aa r X i v : . [ m a t h . OA ] A p r FREE MONOTONE TRANSPORT WITHOUT A TRACE
BRENT NELSON
Abstract.
We adapt the free monotone transport results of Guionnet and Shlyakhtenko to the type IIIcase. As a direct application, we obtain that the q -deformed Araki-Woods algebras are isomorphic (forsufficiently small | q | ). Introduction
Classically, transport from a probability space (
X, µ ) to a probability space (
Z, ν ) is a measurable map T : X → Z such that T ∗ ( µ ) = ν . In particular, it implies that there is a measure-preserving embeddingof L ∞ ( Z, ν ) into L ∞ ( X, µ ) via f f ◦ T . These generalize to non-commutative probability theory inthat the existence of transport from the law ϕ X of an N -tuple of non-commutative random variables X =( X , . . . , X N ) to the law ϕ Z of Z = ( Z , . . . , Z N ) implies that there is a state-preserving embedding of W ∗ ( Z , . . . , Z N ) into W ∗ ( X , . . . , X N ). Provided there is sufficient control over the transport map, theembedding can be made into an isomorphism.In [5], Guionnet and Shlyakhtenko produced a non-commutative analogue of Brenier’s monotone transporttheorem by solving a free analogue of the Monge-Amp´ere equation. This provided criterion for when an N -tuple of non-commutative random variables generate the free group factor. Their result took place in thecontext of a tracial von Neumann algebra ( M, τ ) where X , . . . , X N ∈ M s.a. are free semi-circular variables.We adapt the result to the context of a von Neumann algebra M with a (not necessarily tracial) state ϕ on M . The random variables X , . . . , X N are no longer assumed to be free; instead their joint law is assumed tobe a free quasi-free state ( cf. [7]). This produces criterion for when an N -tuple of non-commutative randomvariables generate the free Araki-Woods algebra Γ( H R , U t ) ′′ ( cf. [7]).One of the key assumptions that was needed in [5] to produce transport was that the trace satisfiedthe so-called Schwinger-Dyson equation with a potential V (that is close in a precise-sense to the Gaussianpotential P X j ) for non-commutative polynomials in the transport variables. In the commutative case, thisamounts to saying that if η is the semicircle law ( dη ( t ) = χ [ − , ( t ) π √ − t dt ) and V ( t ) = t + W ( t )(with W analytic on a disk of radius R and small k · k ∞ -norm) then Z R V ′ ( t ) f ( t ) dη ( t ) = Z R Z R f ( s ) − f ( t ) s − t dη ( s ) dη ( t )for all f which are analytic on the disk of radius R . In general, a measure satisfying this equation is calleda Gibbs state with potential V .In the free case, the measure η is replaced with a state ϕ on M which we call the free Gibbs statewith potential V ( ϕ = ϕ V ), and W is a non-commutative power series in the X , . . . , X N with radius ofconvergence at least R . For quadratic potentials of the form V = 12 N X j,k =1 (cid:20) A (cid:21) jk X k X j , A ∈ M N ( C )the corresponding free Gibbs state is the free quasi-free state induced by the scalar matrix A ( cf. [7]).Provided W is small enough with respect to a particular Banach norm, then the free Gibbs state withpotential V = V + W is unique in the sense that if the laws of two N -tuples Y = ( Y , . . . , Y N ) and Z = ( Z , . . . , Z N ) both solve the Schwinger-Dyson equation, then in fact W ∗ ( Y , . . . , Y N ) ∼ = W ∗ ( Z , . . . , Z N )( cf. Theorem 2.1 in [4]). In this paper, we produce transport from ϕ V to ϕ V .In the tracial case, starting with the non-commutative Schwinger-Dyson equation, Guionnet and Shlyakht-enko produced an equivalent version which is amenable to a fixed point argument. We prove the uniqueness Research supported by NSF grants DMS-1161411 and DMS-0838680. of the free Gibbs state in our non-tracial setting, which allows us to proceed along very similar lines toGuionnet and Shlyakhtenko. However, we are forced to consider slightly more general potentials as well asnon-tracial states. Consequently, the modular automorphism group plays a significant role.In [6], Hiai developed a generalization of Shlyakhtenko’s algebras Γ( H R , U t ) from [7], called q -deformedAraki-Woods algebras. Letting A be the generator of the one-parameter family of unitary operators { U t } t ∈ R ,Hiai was able to show the von Neumann algebras Γ q ( H R , U t ) ′′ are factors and produced a type classification,but only in the case that A has infinitely many mutually orthogonal eigenvectors. In particular, whenthe Hilbert space H R is finite dimensional the questions of factoriality and type classification remainedunanswered. An application of our result in Section 4 yields Γ q ( H R , U t ) ′′ ∼ = Γ( H R , U t ) ′′ for small | q | , andhence we are able to settle these questions using Theorem 6.1 in [7].We begin the paper by recalling the free Araki-Woods factors and q -deformed Araki-Woods algebras andthen considering several derivations defined on these von Neumann algebras. An introduction to some Banachnorms and additional differential operators follows. The definition of free transport is given, including theconditions for it to be monotone, and the Schwinger-Dyson equations and free Gibbs states are defined.As in [5], several equivalent forms of the Schwinger-Dyson equation for a potential V are produced. Usingestimates relevant to these equivalent forms of the Schwinger-Dyson equation, the existence of a solution isproduced through a fixed point argument. The implications of this solution are discussed as well as how toimprove it so that an isomorphism exists. We conclude the paper with an application of the isomorphismresult to q -deformed Araki-Woods algebras. Acknowledgments.
I would like to thank my advisor, Prof. D. Shlyakhtenko for the initial idea of thepaper, many helpful suggestions and discussions, and his general guidance.2.
Preliminaries
The free Araki-Woods factor and q -deformed Araki-Woods algebras. Let H R be a real Hilbertspace and U t a strongly continuous one-parameter group of orthogonal transformations on H R . Letting H C := H R + i H R be the complexified Hilbert space, the U t can be extended to a one-parameter unitarygroup (still denoted as U t ). Let A be the generator of the U t (i.e. U t = A it and A is a potentially unboundedpositive operator). Let h· , ·i be the inner product on H C which is complex-linear in the second coordinate(as all other inner products will be in this section). Define an inner product h· , ·i U on H C by h x, y i U = (cid:28)
21 + A − x, y (cid:29) , x, y ∈ H C . Let H be the complex Hilbert space obtained by completing H C with respect to h· , ·i U . Note that if we startwith the trivial one-parameter group U t = 1 for all t then A = 1, h· , ·i U = h· , ·i and H = H C . In this case wewill write h· , ·i for h· , ·i U .For − < q <
1, the q -Fock space F q ( H ) is the completion of F finite ( H ) := L ∞ n =0 H ⊗ n , where H ⊗ = C Ωwith vacuum vector Ω, with respect to the sesquilinear form h· , ·i U,q given by h f ⊗ · · · ⊗ f n , g ⊗ · ⊗ g m i U,q = δ n = m X π ∈ S n q i ( π ) (cid:10) f , g π (1) (cid:11) U · · · (cid:10) f n , g π ( n ) (cid:11) U , where i ( π ) denotes the number of inversions of the permutation π ∈ S n . We may at times denote F q ( H R , U t ) = F q ( H ) to emphasize { U t } .For any h ∈ H we can define the left q -creation operator l ( h ) ∈ B ( F q ( H )) by l q ( h )Ω = h ; l q ( h )( f ⊗ · · · ⊗ f n ) = h ⊗ f ⊗ · · · ⊗ f n , then its adjoint is the left q -annihilation operator: l ∗ q ( h )Ω = 0; l ∗ q ( h )( f ⊗ · · · ⊗ f n ) = n X i =1 q i − h h, f i i U f ⊗ · · · ⊗ f i − ⊗ f i +1 ⊗ · · · ⊗ f n . Also define s q ( h ) := l q ( h ) + l ∗ q ( h ) . REE MONOTONE TRANSPORT WITHOUT A TRACE 3
We let Γ q ( H R , U t ) be the C ∗ -algebra generated by { s q ( h ) : h ∈ H R } . The corresponding von Neumannalgebra M q := Γ q ( H R , U t ) ′′ ⊂ B ( F q ( H )) is called a q -deformed Araki-Woods algebra , after [6], except when q = 0 where M = Γ ( H R , U t ) ′′ is called a free Araki-Woods factor , after [7].It was shown in [6] that Ω is a cyclic and separating vector for M q and consequently the vacuum state ϕ q ( · ) = h Ω , · Ω i U,q is faithful. For q = 0, ϕ q is called the q -quasi-free state , or the q -quasi-free state associatedto A . For q = 0, ϕ is called the free quasi-free state , or the free quasi-free state associated to A . Remark 2.1.
For f , . . . , f n ∈ H R , computing ϕ q ( s q ( f ) · · · s q ( f n )) is best done diagrammatically throughnon-crossing (when q = 0) and crossing (when q = 0) pairing diagrams. When q = 0, visualize a rectanglewith the vectors f , . . . , f n arranged in order along the top: f f · · · f n .ϕ ( s ( f ) · · · s ( f n )) counts all the ways to pair the vectors to each other via chords above the rectangle so thatno two chords intersect and if a vector f i is connected to a vector f j (with f i on the left) then that diagramis weighted by a factor of h f i , f j i U . For example the following diagram has the denoted weight: f f f f f f = h f , f i U h f , f i U h f , f i U . Thus ϕ ( s ( f ) s ( f ) s ( f ) s ( f )) = f f f f + f f f f = h f , f i U h f , f i U + h f , f i U h f , f i U . Note that ϕ then clearly takes a value of zero on all monomials of odd degree.When q = 0, the chords may intersect and do so at the cost of a factor of q for each intersection. Revisitingthe previous example in this case we then have ϕ q ( s q ( f ) s q ( f ) s q ( f ) s q ( f )) = f f f f + f f f f + f f f f = h f , f i U h f , f i U + q h f , f i U h f , f i U + h f , f i U h f , f i U . We note that in computing ϕ q ( s q ( f ) s q ( f ) s q ( f ) s q ( f )) = h Ω , s q ( f ) s q ( f ) s q ( f ) s q ( f )Ω i U,q by writing out s q ( f ) s q ( f ) s q ( f ) s q ( f )Ω, the term q h f , f i U h f , f i U comes from when s q ( f ) s q ( f ) acts on f ⊗ f andthe operator l ∗ q ( f ) “skips” over the the first vector in the tensor product (hence the factor of q ).It is a worthwhile exercise to restrict to the case when there is only a single operator s q ( f ) (so that allinner-products are 1) and draw out the diagrams corresponding to ϕ ( s q ( f ) n ) for n = 2 , , , M q is established in Lemma 1.4 of [6], which we recall here for convenience.Let S denote the closure of the map x Ω x ∗ Ω, and let S = J ∆ / be its polar decomposition so that J and ∆ are the modular conjugation and modular operator, respectively. Then for n ≥ S ( f ⊗ · · · ⊗ f n ) = f n ⊗ · · · ⊗ f n for f , . . . , f n ∈ H R ;∆( f ⊗ · · · ⊗ f n ) = ( A − f ) ⊗ · · · ⊗ ( A − f n ) for f , . . . , f n ∈ H R ∩ dom A − ; (1) J ( f ⊗ · · · ⊗ f n ) = ( A − / f n ) ⊗ · · · ⊗ ( A − / f n ) for f , . . . , f n ∈ H R ∩ dom A − / . Denote by σ ϕ q t ( · ) = ∆ it · ∆ − it the modular automorphism group of ϕ q .Henceforth we assume dim( H R ) = N < ∞ . Consequently A and A − are bounded operators and hence { σ ϕ q t } t ∈ R extends to { σ ϕ q z } z ∈ C . In particular for a, b ∈ M , ϕ ( ab ) = h a ∗ Ω , b Ω i U,q = h Sa Ω , b Ω i U,q = D Jb Ω , ∆ a Ω E U,q = D ∆∆ − Jb Ω , a Ω E U,q = h ∆ b ∗ Ω , a Ω i U,q = ϕ ( σ ϕ q i ( b ) a ) . Moreover, the action of ∆ in (1) extends to f , . . . , f n ∈ H . BRENT NELSON
From Remark 2.12 in [7] it follows that for a suitable orthonormal basis { e , . . . , e N } of ( H R , h· , ·i ), thegenerator A can be represented as a matrix of the form A = diag ( A , . . . , A L , , . . . , , (2)where for each k ∈ { , . . . , L } A k = 12 (cid:18) λ k + λ − k − i (cid:0) λ k − λ − k (cid:1) i (cid:0) λ k − λ − k (cid:1) λ k + λ − k (cid:19) ∈ M ( C ) , (3)and λ k >
0. Note that A itk = (cid:18) cos( t log λ k ) − sin( t log λ k )sin( t log λ k ) cos( t log λ k ) (cid:19) , which is a unitary matrix such that ( A itk ) ∗ = ( A itk ) T = A − itk . A has the following properties:1. spectrum( A ) = (cid:8) , λ ± , . . . , λ ± L (cid:9) ;2. A T = A − ;3. (cid:0) A it (cid:1) ∗ = (cid:0) A it (cid:1) T = A − it ; and4. for any fixed i ∈ { , . . . , N } , N X j =1 | [ A ] ij | ≤ max (cid:8) , λ ± , . . . , λ ± L (cid:9) ≤ k A k . For each j = 1 , . . . , N , let X ( q ) j = s q ( e j ) and write X ( q ) = ( X ( q )1 , . . . , X ( q ) N ). Since s q is real linear, it followsthat M q = W ∗ ( X ( q )1 , . . . , X ( q ) N ). We observe that σ ϕ q z ( X ( q ) j ) = N X k =1 [ A iz ] jk X ( q ) k , ∀ z ∈ C , or using the vector notation: σ ϕ q z ( X ( q ) ) = A iz X ( q ) , ∀ z ∈ C . (4)Indeed, using (1) it is easy to see that σ ϕ q z ( l q ( e j )) = l q ( A − iz e j ) σ ϕ q z ( l ∗ q ( e j )) = l ∗ q ( A − i ¯ z e j ) . Equation (4) follows from the above properties of A , the linearity of l q , and the conjugate linearity of l ∗ q .2.2. Derivations on M q . For the remainder of this section we will consider a single fixed q ∈ ( − , q ) notation on X ( q ) j , and write P for the ∗ -subalgebra C h X , . . . , X N i ⊂ M q of non-commutative polynomials in N -variables. We also simplify notation with M := M q , ϕ := ϕ q , and σ z := σ ϕ q z for z ∈ C .For each j ∈ { , . . . , N } we let δ j : P → P ⊗ P op be Voiculescu’s free-difference quotient: δ j ( X i · · · X i n ) = n X k =1 δ j = i k X i · · · X i k − ⊗ (cid:0) X i k +1 · · · X i n (cid:1) ◦ ;that is, δ j is the unique derivation satisfying δ j ( X i ) = δ j = i ⊗
1. We set the following conventions for workingwith elementary tensors in P ⊗ P op : • ( a ⊗ b ◦ ) c ⊗ d ◦ ) := ( ac ) ⊗ ( b ◦ d ◦ ) = ( ac ) ⊗ ( db ) ◦ ; • ( a ⊗ b ◦ ) c = acb ; • ( a ⊗ b ◦ ) ∗ := a ∗ ⊗ ( b ∗ ) ◦ ; • ( a ⊗ b ◦ ) † := b ∗ ⊗ ( a ∗ ) ◦ ; • ( a ⊗ b ◦ ) ⋄ := b ⊗ a ◦ ; • m ( a ⊗ b ◦ ) := ab .We also define the left and right actions of P as: • c · ( a ⊗ b ◦ ) := ( ca ) ⊗ b ◦ ; • ( a ⊗ b ◦ ) · c := a ⊗ ( bc ) ◦ . REE MONOTONE TRANSPORT WITHOUT A TRACE 5
Note that c · ( a ⊗ b ◦ ) = ( c ⊗ ◦ ) a ⊗ b ◦ ) , and( a ⊗ b ◦ ) · c = (1 ⊗ c ◦ ) a ⊗ b ◦ ) . We will usually suppress the notation “ ◦ ” and at times represent tensors of monomials in P diagrammaticallyas follows: X i · · · X i n ⊗ X j · · · X j m = i i · · · i n − i n j m j m − · · · j j . (5)Then multiplication is neatly expressed as: i · · · i n j m · · · j k · · · k p l q · · · l = i · · · i n k · · · k p j m · · · j l q · · · l . We note the involutions ∗ , † , ⋄ amount to horizontal reflection, vertical reflection, and 180 ◦ rotation of thediagrams, respectively.For j, k ∈ { , . . . , N } , we use the shorthand notation α jk := (cid:20)
21 + A (cid:21) jk = h e k , e j i U . Note that the last equality implies α jk = α kj , α jj = 1, and | α jk | ≤ j, k ∈ { , . . . , N } .Let Ξ q ∈ HS ( F q ( H )) be the Hilbert-Schmidt operator on F q ( H ) given by the sum P ∞ n =0 q n P n where P n : F q ( H ) → H ⊗ n is the projection onto vectors of length n . We identify the Hilbert space generated by theGNS construction with respect to ϕ ⊗ ϕ op with L ( M ¯ ⊗ M op , ϕ ⊗ ϕ op ) ∼ = HS ( F q ( H )) via a ⊗ b ◦
7→ h Ω , b ·i a Ω( cf. Proposition 5.11 in [10]); in particular, Ξ = P corresponds to 1 ⊗
1. Realize that the involution † defined above corresponds precisely with the adjoint operation in HS ( F q ( H )). Consequently, Ξ † q = Ξ q since,as a real sum of projections, it is a self-adjoint Hilbert-Schmidt operator.For each j = 1 , . . . , N we define the derivation ∂ ( q ) j : P → P ⊗ P op by ∂ ( q ) j ( P ) = N X k =1 α kj δ ( P ) q . That is, ∂ ( q ) j is the unique derivation satisfying ∂ ( q ) j ( X i ) = α ij Ξ q . We shall also consider the derivations¯ ∂ ( q ) j ( P ) := N X k =1 α jk δ k ( P ) q and ˜ ∂ ( q ) j := N X k =1 α jk ( δ k ( P ) q ) ⋄ , which are related to ∂ ( q ) j by ∂ ( q ) j ( P ) † = ¯ ∂ ( q ) j ( P ∗ ) and ∂ ( q ) j ( P ) ∗ = ˜ ∂ ( q ) j ( P ∗ ) . We remark that in the tracial case ( U t = 1 t ), we have ¯ ∂ ( q ) j = [ · , r q ( e j )], where r q ( e j ) is the right q -creationoperator. This is precisely the derivation considered in Lemma 27 of [3].From (4) we see that ∂ ( q ) j ( σ it ( X k )) = N X l =1 [ A − t ] kl α lj Ξ q = (cid:20) A − t A (cid:21) kj Ξ q , and thus ( σ − it ⊗ σ − it ) ◦ ∂ j ◦ σ it defines the unique derivation satisfying X k h A − t A i kj Ξ q . In particular,since (cid:20) A A (cid:21) kj = (cid:20)
21 + A − (cid:21) kj = (cid:20)
21 + A (cid:21) jk , we see that ( σ i ⊗ σ i ) ◦ ∂ ( q ) j ◦ σ − i = ¯ ∂ ( q ) j . (6)The motivation for considering such derivations is precisely the following proposition. BRENT NELSON
Proposition 2.2.
View ∂ ( q ) j and ¯ ∂ ( q ) j as densely defined operators from L ( P , ϕ ) to L ( P ⊗ P op , ϕ ⊗ ϕ op ) .Then ⊗ ∈ dom ∂ ( q ) ∗ j with ∂ ( q ) ∗ j (1 ⊗
1) = X j . (7) Moreover, ⊗ ∈ dom ¯ ∂ ( q ) ∗ j with ¯ ∂ ( q ) ∗ j (1 ⊗
1) = σ − i ( X j ) . (8) Remark 2.3.
The above proposition states that X , . . . , X N (resp. σ − i ( X ) , . . . , σ − i ( X N )) are conjugatevariables to X with respect to the derivations ∂ ( q )1 , . . . , ∂ ( q ) N (resp. ¯ ∂ ( q )1 , . . . , ¯ ∂ ( q ) N ) ( cf. Section 3 of [10]).
Proof.
Consider the monomial P = X i · · · X i n ∈ P . Then, ϕ ( X j P ) = h X j Ω , P Ω i U,q = h P X j Ω , P Ω i U,q = h P X j Ω , P P Ω i U,q , where P ∈ B ( F q ( H )) is the projection onto tensors of length one. As P is a product of the X i k , it is clearthat P P Ω will be a linear combination of e i , . . . , e i n , say P P Ω = P nk =1 c k e i k . We claim that c k = ∞ X l =0 q l (cid:10) P l X i k − · · · X i Ω , P l X i k +1 · · · X i n Ω (cid:11) U,q . Indeed, diagrammatically each term contributing to c k is a pairing of the vectors e i , . . . , e i n with e i k excluded.We can arrange such pairings according to the number of pairs whose connecting chords cross over e i k . Fix l ≥ l chords passing over e i k . Write P = A k X i k B k , then P l B k Ω gives pairingswithin B k that leave l vectors unpaired. Hence h Ω , A k P l B k Ω i counts the pairings in which there are exactly l pairs with one vector coming from A k and one coming from B k . Since the cost of skipping over e i k l timesis q l we see that c k = ∞ X l =0 q l h Ω , A k P l B k Ω i U,q = ∞ X l =0 q l h P l A ∗ k Ω , P l B k Ω i U,q , as claimed. Thus h P X j Ω , P P Ω i U,q = n X k =1 h P X j Ω , e i k i U,q c k = n X k =1 h e j , e i k i U ∞ X l =0 q l h P l A ∗ k Ω , P l B k Ω i U,q . Now, we inductively orthonormalize the monomials X i ∈ P with respect to h· Ω , · Ω i U,q to obtain a basis { r j } | j |≥ so that for each l , span { r j : | j | = l } = span { X i : | i | = l } . Then P l B k = P | j | = l D r j Ω , B Ω E U,q r j andusing our identification with L ( M ¯ ⊗ M op , ϕ ⊗ ϕ op ) we see that P l = P | j | = l r j ⊗ r ∗ j . Thus we have ϕ ( X j P ) = n X k =1 h e j , e i k i U ∞ X l =0 q l h P l A ∗ k Ω , P l B k Ω i U,q = n X k =1 h e j , e i k i U ∞ X l =0 q l X | j | = l D A ∗ k Ω , r j Ω E U,q D r j Ω , B k Ω E U,q = n X k =1 h e j , e i k i U ∞ X l =0 q l X | j | = l ϕ ⊗ ϕ op (cid:16) A k ⊗ B k r j ⊗ r ∗ j (cid:17) = n X k =1 h e j , e i k i U ϕ ⊗ ϕ op ( A k ⊗ B k q )= ϕ ⊗ ϕ op (cid:16) ∂ ( q ) j P (cid:17) , or h X j , P i ϕ = D ⊗ , ∂ ( q ) j P E ϕ ⊗ ϕ op , which implies ∂ ( q ) ∗ j (1 ⊗
1) = X j .Now, h σ − i ( X j ) , P i ϕ = ϕ ( σ i ( X j ) P ) = ϕ ( P X j ) = D P ∗ , ∂ ( q ) ∗ j (1 ⊗ E ϕ = D ¯ ∂ ( q ) j ( P ) † , ⊗ E ϕ = ϕ ⊗ ϕ op ( ¯ ∂ ( q ) j ( P ) ⋄ ) = ϕ ⊗ ϕ op ( ¯ ∂ ( q ) j ( P )) = D ⊗ , ¯ ∂ ( q ) j ( P ) E ϕ , so that 1 ⊗ ∈ dom (cid:16) ¯ ∂ ( q ) j (cid:17) and ¯ ∂ ( q ) ∗ j (1 ⊗
1) = σ − i ( X j ). (cid:3) REE MONOTONE TRANSPORT WITHOUT A TRACE 7
Corollary 2.4.
Viewing ∂ ( q ) j : L ( P , ϕ ) → L ( P ⊗ P op , ϕ ⊗ ϕ op ) as a densely defined operator we have P ⊗ P op ⊂ dom ∂ ( q ) ∗ j . In particular, if η ∈ dom ∂ ( q ) ∗ j and P ∈ P then ∂ ( q ) ∗ j ( η · P ) = ∂ ( q ) ∗ j ( η ) σ − i ( P ) − ⊗ ϕ op (cid:16) η σ ϕi ◦ ¯ ∂ ( q ) j ( P ) ⋄ (cid:17) , and ∂ ( q ) ∗ j ( P · η ) = P ∂ ( q ) ∗ j ( η ) − ϕ ⊗ op (cid:16) ˆ σ i ( η ) ∂ ( q ) j ( P ) ⋄ (cid:17) , where ˆ σ z = σ z ⊗ σ ¯ z with z ∈ C . In particular, for P, Q ∈ P we have ∂ ( q ) ∗ j ( P ⊗ Q ) = [1 ⊗ σ − i ]( P ⊗ Q ) X j − m ◦ (1 ⊗ ϕ ⊗ σ − i ) ◦ (cid:16) ⊗ ¯ ∂ ( q ) j + ¯ ∂ ( q ) j ⊗ (cid:17) ( P ⊗ Q ) , (9) or equivalently (using Equation (6)) ∂ ( q ) ∗ j ( P ⊗ Q ) = [1 ⊗ σ − i ]( P ⊗ Q ) X j − m ◦ (1 ⊗ ϕ ⊗ ◦ (cid:16) ⊗ ∂ ( q ) j + ¯ ∂ ( q ) j ⊗ (cid:17) ◦ [1 ⊗ σ − i ]( P ⊗ Q ) . (10) Proof.
We make the following notational simplifications: h· , ·i ϕ = h· , ·i and h· , ·i ϕ ⊗ ϕ op = h· , ·i ⊗ . First notethat for A, B, C, D ∈ P we have ϕ ⊗ ϕ op ( A ⊗ B C ⊗ D )) = ϕ ⊗ ϕ op (( AC ) ⊗ ( DB )) = ϕ ( AC ) ϕ ( DB ) = ϕ ( σ i ( C ) A ) ϕ ( Bσ − i ( D ))= ϕ ⊗ ϕ op (( σ i ( C ) ⊗ σ − i ( D )) A ⊗ B ) = ϕ ⊗ ϕ op (ˆ σ i ( C ⊗ D ) A ⊗ B ) . Also observe thatˆ σ i (cid:0) ( a ⊗ b ) † (cid:1) = ˆ σ i ( b ∗ ⊗ a ∗ ) = σ i ( b ∗ ) ⊗ σ − i ( a ∗ ) = σ − i ( b ) ∗ ⊗ σ i ( a ) ∗ = ˆ σ i ( a ⊗ b ) † . Now, let Q ∈ P , then D η · P, ∂ ( q ) j ( Q ) E ⊗ = D η, ∂ ( a ) j ( Q ) · P ∗ E ⊗ = D η, ∂ ( q ) j ( QP ∗ ) − Q · ∂ ( q ) j ( P ∗ ) E ⊗ = ϕ (cid:16)h ∂ ( q ) ∗ j ( η ) i ∗ QP ∗ (cid:17) − ϕ ⊗ ϕ op (cid:16) ˆ σ i (cid:16) ∂ ( q ) j ( P ∗ ) (cid:17) η ∗ Q ⊗ (cid:17) = D ∂ ( q ) ∗ j ( η ) σ − i ( P ) , Q E − ϕ ⊗ ϕ op (cid:18) ˆ σ i (cid:16) ¯ ∂ ( q ) j ( P ) (cid:17) † η ∗ Q ⊗ (cid:19) = D ∂ ∗ j ( η ) σ − i ( P ) − ⊗ ϕ op (cid:16) η σ i ◦ ¯ ∂ ( q ) j ( P ) ⋄ (cid:17) , Q E . Similarly, D P · η, ∂ ( q ) j ( Q ) E ⊗ = D η, P ∗ · ∂ ( q ) j ( Q ) E ⊗ = D η, ∂ ( q ) j ( P ∗ Q ) − ∂ ( q ) j ( P ∗ ) · Q E ⊗ = D ∂ ( q ) ∗ j ( η ) , P ∗ Q E − ϕ ⊗ ϕ op (cid:16) ˆ σ i ◦ ∂ ( q ) j ( P ∗ ) η ∗ ⊗ Q (cid:17) = D P ∂ ( q ) ∗ j ( η ) , Q E − D σ − i (cid:16) ϕ ⊗ op (cid:16) η σ i ◦ ¯ ∂ ( q ) j ( P ) ⋄ (cid:17)(cid:17) , Q E = D P ∂ ( q ) ∗ j ( η ) − [ ϕ ⊗ op ] ◦ ˆ σ i (cid:16) η σ i ◦ ¯ ∂ ( q ) j ( P ) ⋄ (cid:17) , Q E = D P ∂ ( q ) ∗ j ( η ) − ϕ ⊗ op (cid:16) ˆ σ i ( η ) ∂ ( q ) j ( P ) ⋄ (cid:17) , Q E . Applying both of these formulas and (7) yields ∂ ( q ) ∗ j ( P ⊗ Q ) = P X j σ − i ( Q ) − m (cid:16) ⊗ h σ − i ( ϕ ⊗ op ) ¯ ∂ ( q ) j i + h (1 ⊗ ϕ op ) ¯ ∂ ( q ) j i ⊗ σ − i (cid:17) ( P ⊗ Q )= [1 ⊗ σ − i ]( P ⊗ Q ) X j − m ◦ (1 ⊗ ϕ ⊗ σ − i ) ◦ (cid:16) ⊗ ¯ ∂ ( q ) j + ¯ ∂ ( q ) j ⊗ (cid:17) ( P ⊗ Q ) . The equivalent form follows easily from Equation (6). (cid:3)
For each j we also define the σ -difference quotient ∂ j : P → P ⊗ P op as ∂ j = N X k =1 α kj δ k , which is the unique derivation satisfying ∂ j ( X k ) = α kj ⊗
1. We see that ∂ ( q ) j ( P ) = ∂ j ( P ) q . BRENT NELSON
For q = 0, we have ∂ j = ∂ (0) j since Ξ = 1 ⊗
1, but otherwise ∂ j = ∂ ( q ) j . We also consider¯ ∂ j ( P ) := N X k =1 α jk δ k ( P ) and ˜ ∂ j ( P ) := N X k =1 α jk δ k ( P ) ⋄ , which are related to ∂ j ( P ) in the expected way. Furthermore, we see that( σ i ⊗ σ i ) ◦ ∂ j ◦ σ − i = ¯ ∂ j , (11)by the same argument that produced (6).These latter derivations do not depend on q and in fact could have been defined on C h t , . . . , t N i wherethe t j are some abstract indeterminates. This “universality” means that they are suitable for stating aSchwinger-Dyson equation ( cf. Subsection 2.10), which is a non-commutative differential equation satisfiedby a unique state under certain restrictions. This uniqueness is precisely what will allow us to to establishthe state-preserving isomorphism M q ∼ = M , for small | q | .2.3. The Banach algebra P ( R,σ ) and norm k · k R,σ . We use the convention that an underline connotesa multi-index: j = ( j , . . . , j n ) ∈ N n for some n . Then | j | gives the length of the multi-index. We write j · k to mean the concatenation of multi-indices j and k : ( j , . . . , j n , k , . . . , k m ). We also allow concatenation ofmulti-indices with single indices: j · l = ( j , . . . , j n , l ). Monomials of the form X j · · · X j n may be denotedby X j when j = ( j , . . . , j n ). Hence an arbitrary P ∈ P may be written as P = deg P X n =0 X | j | = n c ( j ) X j , with c ( j ) ∈ C . Denote the reversed multi-index by j − = ( j n , . . . , j ), then X ∗ j = X j − . For each n ≥
0, welet π n : P → P be the projection onto monomials of degree n : π n ( P ) = X | j | = n c ( j ) X j . For
R > k · k R defined in [5]: k P k R = deg P X n =0 X | j | = n | c ( j ) | R n . Denote the centralizer of ϕ in P by P ϕ = P ∩ M ϕ , where M ϕ = { a ∈ M : σ i ( a ) = a } . Observe that as σ i does not change the degree of a monomial (i.e. [ σ i , π n ] = 0 for each n ), P ∈ P ϕ iff π n ( P ) ∈ P ϕ for every n ≥ ρ ( X j · · · X j n ) = σ − i ( X j n ) X j · · · X j n − , then by letting ρ ( c ) = c for c ∈ C we can extend this to a linear map ρ : P → P . We refer to ρ k ( P ) as a σ -cyclic rearrangement of P . We note that ρ − ( X j · · · X j n ) = X j · · · X j n σ i ( X j ) . We define k P k R,σ := deg P X n =0 sup k n ∈ Z k ρ k n ( π n ( P )) k R ∈ [0 , ∞ ] . Then from the norm properties of k · k R and the subadditivity of the supremum it is easy to see that for P, Q ∈ P and c ∈ C k cP k R,σ = | c |k P k R,σ k P + Q k R,σ ≤ k P k R,σ + k Q k R,σ k P k R,σ = 0 = ⇒ P = 0. REE MONOTONE TRANSPORT WITHOUT A TRACE 9
Hence, k · k
R,σ restricted to the set { P ∈ P : k P k R,σ < ∞} =: P finite is a norm.Observe that ρ k ( σ − im ( X j · · · X j n )) = ρ k + mn ( X j · · · X j n ), so k · k R,σ is invariant under σ im , m ∈ Z .Consequently, P ϕ ⊂ P finite . Indeed, if P ∈ P ϕ then π n ( P ) ∈ P ϕ for all n . Hence ρ k n ( π n ( P )) = ρ l n ( π n ( P )) where k n ≡ l n (mod n ) and l n ∈ { , . . . , n − } . Consequently k P k R,σ = deg P X n =0 max l n ∈{ ,...,n − } k ρ l n ( π n ( P )) k R < ∞ . In fact, since k · k R is a Banach norm and k σ − i ( X j ) k R = N X k =1 | [ A ] jk | R ≤ k A k R, it is easy to see that k ρ l n ( π n ( P )) k R ≤ k A k n − k π n ( P ) k R for n ≥ l n ∈ { , . . . , n − } . Thus wehave the bound k P k R,σ ≤ k A k deg P − k P k R , for P ∈ P ϕ . We let P ( R ) and P ( R,σ ) denote the closures of P and P finite with respect to the norms k · k R and k · k R,σ , respectively. Both can be thought of as non-commutative power series: the former whose radii ofconvergence are at least R and the latter whose radii of convergence for each σ -cyclic rearrangement are atleast R . Note that π n can be extended to both P ( R ) and P ( R,σ ) with k P k R = ∞ X n =0 k π n ( P ) k R and k P k R,σ = ∞ X n =0 k π n ( P ) k R,σ . We claim that P ( R,σ ) is a Banach algebra. It suffices to show k P Q k R,σ ≤ k P k R,σ k Q k R,σ . Initiallywe consider the case P = P | i | = m a ( i ) X i and Q = P | j | = n b ( j ) X j for m, n ≥
0. Fix k ∈ Z and write k = r ( m + n ) + l . We treat the case 0 ≤ l ≤ n , the case n < l < n + m being similar. We also introduce thefollowing notation for | i | = | j | = n and t ∈ R : A t ( i, j ) = n Y u =1 (cid:2) A t (cid:3) i u j u . Now, ρ k ( P Q ) = X | i | = m | j | = n − lk | = l a ( i ) b (cid:0) j · k (cid:1) σ − i ( r +1) ( X k ) σ − ir ( X i X j )= X | i | = m | j | = n − l | k | = l X | ˆ i | = m | ˆ j | = n − l | ˆ k | = l a ( i ) b (cid:0) j · k (cid:1) A r +1 (cid:16) k, ˆ k (cid:17) A r (cid:16) i, ˆ i (cid:17) A r (cid:16) j, ˆ j (cid:17) X ˆ k X ˆ i X ˆ j , hence (cid:13)(cid:13) ρ k ( P Q ) (cid:13)(cid:13) R = X | ˆ i | = m | ˆ j | = n − l | ˆ k | = l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | i | = m | j | = n − l | k | = l a ( i ) b (cid:0) j · k (cid:1) A r +1 (cid:16) k, ˆ k (cid:17) A r (cid:16) i, ˆ i (cid:17) A r (cid:16) j, ˆ j (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R n + m = X | ˆ i | = m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | i | = m a ( i ) A r (cid:16) i, ˆ i (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R m X | ˆ j | = n − l | ˆ k | = l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | j | = n − l | k | = l b (cid:0) j · k (cid:1) A r +1 (cid:16) k, ˆ k (cid:17) A r (cid:16) j, ˆ j (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R n = k ρ rm ( P ) k R k ρ rn + l ( Q ) k R ≤ k P k R,σ k Q k R,σ . Thus k P Q k R,σ ≤ k P k R,σ k Q k R,σ . Now let
P, Q ∈ P ( R,σ ) be arbitrary. Then k P Q k R,σ ≤ ∞ X m,n =0 k π m ( P ) π n ( Q ) k R,σ ≤ ∞ X m,n =0 k π m ( P ) k R,σ k π n ( Q ) k R,σ = ∞ X m =0 k π m ( P ) k R,σ ! ∞ X n =0 k π n ( Q ) k R,σ ! = k P k R,σ k Q k R,σ . Hence P ( R,σ ) is a Banach algebra.Since k·k R is dominated by k·k R,σ , we can embed P ( R,σ ) into P ( R ) . Furthermore, if R ≥ k X k , . . . , k X N k then k · k R dominates the operator norm and hence we can embed P ( R ) into M . From Lemma 4 in [2] wesee that k X j k ≤ −| q | for all j = 1 , . . . , N , so we restrict ourselves to R ≥ −| q | from now on and consider P ( R,σ ) ⊂ P ( R ) ⊂ M as subalgebras. We let P ( R,σ ) ϕ and P ( R ) ϕ denote their respective intersections with M ϕ .We shall also use k · k R,σ to denote the norm on (cid:0) P ( R,σ ) (cid:1) N defined by k ( P , . . . , P N ) k R,σ = max {k P k R,σ , . . . , k P N k R,σ } . The operators N , Σ , S , Π , J σ , J and D . The maps N , Σ, and Π are defined as in [5], but werecall them here for convenience. N is defined on monomials by N ( X i ) = | i | X i , and is linearly extended to a map N : P ( R ) → P ( R ) . Π : P ( R ) → P ( R ) in terms of our present notation issimply 1 − π : it is the projection onto power series with zero constant term. Lastly, Σ is the inverse of N precomposed with Π: Σ( X i ) = 1 | i | X i , if | i | > S ( X i · · · X i n ) = 1 n n − X k =0 ρ k ( X i · · · X i n ) , and on constants as simply S ( c ) = c . For n ≥ P ∈ π n ( P ϕ ), ρ ( S ( P )) = 1 n n − X k =0 ρ k +1 ( P ) = 1 n σ − i ( P ) + n − X k =1 ρ k ( P ) ! = 1 n P + n − X k =1 ρ k ( P ) ! = S ( P ) . And of course ρ ( S ( c )) = c = S ( c ). Thus if we denote the set of σ -cyclically symmetric elements by P ( R,σ ) c.s. = { P ∈ P ( R,σ ) : ρ ( P ) = P } , then S (cid:16) P ( R,σ ) ϕ (cid:17) ⊂ P ( R,σ ) c.s. ⊂ P ( R,σ ) ϕ , with the last inclusion following from the fact that ρ n ( π n ( P )) = σ − i ( π n ( P )) and P ∈ P ( R,σ ) ϕ iff π n ( P ) ∈ P ϕ for each n . Moreover, S is a contraction on P ( R,σ ) ϕ with respect to the k·k R,σ . Indeed, since k Q k R,σ = k Q k R for Q ∈ P ( R,σ ) c.s. , for P ∈ P ( R,σ ) ϕ we have k S ( P ) k R,σ = k S ( P ) k R ≤ ∞ X n =0 n n − X k =0 k ρ k ( π n ( P )) k R ≤ ∞ X n =0 k π n ( P ) k R,σ = k P k R,σ . If f = ( f , . . . , f N ) with f j ∈ P then we write J f, J σ f ∈ M N ( P ⊗ P op ) for the matrices given by[ J f ] ij = δ j f i and [ J σ f ] ij = ∂ j f i . On elements Q ∈ M N ( P ⊗ P op ) we define the adjoint, transpose, and dagger involution as:[ Q ∗ ] ij = [ q ] ∗ ji , [ Q T ] ij = [ q ] ⋄ ji , [ Q † ] ij = [ q ] † ij . REE MONOTONE TRANSPORT WITHOUT A TRACE 11
Thus Q ∗ = ( Q † ) T = ( Q T ) † . Consequently, we define[ ¯ J σ f ] ij = ¯ ∂ j f i and [ ˜ J σ f ] ij = ˜ ∂ i f j , so that ( J σ f ) † = ¯ J σ ( f ∗ ) and ( J σ f ) ∗ = ˜ J σ ( f ∗ ).Recall X = ( X , . . . , X N ) and observe that [ J σ X ] ij = α ij ⊗
1. So after embedding M N ( C ) into M N ( P ⊗ P op ) in the obvious way, J σ X and A can be used interchangeably. Consequently it is clearthat J σ X is self-adjoint (with respect to the adjoint defined above) and invertible with inverse satisfying[ J σ X − ] ij = (cid:2) A (cid:3) ij ⊗ Q, Q ′ ∈ M N ( P ⊗ P op ) and left actions on f = ( f , . . . , f N ) , g =( g , . . . , g N ) ∈ P N by[ Q Q ′ ] i,j = N X k =1 [ Q ] ik Q ′ ] kj ∈ P ⊗ P op , for i, j ∈ { , . . . , N } ,Q g = N X j =1 [ Q ] ij g j Ni =1 ∈ P N , and f g = N X j =1 f j g j ∈ P . For Q ∈ M N ( P ⊗ P op ) we extend the notation of (5) by writing[ Q ] ij = k · · · k n l m · · · l i j when [ Q ] ij = X k · · · X k n ⊗ X l · · · X l m , i, j ∈ { , . . . , N } .Lastly, we define the j th σ -cyclic derivative D j : P → P by D j ( X k · · · X k n ) = n X l =1 α jk l σ − i ( X k l +1 · · · X k n ) X k · · · X k l − . D j can also be written as m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ j . Let D P = ( D P, . . . , D N P ) ∈ P N be the σ -cyclic gradient .We also define ¯ D j ( X k · · · X k n ) = n X l =1 α k l j X k l +1 · · · X k n σ i ( X k · · · X k l − ) , or ¯ D j = m ◦ ⋄ ◦ ( σ i ⊗ ◦ ∂ j . Then ( D j P ) ∗ = ¯ D j ( P ∗ ), and from (11) we also have D j ◦ σ i = ¯ D j .2.5. The norm k · k R ⊗ π R . Following [5], we denote by k · k R ⊗ π R the projective tensor product norm on P ⊗ P op ; that is, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i a i ⊗ b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R ⊗ π R = sup η (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) η X i a i ⊗ b i !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , where the supremum is taken over all maps η valued in a Banach algebra such that η ( a ⊗
1) and η (1 ⊗ b )commute and have norms bounded by k a k R and k b k R , respectively. In particular, letting η be given by leftand right multiplication on P we see that for D ∈ P ⊗ P op and g ∈ P , we have k D g k R ≤ k D k R ⊗ π R k g k R . We extend the norm to ( P ⊗ P op ) N by putting for F = ( F , . . . , F N ) ∈ ( P ⊗ P op ) N k F k R ⊗ π R = max ≤ i ≤ n k F i k R ⊗ π R . The same symbol is used to denote the norm imposed on M N ( P ⊗ P op ) by identifying it with the Banachspace of left multiplication operators on ( P ⊗ P op ) N . In [5] it is noted that this norm is given by k Q k R ⊗ π R = max ≤ i ≤ N N X j =1 k [ Q ] ij k R ⊗ π R . Cyclic derivatives of σ -cyclically symmetric polynomials. Suppose g ∈ π n (cid:16) P ( R,σ ) c.s. (cid:17) and write g = P | j | = n c ( j ) X j . Then the condition ρ l ( g ) = g for l ∈ { , . . . , n − } implies g = ρ l ( g ) = X | j | = n − l | k | = l c ( j · k ) σ − i ( X k ) X j = X | j | = n − l | k | = l c ( j · k ) X | i | = l A ( k, i ) X i X j = X | i | = l | j | = n − l X | k | = l c ( j · k ) A ( k, i ) X i X j . Hence c ( i · j ) = X | k | = l c ( j · k ) A ( k, i ) . (12)A similar computation using l ∈ {− n + 1 , . . . , − , } yields c ( i · j ) = X | k | = l c ( k · i ) A − ( k · j ) . (13)Since ρ n ( g ) = σ − i ( g ) for g ∈ π n ( P ), we can use Equation (12) to characterize the coefficients of g ∈ π n ( P ϕ ): c ( i ) = X | k | = n c ( k ) A ( k, i ) . (14)With these formulas in hand, the following lemmas are easily obtained. Lemma 2.5.
For P = P | i | = n c ( i ) X i ∈ π n ( P c.s. ) and each t ∈ { , . . . , N } we have D t Σ P = X | i | = n α ti n c ( i ) X i · · · X i n − . (15) Moreover, D Σ can be extended to a bounded operator D Σ : P ( R,σ ) c.s. → (cid:0) P ( R ) (cid:1) N with k D Σ k ≤ R . Addition-ally, for < S < R , D can be extended to a bounded operator D : P ( R,σ ) c.s. → (cid:0) P ( S ) (cid:1) N with k D k ≤ C (cid:0) RS (cid:1) depending only on the ratio RS .Proof. Let P = P | i | = n c ( i ) X i . Equation (15) follows easily from Equation (12), which then implies k D t Σ P k R = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | i | = n α ti n c ( i ) X i · · · X i n − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R ≤ X | i | = n | c ( i ) | R n − = 1 R k P k R = 1 R k P k R,σ . So for arbitrary P ∈ P c.s. we have k D Σ P k R = max t ∈{ ,...,N } k D t Σ P k R ≤ deg P X n =0 max t ∈{ ,...,N } k D t Σ π n ( P ) k R ≤ deg P X n =0 R k π n ( P ) k R,σ = 1 R k P k R,σ , and so D Σ extends to P ( R,σ ) c.s. with the claimed bound on its norm.Considering only D , (15) implies D t P = n X | i | = n α ti k c ( i ) X i · · · X i n − REE MONOTONE TRANSPORT WITHOUT A TRACE 13 for P ∈ π n ( P c.s. ). Hence k D P k S ≤ n X | i | = n | c ( i ) | S n − = n (cid:18) SR (cid:19) n − X | i | = n | c ( i ) | R n − = nS n − R n k P k R,σ . A routine computation shows for each nnS n − R n ≤ cS c − R c ≤ c (cid:18) SR (cid:19) c =: C (cid:18) RS (cid:19) , where c = R/S ) . The rest of the argument then proceeds as in the previous case. (cid:3) Lemma 2.6.
For P ∈ P ( R,σ ) ϕ DS Π P = D P. Proof.
Suppose P ∈ π n ( P ϕ ). The cases n = 0 , n ≥
2. Write P = P | i | = n c ( i ) X i , then S Π P = 1 n n − X l =0 X | j | = n − l | k | = l c ( j · k ) σ − i ( X k ) X j = 1 n n − X l =0 X | i | = l | j | = n − l X | k | = l c ( j · k ) A ( k, i ) X i X j = 1 n X | i | = n n − X l =0 X | k | = l c (( i l +1 , . . . , i n ) · k ) A ( k, ( i , . . . , i l )) X i =: 1 n X | i | = n b ( i ) X i . So if we let Q = P | i | = n b ( i ) X i , then S Π P = Σ Q and using Equation (15) we obtain D t S Π P = X | i | = n α ti n b ( i ) X i · · · X i n − , for each t ∈ { , . . . , N } . It is then a straightforward computation to show that the above equals D t P . Thecase for general P ∈ P ( R,σ ) ϕ then follows from linearity. (cid:3) Notation.
We use the same notation as in [5], adjusted slightly to accommodate our new operators.For Q ∈ M N ( P ⊗ P op ) we writeTr( Q ) = N X i =1 [ Q ] ii ∈ P ⊗ P op , Tr A ( Q ) = Tr( A Q ) = N X i,j =1 [ A ] ij [ Q ] ji ∈ P ⊗ P op , Tr A − ( Q ) = Tr( A − Q ) = N X i,j =1 [ A − ] ij [ Q ] ji ∈ P ⊗ P op . By Corollary 2.4 P ⊗ P op ⊂ dom ∂ ∗ j , so we note J ∗ σ ( Q ) = X i ∂ ∗ i ([ Q ] ji ) ! Nj =1 ∈ L ( P , ϕ ) N . where J σ is viewed as a densely defined operator from L ( P N , ϕ ) to L ( M N ( P ⊗ P op ) , ϕ ⊗ ϕ op ⊗ Tr) andthe above is its adjoint.2.8.
Transport and invertible power series.
Let ( M , ψ ) be a von Neumann algebra with state ψ andlet T , . . . , T N ∈ M be self-adjoint elements which generate M . Then, after [5], M can be thought ofas a completion of the algebra C h T , . . . , T N i , and ψ induces a linear functional ψ T on C h t , . . . , t N i , thenon-commutative polynomials in abstract indeterminates t , . . . , t N , via ψ T ( t k · · · t k n ) = ψ ( T k · · · T k n ), k , . . . , k n ∈ { , . . . , N } . ψ T is called the non-commutative law of T , . . . , T N and we write W ∗ ( ψ T ) ∼ = M .Let S , . . . , S N ∈ N be self-adjoint elements generating another von Neumann algebra N with state χ andlet χ S be their law so that W ∗ ( χ S ) ∼ = N . Definition 2.7. By transport from ψ T to χ S we mean an N -tuple of self-adjoint elements Y , . . . , Y N ∈ M having the same law as S , . . . , S N : χ ( P ( S , . . . , S N )) = ψ ( P ( Y , . . . , Y N )) , for all non-commutative polynomials P in N variables. If such an N -tuple exists then there is a state-preserving embedding N ∼ = W ∗ ( Y ) ⊂ M .Let M = W ∗ ( X , . . . , X N ) be as before. Suppose L is a von Neumann algebra generated by self-adjoint Z , . . . , Z N with state ψ and there exists transport Y = ( Y , . . . , Y N ) from ϕ X to ψ Z such that Y = G ( X ) ∈ ( P ( R ) ) N . That is, Y j = G j ( X ) is a power series in terms of X , . . . , X N . If we can invert this power seriesso that X = H ( Y ), then H ( Z ) ∈ L N is transport from ψ Z to ϕ X . It would then follow that we have astate-preserving isomorphism L ∼ = M . The following lemma, which is presented as Corollary 2.4 in [5], showsthat such inverses can be found. Lemma 2.8.
Let
R < S and consider the equation Y = X + f ( X ) with f ∈ ( P ( S ) ) N and k Y k R < S . Thenthere exists a constant C > , depending only on S and R so that whenever k f k S < C , then there exists H ∈ ( P ( R ) ) N so that X = H ( Y ) .Proof. Fix S ′ ∈ ( k Y k R , S ) and define C ( S ′ ) = k f k S max k ≥ k ( S ′ ) k − S − k . Since k f k S < C , we can choose C sufficiently small so that C ( S ′ ) < k Y k R + C − C ( S ′ ) ≤ S ′ . We define a sequence of N -tuples H k ( Y ) = Y − f ( H k − ( Y )) ∈ ( P ( R ) ) N , with H ( Y ) = Y . Denote the component functions of H k ( Y ) by [ H k ( Y )] j , j ∈ { , . . . , N } . We claimthat k H k ( Y ) k R ≤ S ′ for all k ≥
0. This clearly holds for H ( Y ), so assume it holds for 1 , . . . , k −
1. Fix j ∈ { , . . . , N } and suppose f j ( X , . . . , X N ) = X | i |≥ c ( i ) X i . Then for any 0 ≤ l ≤ k − k [ H l +1 ( Y )] j − [ H l ( Y )] j k R = k f j ( H l ( Y )) − f j ( H l − ( Y )) k R ≤ ∞ X n =0 X | i | = n | c ( i ) | n X u =1 k H l ( Y ) k u − R k [ H l ( Y )] i u − [ H l − ( Y )] i u k R k H l − ( Y ) k n − uR ≤ k H l ( Y ) − H l − ( Y ) k R ∞ X n =0 n ( S ′ ) n − S − n X | i | = n | c ( i ) | S n ≤ k H l ( Y ) − H l − ( Y ) k R C ( S ′ ) . As j was arbitrary, we obtain through iteration k H l +1 ( Y ) − H l ( Y ) k R ≤ k H ( Y ) − H ( Y ) k R C ( S ′ ) l = k f ( Y ) k R C ( S ′ ) l ≤ CC ( S ′ ) l , and thus k H k ( Y ) k R ≤ k H ( Y ) k R + k H k ( Y ) − H ( Y ) k R ≤ k Y k R + k − X l =0 k H l +1 ( Y ) − H l ( Y ) k R ≤ k Y k R + k − X l =0 CC ( S ′ ) l ≤ k Y k R + C − C ( S ′ ) ≤ S ′ , by our assumption on C . So the claim holds and by induction we have the bound k H k ( Y ) k R ≤ S ′ for all k ≥
0. Moreover, by a standard argument we can see that { H k ( Y ) } k ≥ is Cauchy and so converges to some REE MONOTONE TRANSPORT WITHOUT A TRACE 15 H ( Y ) ∈ ( P ( R ) ) N which satisfies H ( Y ) = Y − f ( H ( Y )) and k H ( Y ) k R ≤ S ′ . Since k X k R = R ≤ S ′ and Y = X + f ( X ) we obtain (via the same argument as above) k X − H ( Y ) k R = k Y − f ( X ) − Y + f ( H ( Y )) k R ≤ k X − H ( Y ) k R C ( S ′ ) . As C ( S ′ ) <
1, this implies H ( Y ) = X . (cid:3) Monotonicity of transport.
We introduce a definition for what it means for transport to be “mono-tone.’’ Note that in the tracial case ( A = 1) this coincides with Definition 2.1 in [5]. Definition 2.9.
We say that transport from ϕ X to ψ Z via the N -tuple Y = ( Y , . . . , Y N ) is monotone if Y = D G for some G ∈ P ( R ) , R ≥ p k A k , such that ( σ i ⊗ J σ D G ) ≥ L ( P ⊗ P op , ϕ ⊗ ϕ op ) N .Suppose ( M , ψ ) is a von Neumann algebra with a faithful normal state ψ . Let H ψ = L ( M , ψ, ξ ) be theHilbert space obtained via the GNS construction with a cyclic vector implementing ψ . Let S ψ be the Tomitaconjugation for the left Hilbert algebra M ξ , and let ∆ ψ and J ψ be the modular operator and conjugation(respectively). Recall ( cf. [9], Chapter IX, §
1) that there is a canonical pointed convex cone P ψ = { ∆ / ψ xξ : x ∈ M + } k·k ψ , which is self-dual in the sense that if η ∈ H ψ satisfies h η, ξ i ψ ≥ ξ ∈ P ψ then η ∈ P ψ . The embedding x ∆ ψ xξ of M into H ψ then has the benefit of sending positive elements in M into P ψ .In particular, if M = M N ( M ¯ ⊗ M op ) and ψ = ϕ ⊗ ϕ op ⊗ Tr A then∆ ψ qξ = ( σ − i ⊗ σ i )( A q A − ) ξ . We shall see in Lemma 3.1.(iv) that if G ∈ P ( R,σ ) ϕ then A s J σ D G A − s = ( σ − is ⊗ σ − is )( J σ D G ). Henceif Y = D G for such G , then ( σ i ⊗ J σ Y ) embeds into H ψ as( σ − i ⊗ σ i )( A σ i ⊗ J σ Y ) A − ) ξ = J σ Y ξ . The Schwinger-Dyson equation and free Gibbs state.
Our construction of the transport Y willexploit the condition that ϕ Y satisfies the so-called Schwinger-Dyson equation: Definition 2.10.
Given V ∈ P ( R,σ ) c.s. , we say a linear functional ϕ V on P satisfies the Schwinger-Dysonequation with potential V if ϕ V ( D ( V ) P ) = ϕ V ⊗ ϕ opV (Tr( J σ P )) , ∀ P ∈ P . (16)The law ϕ V is called the free Gibbs state with potential V .Note that when J σ is viewed as a densely defined operator from L ( P N , ϕ ) to L ( M N ( P ⊗ P op ) , ϕ ⊗ ϕ op ⊗ Tr), (16) is equivalent to J ∗ σ (1) = D V, (17)where 1 ∈ M N ( P ⊗ P op ) is the identity matrix.Consider the potential V = 12 N X j,k =1 (cid:20) A (cid:21) jk X k X j . (18)Then ρ ( V ) = 12 N X j,k =1 (cid:20) A (cid:21) jk σ − i ( X j ) X k = 12 N X j,k,l =1 (cid:20) A − (cid:21) kj [ A ] jl X l X k = 12 N X k,l =1 (cid:20) A (cid:21) kl X l X k = V , and hence V ∈ P ( R,σ ) c.s. . Also, D l ( V ) = 12 X i,j (cid:20) A (cid:21) ij ( α lj σ − i ( X i ) + α li X j )= 12 N X i,j,k =1 (cid:20)
21 + A (cid:21) lj (cid:20) A − (cid:21) ji [ A ] ik X k + 12 N X i,j =1 (cid:20)
21 + A (cid:21) li (cid:20) A (cid:21) ij X j = 12 X l + 12 X l = X l , so that D V = X . Using A = A ∗ it is also easy to see that V ∗ = V .Now, (17) for V = V states J ∗ σ (1) = X , or ∂ ∗ j (1 ⊗
1) = X j for each j = 1 , . . . , N , where the the adjointis with respect to ϕ V . However, from (7) we know this relation holds when the adjoint of ∂ j = ∂ (0) j is takenwith respect to the free quasi-free state ϕ . We therefore immediately obtain the following result. Theorem 2.11.
The free Gibbs state with potential V is the free quasi-free state ϕ on M = Γ( H R , U t ) ′′ . It is clear that the ϕ V is unique since (16) for V = V recursively defines ϕ V for all monomials. However,even for small perturbations (in the k · k R,σ -norm) V = V + W of V the free Gibbs state with potential V is unique, which we demonstrate below. Consequently, if ψ Z satisfies the Schwinger-Dyson equation for a V ,then to find transport from ϕ X it suffices to produce Y ∈ M N whose law ϕ Y (determined by ϕ ) satisfies theSchwinger-Dyson equation with the same potential V . The proof of uniqueness presented here differs fromthe proof of Theorem 2.1 in [4] only in the differential operators considered. Theorem 2.12.
Fix R ≥ p k A k . Let V = V + W ∈ P ( R,σ ) c.s. . Then for sufficiently small k W k R,σ , theSchwinger-Dyson equation has a unique solution amongst states that satisfy | ϕ ( X j ) | ≤ | j | (19) for any multi-index j .Proof. Suppose two states ϕ and ϕ ′ both solve the Schwinger-Dyson equation with potential V . Then ϕ (1) = ϕ ′ (1) = 1 and hence they agree on π ( P ). Fix l ≥ P ∈ π l − ( P ). Then we have( ϕ − ϕ ′ )( X i P ) = (( ϕ − ϕ ′ ) ⊗ ϕ )( ∂ i P ) + ( ϕ ′ ⊗ ( ϕ − ϕ ′ ))( ∂ i P ) − ( ϕ − ϕ ′ )( D i W P ) . (Note that for l = 1 the first two terms disappear). Define∆ l ( ϕ, ϕ ′ ) := max | j | = l | ( ϕ − ϕ ′ )( X j ) | . In particular ∆ ( ϕ, ϕ ′ ) = 0. Write D W = P i c ( j ) X j . Then we have∆ l ( ϕ, ϕ ′ ) ≤ l − X k =0 ∆ k ( ϕ, ϕ ′ )3 l − − k + ∞ X p =0 X | j | = p | c ( j ) | ∆ p + l − ( ϕ, ϕ ′ ) . For γ >
0, set d γ ( ϕ, ϕ ′ ) = ∞ X l =1 γ l ∆ l ( ϕ, ϕ ′ ) . Since (19) implies ∆ l ( ϕ, ϕ ′ ) ≤ l , we see that d γ ( ϕ, ϕ ′ ) < ∞ so long as γ < . In the above equality wemultiply both sides of the equation by γ l and then sum over l ≥ d γ ( ϕ, ϕ ′ ) ≤ ∞ X l =2 γ l l − X k =0 ∆ k ( ϕ, ϕ ′ )3 l − − k + ∞ X l =1 γ l ∞ X p =0 X | j | = p | c ( j ) | ∆ p + l − ( ϕ, ϕ ′ )= 2 γ ∞ X k =0 γ k ∆ k ( ϕ, ϕ ′ ) ∞ X l = k +2 γ l − − k l − − k + ∞ X p =0 X | j | = p | c ( j ) | γ − p +1 ∞ X l =1 γ p + l − ∆ p + l − ( ϕ, ϕ ′ ) ≤ γ − γ d γ ( ϕ, ϕ ′ ) + γ ∞ X p =0 X | j | = p | c ( j ) | γ − p d γ ( ϕ, ϕ ′ ) . REE MONOTONE TRANSPORT WITHOUT A TRACE 17
Let γ = R . Then γ − < R and R > γ < . Hence d γ ( ϕ, ϕ ′ ) ≤ d γ ( ϕ, ϕ ′ ) (cid:18) k D W k R (cid:19) . Recall from Lemma 2.5, that k D W k R ≤ C k W k R,σ where the constant only depends on the ratio R R/ = . Thus if k W k R,σ < C then d γ ( ϕ, ϕ ′ ) ≤ cd γ ( ϕ, ϕ ′ ) with c < , implying d γ ( ϕ, ϕ ′ ) = 0 and hence ∆ l ( ϕ, ϕ ′ ) = 0 for all l ≥ (cid:3) This theorem implies that if the law ψ Z of Z = ( Z , . . . , Z N ) ⊂ ( L, ψ ) and the law ϕ Y of Y =( Y , . . . , Y N ) ⊂ ( M, ϕ ) both solve the Schwinger-Dyson equation with potential V , then W ∗ ( Z , . . . , Z N ) ∼ = W ∗ ( Y , . . . , Y N ) ∼ = W ∗ ( ϕ V ). In particular, W ∗ ( ϕ V ) is well-defined.2.11. Outline of the paper.
The general outline for the paper is as follows: we begin in Section 3 by fixing q = 0 and a potential V = V + W ∈ P ( R,σ ) c.s. and assuming there exists Y = ( Y , . . . , Y N ) ∈ ( P ( R ) ) N whoselaw (induced by ϕ ) satisfies the Schwinger-Dyson equation with potential V . Several equivalent versionsof this equation will be derived in Sections 3.2 and 3.3 until we arrive at a final version for which a fixedpoint argument can be applied. Several technical estimates will be produced in Section 3.4 for the purposesof this fixed point argument so that in Section 3.5, given certain assumptions regarding V , we can assertthe existence of Y . Having obtained the desired transport, we then use Lemma 2.8 to refine the transportinto an isomorphism in Section 3.6. Finally, in Section 4 we present the main application to q-deformedAraki-Woods algebras.3. Construction of the Non-tracial monotone transport map
For all this section, we consider only q = 0 and maintain the same notational simplifications as above( M = M , ϕ = ϕ , X (0) j = X j , and σ ϕ z = σ z ). Recall that V is defined by (18) and that by Theorem 2.11, ϕ is the free Gibbs state with potential V . Our goal is to construct Y = ( Y , . . . , Y N ) ∈ ( P ( R ) ) N whose lawwith respect to ϕ is the free Gibbs state with potential V = V + W ∈ P ( R,σ ) c.s. , for k W k R,σ sufficiently small.We will need differential operators ∂ j , J σ , J , and D for Y as well as X , so we adopt the followingconvention: differential operators which have no indices or have a numeric index refer to differentiationwith respect to X , . . . , X N . Operators involving differentiation with respect to Y , . . . , Y N shall be labeled ∂ Y j , D Y j , etc. We define these latter operators using the comments at the end of Subsection 2.2; that is, ∂ Y j ( Y k · · · Y k n ) is computed exactly as one would compute ∂ j ( X k · · · X k n ) and exchanging X j ’s for Y j ’s inthe end.Assuming the law ϕ Y of Y = ( Y , . . . , Y N ) is the free Gibbs state with potential V and 1 ⊗ ∈ dom ∂ ∗ Y j ,(17) implies ∂ ∗ Y j (1 ⊗
1) = D Y j ( V ( Y ) + W ( Y )) = Y j + D Y j ( W ( Y )) , or, in short ( J σ ) ∗ Y (1) = Y + ( D W )( Y ) . (20)It will turn out that Y = X + f for some f = D g and g ∈ P ( R,σ ) c.s. , and so we start by considering theimplications of assuming Y is of this form.3.1. Change of variables formula.Lemma 3.1.
Assume Y is such that J σ Y = ( ∂ X j Y i ) ij ∈ M N ( M ¯ ⊗ M op ) is bounded and invertible. (i) Define ˆ ∂ j ( P ) = N X i =1 ∂ X i ( P ) (cid:2) J σ Y − J σ X (cid:3) ij , then ˆ ∂ j = ∂ Y j . (ii) ∂ ∗ Y j (1 ⊗
1) = P l ∂ ∗ X l ◦ ˆ σ − i (cid:16)(cid:2) J σ Y − J σ X (cid:3) ∗ lj (cid:17) . Hence ( J σ ) ∗ Y (1) = J ∗ σ (cid:16) ˆ σ − i (cid:16) J σ X (cid:0) J σ Y − (cid:1) ∗ (cid:17)(cid:17) , (21) where ∈ M N ( M ¯ ⊗ M op ) is the identity matrix. (iii) Assume in addition that Y j = D j G for some G ∈ P ( R,σ ) ϕ with G = G ∗ . Then ( J σ Y ) ∗ = ( σ i ⊗ J σ Y ) and (cid:0) J σ Y − (cid:1) ∗ = ( σ i ⊗ J σ Y − ) and hence Equation (21) becomes ( J σ ) ∗ Y (1) = J ∗ σ ◦ (1 ⊗ σ i ) (cid:0) J σ X J σ Y − (cid:1) . (22)(iv) For G ∈ P ( R,σ ) ϕ , ( σ − is ⊗ σ − is )( J σ D G ) = A s J σ D G A − s , ∀ s ∈ R . Proof.
Let Q = J σ Y .(i): We verify ˆ ∂ i Y k = N X i =1 ∂ X i Y k Q − J σ X ] ij = N X i =1 Q ki Q − J σ X ] ij = [ Q Q − J σ X ] kj = [ J σ X ] kj = ∂ X j X k = α kj ⊗ ∂ Y j Y k . (ii): We compute D ∂ ∗ Y j (1 ⊗ , X k · · · X k p E ϕ = (cid:10) ⊗ , ∂ Y j ( X k · · · X k p ) (cid:11) ϕ ⊗ ϕ op = N X l =1 (cid:10) ⊗ , ∂ X l ( X k · · · X k p ) Q − J σ X ] lj (cid:11) ϕ ⊗ ϕ op = N X l =1 (cid:10) ˆ σ − i (cid:0) [ Q − J σ X ] ∗ lj (cid:1) , ∂ X l ( X k · · · X k p ) (cid:11) ϕ ⊗ ϕ op = * N X l =1 ∂ ∗ X l ◦ ˆ σ − i (cid:0) [ Q − J σ X ] ∗ lj (cid:1) , X k · · · X k p + ϕ . Recalling that J σ X = J σ X ∗ , the definition of J ∗ σ implies (3.1).(iii): Suppose G = G ∗ ∈ P ( R,σ ) ϕ . Then[( J σ D G ) ∗ ] jk = [ J σ D G ] ∗ kj = ∂ j ◦ D k ( G ) ∗ = ˜ ∂ j ◦ ¯ D k ( G ∗ ) = ˜ ∂ j ◦ ¯ D k ( G ) . A computation on monomials shows that h ˜ ∂ j ◦ ¯ D k − ( σ i ⊗ ◦ ∂ k ◦ D j i ◦ σ it ( P ) = H t ( P ) − H t − ( P ) , where H t ( P ) = N X a,b =1 X P = AX b BX a C (cid:20) A t A − (cid:21) ka (cid:20) A t +1 A (cid:21) jb σ i ( t +1) ( B ) ⊗ σ it ( C ) σ i ( t +1) ( A ) . We claim that H t − ( P ) = H t ( σ − i ( P )). Indeed H t − ( P ) = N X a,b =1 X P = AX b BX a C N X p,q =1 (cid:20) A t A − (cid:21) kp [ A − ] pa (cid:20) A t +1 A (cid:21) jq [ A − ] qb × σ i ( t +1) ( σ − i ( B )) ⊗ σ it ( σ − i ( C )) σ i ( t +1) ( σ − i ( A ))= N X a,b,p,q =1 [ A − ] pa [ A − ] qb X P = σ i ( A ) X l σ i ( B ) X k σ i ( C ) (cid:20) A t A − (cid:21) kp (cid:20) A t +1 A (cid:21) jq × σ it ( t +1) ( B ) ⊗ σ it ( C ) σ i ( t +1) ( A ) . REE MONOTONE TRANSPORT WITHOUT A TRACE 19
Note that σ i ( X p ) = P Na =1 [ A − ] pa X a and σ i ( X q ) = P Nb =1 [ A − ] qb X b . So continuing the above com-putation we have H t − ( P ) = N X p,q =1 X P = σ i ( AX p BX q C ) (cid:20) A t A − (cid:21) kp (cid:20) A t +1 A (cid:21) jq σ it ( t +1) ( B ) ⊗ σ it ( C ) σ i ( t +1) ( A )= N X p,q =1 X σ − i ( P )= AX p BX q C (cid:20) A t A − (cid:21) kp (cid:20) A t +1 A (cid:21) jq σ it ( t +1) ( B ) ⊗ σ it ( C ) σ i ( t +1) ( A )= H t ( σ − i ( P )) . Thus from G = σ − i ( G ) we obtain h ˜ ∂ j ◦ ¯ D k − ( σ i ⊗ ◦ ∂ k ◦ D j i ◦ σ it ( G ) = 0 , and hence ( J σ D G ) ∗ = ( σ i ⊗ J σ D G ) . Now, if Y = D G for such G then J σ Y ∗ = ( σ i ⊗ J σ Y ) and J σ Y − satisfies this formula as wellbecause σ i ⊗ J σ X = ( σ it ⊗ σ is )( J σ X ) for all t, s ∈ R (since the entries of J σ X are merely scalarsmultiplied with 1 ⊗ σ − it ⊗ σ − it ) ◦ ∂ j ◦ σ it = N X k =1 [ A − t ] kj ∂ k . Also, ¯ ∂ j = ( σ i ⊗ σ i ) ◦ ∂ j ◦ σ − i , so that ( σ − it ⊗ σ − it ) ◦ ¯ ∂ j ◦ σ it = N X k =1 [ A − t ] kj ¯ ∂ k . Using these identities we have( σ − is ⊗ σ − is ) ◦ ∂ k ◦ D j = ( σ − is ⊗ σ − is ) ◦ ∂ k ◦ m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ j = N X a,b =1 [ A − s ] ak [ A − s ] bj ∂ a ◦ m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ◦ ¯ ∂ b ◦ σ − is = N X a,b, =1 [ A s ] jb [ A − s ] ak ∂ a ◦ D b ◦ σ − is . Hence for G ∈ P ( R,σ ) ϕ we have[( σ − is ⊗ σ − is )( J σ D G )] jk = ( σ − is ⊗ σ − is ) ◦ ∂ k ◦ D j ( G )= N X a,b, =1 [ A s ] jb [ A − s ] ak ∂ a ◦ D b ◦ σ − is ( G )= N X a,b, =1 [ A s ] jb [ A − s ] ak [ J σ D G ] ba = [ A s J σ D G A − s ] jk , for each j, k = 1 , . . . , N . (cid:3) Corollary 3.2.
Assume g = g ∗ ∈ P ( R,σ ) ϕ and put G = V + g and f j = D j g . Let Y j = X j + f j so that Y = D G . Define B = J σ f J σ X − and assume B is invertible. Then Equation (20) is equivalent tothe equation J ∗ σ ◦ (1 ⊗ σ i ) (cid:18)
11 + B (cid:19) = X + f + ( D W )( X + f ) . (23) Proof.
Since J σ X + J σ f = (1 + B ) J σ X , J σ Y = J σ X + J σ f is invertible as a consequence of 1 + B and J σ X both being invertible. Then upon noting that J σ X J σ X + J σ f ) − = 1 B ) − = 11 + B , the corollary follows immediately from Lemma 3.1, (ii) and (iii). (cid:3)
An equivalent form of Equation (23).Lemma 3.3.
Assume that the map ξ (1 + B ) ξ is invertible on (cid:0) P ( R ) (cid:1) N , and that f = D g for someself-adjoint g ∈ P ( R,σ ) ϕ . Let K ( f ) = − J ∗ σ ◦ (1 ⊗ σ i )( B ) − f. Then Equation (23) is equivalent to K ( f ) = D ( W ( X + f )) + (cid:20) B f + B J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) − J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19)(cid:21) . Proof.
Using x = 1 − x x and J ∗ σ ◦ (1 ⊗ σ i )(1) = J ∗ σ (1) = X , we see that Equation (23) is equivalent to0 = J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) + f + ( D W )( X + f ) . By the assumed invertibility of multiplying by (1 + B ), this is then equivalent to0 = J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) + f + ( D W )( X + f )+ B J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) + B f + B D W )( X + f ) . Using x x = x − x x , we then obtain K ( f ) =( D W )( X + f ) + B D W )( X + f )+ (cid:20) B f + B J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) − J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19)(cid:21) . Thus it remains to show D j ( W ( X + f )) = [(1 + B ) D W )( X + f )] j , for each j = 1 , . . . , N . Initially suppose W = X k · · · X k n (the general case will follow via linearity), then W ( X + f ) = ( X k + f k ) · · · ( X k n + f k n ) . For notational convenience, if we are focusing on the k l th factor then we will write W ( X + f ) = A l ( X k l + f k l ) B l . Using the derivation property of ¯ ∂ j in D j = m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ j we have D j ( W ( X + f )) = n X l =1 m ◦ ⋄ ◦ (1 ⊗ σ − i ) (cid:2) A l ( α jk l ⊗ ∂ j ( f k l )) B l (cid:3) = n X l =1 α jk l σ − i ( B l ) A l + (1 ⊗ σ − i ) ◦ ¯ ∂ j ( f k l ) ⋄ σ − i ( B l ) A l =( D W )( X + f ) + n X l =1 ∂ k l ( f j ) σ − i ( B l ) A l , REE MONOTONE TRANSPORT WITHOUT A TRACE 21 where we have used ( J σ f ) ∗ = ( J σ D g ) ∗ = ( σ i ⊗ J σ D g ) = ( σ i ⊗ J σ f ). Now[ B D W )( X + f )] j = N X k =1 [ B ] jk D k W )( X + f ) = N X k =1 N X l =1 [ J σ f ] jl J σ X − ] lk D k W )( X + f )= N X l =1 [ J σ f ] jl N X k =1 [ J σ X − ] lk N X p =1 α kp m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ δ p ( W )( X + f )= N X l =1 [ J σ f ] jl m ◦ ⋄ ◦ (1 ⊗ σ − ) ◦ δ l ( W )( X + f ) = n X l =1 [ J σ f ] jk l σ − i ( B l ) A l , which is precisely the second term in our above computation of D j ( W ( X + f )). (cid:3) Some identities involving J σ and D .Lemma 3.4. Let g ∈ P ( R,σ ) ϕ and let f = D g . Then for any m ≥ − we have: − J ∗ σ ◦ (1 ⊗ σ i )( B m +2 ) + B J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) = 1 m + 2 D [( ϕ ⊗ ◦ Tr A − + (1 ⊗ ϕ ) ◦ Tr A ] ( B m +2 ) Proof.
We prove the identity weakly. Let P ∈ ( P ( R ) ) N be a test function and denote φ = ϕ ⊗ ϕ op ⊗ Tr.Then h P , − J ∗ σ ◦ (1 ⊗ σ i )( B m +2 ) + B J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) (cid:11) ϕ = − (cid:10) J σ P, (1 ⊗ σ i )( B m +2 ) (cid:11) φ + ϕ N X i,j =1 P ∗ i B ij (cid:2) J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) (cid:3) j = − (cid:10) J σ P, (1 ⊗ σ i )( B m +2 ) (cid:11) φ + ϕ N X i,j =1 ( σ i ⊗ B ⋄ ij ) P ∗ i (cid:2) J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) (cid:3) j = − (cid:10) J σ P, (1 ⊗ σ i )( B m +2 ) (cid:11) φ + N X i,j =1 D (1 ⊗ σ − i )( B ∗ ij ) P i , (cid:2) J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) (cid:3) j E ϕ = − (cid:10) J σ P, (1 ⊗ σ i )( B m +2 ) (cid:11) φ + (cid:10) J σ { (1 ⊗ σ − i )( B ∗ ) P } , (1 ⊗ σ i )( B m +1 ) (cid:11) φ = − (cid:10) J σ P, (1 ⊗ σ i )( B m +2 ) (cid:11) φ + (cid:10) J σ X − J σ { ˆ σ i ( J σ f ) P } , (1 ⊗ σ i )( B m +1 ) (cid:11) φ , where we have used ( J σ f ) ∗ = ( σ i ⊗ J σ f ) from Lemma 3.1.(iii). Now we focus on the term J σ { ˆ σ i ( J σ f ) P } :[ J σ { ˆ σ i ( J σ f ) P } ] jk = N X l =1 ( ∂ k ⊗ ◦ ˆ σ i ◦ ∂ l ( f j ) P l + (1 ⊗ ∂ k ) ◦ ˆ σ i ◦ ∂ l ( f j ) P l + ˆ σ i ◦ ∂ l ( f j ) ∂ k ( P l ) , where a ⊗ b ⊗ c ξ = aξb ⊗ c and a ⊗ b ⊗ c ξ = a ⊗ bξc . Define Q Pjk = N X l =1 ( ∂ k ⊗ ◦ ˆ σ i ◦ ∂ l ( f j ) P l + (1 ⊗ ∂ k ) ◦ ˆ σ i ◦ ∂ l ( f j ) P l , so that J σ { ˆ σ i ( J σ f ) P } = Q P + ˆ σ i ( J σ f ) J σ P. Continuing our initial computation we obtain h P, − J ∗ σ ◦ (1 ⊗ σ i ) ( B m +2 ) + B J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) (cid:11) ϕ = − φ (cid:0) ( J σ P ) ∗ ⊗ σ i )( B m +1 ) (cid:1) + φ (cid:0) ( Q P ) ∗ J σ X − ⊗ σ i )( B m +1 ) (cid:1) + φ (cid:0) ( J σ P ) ∗ σ − i (( J σ f ) ∗ ) J σ X − ⊗ σ i )( B m +1 ) (cid:1) = (cid:10) Q P , J σ X − ⊗ σ i )( B m +1 ) (cid:11) φ . Hence h− J ∗ σ ◦ (1 ⊗ σ i )( B m +2 ) + B J ∗ σ ◦ (1 ⊗ σ i )( B m +1 ) , P (cid:11) ϕ = (cid:10) J σ X − ⊗ σ i )( B m +1 ) , Q P (cid:11) φ = φ ((1 ⊗ σ − i )(( Bj ) m +1 ) J σ X − Q P )= φ ((1 ⊗ σ − i )( J σ X − ( J σ f ) ∗ · · · J σ X − ( J σ f ) ∗ | {z } m +1 J σ X − ) Q P )= φ ((1 ⊗ σ − i )( J σ X − ( σ i ⊗ J σ f ) · · · J σ X − ( σ i ⊗ J σ f ) | {z } m +1 J σ X − ) Q P )= φ ( J σ X − ˆ σ i ( B m +1 ) Q P ) = φ ( Q P J σ X − B m +1 ) . We break from the present computation to consider the terms on the other side of the desired equality.For each u = 1 , . . . , m + 2 let R u be the matrix such that [ R u ] i u j u = a u ⊗ b u for some i u , j u ∈ { , . . . , N } and all other entries are zero. ThenTr A − ( R · · · R m +2 ) = Tr( A − R · · · R m +2 ) = [ A − ] j m +2 i m +2 Y u =1 δ j u = i u +1 a · · · a m +2 ⊗ b m +2 · · · b . Denote C = [ A − ] j m +2 i Q m +2 u =1 δ j u = i u +1 . Then X k ϕ (cid:0) ¯ D k ( ϕ ⊗ A − ( R · · · R m +2 ) P k (cid:1) = X k Cϕ ( a · · · a m +2 ) ϕ ( ¯ D k ( b m +2 · · · b ) P k )= X k,u Cϕ ( σ i ( a u · · · a m +2 ) a · · · a u − ) ϕ ( b u − · · · b σ i ( b m +2 · · · b u +1 ) · ˆ σ i ◦ ∂ k ( b u ) P k )= X u ϕ ⊗ ϕ op ⊗ Tr(∆ (1 ,P ) ( R u )( σ i ⊗ σ i )( R u +1 · · · R m +2 ) A − R · · · R u − ) , where for an arbitrary matrix O [∆ (1 ,P ) ( O )] ij = X k σ i ⊗ (ˆ σ i ◦ ∂ k )([ O ] ij ) P k . Using linearity, replace R u with B for each u . From Lemma 3.1.(iv) we know ( σ i ⊗ σ i )( J σ f ) A − = A − J σ f .As [ A, J σ X − ] = 0, we also have ( σ i ⊗ σ i )( B ) A − = A − B and hence X k ϕ (cid:0) ¯ D k ( ϕ ⊗ A − ( B m +2 ) P k (cid:1) = ( m + 2) φ (∆ (1 ,P ) ( B ) A − B m +1 ) . Observe that the left-hand side is (cid:10) D ( ϕ ⊗ A − ( B m +2 , P (cid:11) . Indeed, (cid:10) D ( ϕ ⊗ A − ( B m +2 ) , P (cid:11) = X k ϕ ( ¯ D k ( ϕ ⊗ A − (( B ∗ ) m +2 ) P k ) , and ( ϕ ⊗ A − ( B ∗ ) m +2 ) = ( ϕ ⊗ A − J σ X − ( σ i ⊗ J σ f ) · · · J σ X − ( σ i ⊗ J σ f ) | {z } m +2 )= ( ϕ ⊗ σ i ⊗ A − B m +2 ) = ( ϕ ⊗ A − B m +2 ) , where in the second to last equality we have used the fact that A − and J σ X − commute.So 1 m + 2 (cid:10) D ( ϕ ⊗ A − ( B m +2 ) , P (cid:11) = φ (∆ (1 ,P ) ( B ) A − B m +1 ) , and a similar computation yields1 m + 2 (cid:10) D (1 ⊗ ϕ )Tr A ( B m +2 ) , P (cid:11) = φ (∆ (2 ,P ) ( B ) AB m +1 ) , where for an arbitrary matrix O [∆ (2 ,P ) ( O )] ij = X k (ˆ σ i ◦ ∂ k ) ⊗ σ − i ([ O ] ij ) P k . REE MONOTONE TRANSPORT WITHOUT A TRACE 23
Thus it suffices to show ∆ (1 ,p ) ( B ) A − + ∆ (2 ,P ) ( B ) A = Q P J σ X − . This is easily verified entry-wise using the identities( δ r ⊗ ◦ ˆ σ i ◦ ∂ k = ( σ ⊗ (ˆ σ ◦ ∂ k )) ◦ N X b =1 [ A − ] br δ b ! , (1 ⊗ δ r ) ◦ ˆ σ i ◦ ∂ k = ((ˆ σ i ◦ ∂ k ) ⊗ σ − i ) ◦ N X b =1 [ A ] br δ b ! , and the definitions of Q P , ∆ (1 ,P ) , ∆ (2 ,P ) . (cid:3) Lemma 3.5.
Assume f = D g for g = g ∗ ∈ P ( R,σ ) ϕ and that k B k R ⊗ π R < . Let Q ( g ) = [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] ( B − log(1 + B )) . Then D Q ( g ) = B J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) − J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) . Proof.
Using the previous lemma this follows from comparing the convergent power series of each side. (cid:3)
Lemma 3.6.
Let K ( f ) = − J ∗ σ ◦ (1 ⊗ σ i )( B ) − f. Assume that f = D g for g = g ∗ ∈ P ( R,σ ) ϕ . Then K ( f ) = D { [( ϕ ⊗ ◦ Tr A − + (1 ⊗ ϕ ) ◦ Tr A ] ( B ) − N g } . Proof.
When m = −
1, the equality in Lemma 3.4 becomes D [( ϕ ⊗ ◦ Tr A − + (1 ⊗ ϕ ) ◦ Tr A ] ( B ) = − J ∗ σ ◦ (1 ⊗ σ i )( B ) + B J ∗ σ ◦ (1 ⊗ σ i )(1) . Since X = J ∗ σ (1) = J ∗ σ ◦ (1 ⊗ σ i )(1), the last term becomes B X = J f X = N f . Since DN g =( N + 1) D g = N f + f , we have D { [( ϕ ⊗ ◦ Tr A − + (1 ⊗ ϕ ) ◦ Tr A ] ( B ) − N g } = − J ∗ σ ◦ (1 ⊗ σ i )( B ) + N f − DN g = K ( f ) , as claimed. (cid:3) Lemma 3.7.
Assume f = D g for g = g ∗ ∈ P ( R,σ ) ϕ and k J D g k R ⊗ π R < . Let Q ( g ) be as before. ThenEquation (23) is equivalent to D { [( ϕ ⊗ ◦ Tr A − + (1 ⊗ ϕ ) ◦ Tr A ] ( J D g ) − N g } = D ( W ( X + D g )) + D Q ( g ) + J σ D g J σ X ) − D g. (24) Proof.
By Lemma 3.6, the left-hand side is K ( f ). Then using Lemmas 3.3 and 3.5 we have K ( f ) = D ( W ( X + f )) + B J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) − J ∗ σ ◦ (1 ⊗ σ i ) (cid:18) B B (cid:19) + B f = D ( W ( X + D g )) + D Q ( g ) + J σ D g J σ X ) − D g. Note that the hypothesis in Lemma 2.5 that the map ξ (1+ B ) ξ is invertible is satisfied since k B k R ⊗ π R = k J D g k R ⊗ π R < (cid:3) To prove the existence of a g satisfying the equation above we use a fixed point argument and thereforerequire some preliminary estimates. Technical estimates.
Recall that k X j k ≤ j = 1 , . . . , N . Since ϕ is a state it then followsthat | ϕ ( X i · · · X i n ) | ≤ n . (25) Lemma 3.8.
For g , . . . , g m ∈ P ϕ [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] ( J D g · · · J D g m ) ∈ P ϕ . Proof.
Recall A − t J D g A t = ( σ it ⊗ σ it )( J D g ) for g ∈ P ( R,σ ) ϕ by Lemma 3.1.(iv). Given this identity,for g , . . . , g m ∈ P ϕ we have(1 ⊗ ϕ ) ◦ Tr( A J D g · · · J D g m ) = (1 ⊗ ϕ ) ◦ Tr (( σ − i ⊗ σ − i )( J D g · · · J D g m ) A )= (1 ⊗ ϕ ) ◦ Tr ( A σ − i ⊗ σ − i )( J D g · · · J D g m ))= σ − i ◦ (1 ⊗ ϕ ) ◦ Tr( A J D g · · · J D g m ) , implying (1 ⊗ ϕ ) ◦ Tr A ( J D g · · · J D g m ) ∈ P ϕ . Similarly( ϕ ⊗ ◦ Tr( A − J D g · · · J D g m ) = σ i ◦ ( ϕ ⊗ ◦ Tr( A − J D g · · · J D g m ) , implying ( ϕ ⊗ ◦ Tr A − ( J D g · · · J D g m ) ∈ P ϕ . (cid:3) Using Equation (15) we see that for g ∈ π n (cid:16) P ( R,σ ) c.s. (cid:17) J D Σ g = N X j =1 n − X l =1 X | i | = n c ( i ) α ji n i · · · i l − i n − · · · i l +1 j i l . Lemma 3.9.
Let g , . . . , g m ∈ Π ( P c.s. ) . Set Q m ( g , . . . , g m ) = [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] ( J D g · · · J D g m ) . Assume R ≥ , so that R ≤ . Then k Q m (Σ g , . . . , Σ g m ) k R,σ ≤ k A k m +1 R m m Y u =1 k g u k R,σ . In particular, Q m extends to a bounded multilinear operator on P ( R,σ ) c.s. with values in P ( R,σ ) ϕ .Proof. First, for each u = 1 , . . . , m assume g u ∈ π n u ( P c.s. ) and write g u = P | i ( u ) | = n u c u ( i ( u ) ) X i ( u ) . By thecomputation preceding the statement of the lemma we can see thatTr A − ( J D Σ g · · · J D Σ g m ) = N X p ,...,p m =1 [ A − ] p m p [ J D Σ g ] p p · · · [ J D Σ g m ] p m − p m = N X j =1 n − X l =1 · · · n m − X l m =1 X | i (1) | = n · · · X | i ( m ) | = n m m Y u =1 c u ( i ( u ) ) α i ( u − lu − i ( u ) nu × i (1)1 · · · i (1) l − i (1) n − · · · i (1) l +1 i (0) l i (1) l · · · i ( m )1 · · · i ( m ) l m − i ( m ) n m − · · · i ( m ) l m +1 i ( m − l m − i ( m ) l m [ A − ] i ( m ) lm j ,where i (0) l = j . Hence( ϕ ⊗ ◦ Tr A − ( J D Σ g · · · J D Σ g m ) = N X j =1 X l ,...,l m X i (1) ,...,i ( m ) m Y u =1 c u ( i ( u ) ) α i ( u − lu − i ( u ) nu × ϕ ( X i (1)1 · · · X i (1) l − · · · X i ( m )1 · · · X i ( m ) lm − ) × X i ( m ) lm +1 · · · X i ( m ) nm − · · · X i (1) l · · · X i (1) n − · [ A − ] i ( m ) lm j . REE MONOTONE TRANSPORT WITHOUT A TRACE 25
Fix l , . . . , l m in the above quantity, then the sum over i (0) l and the multi-indices i (1) , . . . , i ( m ) is a sum ofmonomials all with the same degree: P u n u − l u − n . By Lemma 3.8, it suffices to bound k ρ k ( · ) k R for k ∈ {− n + 1 , . . . , − , } . For k = 0 we have k ( ϕ ⊗ ◦ Tr A − ( J D Σ g · · · J D Σ g m ) k R ≤ N X j =1 X l ,...,l m X i (1) ,...,i ( m ) m Y u =1 (cid:12)(cid:12)(cid:12) c u (cid:16) i ( u ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) [ A − ] i ( m ) m j (cid:12)(cid:12)(cid:12) R n − l − ··· + n m − l m − l − ··· + l m − ≤ X l ,...,l m X i (1) ,...,i ( m ) m Y u =1 (cid:12)(cid:12)(cid:12) c u (cid:16) i ( u ) (cid:17)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13) A − (cid:13)(cid:13) R n + ··· + n m − m (cid:18) R (cid:19) l + ··· + l m − m = k A k m Y u =1 R k g u k R n m − X l u =1 (cid:18) R (cid:19) l u − ≤ k A k m Y u =1 R k g u k R = k A k m Y u =1 R k g u k R,σ , where we have used k g u k R = k g u k R,σ .Next, let k ∈ {− n + 1 , . . . , − } and suppose ρ k (cid:16) X i ( m ) lm +1 · · · X i ( m ) nm − · · · X i (1) l · · · X i (1) n − (cid:19) = X i ( v ) a +1 · · · X i ( v ) nv − · · · X i (1) l · · · X i (1) n − σ i (cid:16) X i ( m ) lm +1 · · · X i ( m ) nm − · · · X i ( v ) lv +1 · · · X i ( v ) a (cid:17) , for some v ∈ { , . . . , m } and some a ∈ { l v + 1 , . . . , n v − } . The corresponding ϕ output is ϕ (cid:18) X i (1)1 · · · X i (1) l − · · · X i ( m )1 · · · X i ( m ) lm − (cid:17) = ϕ (cid:18) σ i (cid:16) X i ( v )1 · · · X i ( v ) lv − · · · X i ( m )1 · · · X i ( m ) lm − (cid:17) X i (1)1 · · · X i (1) l − · · · X i ( v − · · · X i ( v − lv − − (cid:19) . Using Lemma 3.8 we can in this case replace Tr( A − J D Σ g · · · J D Σ g m ) withTr( J D Σ g · · · J D Σ g v A − σ − i ⊗ σ − i )( J D Σ g v +1 · · · J D Σ g m ))so that output of ρ k changes to X i ( v ) a +1 · · · X i ( v ) nv − · · · X i (1) l · · · X i (1) n − X i ( m ) lm +1 · · · X i ( m ) nm − · · · σ i (cid:16) X i ( v ) lv +1 · · · X i ( v ) a (cid:17) , and the output of ϕ changes to ϕ (cid:18) σ i (cid:16) X i ( v )1 · · · X i ( v ) lv − (cid:17) · · · X i ( m )1 · · · X i ( m ) lm − X i (1)1 · · · X i (1) l − · · · X i ( v − · · · X i ( v − lv − − (cid:19) . Hence it suffices to consider when v = m . In this case we further fix i (1) , . . . , i ( m − and denote F u := X i ( u )1 · · · X i ( u ) lu − and G u := X i ( u ) lu +1 · · · X i ( u ) nu − . Consider N X j =1 X i ( m ) c m (cid:16) i ( m ) (cid:17) α i ( m − lm − i ( m ) nm [ A − ] i ( m ) lm j × ϕ (cid:16) σ i (cid:16) X i ( m )1 · · · X i ( m ) lm − (cid:17) F · · · F m − (cid:17) X i ( m ) a +1 · · · X i ( m ) nm − G · · · G m − σ i (cid:16) X i ( m ) lm +1 · · · X i ( m ) a (cid:17) = N X j =1 X i ( m ) N X ˆ i ( m )1 ,..., ˆ i ( m ) lm − =1 N X ˆ i ( m ) lm +1 ,..., ˆ i ( m ) a =1 c m (cid:16) i ( m ) (cid:17) α i ( m − lm − i ( m ) nm Y t = l m [ A − ] i ( m ) t ˆ i ( m ) t · [ A − ] i ( m ) lm j × ϕ (cid:16) X ˆ i ( m )1 · · · X ˆ i ( m ) lm − F · · · F m − (cid:17) X i ( m ) a +1 · · · X i ( m ) nm − G · · · G m − X ˆ i ( m ) lm +1 · · · X ˆ i ( m ) a = X i ( m ) X | ˆ i ( m ) | = a c m (cid:16) i ( m ) (cid:17) α i ( m − lm − i ( m ) nm a Y t =1 [ A − ] i ( m ) t ˆ i ( m ) t ϕ (cid:16) X ˆ i ( m )1 · · · X ˆ i ( m ) lm − F · · · F m − (cid:17) × X i ( m ) a +1 · · · X i ( m ) nm − G · · · G m − X ˆ i ( m ) lm +1 · · · X ˆ i ( m ) a = X j ( m ) c m (cid:16) j ( m ) (cid:17) α i ( m − lm − j ( m ) nm − a ϕ (cid:16) X j ( m ) nm − a +1 · · · X i ( m ) nm − a + lm − F · · · F m − (cid:17) × X j ( m )1 · · · X j ( m ) nm − a − G · · · G m − X i ( m ) nm − a + lm +1 · · · X j ( m ) nm , where in the final equality we have used the characterization of the coefficients of elements of P c.s. givenby (13). We note that while the multi-index has changed to j ( m ) , there are still l m − ϕ and n m − l m − (cid:13)(cid:13) ρ k ◦ ( ϕ ⊗ ◦ Tr A − ( J D Σ g · · · J D Σ g m ) k R ≤ X l ,...,l m X i (1) ,...,i ( m ) m Y u =1 (cid:12)(cid:12)(cid:12) c u (cid:16) i ( u ) (cid:17)(cid:12)(cid:12)(cid:12) R n − l − ··· + n m − l m − l − ··· + l m − = X l ,...,l m X i (1) ,...,i ( m ) m Y u =1 (cid:12)(cid:12)(cid:12) c u (cid:16) i ( u ) (cid:17)(cid:12)(cid:12)(cid:12) R n + ··· + n m − m (cid:18) R (cid:19) l − ··· + l m − = m Y u =1 R k g u k R n u − X l u =1 (cid:18) R (cid:19) l u − ≤ m Y u =1 R k g u k R,σ ≤ k A k m Y u =1 R k g u k R,σ . Thus k ( ϕ ⊗ ◦ Tr A − ( J D Σ g · · · J D Σ g m ) k R,σ ≤ k A k m R m m Y u =1 k g u k R,σ , and similar estimates show k ( ϕ ⊗ ◦ Tr A ( J D Σ g · · · J D Σ g m ) k R,σ ≤ k A k m R m m Y u =1 k g u k R,σ . Now let g , . . . , g m ∈ P c.s. be arbitrary. We note that π n u ( g u ) ∈ P c.s. for each n u ≥ ρ, π n u ] = 0.Then since Q m is multi-linear we have Q m (Σ g , . . . , Σ g m ) = ∞ X n ,...,n m =0 Q m (Σ π n ( g ) , . . . , Σ π n m ( g m )) , REE MONOTONE TRANSPORT WITHOUT A TRACE 27 and hence k Q m (Σ g , . . . , Σ g m ) k R,σ ≤ X n ,...,n m k A k m +1 R m m Y u =1 k π n u ( g u ) k R,σ = k A k m +1 R m m Y u =1 ∞ X n u =0 k π n u ( g u ) k R,σ = k A k m +1 R m m Y u =1 k g u k R,σ . Thus Q m extends to a bounded multilinear operator on P ( R,σ ) c.s. . That Q m takes values in P ( R,σ ) ϕ followsfrom Lemma 3.8. (cid:3) Lemma 3.10.
For f, g ∈ P ( R,σ ) c.s. set Q m (Σ g ) = Q m (Σ g, . . . , Σ g ) and assume R ≥ . Then k Q m (Σ g ) − Q m (Σ f ) k R,σ ≤ k A k m +1 R m m − X k =0 k g k kR,σ k f k m − k − R,σ k f − g k R,σ . In particular, k Q m (Σ g ) k R,σ ≤ k A k m +1 R m k g k mR,σ .Proof. Using a telescoping sum we have k Q m (Σ f ) − Q m (Σ g ) k R,σ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m − X k =0 Q m (Σ g, . . . , Σ g | {z } k , Σ f, . . . , Σ f | {z } m − k ) − Q m (Σ g, . . . , Σ g | {z } k +1 , Σ f, . . . , Σ f | {z } m − k − ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R,σ ≤ m − X k =0 k Q m (Σ g, . . . Σ g | {z } k , Σ f − Σ g, Σ f, . . . , Σ f | {z } m − k − ) k R,σ ≤ k A k m +1 R m m − X k =0 k g k kR,σ k f k m − k − R,σ k f − g k R,σ . (cid:3) Lemma 3.11.
Assume R ≥ . Let g ∈ P ( R,σ ) c.s. be such that k g k R,σ < R , and set Q (Σ g ) = X m ≥ ( − m m + 2 Q m +2 (Σ g ) . Then this series converges in k ·k
R,σ . Moreover, in the sense of analytic functional calculus on M N ( W ∗ ( P ⊗ P op , ϕ ⊗ ϕ op )) , we have the equality Q (Σ g ) = [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] { J D Σ g − log(1 + J D Σ g ) } . Furthermore, the function Q satisfies the local Lipschitz condition on n g ∈ P ( R,σ ) c.s. : k g k R,σ < R / o k Q (Σ g ) − Q (Σ f ) k R,σ ≤ k f − g k R,σ k A k R (cid:16) − k f k R,σ R (cid:17) (cid:16) − k g k R,σ R (cid:17) − , and the bound k Q (Σ g ) k R,σ ≤ k A kk g k R,σ R − R k g k R,σ . Proof.
Let κ = R / λ = k g k R,σ . From Lemma 3.10 we know k Q m +2 (Σ g ) k R,σ ≤ k A k (cid:0) λκ (cid:1) m +2 . Since λ < κ , the series defining Q converges. The functional calculus equality then follows from log(1 + x ) = − P m ≥ − x ) m m . Finally, since m + 2 ≥ k Q (Σ g ) − Q (Σ f ) k R,σ ≤ X m ≥ m + 2 k Q m +2 (Σ g ) − Q m +2 (Σ f ) k R,σ ≤ k f − g k R,σ k A k X m ≥ m +1 X k =0 κ − m − k f k m − k +1 R,σ k g k kR,σ ≤ k f − g k R,σ k A k κ X l ≥ X k ≥ κ − l k f k lR,σ κ − k k g k kR,σ − , where we have written m = l + k − l and k are not both zero. Using k f k R,σ , k g k R,σ < κ we see that k Q (Σ g ) − Q (Σ f ) k R,σ ≤ k f − g k R,σ k A k R (cid:16) − k f k R,σ R (cid:17) (cid:16) − k g k R,σ R (cid:17) − . Setting f = 0 yields the bound k Q (Σ g ) k R,σ ≤ k g k R,σ k A k R k g k R,σ R − k g k R,σ = 4 k A kk g k R,σ R − R k g k R,σ , as claimed. (cid:3) The proof of the following lemma is purely computational and left to the reader.
Lemma 3.12. If f = D g for g ∈ P ( R,σ ) ϕ then A − σ − i ( f ) = f. (26) Moreover, if g = g ∗ then D (cid:18) J σ X − f f (cid:19) = J σ f J σ X − f = J f f. (27) Lemma 3.13.
Suppose f ( i ) = D Σ g i with g i ∈ P c.s. for i = 1 , . Then (1 + A ) f (1) f (2) ∈ P ϕ . Further-more, (cid:13)(cid:13)(cid:13) (1 + A ) f (1) f (2) (cid:13)(cid:13)(cid:13) R,σ ≤ N k A k R k g k R,σ k g k R,σ . Proof.
From (26) it is easy to see that (1 + A ) f (1) f (2) ∈ P ϕ . Now, write g = P ∞ m =1 P | i | = m c ( i ) X i and g = P ∞ n =1 P | j | = n c ( j ) X j . Then (15) implies f (1) j = ∞ X m =1 X | i | = m − a ∈{ ,...,N } α ja c ( i · a ) X i and f (2) i = ∞ X n =1 X | j | = n − b ∈{ ,...,N } α ib c ( j · b ) X j . Hence (1 + A ) f (1) f (2) = ∞ X m,n =1 N X i,j =1 [1 + A ] ij N X a,b =1 X | i | = m − | j | = n − α ja α ib c ( i · a ) c ( j · b ) X i X j . REE MONOTONE TRANSPORT WITHOUT A TRACE 29
It suffices to bound k ρ k ( · ) k R for k ∈ {− m − n + 1 , . . . , } . First, for k = 0 we simply have k (1 + A ) f (1) f (2) k R ≤ N X i,j =1 | [1 + A ] ij | ∞ X m,n =1 X | i | = m − ,a | j | = n − ,b | c ( i · a ) c (cid:0) j · b (cid:1) | R m + n − ≤ N (1 + k A k ) 1 R ∞ X m =1 X i,a | c ( i · a ) | R m ∞ X n =1 X j,b | c ( j · b ) R n ≤ N k A k R k g k R,σ k g k R,σ . For − m + 1 ≤ k ≤ −
1, we further fix i, j, a, b . Then using (13) we have X | i | = m − | j | = n − c ( i · a ) c ( j · b ) ρ k (cid:16) X i X j (cid:17) = X | ˆ l | = k | i | = m − k − | j = n − c ( j · b ) c (cid:16) i · a · ˆ l (cid:17) X j X ˆ l . Thus ∞ X m,n =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X i,j =1 [1 + A ] ij N X a,b =1 X | i | = m − | j | = n − α ja α ib c ( i · a ) c ( j · b ) ρ k (cid:16) X i X j (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R ≤ ∞ X m,n =1 N X i,j =1 | [1 + A ] ij | X | i | = m | j | = n | c ( i ) c ( j ) | R n + m − ≤ N (1 + k A k ) 1 R ∞ X m =1 X i,a | c ( i · a ) | R m ∞ X n =1 X j,b | c ( j · b ) R n ≤ N k A k R k g k R,σ k g k R,σ . The cases for − m − n + 1 ≤ k ≤ − m are similar after using σ i ( g ) = g . Thus the claimed bound holds. (cid:3) Corollary 3.14.
Assume R ≥ . Let g ∈ P ( R,σ ) c.s. and assume that k g k R,σ < R / . Let S ≥ R + k g k R,σ andlet W ∈ P ( S ) c.s. . Let F ( g ) = − W ( X + D Σ g ) − { (1 + A ) D Σ g } D Σ g + [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] ◦ log(1 + J D Σ g )= − W ( X + D Σ g ) − { (1 + A ) D Σ g } D Σ g + [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] ( J D Σ g ) − Q (Σ g ) . Then F ( g ) is a well-defined function from P ( R,σ ) c.s. to P ( R,σ ) ϕ . Moreover, g F ( g ) is locally Lipschitz on { g : k g k R,σ < R / } : k F ( g ) − F ( f ) k R,σ ≤≤ k f − g k R,σ k A k R (cid:16) − k f k R,σ R (cid:17) (cid:16) − k g k R,σ R (cid:17) + 1 + N k g k R,σ + k f k R,σ ) + N X j =1 k δ j ( W ) k S ⊗ π S , and bounded: k F ( g ) k R,σ ≤ k g k R,σ k A k R − k g k R,σ + 2 k A k R + N k A k R k g k R,σ + N X j =1 k δ j ( W ) k S ⊗ π S + k W k R,σ . In particular, if R ≥ p k A k , < ρ ≤ k W k R,σ < ρ N P j k δ j ( W ) k ( R + ρ ) ⊗ π ( R + ρ ) < , (28) then F takes the ball E = n g ∈ P ( R,σ ) c.s. : k g k R,σ < ρN o to the ball E = n g ∈ P ( R,σ ) ϕ : k g k R,σ < ρN o and isuniformly contractive with constant λ ≤ on E .Proof. From Lemma 2.5 we know S ≥ R + k g k R,σ > R + k D Σ g k R , and thus W ( X + D Σ g ) is well-definedas an element of P ( R ) . We claim that in fact, W ( X + D Σ g ) ∈ P ( R,σ ) ϕ . Indeed, first note that from (26) wehave σ − i ( X + D Σ g ) = A X + D Σ g ). Hence σ − i ( W ( X + D Σ g )) = W ( σ − i ( X + D Σ g )) = W ( A X + D Σ g )) = σ − i ( W )( X + D Σ g ) = W ( X + D Σ g ) , where we have used (14). Then, using (12) it is not hard to see that k W ( X + D Σ g ) k R,σ = k W ( X + D Σ g ) k R < ∞ . That F ( g ) ∈ P ( R,σ ) ϕ follows from Lemmas 3.8 and 3.13 and W ( X + D Σ g ) ∈ P ( R,σ ) ϕ .Using Lemmas 3.10 and 3.11 we have k Q (Σ g ) − Q (Σ f ) k R,σ + k [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] ( J D Σ( g − f )) k R,σ ≤k f − g k R,σ k A k R (cid:16) − k f k R,σ R (cid:17) (cid:16) − k g k R,σ R (cid:17) − + k A k R k f − g k R,σ = k f − g k R,σ k A k R (cid:16) − k f k R,σ R (cid:17) (cid:16) − k g k R,σ R (cid:17) + 1 . The following estimate is essentially identical to the one produced in Corollary 3.12 in [5]: k W ( X + D Σ g ) − W ( X + D Σ f ) k R,σ = k W ( X + D Σ g ) − W ( X + D Σ f ) k R ≤ X j k δ j ( W ) k S ⊗ π S k D Σ g − D Σ f k R ≤ X j k δ j ( W ) k S ⊗ π S k f − g k R,σ . Finally, from Corollary 3.13, we know14 k { (1 + A ) D Σ g } D Σ g − { (1 + A ) D Σ f } D Σ f k R,σ ≤ k{ (1 + A ) D Σ( g − f ) } D Σ g k R,σ + 14 k{ (1 + A ) D Σ f } D Σ( g − f ) k R,σ ≤
14 2 N k A k R k f − g k Rσ k g k R,σ + 14 2 N k A k R k f k Rσ k f − g k R,σ = N k A k R k f − g k R,σ ( k g k R,σ + k f k R,σ ) . Combining the previous three estimates yields k F ( f ) − F ( g ) k R,σ ≤ k f − g k R,σ k A k R (cid:16) − k f k R,σ R (cid:17) (cid:16) − k g k R,σ R (cid:17) + 1 + N k g k R,σ + k f k R,σ + N X j =1 k δ j ( W ) k S ⊗ π S , as claimed. The estimate on k F ( g ) k R,σ then follows from the above and F (0) = − W ( X ).Now, suppose (28) holds and let f, g ∈ E . Note that R ≥ k f k R,σ , k g k R,σ < N ≤
1. Hence theLipschitz property implies k F ( f ) − F ( g ) k R,σ ≤ k f − g k R,σ (cid:26) (cid:18) (cid:19) + 18 (cid:27) = k f − g k R,σ (cid:26)
849 + 516 (cid:27) < k f − g k R,σ . REE MONOTONE TRANSPORT WITHOUT A TRACE 31
The bound on F then implies k F ( g ) k R,σ ≤ ρN (cid:26)
17 + 18 + 132 + 18 (cid:27) + ρ N < ρ N + ρ N = ρN , and so F maps E into E . (cid:3) Existence of g .Proposition 3.15. Assume that for some R ≥ p k A k and some < ρ ≤ , W ∈ P ( R + ρ,σ ) c.s. ⊂ P ( R,σ ) c.s. andthat (cid:26) k W k R,σ < ρ N P j k δ j ( W ) k ( R + ρ ) ⊗ π ( R + ρ ) < . (29) Then there exists ˆ g and g = Σˆ g with the following properties: (i) ˆ g, g ∈ P ( R,σ ) c.s. (ii) ˆ g satisfies the equation ˆ g = S Π F (ˆ g )(iii) g satisfies the equation N g = S Π (cid:20) − W ( X + D g ) − { (1 + A ) D g } D g + [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] ◦ log(1 + J D Σ g ) (cid:21) , (30) or, equivalently, S Π [(1 ⊗ ϕ ) ◦ Tr A + ( ϕ ⊗ ◦ Tr A − ] ( J D g ) − N g = S Π (cid:26) W ( X + D g ) + Q ( g ) + 14 { (1 + A ) D g } D g (cid:27) . (31)(iv) If W = W ∗ , then ˆ g = ˆ g ∗ and g = g ∗ . (v) ˆ g and g depend analytically on W , in the following sense: if the maps β W β are analytic, thenalso the maps β ˆ g ( β ) and β g ( β ) are analytic, and g → if k W k R,σ → .Proof. We remark that Equation (30) is equivalent to N g = S Π F ( N g ) , with F as in Corollary 3.14. Under our current assumptions, the hypotheses of the corollary are satisfied.We set ˆ g = W ( X , . . . , X N ) ∈ E and for each k ∈ N ,ˆ g k := S Π F (ˆ g k − ) . Since F maps into P ( R,σ ) ϕ , on which S Π is a linear contraction, and S Π E ⊂ E , the final part of Corollary3.14 implies that S Π F is uniformly contractive with constant on E and takes E to itself. Thus ˆ g k ∈ E for all k and k ˆ g k − ˆ g k − k R,σ = k S Π F (ˆ g k − ) − S Π F (ˆ g k − ) k R,σ < k ˆ g k − − ˆ g k − k R,σ , implying that ˆ g k → ˆ g in k · k R,σ , with ˆ g a fixed point of S Π F . We note that ˆ g = 0 as S Π F (0) = S Π( W ) = W = 0. Since ˆ g ∈ P ( R,σ ) c.s. , we also have g := Σˆ g ∈ P ( R,σ ) c.s. . This proves (i) and (ii), and (iii) simply followsfrom the relation ˆ g = N g and the definition of F .It is not hard to see that for h = h ∗ , S Π F ( h ) ∗ = S Π F ( h ). Hence if we assume ˆ g = W is self-adjoint,then each successive ˆ g k will be self-adjoint. Consequently so will their limit ˆ g since k · k R (which is invariantunder ∗ ) is dominated by k · k R,σ . It follows that g = Σˆ g is self-adjoint as well.Assume β W β is analytic. Then each iterate ˆ g k ( β ) is clearly analytic as well, and the convergence toˆ g ( β ) is uniform on any compact disk inside | β | < β . Thus the Cauchy integral formula implies the limitˆ g ( β ) is analytic as well, and clearly so is g ( β ) = Σˆ g ( β ).Finally, we remark that k g k R,σ is bounded by k W k R,σ . Indeed, k ˆ g − W k R,σ = k ˆ g − ˆ g k R,σ ≤ k ˆ g − ˆ g k R,σ ≤ (cid:18)(cid:20) k ˆ g k R,σ (cid:26) (cid:27) + k W k R,σ (cid:21) + k ˆ g k Rσ (cid:19) = 5 k W k R,σ , or k ˆ g k R,σ ≤ k W k R,σ . Since k g k R,σ = k Σˆ g k R,σ ≤ k ˆ g k R,σ , it follows that g k W k R,σ (cid:3) Theorem 3.16.
Let R ′ > R ≥ p k A k . Then there exists a constant C > depending only on R , R ′ , and N so that whenever W = W ∗ ∈ P ( R ′ +1) c.s. satisfies k W k R ′ +1 ,σ < C , there exists f ∈ P ( R ) which satisfiesEquation (23). In addition, f = D g for g ∈ P ( R,σ ) c.s. . The solution f = f W satisfies k f W k R → as k W k R ′ +1 ,σ → . Moreover, if W β is a family which is analytic in β then also the solutions f W β are analyticin β .Proof. Fix S ∈ ( R, R ′ ). Using the bounds in the proof of Theorem 3.15 in [5] we have N X j =1 k δ j ( W ) k ( S +1) ⊗ π ( S +1) ≤ c ( S + 1 , R ′ + 1) k W k R ′ +1 , where c ( S, R ) = sup α ≥ αS − ( R/S ) − α . Also,
S < R ′ + 1 implies k W k S,σ ≤ k W k R ′ +1 ,σ . Hence, by choosing C > k W k R ′ +1 ,σ < C will imply the hypothesis of Proposition 3.15 are satisfied with ρ = 1 and R replaced with S . Thus thereexists g = g ∗ ∈ P ( S,σ ) c.s. satisfying (31). Let f = D g , then from Lemma 2.5 we know f ∈ (cid:0) P ( R ) (cid:1) N . Also,using the bounds from the proof of Theorem 3.15 in [5] again we have k J f k R ⊗ π R ≤ c ′ ( R, S ) k g k S = c ′ ( R, S ) k g k S,σ , where c ′ ( R, S ) = sup α ≥ α R − ( S/R ) − α . Hence by the proof of Proposition 3.15.(v) we can (by possibly choosing a smaller C ) assume k J f k R ⊗ π R < g ∈ P ( R,σ ) c.s. ⊃ P ( S,σ ) c.s. .Recall from Lemma 2.6 that DS Π = D on P ( S,σ ) ϕ . Hence applying D to both sides of (31) yields D { [( ϕ ⊗ ◦ Tr A − + (1 ⊗ ϕ ) ◦ Tr A ] ( J D g ) − N g } = D ( W ( X + D g )) + D Q ( g ) + D (cid:18) { (1 + A ) D g } D g (cid:19) . The final term is equivalent to D (cid:18) { (1 + A ) D g } D g (cid:19) = D (cid:18) J σ X − f f (cid:19) = J f f = J σ f J σ X − f, where we have used (27). Thus f = D g satisfies Equation (24) which, according to Lemma 3.7 is equivalentto Equation (23).The final statements follow from Lemma 2.5 and Proposition 3.15.(v). (cid:3) Summary of results.
We aggregate the results of this section in the following theorem.
Theorem 3.17.
Let ( M, ϕ ) = ( M , ϕ V ) be a free Araki-Woods factor with free quasi-free state ϕ correspond-ing A , and generators X , . . . , X N ∈ M so that the matrix form of A with respect to the basis { X j Ω } Nj =1 isgiven by (2) and (3). Let R ′ > R ≥ p k A k . Then there exists a constant C > depending only on R , R ′ ,and N so that whenever W = W ∗ ∈ P ( R ′ +1 ,σ ) c.s. satisfies k W k R ′ +1 ,σ < C , there exists G ∈ P ( R,σ ) c.s. so that ( Y , . . . , Y N ) = ( D G, · · · , D N G ) ∈ P ( R ) has the law ϕ V , V = P Nj,k =1 (cid:2) A (cid:3) jk X k X j + W , which is the unique free Gibbs state with potential V .If R ′ > R k A k then the transport can be taken to be monotone: ( σ i ⊗ J σ D G ) ≥ as an operator on L ( P ⊗ P op , ϕ ⊗ ϕ op ) N .In particular, there are state-preserving injections C ∗ ( ϕ V ) ⊂ C ∗ ( ϕ V ) and W ∗ ( ϕ V ) ⊂ W ∗ ( ϕ V ) .If the map β W β is analytic, then Y , . . . , Y n are also analytic in β . Furthermore, k Y j − X j k R vanishesas k W k R ′ +1 ,σ goes to zero. REE MONOTONE TRANSPORT WITHOUT A TRACE 33
Proof.
Note for Y j = X j + f j we have k Y j k ≤ k f j k R . By requiring C be small enough so that k f j k R ≤ | ϕ ( Y j ) | ≤ | j | . So by Theorem 2.12, and further shrinking C if necessary, we see that ϕ Y is the unique free Gibbs statewith potential potential V . The only remaining part of this theorem not covered by Theorem 3.16 is thepositivity of ( σ i/ ⊗ J σ f ), so we merely verify this condition when R ′ > R k A k .Recall from Lemma 3.1.(iv),( σ i ⊗ J σ f ) = A σ i ⊗ σ − i )( J σ f ) A − . Hence if S ′ = k A k R then k ( σ i ⊗ J σ f ) k R ⊗ π R ≤ k A k k ( σ i ⊗ σ − i )( J f ) k R ⊗ π R k J σ X k R ⊗ π R ≤ k A k k J f k S ′ ⊗ π S ′ k J σ X k R ⊗ π R . Thus in the proof of Theorem 3.16 we can choose S ∈ ( S ′ , R ′ ) so that k J f k S ′ ⊗ π S ′ ≤ c ′ ( S ′ , S ) k g k S,σ . Inparticular, we can make k J f k S ′ ⊗ π S ′ < k A k − so that k ( σ i ⊗ J σ f ) k R ⊗ π R < k J σ X k R ⊗ π R . Noting that ( σ i ⊗ J σ Y ) = J σ X + ( σ i ⊗ J σ f ), J σ X ≥
0, and ( σ i ⊗ J σ f ) ∗ = ( σ i ⊗ J σ f )(via Lemma 3.1.(iii)) we have that ( σ i ⊗ J σ Y ) ≥ (cid:3) By shrinking the constant further if needed, we can use Lemma 2.8 to turn the state-preserving injectionsinto isomorphisms:
Corollary 3.18.
Let ( M, ϕ ) = ( M , ϕ V ) be a free Araki-Woods factor with free quasi-free state ϕ corre-sponding A , and generators X , . . . , X N ∈ M so that the matrix form of A with respect to the basis { X j Ω } Nj =1 is given by (2) and (3). Let R ′ > R ≥ p k A k . Then there exists C > depending only on R , R ′ , and N sothat whenever W = W ∗ ∈ P ( R ′ +1 ,σ ) c.s. satisfies k W k R ′ +1 ,σ < C , there exists G ∈ P ( R,σ ) c.s. so that: (1) if we set Y j = D j G , then Y , . . . , Y N ∈ P ( R ) has law ϕ V , with V = P Nj,k =1 (cid:2) A (cid:3) jk X k X j + W ; (2) X j = H j ( Y , . . . , Y N ) for some H j ∈ P ( R ) ; and (3) if R ′ > R k A k then ( σ i ⊗ J σ D G ) ≥ as an operator on L ( P ⊗ P op ) N .In particular there are state-preserving isomorphisms C ∗ ( ϕ V ) ∼ = Γ( H R , U t ) , W ∗ ( ϕ V ) ∼ = Γ( H R , U t ) ′′ . Proof.
By Theorem 3.17, it suffices to show the existence of H = ( H , . . . , H N ) ∈ ( P ( R ) ) N . From Theorem3.16, we know that Y = X + f ( X ), and that k f k R → k W k R ′ +1 ,σ →
0. In fact, from Lemma 2.5 we knowthat f ∈ ( P ( S ) ) N for any S ∈ ( R, R ′ ), and k f k S still tends to zero. Set S = ( R + R ′ ) /
2, then by shrinkingthe constant C in the statement of the corollary further if necessary, we may assume that hypothesis ofLemma 2.8 are satisfied. Thus we obtain the desired inverse mapping H ( Y ) = X . (cid:3) Application to the q -deformed Araki-Woods algebras We saw in Theorem 2.11 that ϕ is the free Gibbs state with potential V = 12 N X j,k =1 (cid:20) A (cid:21) jk X (0) k X (0) j . In this section we will show that for small | q | , ϕ q is the free Gibbs state with potential V = 12 N X j,k =1 (cid:20) A (cid:21) jk X ( q ) k X ( q ) j + W ∈ P ( R,σ ) c..s. , and that k W k R,σ → | q | →
0. Hence it will follow from Corollary 3.18 that M q ∼ = M for sufficientlysmall | q | . We now let M = M q for arbitrary (but fixed) q ∈ ( − , Invertibility of Ξ q . Let Ψ : M Ω → M be the inverse of canonical embedding of M into F q ( H ) via x x Ω for x ∈ M , which we note is injective from the fact that Ω is separating. Hence for ξ ∈ M Ω wehave that Ψ( ξ ) is the unique element in M such that Ψ( ξ )Ω = ξ . The uniqueness then implies the complexlinearity of Ψ: Ψ( P i α i ξ i ) = P i α i Ψ( ξ i ). We also note that by the formulas (1) we haveΨ( Sξ )Ω = Sξ = S (Ψ( ξ )Ω) = Ψ( ξ ) ∗ Ω; andΨ(∆ iz ξ )Ω = ∆ iz ξ = ∆ iz Ψ( ξ )∆ − iz Ω = σ z (Ψ( ξ ))Ω , (32)so that the uniqueness implies Ψ( Sξ ) = Ψ( ξ ) ∗ and Ψ(∆ iz ξ ) = σ z (Ψ( ξ )).Recall that Ξ q = P q n P n , where P n ∈ HS ( F q ( H )) is the projection onto tensors of length n . We claimthat (32) implies each P n , when identified with an element in L ( M ¯ ⊗ M op , ϕ ⊗ ϕ op ), is fixed by σ it ⊗ σ it forall t ∈ R . Indeed, fix t ∈ R and let { ξ i } | i | = n be an orthonormal basis for H ⊗ n . Then P n is identified with P | i | = n Ψ( ξ i ) ⊗ Ψ( ξ i ) ∗ since for η ∈ F q ( H ) X | i | = n (cid:10) Ψ( ξ i )Ω , η (cid:11) U,q Ψ( ξ i )Ω = X | i | = n (cid:10) ξ i , η (cid:11) U,q ξ i = P n η. Now, using (32), we see that( σ it ⊗ σ it )( P n ) = X | i | = n Ψ(∆ t ξ i ) ⊗ Ψ(∆ − t ξ i ) ∗ = X | i | = n Ψ (cid:16)(cid:0) A − t (cid:1) ⊗ n ξ i (cid:17) ⊗ Ψ (cid:16)(cid:0) A t (cid:1) ⊗ n ξ i (cid:17) ∗ . Let Q n ∈ HS ( F q ( H )) be the element associated with ( σ it ⊗ σ it )( P n ). That is, for η ∈ F q ( H ) we have Q n η = X | i | = n D(cid:0) A t (cid:1) ⊗ n ξ i , η E U,q (cid:0) A − t (cid:1) ⊗ n ξ i , and so D(cid:0) A t (cid:1) ⊗ n ξ j , Q n η E U,q = X | i | = n D(cid:0) A t (cid:1) ⊗ n ξ i , η E U,q
D(cid:0) A t (cid:1) ⊗ n ξ j , (cid:0) A − t (cid:1) ⊗ n ξ i E U,q = X | i | = n D(cid:0) A t (cid:1) ⊗ n ξ i , η E U,q D ξ j , ξ i E U,q = D(cid:0) A t (cid:1) ⊗ n ξ j , η E U,q = D(cid:0) A t (cid:1) ⊗ n ξ j , P n η E U,q . From Lemma 1.2 of [6], A t > A t ) ⊗ n >
0. Thus n ( A t ) ⊗ n ξ i o | i | = n is a basis for H ⊗ n and hence P n = Q n = ( σ it ⊗ σ it )( P n ) as claimed.It follows that for any t ∈ R we have ( σ it ⊗ σ it )(Ξ q ) = Ξ q , and more generally( σ it ⊗ σ is )(Ξ q ) = ( σ i ( t − s ) ⊗ q ) = (1 ⊗ σ i ( s − t ) )(Ξ q ) ∀ t, s ∈ R . (33)We remind the reader that the norm k · k R ⊗ π R is defined in Section 2.5. Denote the closure of P ⊗ P op with respect to this norm by ( P ⊗ P op ) ( R ) . We now prove an estimate analogous to those in Corollary 29in [3] for the non-tracial case. Proposition 4.1.
Let R = (cid:0) c (cid:1) −| q | > k X i k for some c > . Fix t ∈ R , then for sufficiently small | q | and all | t | ≤ | t | , ( σ it ⊗ q ) ∈ ( P ⊗ P op ) ( R ) with k ( σ it ⊗ q ) − k R ⊗ π R ≤ k A t k (3 + c ) (1 + k A k ) N | q | − (4 + k A t k (3 + c ) (1 + k A k ) N ) | q | =: π ( q, N, A, t ) . Moreover, π ( q, N, A, t ) → as | q | → and π ( q, N, A, s ) ≤ π ( q, N, A, t ) for | s | ≤ | t | . Finally, for π ( q, N, A, t ) < and | t | ≤ | t | , ( σ it ⊗ q ) is invertible with ( σ it ⊗ q ) − = ( σ it ⊗ − q ) ∈ ( P ⊗ P op ) ( R ) and (cid:13)(cid:13) ( σ it ⊗ − q ) − (cid:13)(cid:13) R ⊗ π R ≤ π ( q, N, A, t )1 − π ( q, N, A, t ) −→ as | q | → . Proof.
We first construct the operators Ψ( ξ i ) =: r i from the remarks preceding the proposition (for asuitable orthonormal basis). However, in order to control their k · k R -norms we must build these operators REE MONOTONE TRANSPORT WITHOUT A TRACE 35 out of { Ψ( e i ) } since this latter set is easily expressed as polynomials in the X i . Indeed, for a multi-index j = { j , . . . , j n } let ψ j ∈ P be the non-commutative polynomial defined inductively by ψ j = X j ψ j ,...,j n − X k ≥ q k − h e j , e j k i U ψ j ,..., ˆ j k ,...,j n , (34)where ψ ∅ = 1. From a simple computation it is clear that ψ j = Ψ( e j ⊗ · · · ⊗ e j n ).Fix n ≥
0, then, following [3], we let B = B ∗ ∈ M N n ( C ) be the matrix such that B = π q,N,n (cid:16) P ( n ) − q (cid:17) .In other words, given h , . . . , h n ∈ H if we define g i = P | j | = n B i,j h j then D g i , g j E U,q = D h i , h j E U, = n Y k =1 h h i k , h j k i U . Define p i = P | j | = n B i,j ψ j . Then the p i satisfy D p i , p j E ϕ = D p i Ω , p j Ω E U,q = D e i , e j E U, . Let α ∈ M N ( C ) have entries α ij = h e j , e i i U , and recall that by a previous computation this implies α = A . We note that the eigenvalues of α are contained in the interval h k A k , k A k − i . Lemma 1.2 in [6]implies that α ⊗ n is strictly positive, so let D = D ∗ ∈ M N n ( C ) be such that D = ( α ⊗ n ) − .We claim that k D k ≤ (cid:16) k A k (cid:17) n . Indeed, it suffices to show that the eigenvalues of α ⊗ n are bounded below by (cid:16) k A k (cid:17) n .Suppose λ is an eigenvalue with eigenvector h ⊗ · · · ⊗ h n ∈ H ⊗ n R . Upon renormalizing, we may assume k h i k = 1 for each i . Thus λ = (cid:10) h ⊗ · · · ⊗ h n , α ⊗ n h ⊗ · · · ⊗ h n (cid:11) , = Y i h h i , αh i i ≥ (cid:18)
21 + k A k (cid:19) n , and the claim follows. Setting r i = P k D i,k p k we have D r i , r j E ϕ = X k,l D i,k D j,l (cid:10) p k , p l (cid:11) ϕ = X k,l D k,i D j,l (cid:10) e k , e l (cid:11) U, = X k,l D k,i D j,l *(cid:18)
21 + A − (cid:19) ⊗ n e k , e l + , = X k,l D j,l "(cid:18)
21 + A − (cid:19) ⊗ n k,l D k,i = X k,l D j,l (cid:2) α ⊗ n (cid:3) l,k D k,i = [ Dα ⊗ n D ] j,i = δ i = j . Noting that r i is a linear combination of the ψ j with | j | = n , we see that r i Ω ∈ H ⊗ n . Hence { r i Ω } | i | = n isan orthonormal basis for H ⊗ n and P n can be identified with P | i | = n r i ⊗ r ∗ i ∈ P ⊗ P op .Repeat this construction for each n ≥ i of arbitrary length we have a corre-sponding r i and consequently a representation of P n in P ⊗ P op for every n . Then by definition we haveΞ q = P n ≥ q n P | i | = n r i ⊗ r ∗ i , provided this sum converges. Let C n ( t ) = sup | i | = n k σ it ( ψ i ) k R , then we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | i | = n σ it ( r i ) ⊗ r ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R ⊗ π R ≤ X i,j,k,l,m (cid:12)(cid:12)(cid:12) D i,j B j,l D i,k B k,m (cid:12)(cid:12)(cid:12) (cid:13)(cid:13) σ it ( ψ l ) (cid:13)(cid:13) R k ψ m k R ≤ X m,l (cid:12)(cid:12) ( BD B ) m,l (cid:12)(cid:12) C n ( t ) C n (0) ≤ N n k BD B k C n ( t ) C n (0) ≤ N n (cid:18) k A || (cid:19) n k B k C n ( t ) C n (0) ≤ N n (cid:18) k A k (cid:19) n (1 − | q | ) ∞ Y k =1 | q | k − | q | k ! n C n ( t ) C n (0) , where we have used the bound on k B k from [3]. From Equation (34) and (6), C n ( t ) ≤ k A − t X k R C n − ( t ) + C n − ( t ) / (1 − | q | ). But k A − t X k R ≤ k A − t k R = k A t k R (see property 4 of A in section 2.1), so that C n ( t ) ≤ k A t k n (cid:16) R + −| q | (cid:17) n = k A t k n (cid:16) c −| q | (cid:17) n . Also, we use the bound(1 − | q | ) ∞ Y k =1 | q | k − | q | k ≤ (1 − | q | ) − | q | , from Lemma 13 in [8]. Thus (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | i | = n σ it ( r i ) ⊗ r ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R ⊗ π R ≤ N n (cid:18) k A k (cid:19) n (cid:18) (1 − | q | ) − | q | (cid:19) n k A t k n (cid:18) c − | q | (cid:19) n = (cid:20) k A t k N k A k c ) − | q | (cid:21) n . Thus choosing | q | small enough so that | q |k A t k N k A k c ) − | q | < , we can use k A t k ≤ k A t k for | t | ≤ | t | to obtain k ( σ it ⊗ q ) − ⊗ k R ⊗ π R ≤ ∞ X n =1 (cid:20) | q |k A t k N k A k c ) − | q | (cid:21) n = k A t k (3 + c ) (1 + k A k ) N | q | − (4 + k A t k (3 + c ) (1 + k A k ) N ) | q | . The limit π ( q, N, A, t ) → | q | → π ( q, N, A, t ), and the ordering π ( q, N, A, s ) ≤ π ( q, N, A, t ) for | s | ≤ | t | simply follows from k A s k ≤ k A t k . The final statements are thensimple consequences of the formula x = P ∞ n =0 (1 − x ) n . (cid:3) Remark 4.2.
We note that π ( q, N, ,
0) = π ( q, N ) in [3].4.2. The conjugate variables ξ j . Recall that ˆ σ z = σ z ⊗ σ ¯ z . We will show that ∂ ( q ) ∗ j ◦ ˆ σ − i (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) definesthe conjugate variables for ∂ j , but first we require some estimates relating to ∂ ( q ) ∗ j .Fix c > R = (cid:0) c (cid:1) −| q | . For now, we only assume | q | is small enough that Ξ q ∈ ( P ⊗ P op ) ( R ) . Lemma 4.3.
For each j = 1 , . . . , N , the maps ( ϕ ⊗ ◦ ∂ ( q ) j and (1 ⊗ ϕ ) ◦ ¯ ∂ ( q ) j are bounded operators from P ( R ) to itself with norms bounded by −| q | c k Ξ q k R ⊗ π R . Consequently the maps m ◦ (1 ⊗ ϕ ⊗ ◦ (cid:16) ⊗ ∂ ( q ) j + ¯ ∂ ( q ) j ⊗ (cid:17) are bounded from ( P ⊗ P op ) ( R ) to P ( R ) with norm bounded by −| q | ) c k Ξ q k R ⊗ π R .Proof. Recall that ϕ is a state and k X i k ≤ −| q | and therefore ϕ satisfies (25) with C = −| q | . For P ∈ P ( R ) write P = P i a ( i ) X i and denote k Ξ q k R ⊗ π R = Q . Then (cid:13)(cid:13)(cid:13) ( ϕ ⊗ ◦ ∂ ( q ) j ( P ) (cid:13)(cid:13)(cid:13) R = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i a ( i )( ϕ ⊗ | i | X k =1 α i k j X i · · · X i k − ⊗ X i k +1 · · · X i | i | q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R ≤ X i | a ( i ) | | i | X k =1 (cid:18) − | q | (cid:19) k − R n − k Q = X i | a ( i ) | R n − Q i X k =1 (cid:18)
11 + c/ (cid:19) k − ≤ X i a ( i ) R n − Q − c/ = k P k R Q R c/ c/ k P k R Q − | q | c . The estimate for (1 ⊗ ϕ ) ◦ ¯ ∂ ( q ) j is similar.Define η ( P ⊗
1) to be left multiplication by P on P ( R ) and define η (1 ⊗ P ) to be right multiplication by c −| q | Q − ( ϕ ⊗ ◦ ∂ ( q ) j ( P ) on P ( R ) . Let Q ∈ P ⊗ P op , then by the above computations and the definition REE MONOTONE TRANSPORT WITHOUT A TRACE 37 of k · k R ⊗ π R we have (cid:13)(cid:13)(cid:13) m ◦ (1 ⊗ ϕ ⊗ ◦ (1 ⊗ ∂ ( q ) j )( Q ) (cid:13)(cid:13)(cid:13) R = Q − | q | c k η ( Q )(1) k R ≤ Q − | q | c k Q k R ⊗ π R . Similarly, k m ◦ (1 ⊗ ϕ ⊗ ◦ ( ¯ ∂ ( q ) j ⊗ k ≤ Q −| q | c and so the final statement holds. (cid:3) Now let | q | be sufficiently small that π ( q, N, A, − <
1. Then by Proposition 4.1 and the statementspreceding it, ˆ σ i (Ξ − q ) = ( σ i ⊗ − q ) and ( σ i ⊗ − q ) exist as elements of ( P ⊗ P op ) ( R ) , as do theiradjoints ˆ σ − i (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) and ( σ − i ⊗ (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) . So by the preceding lemma the following defines an elementof P ( R ) for each j = 1 , . . . , N : ξ j := ( σ − i ⊗ (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) X j − m ◦ (1 ⊗ ϕ ⊗ ◦ (cid:16) ⊗ ∂ ( q ) j + ¯ ∂ ( q ) j ⊗ (cid:17) ◦ ( σ − i ⊗ (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) , (35)and k ξ j k R ≤ (cid:13)(cid:13) ( σ i ⊗ (cid:0) Ξ − q (cid:1)(cid:13)(cid:13) R ⊗ π R R + 2(1 − | q | ) c k Ξ q k R ⊗ π R (cid:13)(cid:13) ( σ i ⊗ (cid:0) Ξ − q (cid:1)(cid:13)(cid:13) R ⊗ π R . (36)Now, using (10) we see that ∂ ( q ) ∗ j ◦ ˆ σ − i (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) = ( σ − i ⊗ (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) X j − m ◦ (1 ⊗ ϕ ⊗ σ − i ) ◦ (cid:16) ⊗ ¯ ∂ ( q ) j + ¯ ∂ ( q ) j ⊗ (cid:17) ◦ ( σ − i ⊗ (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) , which is equivalent to ξ j defined above. Hence h ξ j , P i = D ˆ σ − i (cid:16)(cid:2) Ξ − q (cid:3) ∗ (cid:17) , ∂ ( q ) j ( P ) E = ϕ ⊗ ϕ op (cid:16) ˆ σ i (cid:0) Ξ − q (cid:1) ∂ ( q ) j ( P ) (cid:17) = ϕ ⊗ ϕ op (cid:16) ∂ ( q ) j ( P ) − q (cid:17) = ϕ ⊗ ϕ op ( ∂ j ( P )) = h ⊗ , ∂ j ( P ) i . Thus ξ j = ∂ ∗ j (1 ⊗
1) is the conjugate variable of X , . . . , X N with respect to the σ -difference quotient ∂ j . Italso holds that ξ j = ξ ∗ j : (cid:10) ξ ∗ j , P (cid:11) = ϕ ( σ i ( P ) ξ j ) = h ξ j , σ − i ( P ∗ ) i = ϕ ⊗ ϕ op ( ∂ j ◦ σ − i ( P ∗ ))= ϕ ⊗ ϕ op (cid:0) ¯ ∂ j ( P ∗ ) (cid:1) = ϕ ⊗ ϕ op ( ∂ j ( P )) = h ξ j , P i . We remark that this could also be observed directly from the definition of ξ j in (35) using a combination of(33) and the fact that Ξ † q = Ξ q .We claim that there exists V ∈ P ( R,σ ) c.s. ⊂ M such that D j V = ξ j . We first require a technical lemmawhich will lead to what is essentially the converse of Lemma 3.1.(iii) in the case Y = ( ξ , . . . , ξ N ). Lemma 4.4.
Let ξ , . . . , ξ N be as defined above. Then for j, k ∈ { , . . . , N } , ∂ k ( ξ j ) = (1 ⊗ σ − i ) ◦ ¯ ∂ j ( ξ k ) ⋄ (37) as elements of L ( M ¯ ⊗ M op , ϕ ⊗ ϕ op ) . Furthermore, σ − i ( ξ j ) = N X k =1 [ A ] jk ξ k . (38) Proof.
It suffices to check h ∂ i ( ξ j ) , a ⊗ b i = (cid:10) (1 ⊗ σ − i ) ◦ ¯ ∂ j ( ξ i ) ⋄ , a ⊗ b (cid:11) for elementary tensors a ⊗ b ∈ L ( M ¯ ⊗ M op , ϕ ⊗ ϕ op ). So using (9) we compute h ∂ k ( ξ j ) , a ⊗ b i = ϕ ( ξ j aξ k σ − i ( b )) − ϕ ( ξ j a [( ϕ ⊗ σ − i ) ◦ ¯ ∂ k ( b )]) − ϕ ( ξ j [(1 ⊗ ϕ ) ◦ ¯ ∂ k ( a )] σ − i ( b ))= (cid:10) ∂ ∗ j (cid:0) ( σ − i ( b ) ⊗ a ) † (cid:1) , ξ k (cid:11) + ϕ ( (cid:8) a ∗ [( ϕ ⊗ σ − i ) ◦ ¯ ∂ j ◦ σ i ( b ∗ )] (cid:9) ∗ ξ k ) + ϕ ( { [(1 ⊗ ϕ ) ◦ ¯ ∂ j ( a ∗ )] b ∗ } ∗ ξ k ) − ϕ ([( ϕ ⊗ ◦ ∂ j ( a )][( ϕ ⊗ σ − i ) ◦ ¯ ∂ k ( b )]) − ϕ ( a [(1 ⊗ ϕ ) ◦ ∂ j ◦ ( ϕ ⊗ σ − i ) ◦ ¯ ∂ k ( b )]) − ϕ ([( ϕ ⊗ ◦ ∂ j ◦ (1 ⊗ ϕ ) ◦ ¯ ∂ k ( a )] σ − i ( b )) − ϕ ([(1 ⊗ ϕ ) ◦ ¯ ∂ k ( a )][(1 ⊗ ϕ ) ◦ ∂ j ◦ σ − i ( b )]) . We note that ϕ ( P ∗ ξ k ) = h ξ k , P i = ϕ ⊗ ϕ op ( ∂ k ( P )) = ϕ ⊗ ϕ op ( ∂ k ( P ) † ) = ϕ ⊗ ϕ op ( ¯ ∂ k ( P ∗ )) . Applying this to the second and third terms in the above computation yields h ∂ k ( ξ j ) , a ⊗ b i = (cid:10) ∂ ∗ j (cid:0) ( σ − i ( b ) ⊗ a ) † (cid:1) , ξ k (cid:11) + ϕ ⊗ ϕ op ( ¯ ∂ k { [( σ i ⊗ ϕ ) ◦ ∂ j ◦ σ − i ( b )] a } ) + ϕ ⊗ ϕ op ( ¯ ∂ k { b [( ϕ ⊗ ◦ ∂ j ( a )] } ) − ϕ ([( ϕ ⊗ ◦ ∂ j ( a )][( ϕ ⊗ σ − i ) ◦ ¯ ∂ k ( b )]) − ϕ ( a [(1 ⊗ ϕ ) ◦ ∂ j ◦ ( ϕ ⊗ σ − i ) ◦ ¯ ∂ k ( b )]) − ϕ ([( ϕ ⊗ ◦ ∂ j ◦ (1 ⊗ ϕ ) ◦ ¯ ∂ k ( a )] σ − i ( b )) − ϕ ([(1 ⊗ ϕ ) ◦ ¯ ∂ k ( a )][(1 ⊗ ϕ ) ◦ ∂ j ◦ σ − i ( b )])= (cid:10) [ σ − i ( b ) ⊗ a ] † , ∂ j ( ξ k ) (cid:11) + ϕ ([( ϕ ⊗ ◦ ¯ ∂ k ◦ ( σ i ⊗ ϕ ) ◦ ∂ j ◦ σ − i ( b )] a ) − ϕ ( a [(1 ⊗ ϕ ) ◦ ∂ j ◦ ( ϕ ⊗ σ − i ) ◦ ¯ ∂ k ( b )])+ ϕ ( b [(1 ⊗ ϕ ) ◦ ¯ ∂ k ◦ ( ϕ ⊗ ◦ ∂ j ( a )]) − ϕ ([( ϕ ⊗ ◦ ∂ j ◦ (1 ⊗ ϕ ) ◦ ¯ ∂ k ( a )] σ − i ( b )) . Now, applying (6) to the second line in the last equality above yields ϕ ([( ϕ ⊗ ◦ ( ¯ ∂ k ⊗ ϕ ) ◦ ¯ ∂ j ( b )] a ) − ϕ ([(1 ⊗ ϕ ) ◦ ( ϕ ⊗ ¯ ∂ j ) ◦ ¯ ∂ k ( b )] a ) . This is zero if ( ϕ ⊗ ◦ ( ¯ ∂ k ⊗ ϕ ) ◦ ¯ ∂ j = (1 ⊗ ϕ ) ◦ ( ϕ ⊗ ¯ ∂ j ) ◦ ¯ ∂ k , but this is easily verified by computing onmonomials. Finally, the final line in the last equality of the computation is equivalent to ϕ ( b [(1 ⊗ ϕ ) ◦ ¯ ∂ k ◦ ( ϕ ⊗ ◦ ∂ j ( a )]) − ϕ ( b [( ϕ ⊗ ◦ ∂ j ◦ (1 ⊗ ϕ ) ◦ ¯ ∂ k ( a )]) . This is zero if (1 ⊗ ϕ ) ◦ ( ϕ ⊗ ¯ ∂ k ) ◦ ∂ j = ( ϕ ⊗ ◦ ( ∂ j ⊗ ϕ ) ◦ ¯ ∂ k , but again this is easily checked on monomials.Thus h ∂ k ( ξ j ) , a ⊗ b i = (cid:10) [ σ − i ( b ) ⊗ a ] † , ∂ j ( ξ k ) (cid:11) = ϕ ⊗ ϕ op ( a ⊗ σ − i ( b ) ∂ j ( ξ k ))= ϕ ⊗ ϕ op (( σ i ⊗ ◦ ∂ j ( ξ k ) a ⊗ b ) = (cid:10) (1 ⊗ σ − i ) ◦ ¯ ∂ j ( ξ ∗ k ) ⋄ , a ⊗ b (cid:11) , showing (37).Towards verifying (38), we note that N X k =1 (cid:2) A − (cid:3) jk ∂ k = ¯ ∂ j . Hence for P ∈ P we have * N X k =1 [ A ] jk ξ k , P + = N X k =1 [ A ] kj ϕ ⊗ ϕ op ( ∂ k ( P )) = ϕ ⊗ ϕ op (cid:0) ¯ ∂ j ( P ) (cid:1) = ϕ ⊗ ϕ op ( ∂ j ( P ∗ )) = h ξ j , P ∗ i = ϕ ( P ξ j ) = ϕ ( σ i ( ξ j ) P ) = h σ − i ( ξ j ) , P i , which establishes (38). (cid:3) Define V = Σ N X j,k =1 (cid:20) A (cid:21) jk ξ k X j . Note that (36) implies V ∈ P ( R ) . We further claim that D j V = ξ j and V ∈ P ( R,σ ) c.s. . The former is equivalentto D j ( N V ) = (1 + N ) D j V = (1 + N ) ξ j = ξ j + n X k =1 δ k ( ξ j ) X k . To show this, we first note that D j = m ◦ ⋄ ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ j and so by the derivation property of ¯ ∂ j we have D j ( P Q ) = (1 ⊗ σ − i ) ◦ ¯ ∂ j ( P ) ⋄ σ − i ( Q ) + (1 ⊗ σ − i ) ◦ ¯ ∂ j ( Q ) ⋄ P. REE MONOTONE TRANSPORT WITHOUT A TRACE 39
Thus using (37) and σ − i ( X j ) = [ AX ] j from (4) we have D t ( N V ) = N X j,k =1 (cid:20) A (cid:21) jk (cid:0) (1 ⊗ σ − i ) ◦ ¯ ∂ t ( ξ k ) σ − i ( X j ) + α tj ξ k (cid:1) = N X j,k,l =1 (cid:20) A (cid:21) jk ∂ k ( ξ t ) A ] jl X l + N X j,k =1 (cid:20)
21 + A (cid:21) tj (cid:20) A (cid:21) jk ξ k = ξ t + N X l =1 δ l ( ξ t ) X l , as claimed.Now, in order to show V ∈ P ( R,σ ) c.s. we will show that V is invariant under σ − i and that S ( V ) = V Together, these imply that V is invariant under ρ and hence V ∈ P ( R,σ ) c.s. (that V has finite k · k R,σ -normfollows from the fact that for ρ invariant elements this norm agrees with the k · k R -norm). Using (38) and σ − i ( X j ) = [ AX ] j we see that σ − i ( V ) = Σ N X j,k =1 (cid:20) A (cid:21) jk N X l =1 [ A ] kl ξ l N X m =1 [ A ] jm X m = Σ N X j,k,l,m =1 (cid:2) A − (cid:3) mj (cid:20) A (cid:21) jk [ A ] kl ξ l X m = V. Towards seeing S ( V ) = V , we note that S ( X i · · · X i n ) = 1 n n − X l =0 ρ l ( X i · · · X i n ) = 1 n N X l =1 [ m ◦ (1 ⊗ σ − i ) ◦ δ l ( X i · · · X i n ) ⋄ ] X l = Σ N X l =1 [ m ◦ (1 ⊗ σ − i ) ◦ δ l ( X i · · · X i n ) ⋄ ] X l ! , and by linearity this extends to general polynomials P . Hence N X l,m =1 (cid:20) A (cid:21) lm (cid:2) m ◦ (1 ⊗ σ − i ) ◦ ¯ ∂ m ( P ) ⋄ (cid:3) X l = N X l =1 [ m ◦ (1 ⊗ σ − i ) ◦ δ l ( P ) ⋄ ] X l = N S ( P ) = S ( N P ) . Consequently (37) implies S ( N V ) = N X l,m =1 (cid:20) A (cid:21) lm (cid:2) (1 ⊗ σ − i ) ◦ ¯ ∂ m ( N V ) ⋄ (cid:3) X l = N X j,k,l,m =1 (cid:20) A (cid:21) lm (cid:20) A (cid:21) jk (cid:2) (1 ⊗ σ − i ) ◦ ¯ ∂ m ( ξ k ) ⋄ σ − i ( X j ) + α mj ξ k (cid:3) X l = N X j,k,l,m,a =1 (cid:20) A (cid:21) lm (cid:20) A (cid:21) jk [ A ] ja [ ∂ k ( ξ m ) X a ] X l + N X k,l =1 (cid:20) A (cid:21) lk ξ k x l = N X l,m =1 (cid:20) A (cid:21) lm [ N −
1] ( ξ m X l ) + N V = N V. Thus S ( V ) = V , and V ∈ P ( R,σ ) c.s. as claimed.Note V = 12 N X j,k =1 (cid:20) A (cid:21) jk X k X j = Σ N X j,k =1 (cid:20) A (cid:21) jk X k X j , and define W := V − V . Then W ∈ P ( R,σ ) c..s and k W k R,σ = k W k R ≤ N X j,k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) A (cid:21) jk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ξ k − X k k R R. We claim that k ξ k − X k k R → | q | →
0, and consequently k W k R,σ →
0. Indeed, we can write X k = (cid:0) [1 ⊗ ∗ (cid:1) X k − m ◦ (1 ⊗ ϕ ⊗ ◦ (cid:16) ⊗ ∂ ( q ) k + ¯ ∂ ( q ) k ⊗ (cid:17) (cid:0) [1 ⊗ ∗ (cid:1) , and so using (35) and Lemma 4.3 we have k ξ k − X k k R ≤ (cid:13)(cid:13) ( σ i ⊗ − q ) − ⊗ (cid:13)(cid:13) R ⊗ π R R + 2(1 − | q | ) c k Ξ q k R ⊗ π R (cid:13)(cid:13) ( σ i ⊗ − q ) − ⊗ (cid:13)(cid:13) R ⊗ π R . From the final remark in Proposition 4.1, we see that this tends to zero as | q | →
0. Thus we are in a positionto apply our transport results from Section 3. Using Corollary 3.18 we obtain the following result.
Theorem 4.5.
There exists ǫ > such that | q | < ǫ implies Γ q ( H R , U t ) ∼ = Γ ( H R , U t ) and Γ q ( H R , U t ) ′′ ∼ =Γ ( H R , U t ) . Using the classification of Γ ( H R , U t ) ′′ in Theorem 6.1 of [7] we obtain the following classification result. Corollary 4.6.
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