BMO spaces of σ -finite von Neumann algebras and Fourier-Schur multipliers on S U q (2)
aa r X i v : . [ m a t h . OA ] N ov BMO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS ANDFOURIER-SCHUR MULTIPLIERS ON SU q (2) MARTIJN CASPERS AND GERRIT VOS
Abstract.
We consider semi-group BMO spaces associated with an arbitrary σ -finite von Neu-mann algebra ( M , ϕ ). We prove that BMO always admits a predual, extending results from thefinite case. Consequently, we can prove - in the current setting of BMO - that they are Banachspaces and they interpolate with L p as in the commutative situation, namely [BMO( M ) , L ◦ p ( M )] /q ≈ L ◦ pq ( M ). We then study a new class of examples. We introduce the notion of Fourier-Schur mul-tiplier on a compact quantum group and show that such multipliers naturally exist for SU q (2). Introduction
Spaces of functions with Bounded Means of Oscillation (BMO spaces) play an eminent rolein the theory of harmonic analysis. They serve as so-called ‘end-point spaces’ for many naturaloperators in harmonic analysis including singular integral operators and Fourier multipliers, see[Gra09]. More precisely, many singular integral operators and Fourier multipliers like the Riesz orHilbert transform act boundedly as operators L p → L p , < p < ∞ and at the boundary extendto bounded maps L ∞ → BMO. We call the latter bound an end-point estimate. Such endpointestimates have several applications; one of the most important ones being that after interpolationthey immediately yield L p -boundedness with sharp constants.For some singular integrals, like the Riesz and Hilbert transform, BMO spaces even provideoptimal endpoint spaces. We mean this in the following sense (see [FS72], [Ste70]). Consider theHardy-space H . By the celebrated Fefferman-Stein duality we have ( H ) ∗ ≈ BMO. Then theHilbert transform is bounded H → L . Moreover, the graph norm of the Hilbert transform asan unbounded map L → L is equivalent to the H -norm (see [Gra09, Section 6.7.4]). The sameholds for the Riesz transform(s) if one takes all possible coordinates into account.These and other results show that BMO and Hardy spaces occur naturally in the theory ofsingular integrals and their duality is of fundamental importance.In the current paper we take a non-commutative viewpoint on BMO and Hardy spaces. Inthis case the classical approach to BMO using cubes to measure the oscillation is replaced by ananalysis of Markov semi-groups (in the commutative case diffusion semi-groups). In the commu-tative situation these ideas go back (at least) to [Var85], [SV74]. Much more recently an analysisof duality and comparison of several such BMO-spaces was carried out in [DY05a], [DY05b].The introduction of non-commutative semi-group BMO spaces was done by Mei [Mei08] andfurther developed by Junge-Mei in [JM12]. Their work is precedented by the theory of martingaleBMO spaces [PX97], [Pop00], [Mus03], [JM07] and [JP14]. Most notably in the appendix of [PX97]a duality ( H ) ∗ = BMO is proven for a suitable notion of a Hardy space. Such martingale BMOspaces require the existence of a filtration of the von Neumann algebra. Many of the concrete cases Date : December 1, 2020. MSC2020: 43A15, 46L67, 46L51. Keywords: compact quantum groups, BMO-spaces,Hardy spaces, Fourier and Schur multipliers. MC and GV are supported by the NWO Vidi grant ‘Noncommutativeharmonic analysis and rigidity of operator algebras’, VI.Vidi.192.018. of martingale BMO spaces concern semi-classical von Neumann algebras (i.e. tensor products witha commutative von Neumann algebra) or a vector-valued situation where the filtration still comesfrom a commutative space. For some applications this structure is insufficient, see e.g. [JMP14],[Mei17], [Cas19], [CJSZ20] and one requires a true non-commutative version of BMO.Here we shall take the approach to BMO from [Mei08], [JM12] as a starting point. It assumesthe existence of a Markov semi-group Φ = (Φ t ) t ≥ on a finite (or semi-finite) von Neumannalgebra ( M , τ ), see Definition 4.1. [JM12] considers various BMO-norms associated with this andits subordinated Poisson semigroup. We only consider the norm k · k BMO Φ (or k · k BMO(Φ) in thenotation of [JM12]). For x ∈ L ( M ) the column BMO-seminorm is then defined as k x k c Φ = sup t ≥ k Φ t ( | x − Φ t ( x ) | ) k ∞ , (1.1)where the Markov maps Φ t extend naturally to L ( M ) and L ( M ). Then BMO c ( M , Φ) is definedas the space of elements from L ( M ) (minus some degenerate part) where the norm (1.1) is finite.Finally, BMO( M , Φ) is the intersection of BMO c ( M , Φ) and its adjoint space.[JM12] establishes the natural interpolation results between BMO and L p by making use ofMarkov dilations and interpolation results for martingale BMO spaces. In the more generalcontext of σ -finite von Neumann algebras a parallel study was carried out in [Cas19] which againobtains such interpolation results through the Haagerup reduction method [HJX10] and the finitecase [JM12]. Both papers do this for several of the various BMO-norms defined in [JM12]. Themain advantage of considering the BMO-norm (1.1) as opposed to the norm k · k bmo Φ is thatthe Markov dilation is not required to have a.u. continuous path in order to apply complexinterpolation.There is a very subtle but important point that makes a difference between the current paperand [Cas19]. In [Cas19] BMO is defined by only considering x in M and then taking an abstractcompletion with respect to the norm (1.1) (or one of the other BMO-norms). This ‘smaller BMOspace’ has the benefit that basic properties like the triangle inequality and completeness followrather easily. Here we stay closer to the ‘larger BMO space’ of L -elements with finite BMO-norm as defined above, and show that these basic properties still hold. We do this by proving aFefferman-Stein duality result.The contribution of this paper is twofold. Firstly, we study abstract BMO spaces of σ -finitevon Neumann algebras and prove that again an H -BMO duality theorem holds as for the tracialsetting. The proof parallels the tracial proof in [JMP14]. The main difficulty lies in the fact that L p spaces beyond tracial von Neumann algebras do not naturally intersect and we must deal withTomita-Takesaki modular theory to define suitable compatible couples.It should be mentioned that the H Hardy space we construct here is abstract in nature andthe question of whether every BMO space has a natural Hardy space as its predual remains open.We refer to [Mei08] and [JM12, Open problems, p. 741] for details about this question, where itwas resolved under additional assumptions on the semi-group.
Theorem 1.1.
There exists a Banach space h ( M , Φ) such that BMO( M , Φ) ∼ = h ( M , Φ) ∗ . Within the construction of the predual we need some L p -module theory - see [Pas73] and [JS05].In particular, we need to extend some results to the σ -finite case. We give an introduction to thetheory and prove the necessary results in Section 3.The existence of this predual then settles important basic properties of the ‘larger BMO space’,namely the triangle inequality and completeness of the normed space. MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 3 Corollary 1.2.
BMO( M , Φ) is a Banach space. Finally, we show that the interpolation result of [Cas19] still holds for the larger BMO spaceand extends [JM12] beyond the tracial case. We refer to Appendix B and [JM12], [Cas19] for thedefinition of a standard Markov dilation.
Theorem 1.3. If Φ is ϕ -modular and admits a ϕ -modular standard Markov dilation, then for all ≤ p < ∞ , < q < ∞ , [BMO( M , Φ) , L ◦ p ( M )] /q ≈ pq L ◦ pq ( M ) . We note that the modularity assumptions are only needed to carry out the Haagerup reductionmethod as in [Cas19]. Many natural Markov semi-groups are modular or can be averaged to amodular Markov semi-group in case ϕ is almost periodic, see [CS15, Proposition 4.2], [OT15,Theorem 4.15].The second contribution we make consists of concrete examples for compact quantum groups.Theorem 1.3 as well as our construction of the predual h ( M , Φ) open the way for L p -boundednessresults on a wider range of multipliers. We give an application for multipliers on the quantumgroup SU q (2). In Section 5, we define Fourier-Schur multipliers on quantum groups which is ananalogue of Fourier multipliers on group von Neumann algebras. Definition 1.4.
Let G be a compact quantum group and T : Pol( G ) → Pol( G ) a linear map. Wecall T a Fourier-Schur multiplier if the following condition holds. Let u be any finite dimensionalcorepresentation on H . Then there exists an orthogonal basis e i such that if u i,j are the matrixcoefficients with respect to this basis, then there exist numbers c i,j := c ui,j ∈ C such that T u i,j = c i,j u i,j . In this case ( c ui,j ) i,j,u is called the symbol of T .Basically, Fourier-Schur multipliers are Schur multipliers acting on the Fourier domain. Weconsider Fourier-Schur multipliers on G q := SU q (2) , q ∈ ( − , \{ } associated with completelybounded Fourier multipliers on the torus T .The semigroups we use to define BMO are the Heat semi-group on T and the Markov semigroupΦ on G q constructed in Section 5.6. We use the shorthand notation BMO( T ), BMO( G q ) for theassociated BMO spaces; see again Section 5.6. Theorem 1.5.
Let m ∈ ℓ ∞ ( Z ) be such that the Fourier multiplier T m : L ∞ ( T ) → BMO( T ) iscompletely bounded. Let ˜ T m : Pol( G q ) → Pol( G q ) be the Fourier-Schur multiplier with symbol ( m ( − i − j )) i,j with respect to the basis described in (5.5) . Then ˜ T m extends to a bounded map ˜ T ( ∞ ) m : L ∞ ( G q ) → BMO( G q ) . Moreover k ˜ T ( p ) m : L ∞ → BMO( G q ) k ≤ k T m : L ∞ ( T ) → BMO( T ) k cb . Using the interpolation results of Section 4.3, i.e. Theorem 1.3, also the corresponding L p → L p follow. This is proved in Theorem 5.16.In the proof we use our H -BMO duality principle to show that Fourier-Schur multipliers extendfrom the weak- ∗ dense subalgebra of matrix coefficients of irreducible unitary corepresentations.The other important ingredient is a transference principle.In the appendix, we give some comments on the operator space structures on BMO. Also, weprove that the semigroup we use for the definition of our BMO space has a Markov dilation. MARTIJN CASPERS AND GERRIT VOS
Structure of the paper.
In Section 2 we fix preliminary notation and introduce non-commutative L p -spaces associated with σ -finite von Neumann algebras. Section 3 is devoted to L p -moduletheory. We generalize some of the existing results from the tracial to the σ -finite case in order toapply them in the subsequent sections. Section 4 introduces BMO-spaces of σ -finite von Neumannalgebras. We prove that they have a predual and gather its corollaries. In other words, we proveTheorem 1.1 and Corollary 1.2. In Section 4.3 we prove the interpolation result of Theorem1.3. The proof is the same as in [Cas19] provided that we can prove that an inclusion of avon Neumann algebra with expectation yields a 1-complemented BMO-subspace (this point wasalready surprisingly subtle in [Cas19]). We give full details of this fact in Section 4.3. In Section5 we turn to the examples. We introduce Fourier-Schur multipliers and show how to constructthem on SU q (2). Finally, in the Appendix we gather results on operator space structures andMarkov dilations. 2. Preliminaries
General notation.
We use the convention N = Z ≥ . Following the convention in theliterature for L p -modules, inner products are linear in the second component and antilinear inthe first. Dual actions are sometimes linear and sometimes antilinear (namely in the case of L p -modules); whenever something is antilinear this will be explicitly mentioned.2.2. Operator theory.
We use the following notation for tensor products: • A ⊗ B for the algebraic tensor product of vector spaces. • M ¯ ⊗N for the von Neumann algebraic tensor product. • A ⊗ min B for the minimal tensor product of C ∗ -algebras. • H ⊗ K for the Hilbert space tensor product.For general von Neumann algebra theory we refer to [Mur90] or Takesaki’s books [Tak02],[Tak03a], [Tak03b]. For the theory of operator spaces, see [ER00] and [Pis03]. The followingstandard result shall be used several times in this paper. The proof follows directly from thedefinitions. Proposition 2.1 (See [Con90]) . Let
X, Y be Banach spaces and T : X → Y a bounded linearmap. Then T ∗ : Y ∗ → X ∗ is weak- ∗ /weak- ∗ continuous, i.e. normal. Using this (and [Sak71, Chapter 1.22]) one proves that tensoring with the identity preservesnormality. More precisely, for von Neumann algebras M , N and a completely bounded normaloperator T : M → M , the map 1 N ⊗ T extends to a normal operator on N ¯ ⊗M . Convention:
All von Neumann algebras are assumed to be σ -finite. We will remind the readerof this convention a number of times in this paper.2.3. Compatible couples.
We will need the theory of compatible couples of Banach spaces inour definition of BMO-spaces in section 4. An extensive treatment can be found in [BL76].
Definition 2.2.
A pair of Banach spaces ( A , A ) are called a compatible couple if both arecontinuously embedded in some Banach space A .We will always keep track of the embeddings i : A → A and i : A → A . One can definenorms on the ‘intersection space’ i ( A ) ∩ i ( A ) and ‘sum space’ i ( A ) + i ( A ) by k a k ∩ := max {k a k A , k a k A } , a = i ( a ) = i ( a ) ∈ i ( A ) ∩ i ( A ) k a k + = inf {k a k A + k a k A | i ( a ) + i ( a ) = a } , a ∈ i ( A ) + i ( A ) . MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 5 These norms turn the intersection and sum spaces into Banach spaces.Let ( B , B ) be another compatible couple with embeddings j : B → B , j : B → B (wewill only need the case B = B = B , but we state it here in full generality). A pair of boundedmaps T : A → B , T : A → B is called compatible if they coincide on the intersection, i.e. j ( T ( a )) = j ( T ( a )) , whenever i ( a ) = i ( a ) . If ( T , T ) are compatible morphisms, then there exists a unique map T : i ( A ) + i ( A ) → j ( B ) + j ( B ) ‘extending’ T and T , i.e. T ( i ( a )) = T ( a ) , T ( i ( b )) = T ( b ) , a ∈ A , b ∈ A . If ( A , A ) is a compatible couple as above, then the dual spaces ( A ∗ , A ∗ ) also form a compatiblecouple through their embedding in ( i ( A ) ∩ i ( A )) ∗ say through embeddings ι , ι . Proposition 2.3. [BL76, Theorem 2.7.1]
We have ι ( A ∗ ) ∩ ι ( A ∗ ) ∼ = ( i ( A ) + i ( A )) ∗ ,ι ( A ∗ ) + ι ( A ∗ ) = ( i ( A ) ∩ i ( A )) ∗ , where the first identification is an isometric isomorphism. L p -spaces of σ -finite von Neumann algebras. L p -spaces corresponding to arbitrary vonNeumann algebras have been constructed by Haagerup [Haa79] (see also [Ter81]) and Connes-Hilsum [Con80], [Hil81] (see also Kosaki [Kos84] in the σ -finite case). Here we will use theConnes-Hilsum definition.Essential in the Connes-Hilsum construction is Connes’ spatial derivative - see [Con80], [Ter81].Let M ⊆ B ( H ) be a von Neumann algebra. Let ψ be any fixed normal, semifinite faithful weighton the commutant M ′ . For a normal, semifinite weight φ on M , the spatial derivative is an(unbounded) positive (self-adjoint) operator denoted by D φ := dφ/dψ. Remark 2.4.
The choice of ψ will up to isomorphism not affect any of the constructions below.In particular it will yield isometrically isomorphic non-commutative L p -spaces. We will assumehenceforth that a choice for ψ has been made implicitly and suppress it in the notation. Remark 2.5.
In this paper we only deal with σ -finite von Neumann algebras M : von Neumannalgebras with a normal faithful state. In this case we may assume that M ′ is σ -finite as well, forexample by considering the standard form of M [Tak03a]. This way we may assume that ψ isa faithful normal state and we shall not require the general theory of weights on von Neumannalgebras.The spatial derivative of a faithful normal state φ on M implements the modular automorphismgroup: σ φt ( x ) = D itφ xD − itφ , x ∈ M , t ∈ R . (2.1)We define the Tomita algebra T φ = { x ∈ M | t σ φt ( x ) extends analytically to C } . By [Tak03a, Lemma VIII.2.3] T φ is a σ -weakly dense ∗ -subalgebra of M . Hence it is also σ -strong-* dense. MARTIJN CASPERS AND GERRIT VOS
For 1 ≤ p < ∞ the space L p ( M ) is defined as all operators x on H such that | x | p = D φ forsome φ ∈ M + ∗ . We define a trace on L ( M ) as follows: let x ∈ L ( M ) + and φ ∈ M + ∗ be suchthat x = D φ . Then Tr( x ) := φ (1) . The trace is then extended to L ( M ) through the decomposition of an arbitrary operator into alinear combination of four positive operators. The norm on L p ( M ) is given by k x k p = Tr( | x | p ) /p .Further set L ∞ ( M ) := M .Let a, b ∈ L p ( M ) , c ∈ L q ( M ) with 1 ≤ p, q ≤ ∞ . Then a + b and ac are densely definedand preclosed. Their respective closures are called the strong sum and strong product and willsimply be denoted by a + b and ac . With these conventions a + b ∈ L p ( M ) (turning L p ( M ) into aBanach space) and ac ∈ L r ( M ) for r := p + q with r ≥
1. Moreover, we have the H¨older/Kosakiinequality: k ac k r ≤ k a k p k c k q . In case r = 1 we have the trace property Tr( ac ) = Tr( ca ) [Ter81, Proposition IV.13]. Remark 2.6. L p ( M ) may also be defined in the same way for 0 < p <
1. It is not a normedspace though. All we shall need in the current paper is that for ≤ p < L p ( M ) and the square root of a positive element in L p ( M ) is in L p ( M ).For x ∈ L p ( M ) we have x ∗ ∈ L p ( M ) with k x k p = k x ∗ k p [Ter81, Prop IV.8]. In particular, k x ∗ x k / p/ = k x k p = k x ∗ k p = k xx ∗ k / p/ . (2.2)There exists a duality pairing between L p ( M ) and L q ( M ) given by h x, y i = Tr( xy ) , x ∈ L q ( M ) , y ∈ L p ( M ) , for 1 ≤ p < ∞ and p + q = 1. This induces an isometric isomorphism L p ( M ) ∗ ∼ = L q ( M ).2.5. Compatible couples of L p -spaces. For finite von Neumann algebras, we have inclusions L p ( M ) ⊆ L q ( M ) for ≤ q ≤ p ≤ ∞ . In the σ -finite case, the L p -spaces are not included ineach other as sets of operators on Hilbert spaces. However, they can be turned into a scale ofcompatible couples as follows.Let M be a σ -finite von Neumann algebra. Fix a normal faithful state ϕ on M . Fix − ≤ z ≤ x ∈ M , ≤ p ≤ ∞ we have D ( − z ) p ϕ xD ( + z ) p ϕ ∈ L p ( M ) , and such elements are dense. For ≤ q ≤ p ≤ ∞ there are contractive embeddings κ ( z ) p,q : L p ( M ) → L q ( M ) : D ( − z ) q ϕ xD ( + z ) q ϕ D ( − z ) p ϕ xD ( + z ) p ϕ , x ∈ M . Using the embeddings κ ( z ) p, we may view L p ( M ) as a (dense) subspace of L ( M ) and hence thisturns all L p ( M ) , ≤ p ≤ ∞ simultaneously into a ( z -dependent) scale of compactible couples.For x, y ∈ L q ( M ) and 1 ≤ q ≤ p ≤ ∞ we have κ ( z ) q,p ( x ) ∗ = κ ( − z ) q,p ( x ∗ ) , κ ( − q,p ( x ) κ (1) q,p ( y ) = κ (0) q/ ,p/ ( xy ) . (2.3) MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 7 The embedding κ ( z ) ∞ , is ‘state-preserving’ when we consider the trace on L ( M ):Tr( κ ( z ) ∞ , ( x )) = Tr( xD ϕ ) = ϕ ( x ) , x ∈ M . (2.4)Indeed, for x ∈ M + this follows from [Ter81, Theorem III.14] and then use linearity for general x . The following proposition is a special case of [HJX10, Theorem 5.1, Proposition 5.5]. Proposition 2.7.
Let T : M → M be a unital completely positive (ucp) ϕ -preserving map suchthat T ◦ σ ϕt = σ ϕt ◦ T, t ∈ R . Then T extends to a positive contraction T ( p ) : L p ( M ) → L p ( M ) for ≤ p < ∞ satisfying T ( p ) ( κ ( z ) ∞ ,p ( x )) = κ ( z ) ∞ ,p ( T ( x )) , x ∈ M , which is independent of the choice of − ≤ z ≤ . Additionally, T (1) is trace-preserving.Proof. We prove only the last statement. Consider first x = x ′ D ϕ ∈ L ( M ) for x ′ ∈ M . With(2.4) we have Tr( T (1) ( x )) = Tr( T ( x ′ ) D ϕ ) = ϕ ( T ( x ′ )) = ϕ ( x ′ ) = Tr( x ) . For general x ∈ L ( M ) the statement follows by approximation. (cid:3) We recall that on the unit ball of M the strong topology coincides with the k k -topologygenerated by the GNS inner product h x, y i = ϕ ( x ∗ y ) , x, y ∈ M . The following continuity propertythen follows from [JS05, Lemma 1.3]. Proposition 2.8.
Let a λ ∈ M be a bounded net converging to in the strong topology. Then forany ≤ p < ∞ and x ∈ L p ( M ) : k a λ x k p → . L p -module theory and duality results In this section we recall some L p -module theory as introduced in [JS05]. This theory buildsupon the theory of Hilbert C ∗ -modules, see e.g. [Pas73], [Lan95]. It is also [Pas73] that introducesthe ‘GNS module’ corresponding to a completely positive map. In the second part of this section,we extend some duality results to the σ -finite case; specifically, the duality relations of the L p -module corresponding to the GNS modules. In Section 4, we will use these results to construct apredual for BMO in the σ -finite case.In the entire section M is a σ -finite von Neumann algebra with faithful normal state ϕ .3.1. General theory of L p -modules.Definition 3.1. Let 1 ≤ p < ∞ . A sesquilinear form h· , ·i : X × X → L p/ ( M ) on a right M -module X is called an L p/ -valued inner product if it satisfies for x, y ∈ X and a ∈ M :(i) h x, x i ≥ h x, x i = 0 ⇐⇒ x = 0,(iii) h x, y i = h y, x i ∗ ,(iv) h x, ya i = h x, y i a.X is called an L p M -module if it is complete with respect to the norm k x k := kh x, x ik / p/ .X is called an L ∞ M -module if it has a L ∞ -valued inner product and is complete in the topologygenerated by the seminorms x ω ( h x, x i ) / , ω ∈ M + ∗ . MARTIJN CASPERS AND GERRIT VOS
We call this the STOP topology (after [JM12]).
Lemma 3.2. [JS05, Proposition 2.2]
For x, y ∈ X there exists some T ∈ M with k T k ≤ suchthat h x, y i = h x, x i T h y, y i . This implies the ‘ L p -module Cauchy Schwarz inequality’: kh x, y ik p ≤ k x kk y k . Remark 3.3.
The norms defined here are a priori only quasinorms. However, Theorem 3.6 willshow that they are in fact norms.An important class of L p M -modules are the so-called principal L p -modules . Recall the columnspace L p ( M ; ℓ C ( I )) defined for 1 ≤ p < ∞ as the norm closure of finite sequences x = ( x α ) α ∈ I with respect to the norm k x k L p ( M ; ℓ C ) := k ( X α ∈ I | x α | ) / k p . These spaces are isometrically isomorphic to L p ( M ¯ ⊗B ( ℓ ( I ))) e , , the column subspace of L p ( M ¯ ⊗B ( ℓ ( I ))),via ( x α ) x . . .x . . . ... ... . For p = ∞ , we take the space of all sequences in L ∞ ( M ) such that its image under the abovemap is in L ∞ ( M ¯ ⊗B ( ℓ ( I ))). See [PX97] for more details about the above construction.Now let 1 ≤ p ≤ ∞ be fixed, I be some index set and ( q α ) α ∈ I ∈ M be a set of projections.Consider the closed subspace X p = { ( x α ) α ∈ I : x α ∈ q α L p ( M ) , X α ∈ I x ∗ α x α ∈ L p/ ( M ) } ⊆ L p ( M ; ℓ C ( I )) . We define an L p/ -valued inner product on X p by h x, y i = X α ∈ I ( x α ) ∗ y α . We refer to [JS05] for the fact that this is indeed a well-defined L p/ -valued inner product. Thismakes X p into an L p M -module. We call X p a principal L p -module and denote it by L I q α L p ( M ).Note that we have the isometric isomorphism M I q α L p ( M ) ∼ = QL p ( M ¯ ⊗B ( ℓ ( I ))) e , , Q = q . . . q . . . ... ... . . . . (3.1)This equation combined with the following general lemma (which has nothing to do with L p -modules) will show that the family of principal L p -modules L I q a L p ( M ), 1 ≤ p ≤ ∞ , satisfiesthe expected duality relations (although the identifications become antilinear). Lemma 3.4.
Let N be a σ -finite von Neumann algebra and let P, Q ∈ N projections. Then for ≤ p < ∞ , p + p ′ = 1 we have the following antilinear isomorphism: ( QL p ( N ) P ) ∗ ∼ = QL p ′ ( N ) P. MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 9 Proof.
Let 1 ≤ p < ∞ . Define S p := QL p ( N ) P ⊆ L p ( N ). It follows (see for instance [Con90,Theorem III.10.1]) that S ∗ p ∼ = L p ′ ( N ) /S ⊥ p , where S ⊥ p = { b ∈ L p ′ ( N ) : Tr( S p b ) = 0 } . Hence itsuffices to prove L p ′ ( N ) /S ⊥ p ∼ = QL p ′ ( N ) P .Let a ∈ L p ( N ), b ∈ L p ′ ( N ). Then Tr(( QaP ) b ) = Tr( a ( P bQ )), hence for b ∈ L p ′ ( N ): b ∈ S ⊥ p ⇐⇒ P bQ = 0 ⇐⇒ Qb ∗ P = 0 . Therefore if we define the surjective mapΨ : L p ′ ( N ) → QL p ′ ( N ) P, b Qb ∗ P, then ker Ψ = S ⊥ p and hence the induced map Φ : L p ′ ( N ) /S ⊥ p → QL p ′ ( N ) P is an isomorphism. Ψis contractive, hence Φ is also contractive. Conversely, for b ∈ L p ( N ), we have P ( b − P bQ ) Q = P bQ − P bQ = 0 , hence b − P bQ ∈ S ⊥ p , or in other words P bQ ∈ b + S ⊥ p . Thus k Qb ∗ P k = k P bQ k ≥ k b + S ⊥ p k . This implies that Φ − is also contractive, so Φ is an isometric isomorphism. (cid:3) Corollary 3.5.
Let ( q α ) α ∈ I be some family of projections. Then for ≤ p < ∞ , p + p ′ = 1 , wehave an antilinear isometric identification ( M I q α L p ( M )) ∗ ∼ = M I q α L p ′ ( M ) . The main theorem concerning L p -modules states that every L p -module is in fact isomorphic toa principal L p -module. Theorem 3.6 (Theorem 2.5 of [JS05]) . Let X be a right L p M -module. Then there exists someindex set I and projections ( q α ) α ∈ I ∈ M such that X ∼ = M α ∈ I q α L p ( M ) . The following lemma allows us to transfer the duality results for principal L p -modules to generalfamilies of L p -modules satisfying certain requirements. The lemma is essentially copied from[JP14, Corollary 1.13] with some adjustments to go from the finite to the σ -finite case. It is infact slightly more general to circumvent difficulties with finding an embedding X ∞ ֒ → X p . Lemma 3.7.
Let ( X p ) ≤ p ≤∞ be a family of L p M -modules. Assume that there exist maps I q,p : X q → X p and I ∞ ,p : A → X p for some subset A ⊆ X ∞ satisfying for ≤ p < r < q ≤ ∞ :i) I q,p ( xa ) = I q,p ( x ) σ ϕi ( p − q ) ( a ) for x ∈ X q , a ∈ T ϕ ,ii) I q,r ◦ I r,p = I q,p ,iii) κ (0) q/ ,p/ ( h x, y i X q ) = h I q,p ( x ) , I q,p ( y ) i X p for x, y ∈ X q ,iv) I ∞ ,p ( A ) is dense in X p .Then there exists a family of projections ( q α ) α ∈ I ∈ M such that X p ∼ = L α ∈ I q α L p ( M ) , ≤ p ≤∞ . Proof.
We give details only for those parts that differ from [JP14, Corollary 1.13]. One showsthat the maps I p,q are automatically contractive embeddings. By applying Theorem 3.6 (whichholds for σ -finite von Neumann algebras) to the p = ∞ case we acquire projections ( q α ) suchthat X ∞ ∼ = L α ∈ I q α L ∞ ( M ), say through an isometric isomorphism of L ∞ -modules ϕ ∞ . For1 ≤ p < ∞ , the embeddings I ∞ ,p allow us to ‘transfer’ this map to X p : ϕ p : I ∞ ,p ( A ) → M α ∈ I q α L p ( M ) , ϕ p ( I ∞ ,p ( x )) = M α ∈ I κ (1) ∞ ,p ( ϕ ∞ ( x ) α ) = M α ∈ I ϕ ∞ ( x ) α D /pϕ . We show that ϕ p preserves inner products; for x, y ∈ A : h ϕ p ( I ∞ ,p ( x )) , ϕ p ( I ∞ ,p ( y )) i L q α L p = X α D /pϕ ( ϕ ∞ ( x ) α ) ∗ ϕ ∞ ( x ) α D /pϕ = κ (0) ∞ ,p/ ( h ϕ ∞ ( x ) , ϕ ∞ ( y ) i L q α L ∞ = κ (0) ∞ ,p/ ( h x, y i X ∞ ) = h I ∞ ,p ( x ) , I ∞ ,p ( y ) i X p . Since I ∞ ,p ( A ) is dense in X p , ϕ p extends to an isometric homomorphism on X p . It turns out to bean isomorphism since we can use a similar argument to construct an inverse. Next we show that ϕ p preserves the module structure (this was not an issue in the finite case); for x ∈ A , a ∈ T ϕ : ϕ p ( I ∞ ,p ( x ) a ) = ϕ p ( I ∞ ,p ( xσ ϕ − ip ( a ))) = M α ∈ I ϕ ∞ ( xσ ϕ − ip ( a )) α D /pϕ = M α ∈ I ϕ ∞ ( x ) α σ ϕ − ip ( a ) D /pϕ = M α ∈ I ϕ ∞ ( x ) α D /pϕ a = ϕ p ( I ∞ ,p ( x )) a. (3.2)Now let a ∈ M be arbitrary. By Kaplansky and strong density of T ϕ in M , we may choose abounded net ( a λ ) λ in T ϕ converging to a in the strong topology. Then by Proposition 2.8 we have k I ∞ ,p ( x )( a − a λ ) k X p = k ( a − a λ ) ∗ h I ∞ ,p ( x ) , I ∞ ,p ( x ) i X p ( a − a λ ) k / p/ → k ϕ p ( I ∞ ,p ( x ))( a − a λ ) k L q a L p →
0. Since ϕ p is continuous it follows that (3.2) holdsfor any a ∈ M . (cid:3) The GNS-module.
We now describe the GNS-module as introduced by [Pas73], but inthe context of von Neumann algebras. Let Φ :
M → M be a completely positive map of vonNeumann algebras. We define the L ∞ -valued inner product: h X i a i ⊗ b i , X j a ′ j ⊗ b ′ j i ∞ = X i,j b ∗ i Φ( a ∗ i a ′ j ) b ′ j and set N to be the quotient of M ⊗ M by the set { z ∈ M ⊗ M : h z, z i = 0 } .For 1 ≤ p < ∞ , we define the L p/ -valued inner product by simply taking the inclusion of M into L p/ ( M ) (see Remark 2.6 for the case 1 ≤ p < h z, z ′ i p/ = κ (0) ∞ ,p/ (cid:0) h z, z ′ i ∞ (cid:1) . This L p/ -valued inner product gives rise to a norm k z k p, Φ := kh z, z i p/ k / p/ on N . We define L p ( M ⊗ Φ M ) to be the Banach space completion of N with respect to this norm. MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 11 Next we define a module structure on L p ( M ⊗ Φ M ). For z ∈ M ⊗ M and a ∈ T ϕ , it is givenby z · a := z (1 M ⊗ σ − ip ( a )) . (3.3)Note that this module structure satisfies property (iv) of Definition 3.1. By Kaplansky and strongdensity of T ϕ in M , we can approach a ∈ M by a bounded net ( a λ ) λ ∈ M converging to a in thestrong topology. Setting b λ,µ = a λ − a µ and using Proposition 2.8, we have k z · b λ,µ k p, Φ = kh z · b λ,µ , z · b λ,µ i p/ k / p/ = k b ∗ λ,µ h z, z i p/ b λ,µ k / p/ → . Hence we can extend (3.3) for elements a ∈ M , where the right hand side takes values in L p ( M ⊗ Φ M ). This right action is then strong/ k k p, Φ -continuous on the unit ball of M .By the L p -module Cauchy Schwarz inequality, the L p/ -valued inner product and the modulestructure extend to the space L p ( M ⊗ Φ M ). With this, L p ( M ⊗ Φ M ) turns into a well-defined L p M -module.For p = ∞ , we define L ∞ ( M ⊗ Φ M ) to be the completion with respect to the STOP topology,i.e. the one generated by the seminorms z ω ( h z, z i ) / , ω ∈ M ∗ . h· , ·i ∞ is continuous in bothvariables on M ⊗ M with respect to the STOP topology and hence extends to an M -valuedinner product on L ∞ ( M ⊗ Φ M ); one can see this by writing h z, z ′ i ∞ = h z, z i / ∞ T h z ′ , z ′ i / ∞ as inLemma 3.2 and, for ω ∈ M ∗ , using the classical Cauchy Schwarz inequality on the bilinear form( z, z ′ ) ω ( h z, z ′ i ). The module structure is simply given by z · a := z (1 ⊗ a ). Proposition 3.8.
There exists a family of projections ( q α ) α ∈ I ∈ M such that L p ( M ⊗ Φ M ) ∼ = L I q α L p ( M ) , ≤ p ≤ ∞ .Proof. To use Lemma 3.7, we must construct maps I q,p as in the assumptions of that lemma. Themaps will be extensions of the identity map ι : M ⊗ M → M ⊗ M . For q = ∞ , the space A fromthe lemma will be M ⊗ M and I ∞ ,p is simply the identity ι : A → L p ( M ⊗ Φ M ). For p ≤ q < ∞ ,the extensions exist because of the following estimate for z ∈ M ⊗ M : k z k q, Φ = kh z, z i q/ k / q/ = k κ (0) ∞ ,q/ ( h z, z i ∞ ) k / q/ ≥ k κ (0) q/ ,p/ ( κ (0) ∞ ,q/ ( h z, z i ∞ )) k / p/ = k κ (0) ∞ ,p/ ( h z, z i ∞ ) k / p/ = k z k p, Φ . It follows that ι extends to a contractive map I q,p : L q ( M⊗ Φ M ) → L p ( M⊗ Φ M ). The propertiesi)-iv) all follow from the previous constructions. Now we can apply Lemma 3.7 to deduce theresult. (cid:3) Remark 3.9.
We can deduce in hindsight the existence of the expected embedding L ∞ ( M ⊗ Φ M ) ֒ → L p ( M ⊗ Φ M )through the isomorphism with principal L p -modules where the embedding is clear. We will needthis observation later.Our next goal is to define duality results on the GNS-modules. To define a dual relation,we need to show that the bracket can be extended to a map taking arguments from differentspaces. This follows easily through the isomorphism with principal modules where this extensionis evident. In the GNS-picture, the bracket is given by h x, y i p,q = D /pϕ h x, y i ∞ D /qϕ = κ ( z p,q ) ∞ ,r ( h x, y i ∞ ) . (3.4) For x, y ∈ M ⊗ M and p + q = r with 1 ≤ p, q, r ≤ ∞ but p and q not both ∞ .The (antilinear) duality pairing is then defined as follows:( x, y ) = Tr( h x, y i p,q ) , x ∈ L p ( M ⊗ Φ M ) , y ∈ L q ( M ⊗ Φ M ) (3.5)This duality identifies L p ( M ⊗ Φ M ) as a subspace of L q ( M ⊗ Φ M ) ∗ . Using the isomorphismwith principal modules, we can show that this inclusion is an isomorphism. Corollary 3.10.
For ≤ p < ∞ , p + q = 1 , we have an antilinear isomorphism ( L p ( M ⊗ Φ M )) ∗ ∼ = L q ( M ⊗ Φ M ) . Proof.
This follows from Proposition 3.8 and Corollary 3.5. (cid:3)
Remark 3.11.
The definition of h· , ·i p,p coincides with that of h· , ·i p/ . Both notations makesense; the first refers to the inputs, the second to the output (and it corresponds to the term L p/ -valued inner product). We will mostly be using the latter notation. Remark 3.12.
Due to the tracial property, the embedding we choose to define the duality bracketdoes not matter. In particular, if x ∈ L ( M ⊗ Φ M ) ∩ L ( M ⊗ Φ M ) and y ∈ L ∞ ( M ⊗ Φ M ) ∩ L ( M ⊗ Φ M ) then Tr( h x, y i ) = Tr( h x, y i , ∞ )In the next lemma we check that the inner product behaves as expected when we use, informallyspeaking, elements from L p ( M ) in the first tensor leg as inputs. For this last lemma, we presumethat Φ satisfies the conditions of Proposition 2.7 so that Φ ( p/ exists. Lemma 3.13.
Let ≤ p < ∞ , and let Φ be a unital completely positive (ucp) ϕ -preserving mapsuch that T ◦ σ ϕt = σ ϕt ◦ T for all t ∈ R . The map Ψ p : κ (1) ∞ ,p ( M ) → L p ( M ⊗ Φ M ) , κ (1) ∞ ,p ( x ) x ⊗ extends to a contractive mapping Ψ p : L p ( M ) → L p ( M ⊗ Φ M ) . For x, y ∈ L p ( M ) , z = P j a j ⊗ b j ∈ M ⊗ M , it satisfies h Ψ p ( x ) , Ψ p ( y ) i p/ = Φ ( p/ ( x ∗ y ) , ≤ p < ∞ , h Ψ p ( x ) , z i p/ = X j Φ ( p ) ( x ∗ a j ) b j D /p Φ , ≤ p < ∞ . Proof.
We first note the following identity for x, y ∈ M : h x ⊗ , y ⊗ i p/ = κ (0) ∞ ,p/ (Φ( x ∗ y )) = Φ ( p/ ( κ (0) ∞ ,p/ ( x ∗ y )) (2.3) = Φ ( p/ ( κ (1) ∞ ,p ( x ) ∗ κ (1) ∞ ,p ( y )) (3.6)Hence, by the generalised H¨older inequality k x ⊗ k p, Φ = k Φ ( p/ ( κ (1) ∞ ,p ( x ) ∗ κ (1) ∞ ,p ( x )) k / p/ ≤ k κ (1) ∞ ,p ( x ) ∗ κ (1) ∞ ,p ( x ) k / p/ ≤ k κ (1) ∞ ,p ( x ) ∗ k / p k κ (1) ∞ ,p ( x ) k / p = k κ (1) ∞ ,p ( x ) k p . This shows that Ψ p is contractive on κ (1) ∞ ,p ( M ) and hence extends to a contractive mapping on L p ( M ).Now let x, y ∈ L p ( M ) and take ( x n ) , ( y n ) ∈ M such that κ (1) ∞ ,p ( x n ) → p x and κ (1) ∞ ,p ( y n ) → p y .From Minskovski’s inequality and the generalised H¨older inequality it follows that κ (1) ∞ ,p ( x n ) ∗ κ (1) ∞ ,p ( y n ) → p/ x ∗ y. MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 13 Hence by (3.6) and continuity of Φ ( p/ : h Ψ p ( x ) , Ψ p ( y ) i p/ = lim n →∞ h x n ⊗ , y n ⊗ i p/ = lim n →∞ Φ ( p/ ( κ (1) ∞ ,p ( x n ) ∗ κ (1) ∞ ,p ( y n )) = Φ ( p/ ( x ∗ y ) . The final equality is proved with a very similar method and is left to the reader. (cid:3) BMO spaces and BMO- H duality In this section we construct BMO spaces of σ -finite von Neumann algebras and prove that theyhave a predual. We also prove the interpolation result of Theorem 1.3. M is again a σ -finite vonNeumann algebra with faithful normal state ϕ .4.1. Introduction to Markov semigroups and BMO spaces.Definition 4.1.
A semigroup (Φ t ) t ≥ of linear maps M → M is called a (GNS-symmetric)Markov semigroup if it satisfies the following conditions:i) Φ t is normal ucp, t ≥ ϕ (Φ t ( x ) y ) = ϕ ( x Φ t ( y )), x, y ∈ M , t ≥ t Φ t ( x ) is strongly continuous, x ∈ M .The Markov semigroup is called ϕ -modular if Φ ◦ σ ϕs = σ ϕs ◦ Φ for all s ∈ R .Note that by condition ii), ϕ (Φ t ( x )) = ϕ ( x ); in particular, the Φ t are faithful. If Φ := (Φ t ) is a ϕ -modular Markov semigroup, then by Proposition 2.7 there are extensions Φ ( p ) t : L p ( M ) → L p ( M ),where Φ (1) t is trace-preserving. Note that condition ii) implies, after appropriate approximations,that Φ (2) t is self-adjoint.For the rest of this section we assume Φ = (Φ t ) t ≥ to be a ϕ -modular Markov semigroup. Wedefine closed subspaces of M and L ( M ) as follows M ◦ = { x ∈ M | Φ t ( x ) → σ -weakly as t → ∞} ,L ◦ p ( M ) = { x ∈ L p ( M ) | k Φ ( p ) t ( x ) k p → , t → ∞} . Then [Cas19, Lemma 2.3] assures that the inclusions κ ( z ) q,p restrict to contractive inclusions L ◦ q ( M ) → L ◦ p ( M ) for q ≥ p .For x ∈ M we define the column and row BMO-norm: k x k BMO c Φ = sup t ≥ k Φ t ( | x − Φ t ( x ) | ) k / ∞ ; k x k BMO r Φ = k x ∗ k BMO c Φ . The BMO-norm is defined as k x k BMO Φ = max {k x k BMO c Φ , k x k BMO r Φ } . This defines a seminorm by[JM12, Proposition 2.1].Since Φ is faithful, we see that for x ∈ M , k x k BMO Φ = 0 implies that x = Φ t ( x ) for all t > M ◦ .Next, we turn our attention to defining an analogous BMO-norm on the space L ( M ) such asin [JM12]. This turns out to be more involved in the σ -finite case.The embedding κ (0) ∞ , allows us to define k · k ∞ on L ( M ) (it takes values ∞ outside of κ (0) ∞ , ( M )). We will also denote this by k · k ∞ . Then we can define analogous column and row BMO-(semi)norms on L ( M ) by k x k BMO c Φ = sup t ≥ k Φ (1) t ( | x − Φ (2) t ( x ) | ) k / ∞ ; k x k BMO r Φ = k x ∗ k BMO c Φ (4.1)We will only show later (at the end of this chapter) that these seminorms satisfy the triangleinequality. As with the corresponding norms on M , these seminorms are norms on L ◦ ( M ). Nowwe define the column BMO space asBMO c ( M , Φ) = { x ∈ L ◦ ( M ) | k x k BMO c Φ < ∞} and we define the row BMO space as the adjoint of the column BMO space with norm as in (4.1).We emphasize that we have thus constructed a column (resp. row) BMO-norm both on M ◦ and L ◦ ( M ) which by mild abuse of notation are denoted in the same way. They are identified by theright embedding for the column norm and the left embedding for the row norm: k κ (1) ∞ , ( x ) k BMO c Φ = k xD / ϕ k BMO c Φ = k x k BMO c Φ , k κ ( − ∞ , ( x ) k BMO r Φ = k D / ϕ x k BMO r Φ = k x k BMO r Φ , (4.2)where x ∈ M . These equalities are straightforward to check.The first thought for a definition of the BMO-norm would be max {k x k BMO c Φ , k x k BMO r Φ } , sim-ilarly to the definition on M . However, this is not a suitable definition for the following reason.The equalities (4.2) show how the right and left embeddings of M in L ( M ) preserve the col-umn and row norms respectively. However, there is no embedding of M into L ( M ) that wouldpreserve the maximum of these norms.Instead, we embed BMO c ( M , Φ) and BMO r ( M , Φ) in L ◦ ( M ) through the embeddings κ ( − , and κ (1)2 , respectively. This turns (BMO c ( M , Φ) , BMO r ( M , Φ)) into a compatible couple. Thefollowing diagram commutes: L ◦ ( M )BMO c ( M , Φ) M ◦ L ◦ ( M )BMO r ( M , Φ) L ◦ ( M ) κ ( − , ⊆ κ (1) ∞ , κ (0) ∞ , κ ( − ∞ , ⊆ κ (1)2 , We define BMO( M , Φ) = κ ( − , (BMO c ( M , Φ)) ∩ κ (1)2 , (BMO r ( M , Φ))and for x ∈ BMO( M , Φ) we denote by x c ∈ BMO c ( M , Φ) , x r ∈ BMO r ( M , Φ)the elements such that κ ( − , ( x c ) = x = κ (1)2 , ( x r ). The norm on BMO( M , Φ) is defined as k x k BMO Φ = max {k x c k BMO c Φ , k x r k BMO r Φ } . When no confusion can occur, we omit the reference to the semigroup in the notation of thevarious BMO-norms and just write, for instance, k · k
BMO . MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 15 We check that κ (0) ∞ , is indeed an embedding of M ◦ into BMO( M ) that preserves k · k BMO : k κ (0) ∞ , ( z ) k BMO = max {k κ (1) ∞ , ( z ) k BMO c , k κ ( − ∞ , ( z ) k BMO r } = max {k z k BMO c , k z k BMO r } = k z k BMO . The next estimate shows that L ◦ ( M ) contains the closure of κ (0) ∞ , ( M ◦ ) with respect to k · k BMO ,as expected.
Lemma 4.2.
For x ∈ L ◦ ( M ) , we have k x k ≤ k x k BMO c and k x k ≤ k x k BMO r . Hence for x ∈ BMO( M , Φ) , we have k x k BMO ≥ max {k x c k , k x r k } ≥ k x k . Proof.
Let x ∈ L ◦ ( M ). If k x k BMO c = ∞ then the inequality trivially holds. Now assume that k x k BMO c < ∞ . Then for all t ≥ y t ∈ M such that Φ (1) t | x − Φ (2) t ( x ) | = κ (0) ∞ , ( y t ).Let ε >
0. Then we can find t > k Φ (2) t ( x ) k < ε . Then since Φ (1) t is trace-preserving: k x k ≤ k x − Φ (2) t ( x ) k + ε = Tr( | x − Φ (2) t ( x ) | ) / + ε = Tr(Φ (1) t | x − Φ t ( x ) | ) / + ε = Tr( κ (0) ∞ , ( y t )) / + ε = ϕ ( y t ) / + ε ≤ k y t k / ∞ + ε ≤ k x k BMO c + ε. Since k x k = k x ∗ k , we also get k x k ≤ k x k BMO r . The final statement follows from the definitionof k · k BMO and contractivity of κ ( z )2 , . This finishes the proof. (cid:3) It is not a priori clear whether BMO( M , Φ) is complete. However, this will follow as a corollaryfrom the result of the next subsection, which provides an ‘artificial’ predual to BMO( M , Φ).4.2.
A predual of BMO.
We dedicate this section to proving the following theorem:
Theorem 4.3.
There exists a Banach space h ( M , Φ) such that BMO( M , Φ) ∼ = h ( M , Φ) ∗ . In this part we will suppress the reference to M in the notation of BMO , BMO c , BMO r andtheir preduals h , h r , h c .In the finite case a predual was found in [JM12, Section 5.2.3], see also [JMP14, Appendix A].Our proof mostly follows the lines of [JMP14]. However, our predual of BMO c will instead be h r and vice versa, which makes the identification in Theorem 4.3 linear instead of antilinear.Let us first focus on finding preduals to BMO c and BMO r . Since BMO r lies within L ◦ ( M ),we have at our disposal an inner product that can provide us with a duality bracket. We take theHahn-Banach norm relation as the definition of the norm of h c : k y k h c = sup k x k BMO r ≤ | Tr( xy ) | , y ∈ L ◦ ( M ) . (which would be a well-defined norm even if k · k BMO c wouldn’t satisfy the triangle inequality).Then we note that by Lemma 4.2: k y k h c ≤ sup k x k ≤ | Tr( xy ) | = k y k . Hence we define h c to be the completion of L ◦ ( M ) with respect to k·k h c , and we obtain a contrac-tive inclusion L ◦ ( M ) ⊆ h c . We define h r analogously by taking the sup over x with k x k BMO c ≤ Proposition 4.4.
BMO r ∼ = ( h c ) ∗ , BMO c ∼ = ( h c ) ∗ . Proof.
We will only show that BMO r ∼ = ( h c ) ∗ (the other case follows similarly). It is not hard toshow that BMO r ⊆ ( h c ) ∗ contractively. Conversely, let ψ ∈ ( h c ) ∗ . Then ψ | L ◦ ( M ) ∈ L ◦ ( M ) ∗ byLemma 4.2. Hence by the Riesz representation theorem there exists an x ∈ L ◦ ( M ) such that ψ ( z ) = Tr( x ∗ z )for all z ∈ L ◦ ( M ). What remains to be shown is that x ∗ ∈ BMO r , with k x ∗ k BMO r ≤ k ψ k ( h c ) ∗ (theother inequality follows from the definition of h c ). This is equivalent to requiring that x ∈ BMO c with k x k BMO c ≤ k ψ k ( h c ) ∗ Fix t >
0. We will now use the L p -modules L p ( M ⊗ Φ t M ) corresponding to the ucp map Φ t .Let Ψ p be the embedding of Lemma 3.13. Then we can define the map u t : L ◦ ( M ) → L ( M ⊗ Φ t M ) , u t ( y ) = Ψ ( y − Φ (2) t ( y )) . Now it suffices to show that u t ( x ) ∈ L ∞ ( M ⊗ Φ t M ) and k u t ( x ) k ∞ , Φ t ≤ k ψ k ( h c ) ∗ since then, with the first identity of Lemma 3.13: k x k BMO c = sup t> k u t ( x ) k ∞ , Φ t ≤ k ψ k ( h c ) ∗ . Define ϕ u t ( x ) to be the dual action of u t ( x ) on L ( M ⊗ Φ t M ) restricted to M ⊗ M , i.e. ϕ u t ( x ) ( z ) := Tr( h u t ( x ) , z i )The goal is to prove that u t ( x ) also defines a dual action on L ( M ⊗ Φ t M ). The proof is rathertechnical, so we contain it in a separate lemma. Lemma 4.5.
Let z ∈ M ⊗ M . Then | ϕ u t ( x ) ( z ) | ≤ k ψ k ( h c ) ∗ k z k , Φ t In particular, ϕ u t ( x ) extends to an element of L ( M ⊗ Φ M ) ∗ with k ϕ u t ( x ) k ≤ k ψ k ( h c ) ∗ MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 17 Proof.
Let z = P j a j ⊗ b j . Using the second identity of Lemma 3.13 and the fact that Φ (2) t isself-adjoint we haveTr( h u t ( x ) , z i ) = X j Tr(Φ (2) t (( x − Φ (2) t ( x )) ∗ a j ) b j D / ϕ )= X j Tr(Φ (2) t ( x ∗ a j ) b j D / ϕ ) − Tr(Φ (2) t (Φ (2) t ( x ∗ ) a j ) b j D / ϕ )= X j Tr( x ∗ a j Φ (2) t ( b j D / ϕ )) − Tr(Φ (2) t ( x ∗ ) a j Φ (2) t ( b j D / ϕ ))= X j Tr( x ∗ a j Φ (2) t ( b j D / ϕ )) − Tr( x ∗ Φ (2) t ( a j Φ (2) t ( b j D / ϕ )))= X j Tr( x ∗ [ a j Φ (2) t ( b j D / ϕ ) − Φ (2) t ( a j Φ (2) t ( b j D / ϕ ))])= Tr( x ∗ u ∗ t ( z )) . Thus u ∗ t ( z ):= a j Φ (2) t ( b j D / ϕ ) − Φ (2) t ( a j Φ (2) t ( b j D / ϕ )) ∈ L ( M ).We are done if we can prove that k u ∗ t ( z ) k h c ≤ k z k , Φ t . However, we do not even have u ∗ t ( z ) ∈ L ◦ ( M ) in general, so this will not be possible. To circumvent this, let π be the projection L ( M ) → L ◦ ( M ). Then π is self-adjoint and π ( x ) = x , henceTr( x ∗ u ∗ t ( z )) = Tr( x ∗ π ( u ∗ t ( z ))) . We claim that k π ( u ∗ t ( z )) k h c ≤ k z k , Φ t . Indeed, by (3.4) and Remark 3.12: k π ( u ∗ t ( z )) k h c = sup k y k BMO r ≤ | Tr( yπ ( u ∗ t ( z ))) | = sup k y k BMO c ≤ | Tr( y ∗ π ( u ∗ t ( z ))) | = sup k y k BMO c ≤ | Tr( h u t ( y ) , z i ) | = sup k y k BMO c ≤ | Tr( h u t ( y ) , z i ∞ , ) |≤ sup k y k BMO c ≤ k z k , Φ t k u t ( y ) k ∞ , Φ t = k z k , Φ t . It follows that indeed | ϕ u t ( x ) ( z ) | = | Tr( x ∗ u ∗ t ( z )) | ≤ sup k h k hc ≤ | Tr( x ∗ h ) |k z k , Φ t = k ψ k ( h c ) ∗ k z k , Φ t (cid:3) Now through our duality result of Proposition 3.10, u t ( x ) ∈ L ∞ ( M ⊗ Φ M ) and k x k BMO c = sup t> k u t ( x ) k ∞ , Φ = sup t> sup k z k , Φ t ≤ | Tr( h u t ( x ) , z i ∞ , ) | ≤ k ψ k ( h c ) ∗ . This shows that indeed BMO r ∼ = ( h c ) ∗ . (cid:3) Note that this also proves that k · k
BMO c , k · k BMO r and k · k BMO are actually norms, so(BMO c , BMO r ) is a well-defined compatible couple and BMO is a well-defined normed vectorspace. Proof of Theorem 4.3.
Using Proposition 2.3, we have that BMO ∗ = κ ( − , (BMO c ) ∗ + κ (1)2 , (BMO r ) ∗ ,which contains both h r and h c as isometric subspaces, say through embeddings ι r and ι c respec-tively. This makes ( h r , h c ) into a compatible couple. Define h := ι c ( h r ) + ι r ( h c ). Then anotherapplication of Proposition 2.3 gives( h ) ∗ ∼ = ( h r ) ∗ ∩ ( h c ) ∗ ∼ = k ( − , (BMO c ) ∩ κ (1)2 , (BMO r ) = BMO . This finishes the proof. (cid:3)
It follows from the proof that the various BMO-norms satisfy the triangle inequality. We canalso deduce that the associated BMO spaces are complete
Corollary 4.6.
BMO( M , Φ) , BMO c ( M , Φ) and BMO r ( M , Φ) are Banach spaces. Interpolation for BMO space.
In this section we show that [Cas19, Theorem 4.5] holdsagain for the current definiton of BMO. Similar to how [Cas19, Theorem 4.5] is proved, theproof is a mutatis mutandis copy of the methods in [Cas19, Section 3] provided that conditionalexpectations extend to a contraction on BMO. In other words, we must show that [Cas19, Lemma4.3] still holds in the current setup. This is done in Proposition 4.11 below. We start with someauxiliary lemmas that could be of independent interest.Let us state some preliminary facts. By [Ter81, Theorem II.36], a standard form for M is( M , L ( M ) , J, L +2 ( M )), where J is the conjugation operator. Hence we will consider M as avon Neumann subalgebra of B ( L ( M )) by left multiplication. With an inclusion of von Neumannalgebras M ⊆ M we mean a unital inclusion, meaning that the unit of M equals the unit of M . It is a well known fact that M admits a ϕ -preserving conditional expectation if and onlyif σ ϕt ( M ) = M for all t ∈ R , see [Tak03a, Theorem IX.4.2]. If E is a ϕ -preserving conditionalexpectation, then we can use Proposition 2.7 to extend it to a contraction E ( p ) : L p ( M ) → L p ( M ),which can be checked to land in L p ( M ). Lemma 4.7.
Let M ⊆ M be a von Neumann subalgebra that admits a ϕ -preserving conditionalexpectation E . Then for x ∈ L ( M ) and y ∈ M we have Tr( x E ( y )) = Tr( E (1) ( x ) y ) . Proof. If x = D ϕ x ′ with x ′ ∈ M we have since E (1) is Tr-preserving,Tr( x E ( y )) = Tr( E (1) ( x E ( y ))) = Tr( D ϕ E ( x ′ E ( y ))) = Tr( D ϕ E ( x ′ ) E ( y ))=Tr( D ϕ E ( E ( x ′ ) y )) = Tr( E (1) ( D ϕ E ( x ′ ) y )) = Tr( E (1) ( x ) y ) . For general x ∈ L ( M ) the statement follows by approximation. (cid:3) The following lemma is a variation of the Kadison-Schwarz inequality.
Lemma 4.8.
Let M ⊆ M be a von Neumann subalgebra that admits a ϕ -preserving conditionalexpectation E . Then for x ∈ L ( M ) we have the following inequality in L ( M ) , E (2) ( x ) E (2) ( x ) ∗ ≤ E (1) ( xx ∗ ) . Proof.
Naturally L ( M ) ⊆ L ( M ) is a closed subspace and we have that E (2) : L ( M ) → L ( M ) is the orthogonal projection onto this subspace, see [Tak03a, Proof of Theorem IX.4.2]. L ( M ) is an invariant subspace for M . Therefore M commutes with both E (2) and 1 − E (2) .Hence, for y ∈ M and x ∈ L ( M ) we have hE (2) ( x ) , y E (2) ( x ) i + h (1 − E (2) )( x ) , y (1 − E (2) )( x ) i = h x, yx i . MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 19 And so for y ∈ M + we haveTr( E ( y ) E (2) ( x ) E (2) ( x ) ∗ ) = hE (2) ( x ) , E ( y ) E (2) ( x ) i ≤ h x, E ( y ) x i = Tr( E ( y ) xx ∗ ) . (4.3)We further have by Lemma 4.7, Tr( E ( y ) xx ∗ ) = Tr( y E (1) ( xx ∗ )) , and since E (1) is a projection onto L ( M )Tr( E ( y ) E (2) ( x ) E (2) ( x ) ∗ ) = Tr( y E (1) ( E (2) ( x ) E (2) ( x ) ∗ )) = Tr( y E (2) ( x ) E (2) ( x ) ∗ ) . Therefore (4.3) shows that we have the following Kadison-Schwarz type inquality, E (2) ( x ) E (2) ( x ) ∗ ≤ E (1) ( xx ∗ ) . (cid:3) Lemma 4.9.
Let ω ∈ M + ∗ . The following are equivalent:(1) We have ω ≤ ϕ .(2) There exists x ∈ M with ≤ x ≤ such that D ϕ xD ϕ = D ω .Proof. For (1) ⇒ (2), consider the map T : L ( M ) → L ( M ) : D ϕ x D ω x, x ∈ M . From the fact that ω ≤ ϕ it follows that T is a well-defined contraction. Moreover, we claim that T ∈ M . Indeed, the commutant of M acting on L ( M ) is given by J M J where J : ξ ξ ∗ isthe modular conjugation. Then it follows that for x, y ∈ M we have T J yJ D ϕ x = T D ϕ xy ∗ = D ω xy ∗ = J yJ T ( D ϕ x ) . Now set x = T ∗ T ∈ M so that 0 ≤ x ≤
1. We have
T D ϕ = D ω so that ( D ϕ T ∗ )( T D ϕ ) = D ω .The implication (2) ⇒ (1) follows as for y ∈ M we have ω ( yy ∗ ) = Tr( D ω yy ∗ ) = Tr( y ∗ D ϕ xD ϕ y ) = h D ϕ y, xD ϕ y i≤h D ϕ y, D ϕ y i = Tr( y ∗ D ϕ y ) = ϕ ( yy ∗ ) . (cid:3) Lemma 4.10.
Let a, b ∈ L ( M ) + and suppose that a ≤ b and b = D ϕ x b D ϕ with x b ∈ M + . Thenthere exists x a ∈ M + such that a = D ϕ x a D ϕ . Moreover x a ≤ x b .Proof. Let ϕ a and ϕ b be in M + ∗ such that D ϕ a = a and D ϕ b = b . The assumptions and Lemma4.9 imply that ϕ b ≤ k x b k ϕ . We find that ϕ a ≤ ϕ b ≤ k x b k ϕ . Therefore Lemma 4.9 implies thatthere exists x a ∈ M with 0 ≤ x a ≤ k x b k such that a = D ϕ x a D ϕ . We have moreover x a ≤ x b since a ≤ b implies that for y ∈ M , h D ϕ y, x a D ϕ y i = Tr( y ∗ D ϕ x a D ϕ y ) = Tr( D ϕ x a D ϕ yy ∗ ) = Tr( ayy ∗ ) ≤ Tr( byy ∗ ) = Tr( D ϕ x b D ϕ yy ∗ ) = h D ϕ y, x b D ϕ y i . (cid:3) Proposition 4.11.
Let M ⊆ M be a von Neumann subalgebra that admits a ϕ -preservingconditional expectation E . Let Φ = (Φ t ) t ≥ be a Markov semi-group on M that preserves M .Then we have isometric 1-complemented inclusions BMO( M , Φ) ⊆ BMO( M , Φ) . Proof.
That the isometric inclusion exists is clear from the definitions. We have to prove thatthe inclusion is 1-complemented. For t ≥ x ∈ BMO c Φ ( M ) ⊆ L ◦ ( M ) we have the following(in)equalities in L ( M ) by Lemma 4.8, |E (2) ( x ) − Φ (2) t ( E (2) ( x )) | = E (2) ( x − Φ (2) t ( x )) ∗ E (2) ( x − Φ (2) t ( x )) ≤ E (1) (( x − Φ (2) t ( x )) ∗ ( x − Φ (2) t ( x )))) . As Φ (1) t preserves positivity and commutes with E (1) ,Φ (1) t ( |E (2) ( x ) − Φ (2) t ( E (2) ( x )) | ) ≤ E (1) (Φ (1) t (( x − Φ (2) t ( x )) ∗ ( x − Φ (2) t ( x )))) . (4.4)By assumption we may writeΦ (1) t (( x − Φ (2) t ( x )) ∗ ( x − Φ (2) t ( x )) = κ (0) ∞ , ( x ′ t ) , for some x ′ t ∈ M . So the right hand side of (4.4) equals κ (0) ∞ , ( E ( x ′ t )). By Lemma 4.10 it followsthat there exists x ′′ t ∈ M with 0 ≤ x ′′ t ≤ E ( x ′ t ) such thatΦ (1) t ( |E (2) ( x ) − Φ (2) t ( E (2) ( x )) | ) = κ (0) ∞ , ( x ′′ t ) . Taking norms we have kE (2) ( x ) k BMO c = sup t ≥ k x ′′ t k ∞ ≤ sup t ≥ kE ( x ′ t ) k ∞ ≤ sup t ≥ k x ′ t k ∞ = k x k BMO c . The row BMO-estimate and the BMO-estimate follow similarly. (cid:3)
We may now conclude the following theorem. The proof (based on the Haagerup reductionmethod) follows exactly as in [Cas19, Sections 3 and 4] where [Cas19, Lemma 4.3] needs to bereplaced by Proposition 4.11. Note that in the statement of [Cas19, Theorem 4.5] the standardMarkov dilation must be modular as well (this is a misprint in the text of [Cas19]).
Theorem 4.12.
Let Φ be a ϕ -modular Markov semigroup on a σ -finite von Neumann algebra ( M , ϕ ) admitting a modular standard Markov dilation. Then for all ≤ p < ∞ , < q < ∞ , [BMO( M , Φ) , L ◦ p ( M )] /q ≈ pq L ◦ pq ( M ) . Here ≈ pq means that the Banach spaces are isomorphic and the norm of the isomorphism in bothdirections can be estimated by an absolute constant times pq . L p -boundedness of BMO-valued Fourier-Schur multipliers on SU q (2)In this section we prove that Fourier-Schur multipliers on SU q (2) of a certain form extend tothe non-commutative L p spaces corresponding to SU q (2). We first introduce compact quantumgroups, SU q (2) and give the definition of Fourier-Schur multipliers. Then we prove the endpointestimates we need for complex interpolation. MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 21 BMO spaces of the torus.
Define trigonometric functions ζ k : T → T : z z k , k ∈ Z . Set the ∗ -algebra Pol( T ) := Span { ζ k : k ∈ Z } . For m ∈ ℓ ∞ ( Z ) let T m : L ( T ) → L ( T ) bethe Fourier multiplier defined by T m ( ζ k ) = m ( k ) ζ k , k ∈ Z . For t ≥ h t ∈ ℓ ∞ ( Z ) be givenby h t ( k ) = e − tk . Then the maps T h t are well-known to define a Markov semigroup on the vonNeumann algebra L ∞ ( T ) (as they are restrictions of the Heat semi-group on L ∞ ( R )). We use theshorthand notation BMO( T ) := BMO( L ∞ ( T ) , ( T h t ) t ≥ ) . Let m ∈ ℓ ∞ ( Z ) be such that m (0) = 0. Then as t → ∞ , k T h t ( T m ζ k ) k ∞ = e − tk | m ( k ) |k ζ k k ∞ → . So T m maps Pol( T ) to L ◦∞ ( T ).5.2. Compact quantum groups.
For the theory of compact quantum groups we refer to[Wor98] or the notes [MVD98] which follows the same lines.
Definition 5.1.
A compact quantum group G = ( C ( G ) , ∆) consists of a C ∗ -algebra C ( G ) and aunital ∗ -homomorphism ∆ : C ( G ) → C ( G ) ⊗ min C ( G ) called the comultiplication such that (∆ ⊗ ι ) ◦ ∆ = ( ι ⊗ ∆) ◦ ∆ (coassociativity) and such that both ∆( C ( G ))( C ( G ) ⊗
1) and ∆( C ( G ))(1 ⊗ C ( G ))are dense in C ( G ) ⊗ min C ( G ). Here ι : C ( G ) → C ( G ) is the identity map.A finite dimensional (unitary) corepresentation is a unitary u ∈ C ( G ) ⊗ M n ( C ) such that(∆ ⊗ id)( u ) = u u where u = 1 ⊗ u and u is the flip applied to the first two tensor legsof u . All corepresentations are assumed to be unitary. The elements (id ⊗ ω )( u ) ∈ C ( G ) with ω ∈ M n ( C ) ∗ are called matrix coefficients. The span of all matrix coefficients is a ∗ -algebra calledPol( G ). ∆ maps Pol( G ) to Pol( G ) ⊗ Pol( G ).Here we shall mainly be concerned with the quantum group SU q (2) and we shall introducefurther structure such as Haar states and von Neumann algebras for this case only.5.3. Introduction SU q (2) . Let G q := SU q (2) with q ∈ ( − , \{ } . It was introduced byWoronowicz in [Wor87b]. As a ∗ -algebra, it is the universal ∗ -algebra generated by two elements α, γ subject to the following commutation relations: γ ∗ γ = γγ ∗ , αγ = qγα, αγ ∗ = qγ ∗ α,α ∗ α + γ ∗ γ = I, αα ∗ + q γ ∗ γ = I, together with the comultiplication given by∆( α ) = α ⊗ α − qγ ∗ ⊗ γ, ∆( γ ) = γ ⊗ α + α ∗ ⊗ γ. Consider the representation π on the Hilbert space H = ℓ ( N ) ⊗ ℓ ( Z ) with basis vectors e i ⊗ f j , i ∈ N , j ∈ Z . The generators are represented by π ( α ) e i ⊗ f j = p − q i e i − ⊗ f j ,π ( γ ) e i ⊗ f j = q i e i ⊗ f j +1 . In the sequel we suppress π in the notation and identify α and γ with their representation asoperators on H . We define L ∞ ( G q ) = h α, γ i ′′ ⊆ B ( H ). We also define Pol( G q ) ⊆ L ∞ ( G q ) to be the ∗ -algebragenerated by α, γ . This is equivalent to the definition given in Section 5.2. It is the linear span ofelements α k γ l ( γ ∗ ) m , k ∈ Z , l, m ∈ N , where we set α k = ( α ∗ ) | k | in case k <
0. Obviously, Pol( G q )is weakly (or weak- ∗ ) dense in L ∞ ( G q ).The Haar state on L ∞ ( G q ) is given by the following formula: ϕ ( x ) = (1 − q ) X k ∈ N q k h e k ⊗ f , xe k ⊗ f i . (5.1)See [Wor87a, Appendix A1] for the complete calculation. Note that ϕ ( α k γ l ( γ ∗ ) m ) is non-zero ifand only if k = 0 , l = m . It is also faithful, as follows for instance from (5.1).The modular automorphism group is given by σ ϕt ( α k γ l ( γ ∗ ) m ) = q − itk α k γ l ( γ ∗ ) m . (5.2)This can be derived from [Tak03a, Theorem VIII.3.3], where the u t from the Theorem is equal to( γ ∗ γ ) it and the ψ is a trace.5.4. Fourier-Schur Multipliers on SU q (2) .Definition 5.2. Let G be a compact quantum group and T : Pol( G ) → Pol( G ) a linear map. Wecall T a Fourier-Schur multiplier if the following condition holds. Let u be any finite dimensionalcorepresentation on H . Then there exists an orthogonal basis e i such that if u i,j are the matrixcoefficients with respect to this basis, then there exist numbers c i,j := c ui,j ∈ C such that T u i,j = c i,j u i,j . In this case ( c ui,j ) i,j,u is called the symbol of T . Remark 5.3. If G comes from a classical group G , i.e. if all irreducible corepresentations areone-dimensional, then the above definition coincides with the definition of a classical Fouriermultiplier. In general, we see that T = F S F − , where S is a Schur multiplier. Hence the name‘Fourier-Schur multiplier’.We will construct Fourier-Schur multipliers from Fourier multipliers on the torus T ⊆ C . Weassume henceforth that m ∈ ℓ ∞ ( Z ) with m (0) = 0 such that T m : L ∞ ( T ) → BMO( T ) is completelybounded. In the remainder of this section, we will consider the bounded map˜ T m : Pol( G q ) → Pol( G q ) , α k γ l ( γ ∗ ) m m ( k ) α k γ l ( γ ∗ ) m (5.3)We will see after the next subsection that ˜ T m is indeed a Fourier-Schur multiplier.The final goal is to prove that this map extends boundedly to L p ( G q ) → L ◦ p ( G q ) for all p ≥
2. Wedo this through complex interpolation (Riesz-Torin). This requires 3 steps: (1) a lower endpointestimate; (2) an upper endpoint estimate involving BMO spaces and (3) the construction of aMarkov dilation in order to apply Theorem 4.12.We treat the Markov dilation in Appendix B. The remainder of this section is devoted to theendpoint estimates.Similarly to the torus, we have
Lemma 5.4.
Let ≤ p ≤ ∞ . Then κ (1) ∞ ,p ◦ ˜ T m maps Pol( G q ) to L ◦ p ( G q ) . MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 23 Proof.
Let x = α k γ l ( γ ∗ ) m . For k = 0, we have ˜ T m ( x ) = 0 ∈ L ◦ p ( G q ). Now assume k >
0. Thenfor any 1 ≤ p ≤ ∞ , we have as t → ∞ , k Φ ( p ) t ( κ (1) ∞ ,p ( ˜ T m x )) k p = k κ (1) ∞ ,p (Φ t ( ˜ T m ( x ))) k p = | m ( k ) e − tk |k κ (0) ∞ ,p ( x ) k p → . (cid:3) L -estimate. In this subsection we prove that (5.3) extends to a bounded map L ( G q ) → L ( G q ). At the same time we prove (essentially) that it defines a Fourier-Schur multiplier. Themain ingredient will be the Peter-Weyl decomposition of G q (see [KS97, Theorem 4.17]) we shallsummarize now.A complete set of mutually inequivalent irreducible corepresentations of G q can be constructedas follows. They are labeled by half integers l ∈ N . Consider the vector space of linear combi-nations of the homogeneous polynomials in α, γ of degree 2 l . For some specific constants C q,l,k ,we define basis vectors as follows: g ( l ) k = C q,l,k α l − k γ l + k , k = − l, − l + 1 , . . . , l. (5.4)The precise value of the constant C q,l,k can be found in [KS97, Chapter 4.2.3]; it is of littleimportance to us. Next, we define the matrix u ( l ) ∈ Pol( G q ) ⊗ M l +1 ( C ) by∆( g ( l ) k ) = l X i = − l u ( l ) k,i ⊗ g ( l ) i . The Peter-Weyl theorem now takes the following form from which we derive the main result ofthis subsection in Proposition 5.6.
Lemma 5.5 (Proposition 4.16 and Theorem 4.17 of [KS97]) . The matrix coefficients of u ( l ) ∈ M n ( L ∞ ( G q )) are a linear basis for Pol( G q ) satisfying the orthogonality relations ϕ (( u ( l ) i,j ) ∗ u ( k ) r,s ) = C ( l ) i δ l,k δ i,r δ j,s . for some constants C ( l ) i ∈ C . Proposition 5.6.
The t ( l ) i,j form an orthogonal basis of eigenvectors for the map ˜ T m defined in (5.3) with eigenvalues m ( − i − j ) .Proof. To prove this, we will calculate an explicit expression for the matrix elements u ( l ) i,j . Withour notation αα − = αα ∗ = 1 − q γ ∗ γ . Hence, α k ( α ∗ ) k = α k − (1 − q γ ∗ γ )( α ∗ ) k − = (1 − q k γ ∗ γ ) α k − ( α ∗ ) k − = · · · = (1 − q k γ ∗ γ )(1 − q k − γ ∗ γ ) . . . (1 − q γ ∗ γ ) =: ( q γ ∗ γ ; q ) k . The notation ( a ; b ) k is known as the Pochhammer symbol. Let us define an alternative binomialexpansion with the commutation relation αγ = qγα . We define polynomials P ( k ) i by the formula( α + γ ) k = k X i =0 P ( k ) i ( q − ) α i γ ( k − i ) . We get the same polynomial when using the relation αγ ∗ = qγ ∗ α (and the * of both relations). Be-low we will use this expansion on both tensor legs simultaneously, which means that the argumentof the polynomial becomes q − . Thus: ∆( g ( l ) i ) = C l,i,q ∆( α l − i γ l + i ) = C l,i,q ∆( α ) l − i ∆( γ ) l + i = C l,i,q ( α ⊗ α − qγ ∗ ⊗ γ ) l − i ( γ ⊗ α + α ∗ ⊗ γ ) l + i = C l,i,q l − i X a =0 ( − q ) a P ( l − i ) a ( q − ) α l − i − a ( γ ∗ ) a ⊗ α l − i − a γ a ! × l + i X s =0 P ( l + i ) s ( q − ) γ l + i − s ( α ∗ ) s ⊗ α l + i − s γ s ! = C l,i,q l − i X a =0 l + i X s =0 C ′ a,s,q α l − i − a − s ( γ ∗ ) a γ l + i − s ( q γ ∗ γ ; q ) min( s,l − i − a ) ⊗ α l − a − s γ a + s where C ′ a,s,q = ( − q ) a q ( l + i − s )( s − a )+ as P ( l − i ) a ( q − ) P ( l + i ) s ( q − ). Next, we substitute j = a + s − l and b = a + i − j . This gives:∆( g ( l ) i ) = C l,i,q l X j = − l X a,b ≥ a − b = j − i C ′ a,s,q α − ( i + j ) ( γ ∗ ) a γ b ( q γ ∗ γ ; q ) min( l + j − a,l − i − a ) ⊗ α l − j γ l + j = C l,i,q l X j = − l X a,b ≥ a − b = j − i C ′ a,s,q α − ( i + j ) ( γ ∗ ) a γ b ( q γ ∗ γ ; q ) min( l + j − a,l − i − a ) ⊗ g ( l ) j . Hence we find u ( l ) i,j = α − ( i + j ) X a,b ≥ a − b = j − i C l,i,q C ′ a,s,q ( γ ∗ ) a γ b ( q γ ∗ γ ; q ) min( l + j − a,l − i − a ) . (5.5)Now since the only power of α that occurs in (5.5) is α − ( i + j ) , the u ( l ) i,j are eigenvectors for themaps ˜ T m . (cid:3) Corollary 5.7.
The map (5.3) is a Fourier-Schur multiplier for G q with symbol ( m ( − i − j )) i,j,l where l ∈ N indexes the corepresentation and ≤ i, j ≤ l + 1 . Corollary 5.8.
For every m ∈ ℓ ∞ ( Z ) there is a map ˜ T (2) m : L ( G q ) → L ( G q ) extending (5.3) by ˜ T (2) m ◦ κ (1) ∞ , = κ (1) ∞ , ◦ ˜ T m which is bounded with norm at most k m k ∞ . If m (0) = 0 then ˜ T (2) m : L ( G q ) → L ◦ ( G q ) .Proof. Define the ϕ -GNS inner product on Pol( G q ) by h x, y i = ϕ ( x ∗ y ) and denote the associatedGNS space by H ϕ . By Lemma 5.5 and Proposition 5.6 we see that ˜ T m : Pol( G q ) → Pol( G q ) isbounded with respect to this inner product with bound at most k m k ∞ . Hence it extends to amap ˜ T ϕm : H ϕ → H ϕ . By [Ter82, Section 2.2] we have thatPol( G q ) → L ( G q ) : x xD / ϕ is an isometry with respect to this inner product on the left and hence extends to a unitary map U : H ϕ → L ( G q ). Then the map ˜ T (2) m := U ˜ T ϕm U ∗ : L ( G q ) → L ( G q ) satisfies the conditions.The final statement is Lemma 5.4. (cid:3) MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 25 Transference principle and construction of
BMO( G q ) . In this subsection we constructthe BMO spaces corresponding to G q = SU q (2) , q ∈ ( − , \{ } that we need for the upper end-point estimate. The main tool behind both the construction of the BMO spaces and the proof ofthe actual upper endpoint estimate is the transference principle of Lemma 5.10. The idea is toobtain properties of Fourier-Schur multipliers on L ∞ ( G q ) from properties of Fourier multiplierson L ∞ ( T ).Recall that ζ i : T → T was defined by z z i and let e i,j be the matrix units in B ( ℓ ( N )). Wedefine the unitary U = ∞ X i =0 e i,i ⊗ B ( ℓ ( Z )) ⊗ ζ i ∈ B ( H ) ¯ ⊗ L ∞ ( T ) , and the injective normal ∗ -homomorphism π : B ( H ) → B ( H ) ¯ ⊗ L ∞ ( T ) : x U ∗ ( x ⊗ U. Lemma 5.9.
We have for k ∈ Z , l, m ∈ N that π ( α k γ l ( γ ∗ ) m ) = α k γ l ( γ ∗ ) m ⊗ ζ k . (5.6) Proof.
For ξ ∈ L ( T ), i ∈ N , j ∈ Z , π ( α k γ l ( γ ∗ ) m )( e i ⊗ f j ⊗ ξ )= U ∗ ( α k γ l ( γ ∗ ) m ⊗ id) e i ⊗ f j ⊗ ζ i ξ = U ∗ q (1 − q i )(1 − q i − ) . . . (1 − q i − k +2 ) q i ( l + m ) e i − k ⊗ f j + l − m ⊗ ζ i ξ = q (1 − q i )(1 − q i − ) . . . (1 − q i − k +2 ) q i ( l + m ) e i − k ⊗ f j + l − m ⊗ ζ k ξ =( α k γ l ( γ ∗ ) m ⊗ ζ k )( e i ⊗ f j ⊗ ξ ) . (cid:3) This implies that π maps Pol( G q ) into Pol( G q ) ⊗ L ∞ ( T ). Hence by density, it maps L ∞ ( G q )into L ∞ ( G q ) ¯ ⊗ L ∞ ( T ). We denote ι M for the identity operator M → M on a von Neumannalgebra M , reserving 1 M for the unit of M . The following identity is now immediate. We referto this identity as the ‘transference principle’. Lemma 5.10.
Let m ∈ ℓ ∞ ( Z ) . For k ∈ Z , l, m ∈ N we have ( ι L ∞ ( G q ) ⊗ T m ) π ( α k γ l ( γ ∗ ) m ) = m ( k ) π ( α k γ l ( γ ∗ ) m ) . Set again the Heat multipliers h t ( k ) = e − tk , k ∈ Z , t ≥
0. Let us define a semigroup on L ∞ ( G q ) ¯ ⊗ L ∞ ( T ) by S = ( S t ) t ≥ with S t := ι L ∞ ( G q ) ⊗ T h t . Recall that ( T h t ) t ≥ is a Markovsemigroup (see Section 5.1). By approximation with elements from the algebraic tensor productand the text following Proposition 2.1, one can prove that S is also a Markov semigroup. Fromthis and the transference principle, we can now construct the Markov semigroup on L ∞ ( G q ). Proposition 5.11.
The family of maps given by the assignment Φ t ( α k γ l ( γ ∗ ) m ) = e − tk α k γ l ( γ ∗ ) m , k ∈ Z , l, m ∈ N , t ≥ , extends to a Markov semigroup of Fourier-Schur multipliers Φ := (Φ t ) t ≥ on L ∞ ( G q ) satisfying π ◦ Φ t = S t ◦ π. Moreover, the semi-group is modular.
Proof.
By Lemma 5.10 we have the commutative diagram: L ∞ ( G q ) ¯ ⊗ L ∞ ( T ) L ∞ ( G q ) ¯ ⊗ L ∞ ( T )Pol( G q ) L ∞ ( G q ) S t π Φ t π π is a normal injective ∗ -homomorphism so that we may view L ∞ ( G q ) as a (unital) von Neumannsubalgebra of L ∞ ( G q ) ¯ ⊗ L ∞ ( T ). We find that Φ t , being the restriction of S t to Pol( G q ), is alsoa normal ucp map. This means that Φ t extends to a normal ucp map on L ∞ ( G q ). By thesame argument, we deduce strong continuity of t Φ t ( x ). This shows properties (i) and (iii) ofDefinition 4.1.To show property (ii), we recall (see (5.1)) that the Haar functional ϕ on G q is non-zeroon basis elements α k γ l ( γ ∗ ) m only if k = 0 , l = m . If x = α k γ l ( γ ∗ ) m , y = α k ′ γ l ′ ( γ ∗ ) m ′ , then xy = Cα k + k ′ γ l + l ′ ( γ ∗ ) m + m ′ for some constant C . This shows that ϕ ( x Φ t ( y )) = ϕ (Φ t ( x ) y ) on basiselements x, y , and hence everywhere.Finally, by the formula for the modular automorphism group (5.2), we find that Φ t is ϕ -modular. (cid:3) We define corresponding BMO spaces for this semigroup. We use the shorthand notationBMO( G q ) for BMO( L ∞ ( G q ) , Φ), and similarly for the column and row spaces. We can also definea BMO-norm k · k
BMO S on ( L ∞ ( G q ) ¯ ⊗ L ∞ ( T )) ◦ . We will do some of the estimates within thenormed spaces ( L ◦∞ ( G q ) , k · k BMO Φ ) and (( L ∞ ( G q ) ¯ ⊗ L ∞ ( T )) ◦ , k · k BMO S ) to avoid some technical-ities. Lemma 5.12.
The map π is isometric as a map between normed spaces π : ( L ◦∞ ( G q ) , k · k BMO Φ ) → (( L ∞ ( G q ) ¯ ⊗ L ∞ ( T )) ◦ , k · k BMO S ) . Proof.
This follows from the commutative diagram of Proposition 5.11 and the fact that π isan injective, hence isometric, ∗ -homomorphism L ∞ ( G q ) → L ∞ ( G q ) ¯ ⊗ L ∞ ( T ). Indeed, for x ∈ L ∞ ( G q ) ◦ , we have that k S t ( π ( x )) k ∞ = k ( π ◦ Φ t )( x ) k ∞ → , hence π ( x ) ∈ ( L ∞ ( G q ) ¯ ⊗ L ∞ ( T )) ◦ . Also, k π ( x ) k cS = sup t ≥ k S t ( | π ( x ) − S t ( π ( x )) | ) k = sup t ≥ k S t ( | π ( x ) − π (Φ t ( x )) | ) k = sup t ≥ k S t ( π ( | x − Φ t ( x ) | )) k = sup t ≥ k π (Φ t ( | x − Φ t ( x ) | )) k = sup t ≥ k Φ t ( | x − Φ t ( x ) | ) k = k x k cΦ . Replacing x by x ∗ yields isometry for the row BMO-norm from which it follows that π is isometricon BMO as well. (cid:3) MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 27 L ∞ - BMO estimate.
We proceed to prove an upper end point estimate for ˜ T m . Proposition 5.13.
Let m ∈ ℓ ∞ ( Z ) with m (0) = 0 be such that T m : L ∞ ( T ) → BMO( T ) iscompletely bounded. Then there exists a bounded map ˜ T ( ∞ ) m : L ∞ ( G q ) → BMO( G q ) , satisfying ˜ T ( ∞ ) m ( x ) = κ (0) ∞ , ( ˜ T m ( x )) for x ∈ Pol( G q ) . Moreover, k ˜ T ( ∞ ) m : L ∞ ( G q ) → BMO( G q ) k ≤ k T m : L ∞ ( T ) → BMO( T ) k cb . (5.7)The proof consists of the following two lemmas. We first prove a BMO-norm estimate of ˜ T m for the polynomial algebra, using again the transference principle from Lemma 5.10. Lemma 5.14.
Let m ∈ ℓ ∞ ( Z ) with m (0) = 0 be such that T m : L ∞ ( T ) → BMO( T ) is completelybounded. Then for x ∈ Pol( G q ) : k ˜ T m ( x ) k BMO Φ ≤ k T m : L ∞ ( T ) → BMO( T ) k cb k x k ∞ . (5.8) Proof.
By Lemma 5.4, ˜ T m maps Pol( G q ) to L ◦∞ ( G q ). Note that π sends Pol( G q ) to L ∞ ( G q ) ⊗ Pol( T ) and ι L ∞ ( G q ) ⊗ T m sends L ∞ ( G q ) ⊗ Pol( T ) to L ∞ ( G q ) ⊗ L ◦∞ ( T ) ⊆ ( L ∞ ( G q ) ¯ ⊗ L ∞ ( T )) ◦ (seealso Appendix A). Now Lemma 5.10 gives us a commutative diagram like in Proposition 5.11. L ∞ ( G q ) ⊗ Pol( T ) ( L ∞ ( G q ) ¯ ⊗ L ∞ ( T )) ◦ Pol( G q ) L ◦∞ ( G q ) ι L ∞ ( G q ) ⊗ T m π ˜ T m π Note that in particular the restriction T m : Pol( T ) → (L ◦∞ ( T ) , k · k BMO ) is completely bounded.Now Lemma 5.12 and Proposition A.1 allows us to find a BMO-estimate on ˜ T m for x ∈ Pol( G q ): k ˜ T m ( x ) k BMO Φ = k π ◦ ˜ T m ( x ) k BMO S = k ( ι L ∞ ( G q ) ⊗ T m ) ◦ π ( x ) k BMO S ≤ k T m k cb k π ( x ) k = k T m k cb k x k ∞ . where k T m k cb = k T m : L ∞ ( T ) → BMO( T ) k cb . (cid:3) Recall that κ (0) ∞ , isometrically embeds the normed space ( L ◦∞ ( G q ) , k · k BMO Φ ) into BMO( G q ).Now define ˜ T ( ∞ ) m = κ (0) ∞ , ◦ ˜ T m , which we may consider as a bounded map from Pol( G q ) to BMO( G q )by Lemma 5.14. It remains to prove that this map extends to L ∞ ( G q ). The proof is essentiallythat of [JMP14, Lemma 1.6] together with a number of technicalities that we overcome here. Lemma 5.15. ˜ T ( ∞ ) m has a normal extension to L ∞ ( G q ) → BMO( G q ) .Proof. Let h ( G q ) := h ( L ∞ ( G q ) , Φ) be the predual constructed in Section 4.2. We will constructa map S : h ( G q ) → L ( G q ) such that the adjoint S ∗ : L ∞ ( G q ) → BMO( G q ) is an extension of˜ T ( ∞ ) m . Construction of maps S c and S r . We first construct a map S c : h c ( G q ) → L ( G q ). We do thisby proving that the map κ (1)2 , ◦ ( T (2) m ) ∗ is bounded as a map L ◦ ( G q ) → L ( G q ) with respect to k · k h c ( G q ) on the left. For y ∈ L ◦ ( G q ) and z ∈ Pol( G q ) we find h z, ( ˜ T (2) m ) ∗ ( y ) D / ϕ i = h D / ϕ z, ( ˜ T (2) m ) ∗ ( y ) i = h D / ϕ ˜ T m ( z ) , y i . (5.9) By the Kaplansky density theorem and [Tak02, Theorem II.2.6] the unit ball of Pol( G q ) is weak- ∗ dense in the unit ball of L ∞ ( G q ). Hence for y ∈ L ◦ ( G q ) we find: k κ (1)2 , (( ˜ T (2) m ) ∗ y ) k L ( G q ) = sup z ∈ Pol( G q ) ≤ |h z, ( ˜ T (2) m ) ∗ ( y ) D / ϕ i| = sup z ∈ Pol( G q ) ≤ |h D / ϕ ˜ T m ( z ) , y i| ≤ k T m k cb k y k h c ( G q ) . In the last step we used that k κ ( − ∞ , ( ˜ T m ( z )) k BMO r = k ˜ T m ( z ) k BMO r ≤ k T m k cb k z k ∞ . We concludethat κ (1)2 , ◦ ( ˜ T (2) m ) ∗ extends to a bounded map S c : h c ( G q ) → L ( G q ) . In a similar manner we can prove that the map κ ( − , ◦ ( ˜ T (2) m ) ∗ extends to a bounded map S r : h r ( G q ) → L ( G q ) . Pairing identities.
By taking limits in (5.9), we can prove the following equalities for z ∈ Pol( G q ), y a ∈ h c ( G q ) and y b ∈ h r ( G q ): h z, S c ( y a ) i = h D / ϕ ˜ T m ( z ) , y a i , h z, S r ( y b ) i = h ˜ T m ( z ) D / ϕ , y b i (5.10)Recall that for x ∈ BMO( G q ), we defined x c ∈ BMO c ( G q ) and x r ∈ BMO r ( G q ) as thoseelements satisfying D / ϕ x c = x = x r D / ϕ . Furthermore, we denoted by ι c respectively ι r the embeddings of h c ( G q ) respectively h r ( G q ) intoBMO( G q ) ∗ . For example, for y ∈ h c ( G q ) , x ∈ BMO( G q ), the embedding is given by h x, ι c ( y ) i = h x r , y i . Compatible morphisms.
Using the equations in (5.10), we can prove that S c , S r are compatiblemorphisms. Indeed, let y ∈ ι c ( h c ( G q )) ∩ ι r ( h r ( G q )) and y ∈ h c ( G q ) , y ∈ h r ( G q ) such that ι c ( y ) = y = ι r ( y ). This means that for x ∈ BMO( G q ), h x, y i = h x r , y i = h x c , y i . Hence for z ∈ Pol( G q ) (using the above equality with x = κ (0) ∞ , ( ˜ T m ( z )) = D / ϕ ˜ T m ( z ) D / ϕ ): h z, S c ( y ) i = h D / ϕ ˜ T m ( z ) , y i = h ˜ T m ( z ) D / ϕ , y i = h z, S r ( y ) i . By weak- ∗ density of Pol( G q ) in L ∞ ( G q ), this shows that S c ( y ) = S r ( y ), hence S c and S r arecompatible. This means that there is a unique mapping S : h ( G q ) → L ( G q ) extending S c and S r , i.e. S ( ι c ( y a )) = S c ( y a ) and S ( ι r ( y b )) = S r ( y b ) for y a ∈ h c ( G q ), y b ∈ h r ( G q ). Remainder of the proof.
Now consider the adjoint map S ∗ : L ∞ ( G q ) → BMO( G q ), which is weak-* continuous by Proposition 2.1. We will show that this map extends ˜ T ( ∞ ) m . Let z ∈ Pol( G q ) and y ∈ h ( G q ). Let y ∈ h c ( G q ) and y ∈ h r ( G q ) be such that y = ι c ( y ) + ι r ( y ). Applying (5.10) MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 29 again gives h S ∗ ( z ) , y i = h z, S ( y ) i = h z, S c ( y ) + S r ( y ) i = h D / ϕ ˜ T m ( z ) , y i + h ˜ T m ( z ) D / ϕ , y i = h D / ϕ ˜ T m ( z ) D / ϕ , ι c ( y ) i + h D / ϕ ˜ T m ( z ) D / ϕ , ι r ( y ) i = h ˜ T ( ∞ ) m ( z ) , y i . Hence S ∗ ( z ) = ˜ T ( ∞ ) m ( z ), and so S ∗ is the weak-* continuous extension that we were lookingfor. (cid:3) Proof of Proposition 5.13.
The existence of ˜ T ( ∞ ) m follows from Lemma 5.14 and 5.15. The inequal-ity in (5.7) follows from (5.8) and the Kaplansky density theorem. (cid:3) Consequences for L p -Fourier Schur multipliers.Theorem 5.16. Let m ∈ ℓ ∞ ( Z ) be such that the Fourier multiplier T m : L ∞ ( T ) → BMO( T ) iscompletely bounded. Let ˜ T m : Pol( G q ) → Pol( G q ) be the Fourier-Schur multiplier with symbol ( m ( − i − j )) i,j with respect to the basis described in (5.5) . Then for ≤ p < ∞ , ˜ T m extends to abounded map ˜ T ( p ) m : L p ( G q ) → L ◦ p ( G q ) . Proof.
We can assume without loss of generality that m (0) = 0 by switching to the map T m − m (0)1 = T m − m (0) ι L ∞ ( T ) . This map is still completely bounded. Now Proposition 5.8 and Proposition5.13 show that ˜ T ( ∞ ) m and ˜ T (2) m together are compatible morphisms. Therefore, by Riesz-Torin (seee.g. Theorem 2.5 from [Cas13] and the rest of that paragraph), we get bounded maps on theinterpolation spaces. Since Φ admits a Markov dilation (see Proposition B.4), Theorem 4.12 tellsus that [BMO( G q ) , L ◦ ( G q )] /p ≈ L ◦ p ( G q ) . Also we have by [Kos84] that [ L ∞ ( G q ) , L ( G q )] /p ≈ L p ( G q ) . This proves that for 2 ≤ p < ∞ we can construct bounded maps ˜ T ( p ) m : L p ( G q ) → L ◦ p ( G q ) thatextend ˜ T m - or more precisely, they satisfy ˜ T ( p ) m ( κ (1) ∞ ,p ( x )) = κ (1) ∞ ,p ( ˜ T m ( x )) for all x ∈ Pol( G q ).Now consider 1 ≤ p < p ′ be such that p + p ′ = 1. Then the adjoint map ˜ T ∗ m is simplythe Fourier multiplier with symbol ¯ m , and hence by the above argument ˜ T ∗ m extends to a map on L p ′ ( G q ). Hence the map ˜ T ( p ) m : L p ( G q ) → L p ( G q ) given by the double adjoint is the extension wewere looking for. (cid:3) Remark 5.17.
In [JMP14, Lemma 3.3] classes of completely bounded multipliers L ∞ ( T ) → BMO( T ) have been constructed. Further, in [JMP14, Lemma 1.3] the connection between classicalBMO-spaces and non-commutative semi-group BMO spaces is established giving further examples.This shows that indeed the class of symbols m to which Proposition 5.13 applies is non-emptyand contains a reasonable class of examples. Appendix A. Completely bounded maps with respect to the BMO-norm
Throughout this section, let
M ⊆ B ( K ) be a σ -finite von Neumann algebra with n.f. state ϕ and Markov semigroup Φ = (Φ t ) t ≥ . Fix some n ≥
2. Then the maps ι M n ⊗ Φ t define a Markovsemigroup on M n ( M ). Hence we can define the matrix BMO-norms k · k BMO n on M n ( M ) ◦ withrespect to the semigroup S n := ( ι M n ⊗ Φ t ) t ≥ . Through a straightforward calculation, one alsochecks that M n ( M ) ◦ = M n ( M ◦ ). Hence the above norms define matrix norms on M ◦ . It is nothard to prove that these norms turn M ◦ into an operator space, which we denote by ( M ◦ , k·k BMO ).We leave the details to the reader.Let
N ⊆ B ( H ) be a σ -finite von Neumann algebra. Then N ¯ ⊗M is again a σ -finite von Neu-mann algebra. Similarly as in the matrix case, S := ( ι N ⊗ Φ t ) t ≥ is a semigroup on N ¯ ⊗M . Inline with the main text, we denote k · k BMO S for the corresponding BMO-norm on ( N ¯ ⊗M ) ◦ .Using the fact that N ∗ ⊗ M ∗ is dense in ( N ¯ ⊗M ) ∗ (see [Sak71, Chapter 1.22]) one can showthat N ⊗ M ◦ ⊆ ( N ¯ ⊗M ) ◦ . Proposition A.1.
Let
A ⊆ M be a linear subspace and T : A → ( M ◦ , k · k BMO ) be completelybounded. For x ∈ N ⊗ A , k ( ι N ⊗ T )( x ) k BMO S ≤ k T k cb k x k B ( H⊗ K ) . Proof.
Take x ∈ N ⊗ A and write x = P n x n ⊗ x ′ n . Let z = ( ι N ⊗ T )( x ) ∈ N ⊗ M ◦ . Setting w n = T ( x ′ n ) we have z = P n x n ⊗ w n . For a finite dimensional subspace F ⊆ H let P F be theprojection onto F . Denote x Fn = P F x n P F the truncation of x n to F . Denote z F = P n x Fn ⊗ w n and x F = P n x Fn ⊗ x ′ n .Now we prove the column estimate. Let ξ ∈ H ⊗ K (algebraic tensor product) and write ξ = P k ξ k ⊗ η k . Define F ⊆ H to be F = Span { ξ k , x m ξ k , x ∗ n x m ξ k | n, m, k } . Then we note that F is finite dimensional and ( x Fn ) ∗ x Fm ξ k = x ∗ n x m ξ k , Let t ≥ ι N ⊗ Φ t )( | z − ( ι N ⊗ Φ t )( z ) | ) = X n,m x ∗ n x m ⊗ Φ t (( w n − Φ t ( w n )) ∗ ( w m − Φ t ( w m ))) . Hence, denoting S F := ( ι B ( F ) ⊗ Φ t ) t ≥ , k ( ι N ⊗ Φ t )( | z − ( ι N ⊗ Φ t )( z ) | ) ξ k H⊗ K = k ( ι B ( F ) ⊗ Φ t )( | z F − ( ι B ( F ) ⊗ Φ t )( z F ) | ) ξ k F ⊗K ≤ k ( ι B ( F ) ⊗ Φ t )( | z F − ( ι B ( F ) ⊗ Φ t )( z F ) | ) k B ( F ⊗K ) k ξ k≤ k z F k cSF k ξ k = k ( ι B ( F ) ⊗ T )( x F ) k cSF k ξ k ≤ k T k cb k x k B ( H⊗ K ) k ξ k . In the last step, we used that T is also completely bounded when considering k · k BMO c on theright. Taking the supremum over all ξ ∈ H ⊗ K with k ξ k = 1 and t ≥
0, we conclude k ( ι N ⊗ T )( x ) k BMO cS ≤ k T k cb k x k B ( H⊗ K ) The row BMO estimate follows similarly, from which the BMO estimate follows. (cid:3)
MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 31 Remark A.2.
In the case where M is a finite von Neumann algebra, we can extend the operatorspace structure to BMO( M , Φ). In the σ -finite case however, it seems to be more difficult thanexpected to prove that M n (BMO( M , Φ)) ⊆ BMO( M n ( M ) , ι M n ⊗ Φ).
Appendix B. A Markov dilation of the Markov semigroup Φ Definition B.1.
We say that a Markov semigroup Φ on a σ -finite von Neumann algebra M withfaithful normal state ϕ admits a standard Markov dilation if there exist:(i) a σ -finite von Neumann algebra N with normal faithful state ϕ N ,(ii) an increasing filtration ( N s ) s ≥ with ϕ N -preserving conditional expectations E s : N → N s ,(iii) a ∗ -homomorphisms π s : M → N s such that ϕ N ◦ π s = ϕ and E s ( π t ( x )) = π s (Φ t − s ( x )) , s < t, x ∈ M . A Markov dilation is called ϕ -modular if it additionally satisfies π s ◦ σ ϕt = σ ϕ N t ◦ π s , s ≥ , t ∈ R . In this subsection, we construct a Markov dilation for the semigroup Φ = (Φ t ) t ≥ on L ∞ ( G q )given by Φ t ( α k γ l ( γ ∗ ) m ) = e − tk α k γ l ( γ ∗ ) m , k ∈ Z , l, m ∈ N , as used in Section 5.To construct the Markov dilation, we use the fact that L ∞ ( G q ) can be written as the tensorproduct of two relatively simple von Neumann algebras. This is a well-known fact; we give asketch of the proof for the convenience of the reader. We let L ( Z ) be the group von Neumannalgebra of Z generated by the left regular representation λ . Proposition B.2. L ∞ ( G q ) = B ( ℓ ( N )) ¯ ⊗L ( Z ) . Proof.
Let T m , T ˜ m be multiplication maps on ℓ ( N ) with symbols m ( k ) = q k , ˜ m ( k ) = p − q k .Then we can write γ = T m ⊗ λ , Z , α = ( λ ∗ , N T ˜ m ) ⊗ λ , Z and λ , N for the right shift on ℓ ( Z ) and ℓ ( N ) respectively. From theseexpressions it is immediately clear that L ∞ ( G q ) ⊆ B ( ℓ ( N )) ¯ ⊗L ( Z ). For the other inclusion, notethat the partial isometries in the polar decompositions of α, γ are 1 ⊗ λ , Z and λ ∗ , N ⊗ ⊗ L ( Z ) and B ( ℓ ( N )) ⊗ (cid:3) Through this expression for L ∞ ( G q ) we will show that Φ t can be written as a Schur multiplier.We will need the fact that Schur multipliers are normal. Proposition B.3.
Set H = ℓ ( I ) for some index set I and let T : B ( H ) → B ( H ) be a Schurmultiplier with symbol t = ( t i,j ) i,j , i.e. T ( e i,j ) = t i,j e i,j . Then T is normal.Proof. Denote L ( H ) to be the trace class operators and denote t T to be the adjoint of t . Weclaim that T ∗ | L ( H ) is nothing but the Schur multiplier with symbol t T . Indeed, if x ∈ B ( H ), y ∈ L ( H ) and i ∈ I is fixed, then h T ( x ) ye i , e i i = X k ∈ I t i,k x i,k y k,i = X k ∈ I x i,k t Tk,i y k,i = h x ( t Ti,j y i,j ) i,j e i , e i i . Hence Tr( T ( x ) y ) = X i ∈ I h T ( x ) ye i , e i i = X i ∈ I h x ( t Ti,j y i,j ) i,j e i , e i i = Tr( x ( t Ti,j y i,j ) i,j ) . This shows the claim.Let y ∈ L ( H ). By the above T ∗ ( y ) is an operator on H so that we can define its trace-classnorm. Then by Hahn-Banach k T ∗ y k L ( H ) = sup x ∈B ( H ): k x k≤ |h x, T ∗ y i| = sup x ∈B ( H ): k x k≤ |h T x, y i| ≤ k T kk y k L ( H ) . So T ∗ restricts to an operator L ( H ) → L ( H ). Therefore, since B ( H ) = L ( H ) ∗ , we see that byProposition 2.1 T = ( T ∗ | L ( H ) ) ∗ is normal. (cid:3) Proposition B.4.
The semi-group Φ admits a (standard and reversed) ϕ -modular Markov dila-tion.Proof. We prove first that Φ t can be written as a Schur multiplier on the left tensor leg of L ∞ ( G q ).Let x = α k γ l ( γ ∗ ) m . x acts on basis vectors by e i ⊗ f r x ce i − k ⊗ f r + l − m , c := c q,k,l,m,i,r = q (1 − q i )(1 − q i − ) . . . (1 − q i − k +2 ) q i ( l + m ) . In other words, the matrix elements of x are given by h xe i ⊗ f r , e j ⊗ f s i = c δ j,i − k δ s,r + l − m . Hence if we define Ψ t : B ( ℓ ( N )) → B ( ℓ ( N )) as the Schur multiplier given by Ψ t ( e i,j ) = e − t | i − j | e i,j , then we have Φ t ( x ) = e − tk x = (Ψ t ⊗ id L ( Z ) )( x )Hence Φ t and Ψ t ⊗ id L ( Z ) ( x ) coincide on Pol( G q ). Since both are normal (Proposition 5.11 for Φ t and Proposition B.3 for Ψ t ) they must coincide on L ∞ ( G q ).The proof from now on is essentially that of [Ric08] or [CJSZ20, Proposition 4.2] with the maindifference that the unitary u below only sums over the indices of ℓ ( N ). Let ε > H = Span { e i , i ∈ N } ⊆ H by setting h ξ, η i = X i,j ∈ N e − ε ( j − i ) ξ i η j , ξ, η ∈ H We define H R to be the completion of H with respect to h· , ·i after quotienting out the degeneratepart. Let Γ = Γ( H R ) be the associated exterior algebra with vacuum vector Ω and canonicalvacuum state τ Ω . The dilation von Neumann algebra ( B , ϕ B ) will be given by B = L ∞ ( G q ) ¯ ⊗ Γ ⊗∞ , ϕ B = ϕ ⊗ τ ⊗∞ Ω where the infinite tensor product is taken with respect to τ Ω . Next we describe the dilationhomomorphisms π s . We consider the unitary u = X i ∈ N p i,i ⊗ L ( Z ) ⊗ s ( e i ) ⊗ ⊗∞ Γ ∈ L ∞ ( G q ) ¯ ⊗ Γ ⊗∞ which is defined as a strong limit of sums. Let S : v ⊗ v be the tensor shift on Γ ⊗∞ , and let β : B → B be defined by β ( z ) = u ∗ ( ι L ∞ ( G q ) ⊗ S )( z ) u . The ∗ -homomorphisms π s : L ∞ ( G q ) → B are given by π : x x ⊗ ⊗ . . . , π k : x ( β k ◦ π )( x ) , k ≥ . MO SPACES OF σ -FINITE VON NEUMANN ALGEBRAS AND FOURIER-SCHUR MULTIPLIERS 33 One shows by induction that for x ∈ L ∞ ( G q ) π k ( x ) = X i,j ∈ N p i,i xp j,j ⊗ ( s ( e i ) s ( e j )) ⊗ k ⊗ ⊗∞ Γ . By (5.1) it follows that π k is state-preserving, and by [Tak03b, Proposition XIV.1.11], it is ϕ -modular.Finally, the filtration is given by B m = L ∞ ( G q ) ¯ ⊗ Γ ⊗ m ⊗ ⊗∞ Γ ⊆ B . One checks that the associated conditional expectations satisfy E m ( p i,i xp j,j ⊗ ( s ( e i ) s ( e j )) ⊗ k ⊗ id ⊗∞ Γ )= τ Ω ( s ( e i ) s ( e j )) k − m p i,i xp j,j ⊗ ( s ( e i ) s ( e j )) ⊗ m ⊗ ⊗∞ Γ . From this and the identity τ Ω ( s ( e i ) s ( e j )) = h s ( e j )Ω , s ( e i )Ω i = e − ε ( j − i ) one deduces that indeed ( E m ◦ π k )( x ) = π m (Φ ε ( k − m ) ( x )) . So the semigroup (Φ εn ) n ∈ N admits a Markov dilation for any ε >
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