A simple proof of the complete metric approximation property for q-Gaussian algebras
aa r X i v : . [ m a t h . OA ] F e b A SIMPLE PROOF OF THE COMPLETE METRIC APPROXIMATIONPROPERTY FOR q -GAUSSIAN ALGEBRAS MATEUSZ WASILEWSKI
Abstract.
The aim of this note is to give a simpler proof of a result of Avsec, which statesthat q -Gaussian algebras have the complete metric approximation property. Introduction
Bo˙zejko and Speicher introduced q -Gaussian algebras in [BS91] (see also [BKS97]). Thesevon Neumann algebras are built from operators satisfying a deformation of canonical com-mutation relations. But they can also be viewed as q -deformations of the free group factors.It turns out that q -Gaussian algebras share many properties with the free group factors:they are factors (see [Ric05]), they are non-injective (see [Nou04]), they have the Haagerupapproximation property (folklore; see [Was17] for a proof in a more general setting of q -Araki-Woods algebras), etc. One of more important properties of von Neumann algebras studiedrecently is the notion of strong solidity, first introduced by Ozawa and Popa in [OP10]; inthe same paper they prove that free group factors are strongly solid.In an unpublished manuscript [Avs11] Avsec proved that q -Gaussian algebras possess thecomplete metric approximation property (see Theorem A therein). When combined withdeformation/rigidity techniques, namely using a malleable deformation, it allowed him toalso prove that q -Gaussian algebras are strongly solid (see Theorem B therein). We willreprove the first result, namely we will show the following. Theorem 1.1.
Let H R be a real Hilbert space and let Γ q p H R q be the associated q -Gaussianalgebra. Let P n : Γ q p H R q Ñ Γ q p H R q be the projection onto Wick words of length n . Then } P n } cb ď C p q q n , where C p q q is a positive constant depending only on q . Corollary 1.2.
For any real Hilbert space H R the q -Gaussian algebra Γ q p H R q has the w ˚ -complete metric approximation property. The proof splits into two parts – in the first one we analyse the operator space structureinduced on the image of P n by inclusion into Γ q p H R q and define some completely boundedmaps; we will follow closely the original approach of Avsec. The second part of the proofwill consist of showing that a certain linear combination of these maps is equal to the map,whose complete boundedness we want to prove (cf. [Avs11, Proposition 3.18]). Presentinga different (and much simpler) proof of this combinatorial statement is the main aim of thisnote. We discuss more details in the beginning of Subsection 4.2, once all the necessarytools have been introduced; readers familiar with Avsec’s paper [Avs11] may wish to jumpstraight to this subsection.The structure of the paper is the following: in Section 2 we provide necessary informationabout operator spaces and q -Gaussian, in Section 3 we recall Haagerup’s argument on how o deduce Corollary 1.2 from Theorem 1.1, finally Section 4 contains the proof of Theorem1.1. Acknowledgements.
I would like to thank Adam Skalski for careful reading of the prelim-inary version of this note and useful remarks. I am also thankful to ´Eric Ricard for pointingout that the bound on the cb norm of P n is linear in n , not quadratic, as I wrote in an olderversion of the paper. This was also spotted by the referee, whose helpful comments greatlyimproved the exposition.The author was supported in part by European Research Council Consolidator Grant614195 RIGIDITY and by long term structural funding Methusalem grant of the FlemishGovernment. 2. Notation and preliminaries
All inner products will be linear in the second variable. We will denote the set t , . . . , n u by r n s . For simplicity we assume that the Hilbert spaces are finite dimensional – there is astandard approximation procedure that allows us to do it.2.1. Operator spaces. An operator space is a Banach space X equipped with a sequenceof norms on the matrix spaces M n p X q that satisfy natural compatibility conditions, the so-called Ruan’s axioms; any such sequence comes from an embedding X Ă B p H q . Any C ˚ -algebra A admits a canonical operator space structure induced by any faithful representationon a Hilbert space. For information on operator spaces we refer to the monographs [ER00]and [Pis03]. For a linear map T : X Ñ Y between operator spaces we define its cb norm as } T } cb : “ sup n P N } Id n b T : M n p X q Ñ M n p Y q} . We say that T is completely bounded if } T } cb ă 8 .The theory of operator spaces mimicks the theory of Banach spaces. In particular, it re-tains one of most powerful properties of Banach spaces: the duality. For any operator space X there is a naturally defined operator space structure on its Banach space dual X ˚ . An-other construction is the complex conjugate operator space, defined using the identificationM n p X q » M n p X q . Since for Hilbert spaces we have an isometric identification H ˚ » H , dualspaces and conjugate spaces often appear together. One important fact about conjugatespaces is that for a von Neumann algebra M we have M » M op completely isometrically, viathe map x ÞÑ x ˚ , where M op is the same as M as a vector space but the multiplication isreversed i.e. x op ˛ y op : “ yx . We can define an operator space structure on the spaces L p M q for a finite von Neumann algebra M (equipped with a trace τ ) and S p H , K q (the trace classoperators) and the dualities p L p M qq ˚ » M and p S p H , K qq ˚ » B p K , H q hold; the pairings aregiven by L p M q ˆ M Q p x, y q ÞÑ τ p xy q and S p H , K q ˆ B p K , H q Q p S, T q ÞÑ Tr p ST q . Someauthors prefer to include complex conjugations in these dualities, i.e. consider the pairing L p M q ˆ M Q p x, y q ÞÑ τ p x ˚ y q (and a similar one for the trace-class operators), so that theyresemble the case of the Hilbert spaces, i.e. they are “positive definite”, but we will tryto avoid them to make the notation less cumbersome; both approaches have their pros andcons.In this paper we will encounter two special operator space structures on a Hilbert space. Definition 2.1.
Let H be a complex Hilbert space. i) The column Hilbert space structure H c is given by the identification H » B p C , H q ;(ii) The row Hilbert space structure H r is given by the identification H » B p H , C q . Remark 2.2.
In particular, we have H ˚ c » H r and H ˚ r » H c .These Hilbert spaces are homogeneous , i.e. bounded maps on the underlying Hilbertspaces are automatically completely bounded, with cb norm equal to the norm (see [ER00,Theorem 3.4.1 and Proposition 3.4.2]).We are going to need the notion of a tensor product of operator spaces. The simplest one isobtained by the following procedure: we have two operator spaces X Ă B p H q and Y Ă B p K q and we get an operator space structure on X b Y via embedding X b Y Ă B p H b K q . It turnsout that it does not depend on the embeddings and the completion of X b Y is denoted by X b min Y ; it coincides with the minimal tensor product of C ˚ -algebras, in case X and Y are C ˚ -algebras.There is also a special tensor product of operator spaces, called the Haagerup tensorproduct, which does not have a counterpart for Banach spaces; the Haagerup tensor productof two operator spaces X and Y will be denoted by X b h Y . One of its key properties isself-duality, i.e. p X b h Y q ˚ » X ˚ b h Y ˚ for finite dimensional operator spaces X and Y .For the definition and more information, see [ER00, Section 9]. We just collect here theproperties that will be useful for us in the sequel. Proposition 2.3. [Proposition 9.3.4 and Proposition 9.3.5 in [ER00] ] Let H and K be Hilbertspaces. Then we have the following identifications: (i) H c b h K r » K p K , H q (the compact operators); (ii) K r b h H c » H r b h K c » S p K , H q (the trace class operators); (iii) H r b h K r » p H b K q r and H c b h K c » p H b K q c . q –Gaussian algebras. We will now define q -Gaussian algebras, introduced by Bo˙zejkoand Speicher (see [BS91]).Let H R be a real Hilbert space and let H be its complexification; we will denote thecomplex conjugation by I . We would like to define the q -Fock space , which will be acertain completion of the tensor algebra À n ě H b n (both the direct sum and the tensorproducts are algebraic here, also H b “ C Ω). We will encounter tensors fairly often and forconvenience we will sometimes denote v b ¨ ¨ ¨ b v n by v and for subset A Ă r n s we will usethe notation v A : “ v i b ¨ ¨ ¨ b v i k , where A “ t i ă ¨ ¨ ¨ ă i k u , i.e. i , . . . , i k are all elementsof A arranged in an increasing order. Definition 2.4.
Let H be a complex Hilbert space. Fix q P p´ , q . For each n P N wedefine an operator P nq : H b n Ñ H b n by(2.1) P nq p v b ¨ ¨ ¨ b v n q : “ ÿ π P S n q i p π q v π p q b ¨ ¨ ¨ b v π p n q , where i p π q : “ |tp i, j q : i ă j, π p i q ą π p j qu| is the number of inversions of the permutation π . This operator is injective and positive definite (see [BS91, Proposition 1]) and thereforedefines an inner product on H b n . The P nq ’s combine to give an inner product on À n ě H b n and the completion of this space with respect to this inner product is called the q -Fockspace and is denoted by F q p H q . n order to define q -Gaussian algebras, we have to present an important class of operatorson the q -Fock space. Definition 2.5.
Let ξ P H . We define the creation operator a ˚ q p ξ q : F q p H q Ñ F q p H q bythe formula a ˚ q p ξ qp v b ¨ ¨ ¨ b v n q : “ ξ b v b ¨ ¨ ¨ b v n a ˚ q p ξ qp Ω q : “ ξ. We define also the annihilation operators by a q p ξ q : “ p a ˚ q p ξ qq ˚ . Their action on simpletensors is given by a q p ξ qp Ω q “ a q p ξ qp v b ¨ ¨ ¨ b v n q “ n ÿ i “ q i ´ x ξ, v i y v b . . . p v i ¨ ¨ ¨ b v n , where the hat over v i means that this vector is omitted.These operators extend to bounded operators on F q p H q and satisfy the q -commutationrelations a q p ξ q a ˚ q p η q ´ qa ˚ q p η q a q p ξ q “ x ξ, η y Id.
Definition 2.6.
Let H R be a real Hilbert space with complexification H . We define the q -Gaussian algebra Γ q p H R q to be the von Neumann subalgebra of B p F q p H qq generated bythe set t a ˚ q p ξ q ` a q p ξ q : ξ P H R u .The vector Ω is a cyclic and separating vector for Γ q p H R q , moreover the correspondingvector state is a faithful trace. In particular, the L -space L p Γ q p H R qq can be identified withthe Fock space F q p H q . One can show that for any simple tensor v b ¨ ¨ ¨ b v n P H b n thereexists a (unique!) operator W p v b ¨ ¨ ¨ b v n q such that W p v b ¨ ¨ ¨ b v n q Ω “ v b ¨ ¨ ¨ b v n ;these operators will be called the Wick words . Actually, there is an explicit formula forthem (see [BKS97, Proposition 2.7]).
Lemma 2.7.
We have (2.2) W p v b ¨ ¨ ¨ b v n q “ ÿ A Ăr n s q i p A q a ˚ q p v A q a q p I v r n sz A q , where for A “ t i ă ¨ ¨ ¨ ă i k u and r n sz A “ t j k ` ă ¨ ¨ ¨ ă j n u we have i p A q : “ ř kl “ p i l ´ l q , a ˚ q p v A q : “ a ˚ q p v i q . . . a ˚ q p v i k q , and a q p I v r n sz A q : “ a q p Iv j k ` q . . . a q p Iv j n q . Remark 2.8.
The notations i p A q and i p π q are consistent, if we identify A with a permutation π A of r n s given by π A p l q : “ i l for l ď k and π A p l q : “ j l for l ą k . Some authors identifythese permutations with representatives of cosets S n {p S k ˆ S n ´ k q with the minimal numberof inversions. It is useful to think of i p A q as the cost of moving the set A to the left of r n s .First, you move i to the first spot, so you need to make i ´ i to the second one and the cost is i ´
2; proceed like that for otherelements of A .We will need more information about the operators P nq ; as we mentioned, they are injectiveand positive definite, so invertible in the finite dimensional setting. Actually, it was noted byBo˙zejko (see [Bo˙z98, Theorem 6]) that they are always invertible, with a bound (exponentialin n ) for the norm of the inverse provided. Therefore, whenever we have a partition n “ ` ¨ ¨ ¨ ` n k , we may consider R ˚ n ,...,n k – the unique operator on H b n such that P nq “p P n q b ¨ ¨ ¨ b P n k q q R ˚ n ,...,n k . One can easily check that R ˚ n ,...,n k is really the adjoint of theidentity map R n ,...,n k : H b n q b ¨ ¨ ¨ b H b n k q Ñ H b nq . In the case of R ˚ n ´ k,k we have an explicitformula (see [Bo˙z99, Proof of Theorem 2.1]):(2.3) R ˚ n ´ k,k p v b ¨ ¨ ¨ b v n ` k q “ ÿ A Ăr n s , | A |“ n ´ k q i p A q v A b v r n sz A , with the same notation as in (2.2). Bo˙zejko also proved that } R ˚ n,k } ď C p q q as an operatoron H bp n ` k q and, as a consequence, P n ` kq ď C p q q P nq b P kq . It follows that the norm of R n,k : H b nq b H b kq Ñ H bp n ` k q q is bounded by a C p q q ; the same holds for the norm of R ˚ n,k asan operator from H bp n ` k q q to H b nq b H b kq .For future use, we record here some equalities pertaining to operators R ˚ n,k,l . Lemma 2.9.
We have R ˚ n,k,l “ p Id n b R ˚ k,l q R ˚ n,k ` l “ p R ˚ n,k b Id l q R ˚ n ` k,l .Proof. It follows from equalities R n,k ` l p Id n b R k,l q “ R n ` k,l p R n,k b Id l q “ R n,k,l . (cid:3) We also introduce two complex conjugations on H b n : I n given by the formula I n p v b¨ ¨ ¨ b v n q “ Iv b ¨ ¨ ¨ b Iv n and J n given by J n p v b ¨ ¨ ¨ b v n q : “ Iv n b ¨ ¨ ¨ b Iv (this is themodular conjugation); they are both antiunitaries on H b nq . They are related by J n “ I n σ n ,where σ n p v b ¨ ¨ ¨ b v n q : “ v n b ¨ ¨ ¨ b v is a self-adjoint unitary; usually we will just write I and J without the superscripts. With these two conjugations we can associate two differentpairings:(1) m n : H b n b H b n Ñ C given by m n p ξ b η q : “ x J n ξ , η y q ;(2) r m n : H b n b H b n Ñ C given by r m n p ξ b η q : “ x I n ξ , η y q .We will continue to use the same notation for pairings that involve only part of the tensorproduct, i.e. m j might also mean Id n ´ j b m j b Id n ´ j : H b n b H b n Ñ H b n ´ j b H b n ´ j .We can now write a nice formula for the product of two Wick words (cf. [EP03, Theorem3.3]). Proposition 2.10.
Let ξ P H b n and η P H b k . Then we have (2.4) W p ξ q W p η q Ω “ min p n,k q ÿ j “ m j p R ˚ n ´ j,j p ξ q b R ˚ j,k ´ j p η qq . Proof.
We have W p η q Ω “ η . By linearity of the formula we can assume that both ξ and η are simple tensors, i.e. ξ “ ξ b ¨ ¨ ¨ b ξ n and η “ η b ¨ ¨ ¨ b η k . We can use the Wick formula(2.2) to write W p ξ b ¨ ¨ ¨ b ξ n q “ ř A Ăr n s q i p A q a ˚ q p ξ A q a q p I ξ r n sz A q . The action of a ˚ q p ξ A q is verysimple, so we just need to understand a q p I ξ r n sz A qp η b ¨ ¨ ¨ b η k q . Say that | A | “ n ´ j . Then a q p I ξ r n sz A qp η b ¨ ¨ ¨ b η k q will belong to H b k ´ j ; let µ P H b k ´ j . We will compute the innerproduct x µ , a q p I ξ r n sz A qp η b ¨ ¨ ¨ b η k qy q . Because a q p I ξ r n sz A q ˚ “ a ˚ q p J ξ r n sz A q (adjoint reversesthe order), we get that this is equal to x J ξ r n sz A b µ , η y q . It would be easier to compute thisinner product in H b jq b H b k ´ jq instead of H b kq . Because of the formula P kq “ p P jq b P k ´ jq q R ˚ j,k ´ j ,it is possible to switch between the two, at the expense of applying R ˚ j,k ´ j to η . This givesus x J ξ r n sz A b µ , η y q “ r m k ´ j m j p I µ b ξ r n sz A q b R ˚ j,k ´ j p η q . t follows that a q p I ξ r n sz A q η “ m j p ξ r n sz A b R ˚ j,k ´ j p η qq . We can finish the proof by invoking theformula R ˚ n ´ j,j “ ř A Ăr n s , | A |“ n ´ j q i p A q ξ A b ξ r n sz A , combined with the formula for a ˚ q p ξ A q . (cid:3) We will need one more ingredient, crucial for constructing approximating maps on the q -Gaussian algebras. Proposition 2.11 ([BKS97, Theorem 2.11]) . Let T : H R Ñ H R be a contraction on a realHilbert space. Then there exists a unique map on Γ q p H R q , called the second quantisation of T and denoted by Γ q p T q , which satisfies Γ q p T qp W p ξ b ¨ ¨ ¨ b ξ n qq “ W p T ξ b ¨ ¨ ¨ b T ξ n q .Moreover, this map is unital, completely positive and trace-preserving. CMAP and Haagerup’s argument
In this short section we discuss why Theorem 1.1 implies Corollary 1.2, basing on a clas-sical argument of Haagerup. We first need to define the w ˚ -complete metric approximationproperty. Definition 3.1.
Let M be a von Neumann algebra. We say that it has the w ˚ -completemetric approximation property if there exists a net of maps t T i : M Ñ M u i P I that arefinite rank, completely contractive, and lim i P I T i p x q “ x in the w ˚ -topology for any x P M .We denote by P n : Γ q p H R q Ñ Γ q p H R q the projection onto Wick words of length n , i.e. theoperator P n p W p ξ b ¨ ¨ ¨ b ξ m q “ δ nm W p ξ b ¨ ¨ ¨ b ξ m q ; as we will discuss in the next section,it extends to a continuous map on Γ q p H R q . Proof of Corollary 1.2.
We assume that Theorem 1.1 holds, i.e. } P n } cb ď C p q q n . We need todefine the net of approximating maps. We consider T n,t : “ Γ q p e ´ t q Q n , where Q n : “ ř k ď n P k is the projection onto words of length at most n . These maps are finite rank (recall thatwe assume dim H R ă 8 ) and we would like to check that they are (almost) completelycontractive, if we let n depend on t . We have } T n,t } cb ď } Γ q p e ´ t q} cb ` } Γ q p e ´ t qp Id ´ Q n q} cb by the triangle inequality. Clearly } Γ q p e ´ t q} cb ď q p e ´ t qp Id ´ Q n q “ ř k “ n ` e ´ kt P k ,so } Γ q p e ´ t qp Id ´ Q n q} cb ď C p q q ř k “ n ` e ´ kt k . This is a tail of a convergent series, so we canmake it arbitrarily small if we let n be large enough. It will give a bound } T n,t } cb ď ` ε , sothe maps S n,t : “ T n,t } T n,t } cb are completely contractive and for appropriate choice of n and t theydo not differ much from T n,t . In order to check convergence lim S n,t x “ x it suffices to checkit for T n,t . Note that the operators T n,t induce contractions on the level of the L -space,i.e. on the Fock space F q p H q . Since T n,t x sits inside the unit ball, if it converges in the L -norm, it also converges strongly, hence ultraweakly; it suffices to prove the convergencein the L -norm. But the operators T n,t are uniformly bounded, so it is enough to check thisconvergence on a dense subset, and we can choose tensors of finite rank as such a subspace;it is very simple to check the convergence there. (cid:3) Proof of the main result
We would like to show now that the projection P n : Γ q p H R q Ñ Γ q p H R q is completelybounded, and the bound on the cb norm is polynomial in n . In order to proceed, we need anotation for the image of P n – we will denote the space of Wick words of length n by X n . e will first deal with the operator space theoretic considerations and then go on straightto the proof of the combinatorial formula presented in Proposition 4.6. The consistent useof properties of the operators R ˚ n,k (cf. (2.3)) instead of combinatorics of pair partitions willbe key to obtaining a simple proof.4.1. Operator space of Wick words of length n . We start with a non-commutativeKhintchine inequality obtained by Nou (see [Nou04, Theorem 1]), which provides the oper-ator space structure of X n . Theorem 4.1.
Let ξ P B p K q b H b n . Then we have max ď k ď n }p Id b R ˚ n ´ k,k qp ξ q} ď }p Id b W qp ξ q} (4.1) ď C p q qp n ` q max ď k ď n }p Id b R ˚ n ´ k,k qp ξ q} , where the norm }p Id b R ˚ n ´ k,k qp ξ q} is computed in B p K q b min p H b n ´ kq q c b h p H b kq q r . It means that the map ι n : X n Ñ Y n : “ À nk “ p H b n ´ kq q c b h p H b kq q r given by W p ξ q ÞÑp R ˚ n ´ k,k p ξ qq ď k ď n is a completely isomorphic embedding, with distortion at most C p q qp n ` q .Therefore it suffices to prove that the map ι n ˝ P n is completely bounded. It will actuallybe easier to prove that the predual of this map is completely bounded; it is not immediatelyclear that this map should admit a predual, but we will write it down explicitly. So we willwork with the map r Φ n : ℓ ´ n à k “ p H b n ´ kq q r b h p H b kq q c Ñ L p Γ q p H R qq given by r Φ n p ξ , . . . , ξ n q : “ ř nk “ r Φ n ´ k,k p ξ k q with r Φ n ´ k,k p ξ q : “ p W p R n ´ k,k p ξ qqq ˚ . Recall that R n ´ k,k p ξ b ¨ ¨ ¨ b ξ n ´ k b h ξ n ´ k ` b ¨ ¨ ¨ b ξ n q “ ξ b ¨ ¨ ¨ b ξ n . We can use the identification L pp Γ q p H R q op qq » L p Γ q p H R qq given by x ÞÑ x ˚ and take thecomplex conjugate to obtain a mapΦ n : ℓ ´ n à k “ p H b n ´ kq q r b h p H b kq q c Ñ L pp Γ q p H R qq op q given by Φ n p ξ , . . . , ξ n q “ ř nk “ Φ n ´ k,k p ξ k q , where Φ n ´ k,k p ξ q “ W p R n ´ k,k p ξ qq . Remark 4.2.
In order to prove Theorem 1.1, it is enough to prove that } Φ n ´ k,k } cb ď C p q q for any n and k , because this will imply that } Φ n } cb ď C p q q ; the reason is that Φ n isa map defined on an ℓ -direct sum. Moreover, it implies that } ι n ˝ P n } cb ď C p q q , hence } P n } cb ď C p q qp n ` q ď C p q q n , as ι n is an isomorphic embedding with distortion controlledby a multiple of p n ` q .Since it will simplify the notation, we will work with the maps Φ n,k . We will write themdown in terms of some other maps, whose complete boundedness is easy to check. Lemma 4.3 ([Avs11, Lemma 3.4]) . Let v n,k : p H b nq q r b h p H b kq q c Ñ L pp Γ q p H R qq op q be givenby v n,k p ξ b ¨ ¨ ¨ b ξ n b h ξ n ` b ¨ ¨ ¨ b ξ n ` k q : “ W p ξ b ¨ ¨ ¨ b ξ n q W p ξ n ` b ¨ ¨ ¨ b . . . ξ n ` k q . Then } v n,k } cb ď . roof. We can view v n,k as a restriction of the multiplication map on F q p H q r b h F q p H q c . Itwill be convenient to view F q p H q as the L -space of Γ q p H R q . By Proposition 2.3 we havethe identification ´ L p Γ q p H R qq ¯ r b h p L p Γ q p H R qqq c » S p F q p H qq given by (the extension of) x b h y ÞÑ | x yx y | . It is not hard to check that the map from ´ L p Γ q p H R qq ¯ r b h p L p Γ q p H R qqq c to L p Γ q p H R qq given by x b h y ÞÑ x ˚ y is precisely the predual of the inclusion p Γ q p H R qq op ã Ñ B p F q p H qq , given by the right action (traciality of the vacuum state is important here).We get a map from L p Γ q p H R qq b h L p Γ q p H R qq to L p Γ q p H R qq by using the identification p L p Γ q p H R qqq r » ´ L p Γ q p H R qq ¯ r induced by x ÞÑ x ˚ . (cid:3) As mentioned in the introduction, the important combinatorial input will be a formulapresenting a Wick word in terms of products of shorter Wick words. This aim will be achievedusing the following maps.
Lemma 4.4.
Let w jn,k : p H b nq q r b h p H b kq q c Ñ L pp Γ q p H R qq op q (for j ď min p n, k q ) be given by w jn,k “ v n ´ j,k ´ j ˝ p r m j p R ˚ n ´ j,j b R ˚ j,k ´ j qq . Then } w jn,k } cb ď D p q q .Proof. By Proposition 2.3 we know that ` H b n ´ jq ˘ r b h ` H b jq ˘ r » ` H b n ´ jq b H b jq ˘ r (analogouslyfor the column spaces). It follows that the maps R ˚ n ´ j,j : ` H b nq ˘ r Ñ ` H b n ´ jq ˘ r b h ` H b jq ˘ r and R ˚ j,k ´ j : ` H b kq ˘ c Ñ ` H b jq ˘ c b h ` H b k ´ jq ˘ c are completely bounded with cb norms boundedby C p q q . The map r m j : ` H b jq ˘ r b h ` H b jq ˘ c Ñ C is the duality pairing (2). The conjugation I j : H b jq Ñ H b jq is a linear isometry, so it also a complete isometry between the correspondingrow Hilbert spaces. It follows that the cb norm of r m j is equal to the cb norm of the innerproduct, viewed as a map from ´ H b jq ¯ r b h ` H b jq ˘ c . Since ´ H b jq ¯ r b h ` H b jq ˘ c » S p H b jq q andthe pairing is then equal to the trace, it is completely contractive. The map v n ´ j,k ´ j iscompletely contractive by the previous lemma, so we get } w jn,k } cb ď D p q q : “ C p q q . (cid:3) In the next subsection we will show that Φ n,k “ ř min p n,k q j “ α j w jn,k for an appropriate choiceof scalars α j .4.2. The combinatorial formula.
Before we proceed, it is in order to discuss the maindifferences between our approach and Avsec’s (see [Avs11, Remark 3.7–Claim 3.19] for hisproof). We employ properties of the operators P nq , defining the q -deformed inner products,and especially the related operators R ˚ n,k and R ˚ n,k,l to harness the combinatorics; a simplecase of this procedure can be seen in the formula 2.4, which usually features a sum overa certain type of pair partitions. This replaces the intricate analysis of pair partitionsperformed by Avsec, which included introduction of a new crossing number, based on away in which a given pair partition can be build from smaller pair partitions. We, in turn,use simple algebraic properties of the aforementioned operators to arrive at the formula(4.2), in which dependence on the variable j has been drastically diminished. This allowsus to conclude by using a very elementary Lemma 4.5. We also completely avoid the useof ultraproduct embeddings and “colour” operators (see [Avs11, Definition 3.11]). To sum p, our argument uses less sophisticated tools, and the algebraic manipulations involved arefairly elementary and not too involved; the result is a significantly shorter proof.Let us get back to the study of the maps w jn,k . Previously we had to be careful with certainoperator space theoretic identifications, because we had to ensure complete boundedness ofcertain maps. Now our only concern is an algebraic equality, so we will drop most of thedecorations. So now w jn,k will be treated as a map from H b n ` k to F q p H q , where W p ξ q P L pp Γ q p H R qq op q is identified with the corresponding tensor ξ . Since w jn,k gives as an outputa product of two Wick words, we will use the formula (2.4) to make the result of applying w jn,k to a tensor more explicit: w jn,k p ξ b η q “ ÿ ď s ď min p n,k q´ j m s p R ˚ n ´ j ´ s,s b r m j b R ˚ s,k ´ j ´ s qp R ˚ n ´ j,j p ξ q b R ˚ j,k ´ j p η qq . Now we can move the r m j to the left, so that each summand in our formula is of the form p R ˚ n ´ j ´ s,s b Id j b Id j b R ˚ s,k ´ j ´ s qp R ˚ n ´ j,j p ξ q b R ˚ j,k ´ j p ξ qq followed by the pairing r m j and then m s . By Lemma 2.9 we can therefore write the end resultas w jn,k p ξ b η q “ ÿ ď s ď min p n,k q´ j m s r m j p R ˚ n ´ j ´ s,s,j p ξ q b R ˚ j,s,k ´ j ´ s p η qq . This is already nice, but we can make it even nicer by introducing a new variable p “ j ` s .Our next aim is to transform the two pairings m s and r m j into a single pairing m s ` j . Inorder to do that, we recall the formula r m j “ m j p Id b σ j q , so we can write w jn,k p ξ b η q “ ÿ ď s ď min p n,k q´ j m s m j p R ˚ n ´ j ´ s,s,j p ξ q b p σ j b Id k ´ j q R ˚ j,s,k ´ j ´ s p η qq . Now we want to compare the pairings m s m j and m s ` j . The former is actually computingthe inner product in H b sq b H b jq and the latter in H b s ` jq . We know that the two are relatedby the operator R ˚ s,j . We have to be slightly careful, because the pairings involve not onlythe inner product, but also the complex conjugation J , which reverses the order. Takingthat into account, we obtain a formula m s ` j “ m s m j p R ˚ s,j b Id s ` j q . We can now use Lemma 2.9 to write R ˚ n ´ j ´ s,s,j “ p Id n ´ p b R ˚ s,j q R ˚ n ´ p,p . Similarly, we get R ˚ j,s,k ´ j ´ s “ p R ˚ j,s b Id k ´ p q R ˚ p,k ´ p . Therefore(4.2) w jn,k p ξ b η q “ ÿ j ď p ď min p n,k q m p p R ˚ n ´ p,p p ξ q b pp σ j b Id p ´ j q R ˚ j,p ´ j b Id k ´ p q R ˚ p,k ´ p p η qq . Now the variable j only appears in a few places. Our goal was to obtain a formula Id n ` k “ ř j α j w jn,k ; we can write the right hand side as(4.3) ÿ j α j w jn,k “ ÿ ď p ď min p n,k q m p p Id n b ˜ÿ j ď p α j p σ j b Id p ´ j q R ˚ j,p ´ j ¸ b Id k ´ p qp R ˚ n ´ p,p b R ˚ p,k ´ p q . Note that for different p ’s we get tensors of different length, so it is necessary and sufficientto check that ř j ď p α j p σ j b Id p ´ j q R ˚ j,p ´ j is equal to Id n ` k for p “ α “
1) and equal to 0 for p ě
1. This is what we plan to do. ecall that the operator R ˚ j,p ´ j is a weighted sum of certain permutations, where theweights are of the form q i p π q . Let us be more specific. If A “ t i ă ¨ ¨ ¨ ă i j u then thepermutation is given by π p l q “ i l for l ď j and it is increasing for l ą j . The numberof inversions is then i p π q “ ř jl “ p i l ´ l q . If we then apply σ j b Id p ´ j , the permutation is r π p l q “ i j ´ l ` for l ď j and at stays the same for l ą j . So we get permutations, whoserestrictions to t , . . . , j u are decreasing and restrictions to t j ` , . . . , p u are increasing.It turns out that a permutation of this form can arise only in two ways. Lemma 4.5.
Let S jn be the set of permutations of r n s such that the restriction to t , . . . , j u is decreasing and the restriction to t j ` , . . . , n u is increasing. Let π P S jn . Suppose that σ P S kn (for k ‰ j ) satisfies σ “ π . Then k “ j ´ or k “ j ` and only one of thesesituations occurs.Proof. Suppose that σ P S kn for k ě j `
2. Then σ restricted to t j ` , . . . , n u is not increasing.We argue similarly for k ď j ´ π be given by a subset A “ t i ă ¨ ¨ ¨ ă i j u . If i “ σ P S j ` n such that σ “ π . Indeed, we have σ p j q “
1, so σ p j ` q ą σ p j q , even though σ wassupposed to decrease on t , . . . , j ` u . On the other hand, there is a permutation σ P S j ´ n such that σ “ π ; simply let σ p j q “ t j, . . . , n u . In case i ą σ P S j ` n such that σ “ π ; the proof is similar. (cid:3) What remains to be done is the appropriate choice of coefficients α j . We need cancellations,so the signs must alternate and we will work with powers of q , so it will be easier to work witha coefficient β j such that α j “ p´ q j q β j . Suppose that we have a set A “ t i ă ¨ ¨ ¨ ă i j u with i p A q “ ř jl “ p i l ´ l q . In case i “ A : “ t i ă ¨ ¨ ¨ ă i j u with i p A q “ ř jl “ p i l ´ p l ´ qq “ ř jl “ p i l ´ l q ` p j ´ q “ i p A q ` j ´ β j ´ ` p j ´ q “ β j . Since we know that β “ β j “ ` j ˘ . We just have to check that these coefficients also work in thecase 1 R A . Then the set r A : “ t ă i ă ¨ ¨ ¨ ă i j u yields the same permutation and i p r A q “ ř jl “ p i l ´p l ` qq “ i p A q´ j . So the compatibility condition in this case is β j ` ´ j “ β j ,i.e. β j ` “ β j ` j , which is satisfied by our choice. Thus we have proved the followingproposition. Proposition 4.6.
We have Φ n,k “ ř min p n,k q j “ p´ q j q p j q w jn,k .Proof of Theorem 1.1. We want to find a bound for the cb norm of the projection P n ontoWick words of length n . The Khintchine inequality (4.1) shows that this subspace is com-pletely isomorphic (with distortion linear in n ) to a certain well behaved operator space; adirect sum of Haagerup tensor products of row and column Hilbert spaces. A modificationof the predual of this map is Φ n , which is defined using Φ n,k (see Remark 4.2); it sufficesto prove that } Φ n,k } cb is bounded by a constant depending only on q . By the previousproposition we have } Φ n,k } cb ď min p n,k q ÿ j “ | q |p j q} w jn,k } cb ď ˜ ÿ j “ | q |p j q ¸loooooomoooooon “ C p q q D p q q , where the bound for } w jn,k } cb comes from Lemma 4.4. (cid:3) eferences [Avs11] Stephen Avsec. Strong Solidity of the q -Gaussian Algebras for all ´ ă q ă
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