Radial multipliers on amalgamated free products of II_1-factors
aa r X i v : . [ m a t h . OA ] D ec RADIAL MULTIPLIERS ON AMALGAMATED FREEPRODUCTS OF II -FACTORS SÖREN MÖLLER
Abstract.
Let M i be a family of II -factors, containing a com-mon II -subfactor N , such that [ M i : N ] ∈ N for all i . Further-more, let φ : N → C . We show that if a Hankel matrix related to φ is trace-class, then there exists a unique completely bounded map M φ on the amalgamated free product of the M i with amalgama-tion over N , which acts as an radial multiplier. Hereby we extenda result of U. Haagerup and the author for radial multipliers onreduced free products of unital C ∗ - and von Neumann algebras. Introduction
Let C denote the set of functions φ on the non-negative integers N for which the matrix h = ( φ ( i + j ) − φ ( i + j + 1)) i,j ≥ is of trace-class. Let ( M , ω ) = ¯ ∗ i ∈ I ( M i , ω i ) be the w*-reduced freeproduct of von Neumann algebras ( M i ) i ∈ I with respect to normal states ( ω i ) i ∈ I for which the GNS-representation π ω i is faithful, for all i ∈ I .In [3] U. Haagerup and the author proved that if φ ∈ C , then there isa unique linear completely bounded normal map M φ : M → M suchthat M φ (1) = φ (0)1 and M φ ( a a . . . a n ) = φ ( n ) a a . . . a n whenever a j ∈ ˚ M i j = ker( ω i j ) and i = i = · · · 6 = i n . Moreover, k M φ k cb ≤ k φ k C where k · k C is the norm on C defined in (2.2) below.This result generalized the corresponding result for reduced C ∗ -alge-bras of discrete groups which was proved by J. Wysoczański in [6].In this paper we generalize above result to the case of amalgamatedfree products of II -factors where amalgamation occurs over a integer-index II -subfactor, resulting in the following theorem. Theorem 1.1.
Let ( M , E N ) = ¯ ∗ N ,i ∈ I ( M i , E i N ) be the amalgamatedfree product of II -factors ( M i ) i ∈ I over a II -subfactor N for which [ M i : N ] ∈ N for all i ∈ I . Here E i N : M i → N , respectively, E N : M → N denote the canonical conditional expectations. Date : September 24, 2018.2010
Mathematics Subject Classification.
Primary 46L54; Secondary 46L07. If φ ∈ C , then there is a unique linear completely bounded normalright- N -module map M φ : M → M such that M φ (1) = φ (0)1 and M φ ( b a b a b . . . a n b n ) = φ ( n ) b a b a b . . . a n b n whenever a j ∈ ˚ M i j = ker( E i N ) , i = i = · · · 6 = i n and b , . . . , b n ∈ N .Moreover, k M φ k cb ≤ k φ k C . Recently this result has been extended by S. Deprez to arbitraryamalgamated free products of finite von Neumann algebras in [2] usingdifferent methods related to the construction of relative Fock spaces.It is an open problem if a similar result holds for even more generalamalgamated free products. È. Ricard and Q. Xu investigated in [5,Section 5] the amalgamated setting and the weaker estimate k M φ k cb ≤ | φ (0) | + ∞ X n =1 n | φ ( n ) | can be obtained from [5, Corollary 3.3].We start by investigating some preliminaries in Section 2 followed byconstructing the main building blocks of the radial multiplier in Section3. The main result is then proved in Section 4.2. Preliminaries
Assume that ( N , τ ) is a II -factor equipped with its finite trace τ .Let now I be some index set and let M i , i ∈ I be a family of II -factorscontaining N equipped with the canonical conditional expectations E i N : M i → N . We will assume that the Jones index [ M i : N ] is aninteger for all i ∈ I . Denote by τ i = τ ◦ E i N the state on M i and by H i = L ( M i , τ i ) the Hilbert space associated to M i with respect to τ i .Observe that we can see H i as an N -bimodule. Set ˚ H i = L ( N , τ ) ⊥ and set ˚ M i = M i ∩ ker( E i N ) . Moreover, denote by e i N the projectionfrom L ( M i , τ i ) onto L ( N , τ ) .Let ( M , τ ) = ¯ ∗ N ,i ∈ I M i be the amalgamated free product with re-spect to E i N and let E N : M → N be the conditional expectationcomparable with τ .We have the amalgamated free Fock space F N = L ( N ) ⊕ M n ≥ ,i = ···6 = i n ˚ H i ⊗ N · · · ⊗ N ˚ H i n where ⊗ N fulfills the relation x ⊗ N by = xb ⊗ N y for all x ∈ ˚ H i , y ∈ ˚ H j , i = j and b ∈ N . Note that there are natural left and right actionsof N on F N . We denote by B N ( F N ) the subspace of B ( F N ) of mapsthat commute with the right action of N .Moreover, we will by h· , ·i N denote the N -valued inner product on H i coming from E i N and extended to F . ADIAL MULTIPLIERS ON AMALGAMATED FREE PRODUCTS 3
As M. Pimsner an S. Popa proved in [4] there will exist Pimsner-Popa-bases for these inclusions.
Theorem 2.1 ([4, Proposition 1.3]) . Let M i and N as above andassume [ M i : N ] = N i ∈ N . Set n i = N i − . There exists a set Γ i = { e i , . . . , e in i } ⊂ M i satisfying the properties: • E i N ( e i ∗ k e il ) = 0 for l = k • E i N ( e i ∗ k e ik ) = 1 for ≤ k ≤ n i • P n i j =0 e ij e i N e i ∗ j as an equality in B ( L ( M i )) • For all x ∈ M i we have x = n i X j =0 E i N ( xe ij ) e i ∗ j (2.1)We can without loss of generality choose e i = 1 i N for all i ∈ I ,indeed, in the basis construction of [4], one could choose e i N as thefirst projection and then use well-known facts on projections in II -factors to complete the basis.Set ˚Γ i = Γ i \ { e i } and set ˚Γ = S i ∈ I ˚Γ i . Denote Λ( k ) = { b χ b ⊗ N · · · ⊗ N χ k − b k − ⊗ N χ k b k : χ j ∈ ˚Γ i j , i = i = · · · 6 = i n , b , . . . , b k ∈ N } ⊂ F N and note that the span of Λ( k ) , k ≥ is dense in F N . Definition 2.2.
For γ ∈ ˚Γ i and χ ∈ Λ( k ) with χ ∈ ˚Γ j and χ k ∈ ˚Γ h we define L γ ( χ ) = (cid:26) γ ⊗ N χ if i = j otherwise L ∗ γ ( χ ) = E N ( γ ∗ b χ ) χ ′ if χ = b χ ⊗ N χ ′ R γ ∗ ( χ ) = (cid:26) χ ⊗ N γ ∗ if i = h otherwise R ∗ γ ∗ ( χ ) = χ ′ E N ( χ k b k γ ) if χ = χ ′ ⊗ N χ k b k . Note that if γ ∈ ˚Γ i then L γ , L ∗ γ , R γ and R ∗ γ naturally sit in the copyof B N ( H i ) inside B N ( F N ) . Observation 2.3.
Observe that L γ and L ∗ γ commute with the rightaction of N on F N , hence L γ , L ∗ γ ∈ B N ( F N ) , while R γ ∗ and R ∗ γ ∗ commute with the left action of N on F N . We also have to define sets of operators, which in the amalgamatedsetting replace the idea of B ( H ) in the non-amalgamated case. Definition 2.4.
Denote by L the set L = { b L ξ b · · · L ξ k b k L ∗ η ˜ b · · · L ∗ η l ˜ b l : k, l ∈ N , ξ j , η j ∈ ˚Γ , b i , ˜ b j ∈ N } SÖREN MÖLLER and denote by ˜ L the weakly closed linear span of L in B N ( F N ) .Furthermore, we define L k,l = { b L ξ b · · · L ξ k b k L ∗ η ˜ b · · · L ∗ η l ˜ b l : ξ j , η j ∈ ˚Γ , b i , ˜ b j ∈ N } . Similarly, for i ∈ I denote by L i = { b L ξ b · · · L ξ k b k L ∗ η ˜ b · · · L ∗ η l ˜ b l : k, l ∈ N , ξ j , η j ∈ ˚Γ i , b i , ˜ b j ∈ N } and denote by ˜ L i the weakly closed linear span of L i in B N ( H i ) .Furthermore, we define L k,li = { b L ξ b · · · L ξ k b k L ∗ η ˜ b · · · L ∗ η l ˜ b l : ξ j , η j ∈ ˚Γ i , b i , ˜ b j ∈ N } . Observation 2.5.
Note that by an argument similar to [3, Lemma 3.1] it would be enough to consider terms of the forms b L γ b , b L ∗ γ b and b L γ b L ∗ δ b in the defintions of the spans of L i and L k,li as we can useequation (2.1) to move the b i out of the way. Definition 2.6 ([3, Definition 2.1]) . Let C denote the set of functions φ : N → C for which the Hankel matrix h = ( φ ( i + j ) − φ ( i + j + 1)) i,j ≥ is of trace-class. We will use the following facts about maps from C which are provenin [3, Lemmas 4.6 and 4.8, Remark 4.7, after Definition 2.1]. Here k x k = T r ( | x | ) is the trace-class norm for x ∈ B ( l ( N )) and we usethe notation ( u ⊙ v )( t ) = h t, v i u , for u, v, t ∈ l ( N ) . Lemma 2.7.
Let φ ∈ C . Then k = ( φ ( i + j + 1) − φ ( i + j + 2)) i,j ≥ is of trace-class. Furthermore, c = lim n →∞ φ ( n ) exists and ∞ X n =0 | φ ( n ) − φ ( n + 1) | ≤ k h k + k k k < ∞ . Let now k φ k C = k h k + k k k + | c | (2.2) and set for n ≥ ψ ( n ) = ∞ X i =0 ( φ ( n + 2 i ) − φ ( n + 2 i + 1)) ψ ( n ) = ψ ( n + 1) . Then φ ( n ) = ψ ( n ) + ψ ( n ) + c ADIAL MULTIPLIERS ON AMALGAMATED FREE PRODUCTS 5 and for i, j ≥ the entries h i,j and k i,j of h and k are given by h i,j = ψ ( i + j ) − ψ ( i + j + 2) k i,j = ψ ( i + j ) − ψ ( i + j + 2) . Furthermore, there exist x i , y i , z i , w i ∈ l ( N ) such that h = ∞ X i =1 x i ⊙ y i ∞ X i =1 k x i k k y i k = k h k k = ∞ X i =1 z i ⊙ w i ∞ X i =1 k z i k k w i k = k k k . Moreover, we have ψ ( k + l ) = ∞ X i =1 ∞ X t =0 x i ( k + t ) y i ( l + t ) ψ ( k + l ) = ∞ X i =1 ∞ X t =0 z i ( k + t ) w i ( l + t ) . Technical lemmas
We will start by proving some technical lemmas, which will be usefulin the main proof.
Lemma 3.1.
Let γ ∈ ˚Γ i and b ∈ N then γb = n i X j =0 E i N ( γbe ij ) e i ∗ j = n i X j =1 E i N ( γbe ij ) e i ∗ j . Proof.
The first equality follows directly from equation (2.1). For thesecond equality observe that γ ∈ ˚Γ i implies γ = e ij for some j = 0 hence E i N ( γbe i ) e i ∗ = E i N ( e ij be i ) e i ∗ = E i N ( e ij b E i N ( e ij ) b = 0 b = 0 . (cid:3) Next we have to construct the building blocks for an explicit con-struction of the radial multiplier. Here we closely follow the outline of[3, Section 4] but generalize it to the amalgamated setting.
Lemma 3.2.
The map ρ : ˜ L → B N ( F N ) defined by ρ ( a ) = X γ ∈ ˚Γ R γ ∗ aR ∗ γ ∗ SÖREN MÖLLER is a normal map that commutes with the right action of N on ˜ L ,respectively, B N ( F N ) .Proof. Let a = b L ξ b · · · L ξ k b k L ∗ η ˜ b · · · L ∗ η l ˜ b l ∈ L k,l and χ = d χ d ⊗ N · · · ⊗ N χ m d m ∈ Λ( m ) then we have ρ ( a ) χ = X γ ∈ ˚Γ R γ ∗ aR ∗ γ ∗ χ = X γ ∈ ˚Γ R γ ∗ a ( d χ d ⊗ N · · · ⊗ N χ m − d m − E N ( χ m d m γ )) . Now X γ ∈ ˚Γ R γ ∗ a ( d χ d ⊗ N · · · ⊗ N χ m − d m − E N ( χ m d m γ ))= X γ ∈ ˚Γ R γ ∗ b L ξ b · · · L ξ k b k L ∗ η ˜ b · · · L ∗ η l ˜ b l ( d χ d ⊗ N · · · ⊗ N χ m − d m − E N ( χ m d m γ ))= X γ ∈ ˚Γ R γ ∗ b L ξ b · · · L ξ k b k c l ( d l χ l +1 d l +1 ⊗ N · · · ⊗ N χ m − d m − E N ( χ m d m γ )) where c k ∈ N , k = 1 . . . l , is iteratively defined by c = 1 , c k = E N ( η l − k +1 b l − k +1 c k − d k − χ k ) . This gives us X γ ∈ ˚Γ R γ b L ξ b · · · L ξ k b k c l ( d l χ l +1 d l +1 ⊗ N · · · ⊗ N χ m − d m − E N ( χ m d m γ ))= X γ ∈ ˚Γ R γ ∗ ( b ξ ⊗ N · · · ⊗ N ξ k b k ⊗ N c l d l χ l +1 d l +1 ⊗ N · · · ⊗ N χ m − d m − E N ( χ m d m γ )) if ξ k and χ l +1 are from different ˚Γ i as multiplying with an element from N does not change the Hilbert space and both side vanish if ξ k and ADIAL MULTIPLIERS ON AMALGAMATED FREE PRODUCTS 7 χ l +1 are from the same ˚Γ i . Hence X γ ∈ ˚Γ R γ ∗ ( b ξ ⊗ N · · · ⊗ N ξ k b k ⊗ N c l d l χ l +1 d l +1 ⊗ N · · · ⊗ N χ m − d m − E N ( χ m d m γ ))= X γ ∈ ˚Γ b ξ ⊗ N · · · ⊗ N ξ k b k ⊗ N c l d l χ l +1 d l +1 ⊗ N · · · ⊗ N χ m − d m − E N ( χ m d m γ ) ⊗ N γ ∗ = b ξ ⊗ N · · · ⊗ N ξ k b k ⊗ N c l d l χ l +1 d l +1 ⊗ N · · · ⊗ N χ m − d m − ⊗ N X γ ∈ ˚Γ E N ( χ m d m γ ) γ ∗ = b ξ ⊗ N · · · ⊗ N ξ k b k ⊗ N c l d l χ l +1 d l +1 ⊗ N · · · ⊗ N χ m − d m − ⊗ N χ m d m in the case of l < m − , which does not vanish as χ m − and χ m arefrom different ˚Γ i from the start. If instead l = m − we have X γ ∈ ˚Γ R γ ( b ξ ⊗ N · · · ⊗ N ξ k b k c l d l E N ( χ m d m γ ))= X γ ∈ ˚Γ b ξ ⊗ N · · · ⊗ N ξ k b k c l d l E N ( χ m d m γ ) ⊗ N γ ∗ = b ξ ⊗ N · · · ⊗ N ξ k b k c l d l ⊗ N X γ ∈ ˚Γ E N ( χ m d m γ ) γ ∗ = b ξ ⊗ N · · · ⊗ N ξ k b k c l d l ⊗ N χ m d m if ξ k and χ m are from different ˚Γ i and vanishing otherwise.Hence, in total we have ρ ( a ) χ = b ξ ⊗ N · · · ⊗ N ξ k b k ⊗ N c l d l χ l +1 d l +1 ⊗ N · · · ⊗ N χ m − d m − ⊗ N χ m d m = aχ if l < m − or l = m − and ξ k and χ m are from different ˚Γ i .Now let b ∈ N then ρ ( a )( χb ) = a ( χb ) = a ( χ ) b = ( ρ ( a ) χ ) b as multiplication from the right by b neither changes m and l norchanges if ξ k and χ m are from different ˚Γ i .Normality follows from weak- ∗ -continuity of multiplication and ad-dition and hereby we can extend ρ to all of ˜ L while preserving com-mutation with the right action of N . (cid:3) As the next building block we construct the map D x , closely following[3, Section 4] and omitting the proof as it is a simple calculation. SÖREN MÖLLER
Lemma 3.3.
Let x ∈ l ∞ ( N ) . The map D x : F N → F N defined by D x ( χ ) = x k χ for χ ∈ Λ( k ) and extended by linearity is a N -bimodule map and hasthe adjoint D ¯ x . The last necessary building block is the map ǫ which, following [3,Section 4], is constructed in the next lemma. Lemma 3.4.
Let q i be the projection on the closed linear span in F N of the set n χ ∈ Λ( k ) : k ≥ , χ = d χ d · · · χ k d k , d , . . . , d k ∈ N , χ k ∈ ˚ H i o and define ǫ : ˜ L → B N ( F N ) by ǫ ( a ) = P i ∈ I q i aq i . Then ǫ ( a ) is a nor-mal map that commutes with the right action of N on ˜ L , respectively, B N ( F N ) .Proof. To prove commutation with the right action of N observe thatright multiplication of χ with b ∈ N will not change which ˚ H i χ k d k belongs to hence q i ( χb ) = ( q i χ ) b so q i ∈ B N ( F N ) which implies ǫ ( a ) ∈ B N ( F N ) .Normality follows from weak- ∗ -continuity of multiplication and ad-dition. (cid:3) The following lemma is modelled after [1, Theorem 1.3], which statesthe similar result for the Hilbert space case, from which this follows asa simple observation.
Lemma 3.5.
Let φ : ˜ L → B N ( F N ) be given by φ ( a ) = X k u k av k for some bounded right- N -module maps u k , v k on F N . Then we have k φ k cb ≤ k X k u k u ∗ k k / k X k v ∗ k v k k / . Combining the four preceeding lemmas we now can define maps thatwill be used to construct the radial multiplier explicitely and calculatebounds on its completely bounded norms.
Lemma 3.6.
For x, y ∈ l ( N ) and a ∈ ˜ L set Φ (1) x,y ( a ) = ∞ X n =0 D ( S ∗ ) n x aD ∗ ( S ∗ ) n y + ∞ X n =1 D S n x ρ n ( a ) D ∗ S n y , ADIAL MULTIPLIERS ON AMALGAMATED FREE PRODUCTS 9 respectively, Φ (2) x,y ( a ) = ∞ X n =0 D ( S ∗ ) n x aD ∗ ( S ∗ ) n y + ∞ X n =1 D S n x ρ n − ( ǫ ( a )) D ∗ S n y . Then Φ (1) x,y , Φ (2) x,y : ˜ L → B N ( F N ) are well-defined, normal, completelybounded maps that commute with the right action of N on ˜ L , respec-tively, B N ( F N ) . Furthermore, k Φ ( i ) x,y k cb ≤ k x k k y k , for i = 1 , .Proof. Firstly, we by an argument akin to [3, Lemma 4.2] can observethat ∞ X n =0 D ( S ∗ ) n x D ∗ ( S ∗ ) n x + ∞ X n =1 D S n x Q n D ∗ S n x = k x k B ( F N ) . Here Q n denotes the projection on the span of simple tensors of lengthgreater or equal n in F N .Similarly, if we restrict us to a ∈ L we can observe ρ n ( a ) = X ζ ∈ Λ( n ) R ζ aR ∗ ζ , a ∈ ˜ L ρ n − ( ǫ ( a )) = X ζ ∈ Λ( n − R ζ q i aq i R ∗ ζ , a ∈ ˜ L and ρ n (1) = Q n ρ n − ( ǫ (1)) = ρ n − ( Q ) = Q n . which by normality can be extended to all of ˜ L .This, similarly to [3, Lemma 4.3] gives us ∞ X n =0 D ( S ∗ ) n x D ∗ ( S ∗ ) n x + ∞ X n =1 X ζ ∈ Λ( n ) D S n x R ζ R ∗ ζ D ∗ S n x = k x k B ( F N ) and ∞ X n =0 D ( S ∗ ) n x D ∗ ( S ∗ ) n x + ∞ X n =1 X ζ ∈ Λ( n − X i ∈ I D S n x R ζ q i R ∗ ζ D ∗ S n x = k x k B ( F N ) and hence we from Lemmas 3.2, 3.3 and 3.4 get that Φ ( i ) x,y is a well-defined, normal, completely bounded right- N -module map on ˜ L andfrom Lemma 3.5 that k Φ ( i ) x,y k cb ≤ k x k k y k for i = 1 , and for all x, y ∈ l ( N ) . (cid:3) Definition 3.7. If a = b L ξ b · · · L ξ k b k L ∗ η ˜ b · · · L ∗ η l ˜ b l ∈ L k,l we saythat we are in • Case 1 if k = 0 or l = 0 or k, l ≥ and ξ k ∈ ˚Γ i , η l ∈ ˚Γ j and i = j , i, j ∈ I ,respectively, • Case 2 if k, l ≥ and ξ k , η l ∈ ˚Γ i for some i ∈ I . Lemma 3.8.
We have for all n ≥ and a ∈ L k,l that ρ n ( a ) = aQ l + n and ǫ ( a ) = ρ ( a ) in Case 1, and, respectively, ǫ ( a ) = a in Case 2.Proof. Following the argument of [3, Lemma 4.4] and using that ξb ∈ ˚ H i if ξ ∈ ˚ H i and b ∈ N . (cid:3) Lemma 3.9.
Let k, l ≥ and a ∈ L k,l . Then Φ (1) x,y ( a ) = ∞ X t =0 x ( k + t ) y ( l + t ) ! a and, respectively, Φ (2) x,y ( a ) = (cid:26) P ∞ t =0 x ( k + t ) y ( l + t ) a in Case 1 P ∞ t =0 x ( k + t − y ( l + t − a in Case 2 . Proof.
We prove this by showing that both sides act similarly on allsimple tensors in F N . Following the argument of [3, Lemma 4.5] andusing that ξb ∈ ˚ H i if ξ ∈ ˚ H i and b ∈ N as well as the fact that thebehavior of D only depends on the length of simple tensors in F N butnot on the elements from N contained in them. (cid:3) Proof of the main result
Now we have constructed all the necessary tools to prove the mainresult of this paper.
Lemma 4.1.
Let T : ˜ L → ˜ L be a bounded, linear, normal right- N -module map, and let φ : N → C . The following statements areequivalent.(a) For all n ≥ , i , . . . i n ∈ I with i = i = · · · 6 = i n and a j ∈ ˜ L i j ,we have T (1) = φ (0)1 and T ( a a · · · a n ) = φ ( n ) a a · · · a n .(b) For all k, l ≥ and a ∈ L k,l we have T ( a ) = (cid:26) φ ( k + l ) a in Case 1 φ ( k + l − a in Case 2 . Proof. (a) implies (b) follows by the same argument as in the proof of[3, Lemma 5.2] as L ⊂ ˜ L .To prove (b) implies (a) let a ∈ ˜ L which by definition and right- N -module property is in the weak closure of linear combinations of L . Hence it by normality is enough to check that T (1) = φ (0)1 and T ( a · · · a n ) = φ ( n ) a · · · a n whenever n ≥ and a = a · · · a n ∈ L . ADIAL MULTIPLIERS ON AMALGAMATED FREE PRODUCTS 11
Hence a = a · · · a n = b L γ b · · · L γ k b k L ∗ δ ˜ b · · · L ∗ δ l ˜ b l for some γ j ∈ ˚Γ i j ,δ s ∈ ˚Γ r s , i j = i j +1 , r s = r s +1 and i , . . . , i k , r , . . . , r l ∈ I .If we are in Case 1, we have i k = r l . Hence neighboring elements onthe right hand side are from different L i and thus n = k + l . If we arein Case 2, we have i k = r l . Hence L γ k b k L ∗ δ l ∈ L i k , thus n = k + l − .Now (b) gives the result for k ≥ or l ≥ .Moreover, the k = l = 0 case of (b) gives T ( b ) = φ (0) b for b ∈ N and hence T (1) = φ (0)1 . (cid:3) Next, we explicitly construct such a map T . Definition 4.2.
Let φ ∈ C . Define maps T = ∞ X i =1 Φ (1) x i ,y i and T = ∞ X i =1 Φ (2) z i ,w i where Φ ( · ) x,y are as in Lemma 3.6, ψ , ψ as in Lemma 2.7, and x i , y i , z i , w i as in Remark 2.7. Moreover, define T = T + T + c Id where Id denotesthe identity operator on ˜ L , and c = lim n →∞ φ ( n ) . First we prove that these maps are well-defined, normal, and com-pletely bounded, afterwards we will prove that the maps exhibit theright behavior.
Lemma 4.3.
The maps T , T and T are normal and completely boundedright- N -module maps and k T k cb ≤ k φ k C .Proof. This lemma follows directly from Lemmas 2.7 and 3.6. (cid:3)
Lemma 4.4.
For T , T defined as above, k, l ≥ and a ∈ L k,l wehave T ( a ) = ψ ( k + l ) a , respectively, T ( a ) = (cid:26) ψ ( k + l ) a in Case 1 ψ ( k + l − a in Case 2 . Furthermore, for T defined as above we have T ( a ) = (cid:26) φ ( k + l ) a in Case 1 φ ( k + l − a in Case 2.Proof. Following the argument of [3, Lemma 5.5]. (cid:3)
Note that by Lemma 4.1 this implies that T (1) = φ (0)1 and that for n ≥ , T φ ( a a · · · a n ) = φ ( n ) a a · · · a n .Finitely we have to check that combining Lemmas 4.3 and 4.4 isenough to prove Theorem 1.1. Lemma 4.5.
The von Neumann algebra M i is isomorphic to a subal-gebra of ˜ L i . Proof.
We can interpret both M i and ˜ L i as subalgebras of B ( H i ) . Let a ∈ M i . Now set ˜ a = n i X j,k =0 L e ij E i N ( e i ∗ j ae ik ) L ∗ e ik ∈ ˜ L i . As M i acts on H i = L ( M i , τ i ) we want to check that a and ˜ a actas the same operator on H i . As H i is spanned over N from the rightby e i , . . . , e in i it is enough to check that h ae il , e im i N = h ˜ ae il , e im i N for all l, m = 0 , . . . , n i .But we have h ˜ ae il , e im i N = h n X i,j =2 L e ij E i N ( e i ∗ j ae ik ) L ∗ e ik e il , e im i N = n i X j,k =0 h L e ij E i N ( e i ∗ j ae ik ) L ∗ e ik e il , e im i N = n i X j,k =0 E i N ( e i ∗ m L e ij E i N ( e i ∗ j ae ik ) L ∗ e ik e il )= n i X j,k =0 δ k,l E i N ( e i ∗ m L e ij E i N ( e i ∗ j ae ik ))= n i X j,k =0 δ k,l E i N ( e i ∗ m L e ij E i N ( e i ∗ j ae ik )= n i X j,k =0 δ k,l E i N ( e i ∗ m e ij ) E i N ( e i ∗ j ae ik )= n i X j,k =0 δ k,l δ m,j E i N ( e i ∗ j ae ik )= E i N ( e i ∗ m ae il )= h ae il , e im i N as desired. (cid:3) Lemma 4.6. ˜ L = ∗ N ˜ L i .Proof. By definition of the algebraic amalgamated free product of L i over N , it contains of all words of the form b L † ξ b L † ξ · · · b n − L † ξ n b n where † is either ∗ or nothing and ξ j ∈ ˚Γ i j and i = i = · · · 6 = i n . Butas multiplication with elements from N does not change which ˚ H i an ADIAL MULTIPLIERS ON AMALGAMATED FREE PRODUCTS 13 element belongs to, all terms containing a sub-term of the form L ∗ γ bL δ will vanish, hence it is enough to consider terms of the form b L ξ b L ξ · · · b n − L ξ n b k L ∗ η ˜ b · · · L ∗ η l ˜ b l for neighbouring ξ i , η i from different ˚Γ i (although ξ k and η might befrom the same ˚ H i ) and b i ∈ N . Here we use Observation 2.5 to beable to restrict us to short terms from each L i .And all these terms by definition lie in L . On the other hand thespan over N of those terms clearly lies in the amalgamated free prod-uct and by definition is w ∗ -dense in ˜ L i , hence the two von Neumannalgebras are isomorphic. (cid:3) The preceeding two lemmas now allow us to prove the main theorem.
Proof of Theorem 1.1. ( M i , E i N ) can be realized as subalgebras of ˜ L i by Lemma 4.5 and hence ( M , E i N ) by Lemma 4.6 can be realized asa subalgebra of ˜ L . Then as condition (b) and hence (a) holds for allof ˜ L by Lemma 4.1 it also holds for elements in the subalgebra M asthe length of an operator in the amalgamated free product is preservedwhen restricting to a subalgebra. Hence T restricted to the subalgebrahas the desired behaviour and M φ = T | M can be obtained by restricting T to M and we then have k M φ k cb = k T | M k cb ≤ k T k cb ≤ k φ k C . (cid:3) References [1] E. Christensen and A. M. Sinclair. A survey of completely bounded operators.
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