .The converse implication ( ii ) ⇒ ( i ) can be obtained from the proof of Theorem 0.3 in [15] . Forconvenience of the reader, and in order to obtain a better constant, we include below a slightly modifiedargument.Let u : E × F → C be a bounded bilinear form satisfying (3.14). We will show that k u k ER ≤ κ . ByLemma 2.4 in [15] , given a positive integer n and λ , . . . , λ n > { x , . . . , x n } and { y , . . . , y n } of operators on a Hilbert space H with a unit vector Ω such that the following propertieshold:( a ) For all a , . . . , a n ∈ E and all b , . . . , b n ∈ B , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 a i ⊗ x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i a i a ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ − i a ∗ i a i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 b i ⊗ y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i b i b ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ − i b ∗ i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / ( b ) The von Neumann algebra W ∗ ( x , . . . , x n ) generated by x , . . . , x n commutes with the vonNeumann algebra W ∗ ( y , . . . , y n ) generated by y , . . . , y n .( c ) h x i y j Ω , Ω i H = δ ij , for all 1 ≤ i, j ≤ n .Let now n ∈ N and let λ , . . . , λ n > a , . . . , a n ∈ E and b , . . . , b n ∈ F , that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 u ( a i , b i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i,j =1 u ( a i , b j ) h x i y j Ω , Ω i H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i,j =1 u ( a i , b j ) x i y j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ( H ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i,j =1 u ( a i , b j ) x i ⊗ y j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W ∗ ( x ,... ,x n ) ⊗ max W ∗ ( y ,... ,y n ) ≤ κ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 a i ⊗ x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ⊗ min W ∗ ( x ,... ,x n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 b i ⊗ y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F ⊗ min W ∗ ( y ,... ,y n ) ≤ κ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i a i a ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ − i a ∗ i a i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i b i b ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ − i b ∗ i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ κ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i a i a ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ − i a ∗ i a i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i b i b ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ − i b ∗ i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)! , here we have used the well-known inequality √ α + p β ≤ √ p α + β , α , β ≥ . Since 2 √ αβ ≤ α + β for all α , β ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 u ( a i , b i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ κ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i a ∗ i a i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ − i a i a ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i b i b ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ − i b ∗ i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)! . Using a Pietsch separation argument similar to the one given in the proof of Lemma 3.4 in [9], we inferthe existence of states f , f on A and g , g on B such that for all a ∈ E , b ∈ F and λ > | u ( a, b ) | ≤ κ (cid:0) λf ( aa ∗ ) + λ − f ( a ∗ a ) + λg ( bb ∗ ) + λ − g ( b ∗ b ) (cid:1) . Replacing now a by t / a and b by t − / b , where t > a ∈ E , b ∈ F , t > λ > | u ( a, b ) | ≤ κ (cid:18) tλf ( aa ∗ ) + tλ − f ( a ∗ a ) + 1 t λg ( bb ∗ ) + 1 t λ − g ( b ∗ b ) (cid:19) . By taking the infimum over all t > a ∈ E , b ∈ F and all λ > | u ( a, b ) | ≤ κ ( λf ( aa ∗ ) + λ − f ( a ∗ a )) / ( λg ( bb ∗ ) + λ − g ( b ∗ b )) / . Therefore, for all a ∈ E , b ∈ F and λ > | u ( a, b ) | ≤ (2 κ ) ( f ( aa ∗ ) g ( b ∗ b ) + f ( a ∗ a ) g ( bb ∗ ) + λ f ( aa ∗ ) g ( bb ∗ ) + λ − f ( a ∗ a ) g ( b ∗ b )) . By taking infimum over λ > a ∈ E and b ∈ F , | u ( a, b ) | ≤ (2 κ ) (cid:16) f ( aa ∗ ) g ( b ∗ b ) + f ( a ∗ a ) g ( bb ∗ ) + 2 f ( aa ∗ ) / g ( b ∗ b ) / f ( a ∗ a ) / g ( bb ∗ ) / (cid:17) = (2 κ ) (cid:16) f ( aa ∗ ) / g ( b ∗ b ) / + f ( a ∗ a ) / g ( bb ∗ ) / (cid:17) . This implies that k u k ER ≤ κ , which completes the proof of the implication ( ii ) ⇒ ( i ) and it also provesthe inequality κ ≥ k u k ER . (cid:3) Acknowledgements
This paper was completed during the authors stay at the Fields Institute in the Fall of 2007, whileattending the Thematic Program on Operator Algebras. We would like to thank the Fields Institute andthe organizers of the program for their support and warm hospitality.
References [1]
D. Blecher , Generalizing Grothendieck’s program , Function spaces, Edited by K. Jarosz, Lect. Notes in Pure andApplied Math., Vol. 136, Marcel Dekker, 1992.[2]
E. Christensen, A. Sinclair , A survey of completely bounded operators , Bull. London Math. Soc. (1989), 417-448.[3] A. Connes , Une classification des facteurs de type III , Ann. Scient. ´Ec. Norm. Sup., 4 e s´erie, tome 6 (1973), 133-252.[4] A. Connes , Classification of injective factors. Cases II , II ∞ , III λ , λ = 1 . , Ann. Math. (1976), 73-115.[5] E. Effros and Z.-J. Ruan , A new approach to operator spaces , Canad. Math. Bull. Vol. 34 (3) (1991), 329-337.[6]
E. Effros and Z.-J. Ruan , Self-duality for the Haagerup tensor product and Hilbert space factorization , J. Funct.Analysis (1991), 257-284.[7]
E. Effros and Z.-J. Ruan , Operator Spaces , London Math. Soc. Monographs New Series , Oxford University Press,2000. A. Grothendieck , Resum´e de la th´eorie m´etrique des produits tensorielles topologiques , Bol. Soc. Mat. Sao Paolo (1956), 1-79.[9] U. Haagerup , The Grothendieck inequality for bilinear forms on C ∗ -algebras , Adv. Math., Vol. 56, No. 2 (1985),93-116.[10] U. Haagerup , The injective factors of type III λ , < λ <
1, Pacific J. Math., Vol. 137, No. 2 (1989), 265-310.[11]
J. Lindenstrauss and L. Tzafriri , Classical Banach Spaces, Sequence Spaces , Ergebnisse, Vol. 92, Springer-Verlag,1992.[12]
G. Pisier , Grothendieck’s theorem for non-commutative C ∗ -algebras with an appendix on Grothendieck’s constant , J.Funct. Analysis (1978), 397-415.[13] G. Pisier , The Operator Hilbert Space OH , Complex Interpolation and Tensor Norms , Mem. Amer. Math. Soc.Number , Vol. 122, Providence, RI, 1996.[14] G. Pisier , An Introduction to the Theory of Operator Spaces , London Math. Soc. Lect. Notes Series , CambridgeUniversity Press, Cambridge 2003.[15]
G. Pisier and D. Shlyakhtenko , Grothendieck’s theorem for operator spaces, Invent. Math. (2002), 185-217.[16]
M. Takesaki : The structure of von Neumann algebras with a homogeneous periodic state , Acta Math. (1973),79-121.[17]
M. Takesaki , Theory of Operator Algebras II, III , Springer-Verlag, New-Yord, 1979.[18]
Q. Xu , Operator space Grothendieck inequalities for noncommutative L p -spaces , Duke Math. J. (2006), 525-574. (1) Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230Odense M, Denmark., (2)
Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis,TN, 38152, USA.
E-mail address : (1) [email protected], (2) [email protected]@memphis.edu