Lower bounds in L p -transference for crossed-products
aa r X i v : . [ m a t h . OA ] S e p Lower bounds in L p -transference for crossed-products Adri´an M. Gonz´alez-P´erez ∗ Abstract
Let Γ y Ω be a measure-preserving action and L Γ ֒ → L ∞ (Ω) ⋊ Γ the natural inclusion of thegroup von Neumann algebra into the crossed product. When µ (Ω) = ∞ , we have that this naturalembedding is not trace-preserving and therefore does not extends boundedly to the associatednoncommutative L p -spaces. Nevertheless, we show that when Ω has an invariant mean there is anisometric embedding of L p ( L Γ) into an ultrapower of L p (Ω ⋊ Γ) that intertwines Fourier multipliersand it is L Γ-bimodular. As a consequence we obtain the lower transference bound (cid:13)(cid:13) T m : L p ( L Γ) → L p ( L Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) (id ⋊ T m ) : L p (Ω ⋊ Γ) → L p (Ω ⋊ Γ) (cid:13)(cid:13) , and the same follows for complete norms. The techniques employed are in line with those of [GP18]in which the reverse inequality was proven for measure preserving Zimmer-amenable actions. Bothare preceded by the pioneering works of Neuwirth/Ricard and Caspers/de la Salle [NR11, CdlS15]for amenable groups.The condition of having an invariant mean is quite restrictive. Therefore, we explore whetherother equivariant embeddings Φ : L Γ → L ∞ (Ω) yield a transference result as above. In thiscontext, a map is considered equivariant if it is co-multiplicative with respect to the canonicalco-multiplication of L Γ and the canonical co-action of L Γ on L ∞ (Ω) ⋊ Γ. Those maps are givenby linear extension of Φ( λ g ) = ϕ g ⋊ λ g , for a collection of functions ( ϕ g ) g ∈ Γ ⊂ L ∞ (Ω). We start by noticing that the multiplicative,completely positive and completely bounded equivariant maps can be easily characterized. Inparticular, completely positive equivariant maps are given by the matrix coefficients of unitary1-cocycles κ : Γ × Ω → U ( H ). Then, we show that the transference proof above works verbatimwhenever Φ is completely positive, amenable in the sense of Popa and Anantharaman-Delaroche[AD95] and intertwines Fourier multipliers at the L -level. Although no new transference resultsare obtained, both the classification of equivariant maps and the study their amenability may beof independent interest to some readers. Introduction
Fourier and Herz-Schur multipliers.
Let Γ be a discrete group and let L Γ ⊂ B ( ℓ Γ) be its leftregular von Neumann algebra, that is, the weak- ∗ closure of the group algebra C [Γ] under the leftregular representation λ : Γ → B ( ℓ Γ). Let m ∈ ℓ ∞ (Γ) be a function. The (potentially unbounded)operator T m : C [Γ] ⊂ L Γ → L Γ given by T m (cid:16) X g ∈ Γ a g λ g (cid:17) = X g ∈ Γ a g m ( g ) λ g , is called the Fourier multiplier of symbol m . Whenever T m : L Γ → L Γ is bounded/completely bounded,it is said that m is a bounded/completely bounded Fourier multiplier. Observe that, if ι : L Γ ֒ → B ( ℓ Γ) ∗ The author has been partially funded by the ANR grant HASCON
1s the natural embedding, we have that the following diagram commutes L Γ ι / / T m (cid:15) (cid:15) B ( ℓ Γ) H m (cid:15) (cid:15) L Γ ι / / B ( ℓ Γ) , where H m : M Γ × Γ ( C ) ⊂ B ( ℓ Γ) → B ( ℓ Γ) is the so-called
Herz-Schur multiplier of symbol m , that isthe (again potentially unbounded) operator given by H m (cid:16) X g ∈ Γ a g,h e g,h (cid:17) = X g ∈ Γ a g,h m ( g h − ) e g,h . Since ι is isometric, we have that the norm of T m : L Γ → L Γ is bounded by that of H m : B ( ℓ Γ) →B ( ℓ Γ). The reciprocal was shown to be true by Bozejko/Fendler [BF84] but in that case the completeboundedness of T m is required. In sum, its is known that for every m , the following holds (A) (cid:13)(cid:13) H m : B ( ℓ Γ) → B ( ℓ Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) T m : L Γ → L Γ (cid:13)(cid:13) cb . (B) (cid:13)(cid:13) H m : B ( ℓ Γ) → B ( ℓ Γ) (cid:13)(cid:13) cb = (cid:13)(cid:13) H m : B ( ℓ Γ) → B ( ℓ Γ) (cid:13)(cid:13) , ie: H m is automatically cb. (C) (cid:13)(cid:13) T m : L Γ → L Γ (cid:13)(cid:13) ≤ (cid:13)(cid:13) H m : B ( ℓ Γ) → B ( ℓ Γ) (cid:13)(cid:13) .The fact that H m is automatically continuous holds for general Schur multipliers , ie bounded operatorsgiven by e i,j m i,j e i,j and more general for bimodular operators, see [Smi91]. For Fourier multipliersit is known that boundedness and complete boundedness are not equivalent, since there are examplesof multipliers that fail to be completely bounded over nonamenable groups, see [HK94, HSS09]. Noncommutative L p -bounds and transference. Given a semifinite von Neumann algebra M and a normal, semifinite and faithful trace τ : M + → [0 , ∞ ] their noncommutative L p -spaces, see[Ter81, PX03], can be defined as the spaces of τ -measurable operators x such that L p ( M , τ ) = (cid:8) x : k x k p := τ (cid:0) | x | p (cid:1) p < ∞ (cid:9) . In the case of B ( ℓ ) with its usual trace Tr the associated noncommutative L p -spaces are called the p -Schatten classes and denoted by S p ( ℓ ). The group von Neumann algebra L Γ of a discrete groupΓ admits a normal tracial state τ : L Γ → C , given by τ ( x ) = h δ e , x δ e i . We will denote theirassociated L p -spaces by L p ( L Γ). Asking whether, given a symbol m , their associate Fourier and Herz-Schur multipliers are bounded over L p ( L Γ) and S p ( ℓ Γ) respectively its an extremely difficult questionthat has received attention for its connections with approximations properties [LdlS11], the theory ofMarkovian semigroups [GPJP17, JMP18] as well as the convergence of Fourier series over non-Abeliangroups [HWW20] among other problems. For noncommutative L p -spaces, the analogues of points (B)and (C) are widely open. Nevertheless, point (A) was shown to be true by Neuwirth/Ricard [NR11],which also showed that (A’) (cid:13)(cid:13) T m : L p ( L Γ) → L p ( L Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) H m : S p ( ℓ Γ) → S p ( ℓ Γ) (cid:13)(cid:13) , when Γ is amenable.The same follows from complete norms. Their technique, which is sometimes called transference , worksby using the amenability of Γ to construct, for every 1 ≤ p < ∞ , a completely isometric embedding L p ( L Γ) J p −−−−−→ Y U S p ( ℓ Γ) , where the space in the right hand side is a proper ultrapower of the Schatten classes, that intertwinesthe operators T m and H U m . This technique was later generalized to locally compact groups [CdlS15]2nd to the context of trace-preserving Zimmer-amenable actions θ : Γ → Aut(Ω , µ ) on a semifinitemeasure space [GP18]. Indeed, in [GP18] a (complete) isometry L p (Ω ⋊ θ Γ) J p −−−−−→ Y U L p (Ω) ⊗ p S p ( ℓ Γ) , that intertwines id ⋊ T m and (id ⊗ H m ) U was constructed using the amenability of θ . That constructionimplies that (cid:13)(cid:13) id ⋊ T m : L p (Ω ⋊ θ Γ) → L p (Ω ⋊ θ Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) id ⊗ H m : L p (Ω) ⊗ p S p ( ℓ Γ) → L p (Ω) ⊗ p S p ( ℓ Γ) (cid:13)(cid:13) , (0.1)and the same follows for complete bounds. Although, Zimmer-amenable actions preserving a finitemeasure (or more generally a mean) can only be constructed for amenable groups, there are plenty ofexamples of non-amenable groups which act in a Zimmer-amenable way on a semifinite measure space.Indeed, any exact discrete group admits such an action, see the comments after Remark 2.3.If the reverse inequality of (0.1) were true for some action θ , ie if (cid:13)(cid:13) T m : L p ( L Γ) → L p ( L Γ) (cid:13)(cid:13) cb ≤ (cid:13)(cid:13) id ⋊ T m : L p ( L Γ) → L p ( L Γ) (cid:13)(cid:13) cb (0.2)holds, then using the fact that the complete norm of H m is always bounded by that of T m , we will havethat the complete norms of T m : L p ( L Γ) → L p ( L Γ) and H m : S p ( ℓ Γ) → S p ( ℓ Γ) coincide. Sadly,we have only been able to obtain the reverse inequality when the action admits an invariant mean.Indeed, it holds that
Theorem A.
Let ≤ p ≤ ∞ and U be a proper ultrafilter. If Ω has a θ -invariant mean, there is acomplete isometry L p ( L Γ) J p −−−−−→ Y U L p (Ω ⋊ θ Γ) which satisfies that L p ( L Γ) T m (cid:15) (cid:15) J p / / Y U L p (Ω ⋊ θ Γ) (id ⋊ T m ) U (cid:15) (cid:15) L p ( L Γ) J p / / Y U L p (Ω ⋊ θ Γ) (0.3) and is L Γ -bimodular.As a consequence we have that, for every ≤ p ≤ ∞ , (cid:13)(cid:13) T m : L p ( L Γ) → L p ( L Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) (id ⋊ T m ) : L p (Ω ⋊ θ Γ) → L p (Ω ⋊ θ Γ) (cid:13)(cid:13) , (0.4) the same follows for completely bounded norms. This result does not provide new examples of groups for which both the Fourier and Herz-Schurmultipliers have the same norm in L p since every group Γ admitting a Zimmer-amenable action withan invariant mean is amenable.In the search for examples beyond amenable groups we explore if changing the natural embedding L Γ ֒ → L ∞ (Ω) ⋊ θ Γ by other completely positive maps Φ : L Γ → L ∞ (Ω) ⋊ θ Γ will give weakerconditions than the existence of an invariant mean. Since we want to intertwine the operators T m andid ⋊ T m the map Φ must be of the form λ g Φ ϕ g ⋊ λ g , for some collection of functions ( ϕ g ) g ∈ Γ . Such maps, which we will call equivariant , can be easilyclassified as follows 3 heorem B. Let
Φ : L Γ → L ∞ (Ω) ⋊ θ Γ be a normal and equivariant map of symbol ( ϕ g ) g ∈ Γ then (i) Φ is a ∗ -homomorphism iff ϕ : Γ → L ∞ (Ω; T ) is a multiplicative -cocycle. (ii) Φ is unital and completely positive iff there exists a Hilbert space H and a multiplicative -cocycle κ : Γ → L ∞ (Ω; H ) and a unit vector ξ ∈ H such that ϕ g ( ω ) = h ξ, κ g ( ω ) ξ i (iii) Φ is completely bounded iff there is a Hilbert space H and maps Ξ , Σ ∈ L ∞ (Ω × Γ; H ) suchthat ϕ g h − ( ω ) = (cid:10) Ξ g , Σ h (cid:11) ( θ g ω ) = (cid:10) Ξ g ( θ g ω ) , Σ h ( θ g ω ) (cid:11) . (0.5) Furthermore it holds that (cid:13)(cid:13)
Φ : L Γ → L ∞ (Ω) ⋊ θ Γ (cid:13)(cid:13) cb = inf n k Σ k L ∞ (Ω × Γ; H ) k Ξ k L ∞ (Ω × Γ; H ) o where the infimum is taken over all the Σ and Ξ as in (0.5) . Observe that the result above is a straightforward generalization of [Jol92], in the case of (iii) as wellas the classical correspondence between positive type functions and group representations, see [Fol95,Section 3.3] and [BdlHV08, Appendix C]. Although the theorem above is not surprising it seems thatit is not in the literature, therefore we have chosen to include it in its full generality, although we willjust use the points (i) and (ii).
Weak containment of correspondences and transference.
In Sections 3 and 4 we will connectthe transference technique described above with the theory of weak containment of correspondencesover von Neumann algebras. Given two von Neumann algebras N and M , an N - M -correspondence is aHilbert space H with two normal and commuting representation that turn H into an N - M -bimodule.Those objects form a natural category with the bounded bimodular maps as their morphisms, see[Pop86], [Con94, Appendix B] or [AP17, Chapter 13]. This category behaves in the context of vonNeumann algebras in a way that is reminiscent of the category of representations in the context ofgroups, see [CJ85]. Following this common analogy, the trivial N - N -bimodule L ( N ) is understoodas an analogue of the trivial group representation : Γ → C . Correspondences admit a naturalnotion of weak containment and a N - M correspondence H is said to be (left) amenable iff H ¯ ⊗ M H weakly contains the trivial bimodule, see [AD95]. This notion generalizes the definition of amenablerepresentation introduced in [Bek90] in which ρ is amenable iff ≺ ρ ⊗ ρ . Using a common GNS-typeconstruction, we can associate to each completely positive map Φ : N → M a cannonical N - M corre-spondence H (Φ). The map Φ is said to be left amenable precisely when H (Φ) is. The key observationon Sections 3 and 4 is that the transference techniques used in [NR11, CPPR15, CPPR15, GP18] canbe understood as an extrapolation result by which the existence of a weak containment expressed interms of Hilbert bimodules can be extended to noncommutative L p -bimodules, see Theorem 3.6.In order to explain this idea, in the particular case of ∗ -homomorphism, let us denote N = L Γ. We needto recall that if π : N → M is some normal ∗ -homomorphism into a semifinite von Neumann algebra M , we have that π is left amenable in the sense of [AD95] iff the trivial N -bimodule L ( N ) is weaklycontained in the N -bimodule L ( M ), where the left and right actions are given by multiplication x · ξ · y = π ( x ) ξ π ( y ) . But L ( N ) ≺ L ( M ) iff there is a N -bimodular isometry L ( N ) J −−−−→ Y U L ( M ) U , where the space on the right hand side is the bimodule given by a proper ultrapower. The insightis that this bimodular map at the L -level exists iff there is a N -bimodular isometry J p : L p ( N ) → p ( M ). Similarly, if J satisfy an intertwining identity between multiplier operators, so does J p .Thus, transference theorem for L p -spaces can be seen as extrapolation theorems in which an inclusionof Hilbert N -bimodules is generalize to an inclusion of L p bimodules. Then, in Theorem 3.7 andTheorem 4.4 we classify which equivariant ∗ -homomorphisms and completely positive maps respectivelyare amenable and admit an isometry J intertwining the required multiplier operators. Our quest tofind new examples beyond the need for invariant mean fails, since we obtain conditions that implythe existence of such a mean. Nevertheless, we consider that the resulting theorems may still be ofinterest.An important question left open by the approach of this paper is the necessity of the N -bimodularity onthe extrapolation argument. In principle, just the existence of an intertwining L p -isometry will sufficeto prove a transference theorem. Nevertheless, it seems that all the current proofs in the literature usea certain bimodularity in order to prove the L p -isometric character of J p and it is still open whetherthis condition is necessary or not.
1. Equivariant maps into crossed products
Let Γ be a discrete countable group acting on a σ -finite measure space (Ω , µ ) by measure-preservingtransformations. Denote the action by θ : Γ → Aut(Ω , µ ). Recall that the group von Neumann algebraof
Γ, denoted by L Γ ⊂ B ( ℓ Γ) is the von Neumann algebra given by { λ g : g ∈ Γ } ′′ = span w ∗ { λ g : g ∈ Γ } ⊂ B ( ℓ Γ) , where λ : Γ → B ( ℓ Γ) is the left regular representation given by λ g ( δ h ) = δ g h , where g, h ∈ Γ and( δ h ) h ∈ Γ is the canonical orthonormal base of placeholder functions in ℓ Γ.The crossed-product von Neumann algebra L ∞ (Ω) ⋊ θ Γ, also called the group measure space construc-tion of the action θ : Γ → Aut(Ω , µ ), is given the von Neumann algebra L ∞ (Ω) ⋊ θ Γ ⊂ B ( L Ω ⊗ ℓ Γ)generated by the representation ⊗ λ : Γ → B ( L Ω ⊗ ℓ Γ) and the ∗ -homomorphism π : L ∞ (Ω) →B ( L Ω ⊗ ℓ Γ) π ( f ) = X h ∈ Γ θ h − ( f ) ⊗ e h,h , where θ g ( f )( ω ) = f ( θ g − ω ). Observe that the algebra L ∞ (Ω) ⋊ θ Γ is given by by the weak- ∗ closureof finite sums of the form x = X g ∈ Γ f g ⋊ λ g , where f ⋊ λ g is just shorthand notation for the operator π ( f ) · ( ⊗ λ g ).Let τ : L Γ → C be the canonical trace of L Γ, given by the vector state τ Γ ( x ) = h δ e , xδ e i , that wewill denote simply as τ when no ambiguity is present. The map E : L ∞ (Ω) ⋊ Γ → L ∞ (Ω), given byrestriction to L ∞ (Ω) ⋊ θ Γ of id ⊗ τ is a normal conditional expectation. A straightforward calculationgives that, over finite sums, it takes the form E (cid:16) X g ∈ Γ f g ⋊ λ g (cid:17) = f e Furthermore it satisfies the following properties • It is faithful , ie for every x ≥ E [ x ] = 0 iff x = 0. • It is equivariant , ie E (cid:2) λ g x λ ∗ g (cid:3) = θ g E [ x ]. 5ee [BO08, Proposition 4.1.9] for both facts above. Using those properties we obtain that everymeasure ν : L ∞ (Ω) + → [0 , ∞ ] gives a tracial weight τ ν = ν ⋊ τ Γ : ( L ∞ (Ω) ⋊ Γ) + → [0 , ∞ ] given by τ ν = ν ◦ E , when ν is θ -invariant. We will denote the canonical tracial weight associated with µ by τ ⋊ . The noncommutative L p -spaces associated to both ( L Γ , τ ) and ( L ∞ Ω ⋊ θ Γ , τ ⋊ ) would be denotedby L p ( L Γ) and L p (Ω ⋊ θ Γ) respectively.Observe also that both L Γ and L ∞ (Ω) ⋊ θ Γ admit the following natural comultiplication and coactionmaps.
Proposition 1.1.
There are normal ∗ -homomorphisms ∆ : L Γ −→ L Γ ¯ ⊗L Γ , (1.1)∆ ⋊ : L ∞ (Ω) ⋊ θ Γ −→ (cid:0) L ∞ (Ω) ⋊ θ Γ (cid:1) ¯ ⊗L Γ , (1.2) given by extension of λ g λ g ⊗ λ g and f ⋊ λ g ( f ⋊ λ g ) ⊗ λ g respectively. Those maps satisfy thefollowing coassociativity properties (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ and (∆ ⋊ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ . The proposition above follows easily from the Fell absorption principle for representations and theFell absorption principle for actions, see [BO08, Propositions 4.1.7]. We will use a form of the Fellabsorption principle in the proof of Theorem B (i). Notice that in the case of Γ Abelian, the map∆ is just the pullback over functions of the multiplication of b Γ, the Pontryagin dual of Γ, while themap ∆ ⋊ is the pullback of the dual action b θ : b Γ → Aut( L ∞ Ω ⋊ θ Γ), given by sending χ ∈ b Γ into f ⋊ λ g
7→ h χ, g i f ⋊ λ g .A map T : L Γ → L Γ is ∆-equivariant (id ⊗ ∆) ◦ T = (id ⊗ T ) ◦ ∆ iff it is a Fourier multiplier , ie a mapgiven by extension of λ g m ( g ) λ g for some function m ∈ ℓ ∞ (Γ). That operator is called the Fourier multiplier of symbol m and denote T = T m , is we want to make the dependence on the symbol explicit. A map Φ : L Γ → L ∞ (Ω) ⋊ θ Γis equivariant iff (id ⊗ ∆) ◦ T = ( T ⊗ id) ◦ ∆ ⋊ = (id ⊗ T ) ◦ ∆ ⋊ . Those operators are given by linearextension of the map Φ( λ g ) = ϕ g ⋊ λ g , for some symbol ϕ : Γ → L ∞ (Ω).Let G be a topological group. The map κ : Γ → L ∞ (Ω; G ) is a multiplicative 1-cocycle, or simply a1-cocycle if it satisfies that κ g h = κ g θ g ( κ h ) , where θ g ( κ )( ω ) = κ ( θ − g ω ). We will say that κ is a unitary 1-cocycle if G = U ( H ) for some Hilbertspace H .The following lemma is a trivial application of the embedding L Γ ֒ → L ∞ (Ω) ¯ ⊗B ( ℓ Γ).
Lemma 1.2.
A equivariant map
Φ : L Γ → L ∞ (Ω) ⋊ θ Γ , given by λ g ϕ g ⋊ λ g , is positivity preservingiff for every finite set S = { g , g , ..., g r } ⊂ Γ the matrices (cid:2) θ g − i ( ϕ g i g − j ) (cid:3) i,j ∈ L ∞ (Ω) ⊗ M S ( C ) . are positive definite. In that case, the map Φ is also completely positive. Proof.
Observe that we have that the following diagram commutes6 Γ Φ / / (cid:15) (cid:15) L ∞ (Ω) ⋊ θ Γ (cid:15) (cid:15) B ( ℓ Γ) Ω ⊗ id ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ L ∞ (Ω) ¯ ⊗B ( ℓ Γ) L ∞ (Ω) ¯ ⊗B ( ℓ Γ) H Φ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ where the vertical arrows are the natural inclusions and the map H Φ is the L ∞ (Ω)-valued Schurmultiplier given by X g,k ∈ Γ f g,k ⊗ e g,k X g,k ∈ Γ f g,k θ g − ( ϕ g k − ) ⊗ e g,k . (1.3)Using that Schur multipliers preserve positivity iff they are positive definite gives the result.We will need the following technical lemma which extends Grothendieck/Haagerup decomposition ofbounded Schur multipliers to the L ∞ (Ω)-valued case, see [BF84]. Although the lemma is straightfor-ward we include its proof for the sake of completeness. Lemma 1.3.
Let (Ω , µ ) be a σ -finite measure space as before and S be a countable discrete set. Let m ∈ L ∞ ( S × S × Ω) be a function and B ( ℓ S ) H m −−−−−−→ L ∞ (Ω) ¯ ⊗B ( ℓ S ) the normal and bounded operator given as the L ∞ (Ω) -valued Schur multiplier of symbol mH m (cid:16) X s,t ∈ S a s,t e s,t (cid:17) = X s,t ∈ S m s,t ( ω ) a s,t ⊗ e s,t . Then, there exists a Hilbert space H and Ξ , Σ ∈ L ∞ (Ω × S ; H ) such that m s,t ( ω ) = (cid:10) Ξ s ( ω ) , Σ t ( ω ) (cid:11) and k H m k = inf n k Ξ k L ∞ (Ω; H ) k Σ k L ∞ (Ω; H ) o , Reciprocally, all operators H m with the above decomposition are bounded. Proof.
First observe that if (Ω , µ ) is a countable discrete space, so that L ∞ (Ω) ∼ = ℓ ∞ (Ω), the resultabove follows easily from the scalar case, see [BF84]. Indeed, using that the restriction of H m to C δ ω ⊗ B ( ℓ ∞ S ) gives a normal Schur multiplier H m ( ω ) : B ( ℓ S ) → B ( ℓ S ) of symbol m ( ω ) and normbounded by that of H m . Furthermore, the fact that the norm of ℓ ∞ (Ω) ¯ ⊗B ( ℓ S ) = ℓ ∞ (Ω; B ( ℓ S )), aswell as all its matrix amplifications, can be taken as a supremum in Ω easily yields that (cid:13)(cid:13) H m : B ( ℓ S ) → ℓ ∞ (Ω) ¯ ⊗B ( ℓ S ) (cid:13)(cid:13) cb = max ω ∈ Ω n(cid:13)(cid:13) H m ( ω ) : B ( ℓ S ) → B ( ℓ S ) (cid:13)(cid:13) cb o . For each ω ∈ Ω, there is a Hilbert space H ω such that m s,t = (cid:10) Ξ [ ω ] s , Σ [ ω ] t (cid:11) and satisfying that the complete norm of H m ( ω ) is equal to k Ξ [ ω ] k k Σ [ ω ] k . You can construct H as thedirect sum of all H ω , ie H = ℓ (Ω; { H ω } ω ∈ Ω ), and the maps Ξ : S × Ω → H and Σ : S × Ω → H byΞ s ( ω ) = Ξ [ ω ] s ∈ H ω ⊂ H and a similar formula for Σ. A trivial calculation yields the result.7he case in which Ω is diffuse, although not conceptually different, runs into technical difficulties due tomeasurability. Let us start noticing that, if ( m α ) α is a bounded sequence of symbols and H m α → H m in the weak- ∗ topology, then lim α m αs,t ( ω ) = m s,t ( ω ) µ -almost everywhere. Let S be the σ -algebra of(Ω , µ ). Take a filtration of σ -algebras S α ⊂ S α +1 · · · ⊂ S such that each of the sigma algebras S α isgiven by the power set of a partition Ω = [ j ≥ Ω αj into sets of finite measure. We will also assume that the union of all the partitions S α generated S . Asa consequence, we have that the subalgebras L ∞ (Ω; S α ) ⊂ L ∞ (Ω), are thus isomorphic to ℓ ∞ and theunion of them is a weak- ∗ dense subalgebra of L ∞ (Ω). Let us denote by E α : L ∞ (Ω) → L ∞ (Ω; S α ) ∼ = ℓ ∞ the conditional expectation associated to the sub- σ -algebra S α . We will denote as well by E α thetensor amplification E α ⊗ id : L ∞ (Ω) ¯ ⊗B ( ℓ S ) → ℓ ∞ ¯ ⊗B ( ℓ S ). Clearly we have that E α ◦ H m convergein the weak- ∗ topology to H m . But it also holds that E α ◦ H m = H E α ( m ) and therefore E α ( m s,t )converges almost everywhere to m s,t B ( ℓ S ) H m / / L ∞ (Ω) ¯ ⊗B ( ℓ S ) E α (cid:15) (cid:15) B ( ℓ S ) = O O H E α ( m ) / / ℓ ∞ ¯ ⊗B ( ℓ S ) . But now, by applying the discrete case, we have that each of the symbols m αs,t = E α ( m s,t ) has adecomposition as h Ξ αs ( ω ) , Σ αt ( ω ) i , where Ξ, Σ are S α -measurable functions in L ∞ (Ω × S ; H α ), forsome Hilbert space H α and they reach the norm of H αm . We can take the ultraproduct Hilbert space H = Q α, U H α , for some proper ultrafilter U , and notice that we can construct Ξ and Σ by identifying L ∞ (Ω × S ; H ) with Q α, U L ∞ (Ω × S ; H α ) so that Ξ = (Ξ α ) U α and Σ = (Σ α ) U α . We have that m s,t ( ω ) = (cid:10) X s ( ω ) , Σ t ( ω ) (cid:11) = lim α →U (cid:10) X αs ( ω ) , Σ αt ( ω ) (cid:11) = lim α →U m αs,t ( ω ) = m s,t ( ω )where the last equality follows almost everywhere. The identity for the norms follows similarly. Remark 1.4.(R.1)
Like in the case of Schur multipliers the complete boundedness of the L ∞ (Ω)-valued multipliersis automatic, see [BF84]. This can be obtained from the automatic complete boundednessresults of [Smi91] after noting that the maps above are L ∞ (Ω) ¯ ⊗ ℓ ∞ ( S )-bimodular. Like inthe case of Fourier multipliers, the decomposition above extends to equivariant operatorsΦ : L Γ → L ∞ (Ω) ⋊ θ Γ only when the operator Φ us completely bounded, see the proof ofTheorem B (iii). (R.2)
The proof above works verbatim to prove that the alternative L ∞ (Ω)-valued Herz-Schurmultiplier given by H m (cid:16) X s,t ∈ S f s,t ( ω ) ⊗ e s,t (cid:17) = X s,t ∈ S m s,t ( ω ) f s,t ( ω ) ⊗ e s,t , is bounded/completely bounded if and only if there exists a Hilbert space H and Ξ , Σ ∈ L ∞ (Ω × S ; H ) such that m s,t ( ω ) = (cid:10) Ξ s ( ω ) , Σ t ( ω ) (cid:11) ω ∈ Ω. In that case, its norm is again given by k H m : L ∞ (Ω) ¯ ⊗B ( ℓ ) → L ∞ (Ω) ¯ ⊗B ( ℓ ) k = inf n k Ξ k L ∞ (Ω; H ) k Σ k L ∞ (Ω; H ) o , where the infimum is taken over all such decompositions. (R.3) There is an alternative route to prove the point above that, although much less direct, wouldgive a far reaching result. Such alternative route would be obtained from an intricate combi-nations of scattered results in the literature as follows. Given an hyperfinite von Neumann al-gebra M , the results in [CS92, CS93] imply that the map from the, so called, central Haageruptensor product M⊗ Z ,h M into CB σ ( M , M ) given by extension of x ⊗ y ℓ ( x ) r ( y ), where ℓ, r represent the left and right multiplication operators on M respectively, is isometric. Its im-age is point weak- ∗ dense inside the normal and decomposable maps of M , which by [Haa85]are all completely bounded normal maps. Extending the isometry from M ⊗ Z ,h M to itsextended tensor product version, in the sense of [ER03, BS92] will make the map surjective.Given a sub-von Neumann algebra N ⊂ M , we would have to follows the steps of [Smi91] toconstruct an isomorphism between the normal N ′ - N ′ -bimodular maps CB σ N ′ , N ′ ( M , M ) anda central version of the extended Haagerup tensor N ⊗ Z ,eh N . Taking M = L ∞ (Ω) ¯ ⊗B ( ℓ S )and N = L ∞ (Ω) ¯ ⊗ ℓ ∞ ( S ) will give the result above.We can proceed to prove the main Theorem of the section. Proof (of Theorem B).
For (i), the only if part is trivial. Indeed, it is trivial to see that if the map λ g ϕ g ⋊ λ g is multiplicative, then ϕ : Γ → L ∞ (Ω; T ) is a multiplicative 1-cocycle. To see that everymultiplicative 1-cocycle induces a normal ∗ -homomorphism, we will use the following version of theFell absorption principle. Let W : L (Ω) ⊗ ℓ (Γ) → L (Ω) ⊗ ℓ Γ, be the unitary given by extensionof ξ ⊗ δ h W θ − h ( ϕ h ) ξ ⊗ δ h . The following diagram commutes L (Ω) ⊗ ℓ (Γ) W / / ϕ k ⋊ λ k (cid:15) (cid:15) L (Ω) ⊗ ℓ (Γ) ⊗ λ k (cid:15) (cid:15) L (Ω) ⊗ ℓ (Γ) W / / L (Ω) ⊗ ℓ (Γ) . Indeed, we have that (cid:2) ( ⊗ λ k ) W (cid:3) ( ξ ⊗ δ h ) = ( ⊗ λ k ) ( θ h − ( ϕ h ) ξ ⊗ δ h = θ h − ( ϕ h ) ξ ⊗ δ k h while (cid:2) W ( ϕ k ⋊ λ k ) (cid:3) ( ξ ⊗ δ h ) = W (cid:2) θ − kh ( ϕ k ) ξ ⊗ δ kh (cid:3) = θ − k h ( ϕ k h ) θ − k h ( ϕ k ) ξ ⊗ δ kh = θ − h ( ϕ h ) ξ ⊗ δ kh Therefore W spatially implements the homomorphism λ g ϕ g ⋊ λ g .For (ii) the if side is very similar. Indeed, the map λ g κ g ⋊ λ g extends to a normal ∗ -homomorphism π : L Γ → L ∞ (Ω; B ( H )) ⋊ θ ⊗ id Γ. To see that, just notice that, again by a Fell like absorption principle,there is a unitary W : H ⊗ L (Ω) ⊗ ℓ Γ → H ⊗ L (Ω) ⊗ ℓ Γ such that the following diagramcommutes H ⊗ L (Ω) ⊗ ℓ Γ W / / ( κ g ⊗ λ g ) (cid:15) (cid:15) H ⊗ L (Ω) ⊗ ℓ Γ ( ⊗ λ g ) (cid:15) (cid:15) H ⊗ L (Ω) ⊗ ℓ Γ W / / H ⊗ L (Ω) ⊗ ℓ Γ , (1.4)9here, after identifying H ⊗ L (Ω) with L (Ω; H ), W is given by W ( ξ ( ω ) ⊗ δ h ) = κ h − ( ω ) ξ ( ω ) ⊗ δ h . The map Φ of the form Φ( x ) = V ∗ π ( x ) V , where V : L (Ω) ⊗ ℓ (Ω) → H ⊗ L (Ω) ⊗ ℓ (Γ) is givenby ηδ k ξ ⊗ η ⊗ δ k , is completely positive by Stinespring’s Theorem.To see that any completely positive equivariant map is of this form we will use the theory of W ∗ -Hilbertmodules [MV05, Lan95]. Define the L ∞ (Ω)-valued inner product over X = L ∞ (Ω) ⊗ alg C [Γ] by D X g ∈ Γ F g ⊗ δ g , X k ∈ Γ G k ⊗ δ k E Φ = X g ∈ Γ X k ∈ G θ g − (cid:0) F g G k (cid:1) ϕ g − k . Observe that h· , ·i Φ is positive definite by Lemma 1.2. Let us denote by X the completion of X moduloits nulspace for the seminorms x φ (cid:0) h x, x i (cid:1) , for every φ ∈ L (Ω). Using [JS05, Theorem 2.5.], we have that X embeds as a complemented L ∞ (Ω)-module of L ∞ (Ω; H ) for some Hilbert space H . Observe also that every element x ∈ C 1 Ω ⊗ alg C [Γ]extends to a constant vector x = Ω ⊗ ξ ∈ L ∞ (Ω; H ). For every h ∈ Γ, let L h be the operator givenlinear extension of L h ( F ⊗ δ k ) = F ⊗ δ h k . It holds that (cid:10) L h ( x ) , L h ( y ) (cid:11) Φ = θ h h x, y i Φ . (1.5)Observe that equation (1.5) implies that L h extends to an isometric operator π h over X and to aunitary operator π h acting on the Hilbert space L (Ω; H ) whose inner product is given by µ ◦ h· , ·i Φ .Equation (1.5) also yields that, for every ξ ∈ L ∞ (Ω; H ), [ π h ξ ]( ω ) = κ h ( ω ) ξ ( θ − h ω ), for some unitary κ h ( ω ) ∈ U ( H ). A straightforward computation gives that κ is a unitary 1-cocycle and that ϕ g ( ω ) = h ξ, κ g ( ω ) ξ i , where ξ ∈ H is the vector associated to the constant extension of x = ⊗ δ e .For (iii) first we check that every ϕ satisfying that ϕ g − h = (cid:10) θ − g Ξ g , θ − g Σ h (cid:11) is completely bounded.For that, use that the embedding ι : L ∞ (Ω) ⋊ θ G ֒ → L ∞ (Ω) ¯ ⊗B ( ℓ S ) is given by ι (cid:16) X g ∈ Γ f g ⋊ λ g (cid:17) = X g,k ∈ Γ θ g − ( f g h − ) ⊗ e g,h . (1.6)We have that the following diagram commutes L Γ ι / / Φ (cid:15) (cid:15) B ( ℓ Γ) H m (cid:15) (cid:15) L ∞ (Ω) ⋊ θ Γ ι / / L ∞ (Ω) ¯ ⊗B ( ℓ Γ) , (1.7)where H m is the L ∞ (Ω)-valued Schur multiplier of symbol m g,h ( ω ) = θ g − ( ϕ g h − ) = h Ξ g ( ω ) , Σ h ( ω ) i ,which is completely bounded. For the reciprocal we define the injective and normal ∗ -homomorphism L ∞ (Ω) ¯ ⊗B ( ℓ Γ) π −−−−→ (cid:0) L ∞ (Ω) ⋊ θ Γ (cid:1) ¯ ⊗B ( ℓ Γ)given by π (cid:16) X g,h ∈ Γ f g,h ⊗ e g,h (cid:17) = X g,h ∈ Γ (cid:0) θ g − ( f g,h ) ⋊ λ s − t (cid:1) ⊗ e s,t . In an abuse of notation we will also denote by π the ∗ -homomorphism given π : B ( ℓ Γ) → L Γ ¯ ⊗B ( ℓ Γ)obtained when L ∞ (Ω) = C1 . Those maps satisfy the following intertwining identity10 ( ℓ Γ) π / / H θg ( ϕg − h ) (cid:15) (cid:15) L Γ ¯ ⊗B ( ℓ Γ) Φ ⊗ id (cid:15) (cid:15) L ∞ (Ω) ¯ ⊗B ( ℓ Γ) π / / (cid:0) L ∞ (Ω) ⋊ θ Γ (cid:1) ¯ ⊗B ( ℓ Γ) . Observe that the symbol θ g ( ϕ g − h ) induces a bounded multiplier if Φ ⊗ id is bounded or, equivalently,if Φ is completely bounded. But, by Lemma 1.3, if the L ∞ (Ω)-valued Schur multiplier is bounded, itadmits a decomposition θ g ( ϕ g − h ) = (cid:10) Ξ ′ g , Σ ′ h (cid:11) and by setting Ξ g = Ξ ′ g − and Σ g = Σ ′ g − we obtain a decomposition like that of (0.5). Remark 1.5.
In direct analogy with the case of Fourier multipliers, see [BF84], the complete bounded-ness of the equivariant map Φ : L Γ → L ∞ (Ω) ⋊ θ Γ given by extension of λ g ϕ g ⋊ λ g is only requiredto prove that the factorization (0.5) is necessary. Quantitatively, we have that if m g,h = θ g − ( ϕ g h − )and H m is its associated L ∞ (Ω)-valued Schur multiplier, we have that (cid:13)(cid:13) Φ : L Γ → L ∞ (Ω) ⋊ θ Γ (cid:13)(cid:13) ≤ (cid:13)(cid:13) H m : L ∞ (Ω) ¯ ⊗B ( ℓ Γ) → L ∞ (Ω) ¯ ⊗B ( ℓ Γ) (cid:13)(cid:13) (1.8) (cid:13)(cid:13) H m : L ∞ (Ω) ¯ ⊗B ( ℓ Γ) → L ∞ (Ω) ¯ ⊗B ( ℓ Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) Φ : L Γ → L ∞ (Ω) ⋊ θ Γ (cid:13)(cid:13) cb , where equation (1.8) works also if you use completely bounded norms in both sides of the inequality.The factorization (0.5) follows by applying Lemma 1.3 to m g,h = θ g − ( ϕ g h − ).
2. Transference for actions with an invariant mean
Our proof leans in a key way on the following technical lemma, which follows techniques of almostmultiplicative maps developed in [CPPR15, Theorem B/Corollary 2.3], which generalize both pseu-dolocalization techniques for Fourier multipliers and the noncommutative Powers-Størmer’s inequlity.Some consequences of those result are also obtained in [Ric15]. We recall that the technniques usehere are similar to those in [GP18].
Theorem 2.1 ([CPPR15, Theorem B/Corollaries 2.3/2.4.]) . Let ( M , τ ) be a semifinite von Neumannalgebra with a normal, faithful and semifinite tracial weight and let R : M → M be a positive subunitalmap such that τ ◦ T ≤ τ map, we have that (i) For every ξ ∈ L ( M ) + (cid:13)(cid:13) R ( ξ θ ) − ξ θ (cid:13)(cid:13) θ ≤ C (cid:13)(cid:13) R ( ξ ) − ξ (cid:13)(cid:13) θ k ξ k θ . (ii) For every ξ ∈ L ( M ) (cid:13)(cid:13) R ( u | ξ | θ ) − u | ξ | θ (cid:13)(cid:13) θ ≤ C (cid:13)(cid:13) R ( ξ ) − ξ (cid:13)(cid:13) θ k ξ k θ , where ξ = u | ξ | is the polar decomposition of ξ . We also cite the following lemma, although it can be obtained from Theorem 2.1 above by taking R ( ξ ) = u ξ u ∗ , for every unitary in the spectral calculus of x and using ultraproduct techniques. Lemma 2.2 ([Ric15, Lemma 2.4/ Lemma 2.6]) . Let M be a von Neumann algebra and assume M is semifinite with a n.s.f. trace τ : M + → [0 , ∞ ] . Let ξ ∈ L ( M ) be a unit vector and ξ = u | ξ | beits polar decomposition. Fix ≤ p ≤ ∞ and notice that | ξ | p ∈ L p ( M ) . For every x ∈ M and ε > ,there is a δ > such that i) (cid:13)(cid:13) [ x, ξ ] (cid:13)(cid:13) < δ = ⇒ (cid:13)(cid:13)(cid:2) x, | ξ | p (cid:3)(cid:13)(cid:13) p < ε. (ii) (cid:13)(cid:13)(cid:2) x, ξ (cid:3)(cid:13)(cid:13) < δ = ⇒ (cid:13)(cid:13)(cid:2) x, u | ξ | p (cid:3)(cid:13)(cid:13) p < ε. With that lemma at hand we can prove the following theorem.
Proof (of Theorem B).
Let ι : L Γ ֒ → L ∞ (Ω) ⋊ θ Γ be the natural embedding that sends λ g to ⋊ λ g .Observe that if µ (Ω) = 1 we are already finishes since ι is trace preserving, ie τ ⋊ ◦ ι = τ , and thus ι extends to all L p -space. In the case of µ (Ω) = ∞ we have that, by the fact that L ∞ (Ω) admits aleft-invariant mean m : L ∞ (Ω) → C , there is a unit net of vectors ξ α ∈ L (Ω) such that, for every g ∈ Γ, (cid:13)(cid:13) θ g ( ξ α ) − ξ α (cid:13)(cid:13) → . The vectors also satisfy that m is any weak- ∗ accumulation point of the states f
7→ h ξ α , f x α i . Let usdefine J αp : L p ( L Γ) → L p (Ω ⋊ θ Γ) by J αp ( x ) = u α | ξ α | p ι ( x ) | ξ α | p , where x α = u | ξ n | is the polar decomposition and we are identifying ξ α ∈ L (Ω) with ξ α ⋊ λ e ∈ L (Ω ⋊ θ Γ). We have that (1) (cid:13)(cid:13) J αp : L p ( L Γ) → L p (Ω ⋊ θ Γ) (cid:13)(cid:13) ≤ . (2) lim α (cid:10) J αp ( x ) , J αp ( y ) (cid:11) = h x, y i for every x ∈ L p ( L Γ) and y ∈ L q ( L Γ) with p + q = 1.The proofs of both points are straightforward generalizations of the results in [NR11, CdlS15, GP18].Indeed, (1) is obvious for p = ∞ . By interpolation it suffices to be proven in the case of p = 1. Let x ∈ L ( L Γ) an element of unit norm. We can take y , z ∈ C [Γ] such that k z − y, z k ≤ ǫ , for every ǫ >
0. To see that just notice that there are elements y ′ , z ′ ∈ L ( L Γ) of unit norm and such that x = y z . By density, there are elements y and z in C [Γ] such that k z − z ′ k ≤ δ and k y − y ′ k ≤ δ . Acalculation yields that k x − y z k ≤ k y ( z − z ′ ) k + k ( y − y ′ ) z ′ k ≤ δ + δ , and so, taking δ small enough gives the desired identity. Now, we have that (cid:13)(cid:13) J αp ( x ) (cid:13)(cid:13) = (cid:13)(cid:13) J αp ( y z ) (cid:13)(cid:13) + (cid:13)(cid:13) J αp ( x − y z ) (cid:13)(cid:13) = (cid:13)(cid:13) u α | ξ α | ι ( y z ) | ξ α | (cid:13)(cid:13) + (cid:13)(cid:13) u α | ξ α | ι ( x − y z ) | ξ α | (cid:13)(cid:13) = I + IIThe term II can be bounded by (cid:13)(cid:13) u α | ξ α | ι ( x − y z ) | ξ α | (cid:13)(cid:13) ≤ (cid:13)(cid:13) u α | ξ α | (cid:12)(cid:12) (cid:13)(cid:13) ι ( x − y z ) (cid:13)(cid:13) ∞ (cid:13)(cid:13) | ξ α | (cid:13)(cid:13) ≤ (cid:13)(cid:13) x − y z (cid:13)(cid:13) ≤ ǫ. While for II, we have that II ≤ (cid:13)(cid:13) u α | ξ α | ι ( y ) (cid:13)(cid:13) (cid:13)(cid:13) ι ( z ) | ξ α | (cid:13)(cid:13) = A B.
Both terms, A and B are estimated similarly. Let y be the finite sum P g ∈ Γ y g λ g . We have that A isbounded by A = (cid:13)(cid:13) u α | ξ α | ι ( y ) (cid:13)(cid:13) = (cid:13)(cid:13) | ξ α | ι ( y ) (cid:13)(cid:13) = τ ⋊ (cid:26)(cid:16) X g ∈ Γ y g Ω ⋊ λ g − (cid:17) | ξ α | (cid:16) X g ∈ Γ y h Ω ⋊ λ h (cid:17)(cid:27) = X g ∈ Γ y g y g Z Ω θ g − ( | ξ α | ) dµ = k y k ≤ (1 + ǫ ) .
12 similar estimate holds for B . Joining both and using that ǫ is arbitrary gives (1) . For (2) , we startchoosing x, y ∈ C [Γ], given by x = P g ∈ Γ x g λ g and y = P h ∈ Γ y h λ h and notice thatlim α (cid:10) J αp ( x ) , J αq ( x ) (cid:11) = lim α τ ⋊ (cid:8) | ξ α | p ι ( x ∗ ) | ξ α | p u ∗ α u α | ξ α | q ι ( y ) | ξ | q (cid:9) (2.1)= lim α τ ⋊ (cid:8) ξ ∗ α ι ( x ∗ y ) ξ (cid:9) = lim α τ ⋊ n X g,h ∈ Γ x g y h ξ α θ g − h ( ξ α ) ⋊ λ g − h o = lim α X g ∈ Γ x g y g Z Ω | ξ α ( ω ) | dµ ( ω ) = h x, y i , where we have used Lemma 2.2 in (2.1). The identity above can be extended to elements not in C [Γ]. Indeed, fix y ∈ C [Γ] and choose x ∈ L p ( L Γ), where p < ∞ . Approximating x in the L p -normby elements in C [Γ] gives that the inequality duality bracket extends to L p ( L Γ) × C [Γ], but now,approximating y by elements in C [Γ] in the L q -norm, or in the weak- ∗ topology if q = ∞ gives thatthe bracket above is well defined for pairs in L p ( L Γ) × L q ( L Γ). Now, (1) and (2) imply that theultraproduct J p = J U p , given by J p ( x ) = (cid:0) J αp ( x ) (cid:1) α + n U ∈ Y U L p (Ω ⋊ θ Γ) , is an isometry. Indeed, that it is a contraction follows from (1) . For the isometry, we notice that thedual of L p (Ω ⋊ θ Γ) U contains L q (Ω ⋊ Γ) U isometrically. Therefore (cid:13)(cid:13) j p ( x ) (cid:13)(cid:13) p = sup φ ∈ Ball[( L p (Ω ⋊ Γ) U ) ∗ ] (cid:10) x, y (cid:11) ≥ sup y ∈ Ball[ L q ] (cid:10) j p ( x ) , j p ( y ) (cid:11) = sup y ∈ Ball[ L q ] (cid:10) x, y (cid:11) = k x k p The fact that J p intertwines T m and id ⋊ T m and the L Γ-modularity follow immediately.
Remark 2.3.
Several remarks are in order. (R.1)
First, in the definition of J αp the choice of putting half a power of | ξ α | on the right and halfon the left is arbitrary. We could have chosen instead any map of the form J αp ( x ) = u α | x α | θp ι ( x ) | x α | − θ ) p , where 0 ≤ θ ≤
1, and the proof would work similarly. Indeed, all such map becomeindependent of θ after taking the ultrapower by the asymptotic centrality of the vectors | ξ α | β ∈ L β ( M ) for any β ∈ (0 , u α at the leftor at the right. (R.2) The statement of Theorem A covers only the case of crossed products of an action on a measurespace. But a similar result holds in the case of of trace preserving actions θ : Γ → Aut( M , ϕ ),where M is a semifinite von Neumann algebra and ϕ is a n.s.f trace. In that setting thefollowing two conditions are equivalent • There is a (non necessarily normal) state ̟ : M → C that is θ -invariant. • There is a net of unit vectors ξ α ∈ L ( M ) such that (cid:13)(cid:13) θ g ( ξ α ) − ξ α (cid:13)(cid:13) → J αp ( x ) = u α | ξ α | p ι ( x ) | ξ α | p , yield,after passing to an ultrapower, a complete isometry J p : L p ( L Γ) → L p ( M × θ Γ) U . Thatisometry is also L Γ-bimodular and intertwines the operators T m and id ⋊ T m .13n [GP18] it was proved that, when the action θ : Γ → Aut(Ω , µ ) is µ -preserving and amenablein the sense of Zimmer, see [BO08, Section 4.3], [Zim84] or the original sources of [Zim77, Zim78b,Zim78a, AEG94], then the natural embedding L ∞ (Ω) ⋊ θ Γ ֒ → L ∞ (Ω) ¯ ⊗B ( ℓ Γ) gives rise to a complete L p -isometry L p (Ω ⋊ θ Γ) J p −−−−−→ Y U L p (cid:0) Ω; S p [ ℓ Γ] (cid:1) that intertwines the maps id ⋊ T m and id ⊗ H m , where H m is the Herz-Schur multiplier associated to m ∈ ℓ ∞ (Γ). Therefore, we have that (cid:13)(cid:13) id ⋊ T m : L p (Ω ⋊ θ Γ) → L p (Ω ⋊ θ Γ) (cid:13)(cid:13) cb ≤ (cid:13)(cid:13) id ⊗ H m : L p (Ω; S p [ ℓ Γ]) → L p (Ω; S p [ ℓ Γ]) (cid:13)(cid:13) ≤ (cid:13)(cid:13) H m : S p [ ℓ Γ] → S p [ ℓ Γ] (cid:13)(cid:13) . (2.2)Observe that, although if a group Γ is Zimmer-amenable and has a probability measure preservingaction then Γ itself is amenable, there are plenty of Zimmer-amenable actions that preserve semifinitemeasures. For instance, if Γ is a countable exact group, then it admits an topological action θ : Γ → Aut( X ) on a compact space X that is Zimmer amenable, see [Oza00]. Take a probability measure µ ∈ P ( X ) with total support and such that the action θ is nonsingualr, ie such that for every g ∈ Γ, µ and θ ∗ g µ are mutually absolutely continuous. Then, the Maharam extension of θ , see [Mah53], isboth Zimmer-amenable and measure preserving. Recall that the Maharam extension of θ is given byˆ θ : Γ → Aut( X × R , µ ⊗ ν ), where dν ( t ) = e t dt and the action is defined as( x, s ) ( θ g x, s + log ω g ( x )) , where ω g : X → R is the Radon-Nykodim derivative of θ ∗ g µ with respect to µ . Therefore each exactdiscrete group admits a measure preserving Zimmer-amenable action. If the lower bound (0.4) holdsfor a particular Zimmer-amenable action θ : Γ → Aut( X ) preserving a σ -finite measure, then we willhave that (cid:13)(cid:13) T m : L p ( L Γ) → L p ( L Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) id ⋊ T m : L p (Ω ⋊ θ Γ) → L p (Ω ⋊ θ Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) H m : S p [ ℓ Γ] → S p [ ℓ Γ] (cid:13)(cid:13) and the same would follow from complete norms, thus solving the open problem. Sadly, the conditionsof being Zimmer amenable and having a Γ-invariant mean m : L ∞ (Ω) → C give amenability for Γwhen they hold simultaneously. In summary, the situation that we obtain is the following • The embedding L Γ ֒ → L ∞ (Ω) ⋊ θ Γ yields an L p -isometry L p ( L Γ) → L p (Ω ⋊ θ Γ) U into theultrapower that intertwines T m and (id ⋊ T m ) U and therefore implies (0.4) when Ω has a Γ-invariant mean. • The embedding L ∞ (Ω) ⋊ θ Γ → L ∞ (Ω) ¯ ⊗B ( ℓ Γ) yields an L p -isometry L p (Ω ⋊ θ Γ) → L p (Ω; S p ) U into the ultrapower that intertwines id ⋊ T m and (id ⋊ H m ) U and therefore implies (2.2) when θ : Γ → Aut(Ω) is Zimmer amenable.The intersection of both conditions only contains actions of amenable groups. It is open whether theconditions above are necessary for transference. Therefore the following problems remain widely open.
Problem 2.4.
Let θ : Γ → Aut(Ω , µ ) be a measure-preserving action on a σ -finite measure space andlet A (Γ) be the Fourier algebra of Γ. (P.1) Which actions satisfy that (cid:13)(cid:13) (id ⋊ T m ) : L p (Ω ⋊ θ Γ → L p (Ω ⋊ θ Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) T m : L p ( L Γ) → L p ( L Γ) (cid:13)(cid:13) , for every m ∈ A (Γ)? 14 P.2)
Which actions satisfy that (cid:13)(cid:13) T m : L p ( L Γ) → L p ( L Γ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) (id ⋊ T m ) : L p (Ω ⋊ θ Γ → L p (Ω ⋊ θ Γ) (cid:13)(cid:13) , for every m ∈ A (Γ)?Problem (P.1) has a positive solution when θ is Zimmer amenable, while Problem (P.2) has a pos-itive solution when Ω has a Γ-invariant mean. For actions beyond those cases neither examples norcounterexamples are known. A correction on [GP18].
In the first paragraph after [GP18, Corollary 3.2.] the equality of thenorms of T m and id ⋊ T m is stated for Zimmer-amenable actions. This is a clearly mistaken statementthat did not appear in the ArXiv preprint of that same paper and was added during the process ofproviding more context to the results therein. The question that that paragraph was intending toaddress is whether the following identity holds (cid:13)(cid:13) id ⋊ T m : L p ( N ⋊ θ Γ) → L p ( N ⋊ θ Γ) (cid:13)(cid:13) = (cid:13)(cid:13) id ⋊ T m : L p ( N ⋊ θ Γ) → L p ( N ⋊ θ Γ) (cid:13)(cid:13) cb where θ : Γ → Aut( N , τ ) is any τ -preserving action on a semifinite von Neumann algebra N , providedthat the algebra N is large enough in some sense. For instance, notice that if N absorbs any finitematrix M k ( C ), ie M k ( C ) ⊗ N ∼ = N , and θ is unitarily equivalent to id ⊗ θ , then identity above isautomatic. That sort of equivariant decomposition follow for actions on amenable groups on finiteMcDuff factors by the work of Ocneanu [Ocn85], that generalized to earlier results of Connes [Con75,Con77].Indeed, let us denote by R the hyperfinite II -factor. An von Neumann algebra N is said to be McDuffiff R ¯ ⊗N ∼ = N . Recall similarly that, given two actions θ and β of Γ on a von Neumann algebra N ,they are said to be outer conjugate iff there exists a normal ∗ -isomorphism φ : N → N and a function u : G → U ( N ) such that u g θ g ( x ) u g = (cid:0) φ ◦ β g ◦ φ − (cid:1) ( x ) , for every x ∈ N and g ∈ Γ. Observe that the equation above forces u to be a multiplicative 1-cocycle. The equation above can easily be reinterpreted as saying that the group homomorphisms ¯ θ ,¯ β : Γ → Aut( N ) / Inn( N ) are conjugate by the automorphism φ . Whenever two actions are outerlyconjugate, we have that the map X g ∈ Γ x g ⋊ λ g X g ∈ Γ u g φ ( x g ) ⋊ λ g (2.3)is an equivariant and normal ∗ -isomorphism of M ⋊ θ Γ, Furthermore, if τ is a trace on M , theautomorphism is τ ⋊ = ( τ ◦ E )-preserving iff φ is. The reciprocal is also true and those maps are allequivariant ∗ -automorphisms.We also recall the following definitions related to action on von Neumann algebras, the interestedreader can look in [Ocn85] and references therein for more information. • The action θ : Γ → Aut( N ) is free iff for every g ∈ Γ \{ e } , we have that θ g doesn’t have nontrivialfixed points. Observe that, if θ is free then ¯ θ : Γ → Out( N ) is faithful, otherwise if θ g ( x ) wereinner, then there will be a unitary u ∈ U ( N ) such that θ g ( x ) = u x u ∗ and θ g ( u ) = u . • Let ( x n ) n ∈ ℓ ∞ [ N ] be a centralizing sequence, ie a sequence such that for every ψ ∈ N ∗ lim n (cid:13)(cid:13) [ x n , ψ ] (cid:13)(cid:13) ∗ = 0 . The action θ is centrally trivial iff for every centralizing sequence ( x n ) n we have that θ g ( x n ) − x n → ∗ topology for every g ∈ Γ \ { e } . • The action θ is strongly centrally trivial iff for every Γ-invariant central projection p ∈ P ( N ) wehave that θ | p M p is centrally trivial. 15otice that the last two definitions above collapse if N is a factor. We also have that the notion ofbeing centrally trivial can be interpreted as saying that the ultraproduct action θ g acts trivially overthe relative commutant N U ∩ N ′ . We have the following theorem. Theorem 2.5 ([Ocn85, Section 1.2]) . Let M be a McDuff von Neumann algebra, Γ an amenable groupand θ : Γ → Aut( M ) a centrally free action, then θ and θ ⊗ id R : Γ → Aut( M ¯ ⊗R ) are outer conjugate. With the previous theorem at hand we can easily obtain the automatic complete boundedness ofid ⋊ T m for those families of actions. Indeed, lest Ψ be the normal ∗ -isomorphism N ⋊ θ Γ Ψ −−−−−→ (cid:0) N ¯ ⊗R ) ⋊ θ ⊗ id Γobtained from the outer conjugation of θ and θ ⊗ id R as in (2.3). Observe that, since the map Φ isequivariant, it follows that the following diagram commutes N ⋊ θ Γ id ⋊ T m (cid:15) (cid:15) Ψ / / (cid:0) N ¯ ⊗R ) ⋊ θ ⊗ id Γ id R ⊗ (id ⋊ T m ) (cid:15) (cid:15) N ⋊ θ Γ Ψ / / (cid:0) N ¯ ⊗R ) ⋊ θ ⊗ id Γ . Therefore (cid:13)(cid:13) id ⋊ T m : N ⋊ θ Γ → N ⋊ θ Γ (cid:13)(cid:13) = (cid:13)(cid:13) id R ⊗ (id ⋊ T m ) : (cid:0) N ¯ ⊗R ) × θ ⊗ id Γ → (cid:0) N ¯ ⊗R ) ⋊ θ ⊗ id Γ (cid:13)(cid:13) = (cid:13)(cid:13) id R ⊗ (id ⋊ T m ) : R ¯ ⊗ (cid:0) N × θ Γ (cid:1) → R ¯ ⊗ (cid:0) N × θ Γ (cid:1)(cid:13)(cid:13) = (cid:13)(cid:13) id ⋊ T m : N ⋊ θ Γ → N ⋊ θ Γ (cid:13)(cid:13) cb . Observe that, when M is a McDuff factor and not just an algebra, the uniqueness of the trace impliesthat Ψ is trace preserving and therefore the diagram above passes trivially to L p -spaces. In thenon-factor case we need to use the following lemma whose proof is a trivial application of the Radon-Nykodim theorem Lemma 2.6.
Let τ i , i ∈ { , } be two faithful, normal and θ -invariant tracial states over N , where θ : Γ → Aut( N ) is an action. For every ≤ p ≤ ∞ there is a complete isometry j p such that L p ( N ⋊ θ Γ; τ ⋊ τ Γ ) id ⋊ T m (cid:15) (cid:15) j p / / L p ( N ⋊ θ Γ; τ ⋊ τ Γ ) id ⋊ T m (cid:15) (cid:15) L p ( N ⋊ θ Γ; τ ⋊ τ Γ ) j p / / L p ( N ⋊ θ Γ; τ ⋊ τ Γ ) . Proof.
We just add an sketch for completion. First notice that since both τ and τ are faithful tracialstates we have that τ ( x ) = τ ( δ x ) for some positive central element in L ( Z ( N )) with unit norm. Thefact that both are θ -invariant implies that δ is fixed by the action of θ . Therefore ι ( δ ) = δ ⋊ is alsocentral and we have that τ ⋊ τ Γ ( x ) = τ ⋊ τ Γ ( ι ( δ ) x ), for every x ∈ N ⋊ θ Γ. Let us denote ι ( δ ). Simplyby δ . Notice that it is invertible as an element of L ( Z ( N )) by the faithfulness of τ and τ . We havethat the map j p ( x ) = δ − p x is an L p -isometry, equivariant and that the diagram above commutes. Corollary 2.7.
Let θ : Γ → Aut( M , τ ) be like in Theorem 2.5 and assume that it is trace preserving.For every < p < ∞ , we have that (cid:13)(cid:13) id ⋊ T m : L p ( N ⋊ θ G ) → L p ( N ⋊ θ G ) (cid:13)(cid:13) = (cid:13)(cid:13) id ⋊ T m : L p ( N ⋊ θ G ) → L p ( N ⋊ θ G ) (cid:13)(cid:13) cb Proof.
Since, by Theorem 2.5, θ and θ ⊗ id R are outer conjugate there is a ∗ -isomorphism φ : N →N ¯ ⊗R and a 1-cocycle u : Γ → U ( N ). Denote by Ψ : N ⋊ θ Γ → N ¯ ⊗R ⋊ θ ⊗ id Γ the ∗ -isomorphism given16y sending x ⋊ λ g to φ ( x ) u g ⋊ λ h . Let us denote by τ φ the tracial state ( τ ⊗ τ R ) ◦ φ . We have that τ φ is faithful, since both τ ⊗ τ R are, and θ -invariant. We have that the following diagram commute L p ( N ⋊ θ Γ) id ⋊ T m (cid:15) (cid:15) Ψ / / L p ( R ¯ ⊗N ⋊ id ⊗ θ Γ; τ φ ⋊ τ Γ ) j p / / id R ⊗ id ⋊ T m (cid:15) (cid:15) L p ( R ¯ ⊗N ⋊ id ⊗ θ Γ) id R ⊗ id ⋊ T m (cid:15) (cid:15) L p ( N ⋊ θ Γ) Ψ / / L p ( R ¯ ⊗N ⋊ id ⊗ θ Γ; τ φ ⋊ τ Γ ) j p / / L p ( R ¯ ⊗N ⋊ id ⊗ θ Γ) . The map Ψ extends to an L p -isometry since it is trace preserving by the definition of τ φ . The map j p is L p -isometric by Lemma 2.6. Both intertwine the required multipliers and are equivariant. Thereforethe norm of id ⋊ T m equals that of id R ⊗ (id ⋊ T m ) and the claim follows.The Theorem 2.5 above can be simplified in certain cases or extended to more general contexts. Forinstance, if N = R , then the conditions of strong central triviality can be reduced to freeness. Thereason to require freeness for the action θ is that, if we assume that θ g can be inner for some g = e , then,the it is posible to produce a cohomological obstruction to the outer conjugation. General actions ofamenable groups on R are not all equivalent under outer conjugation and Connes’ standard invariant,see [Con77], is required. Although we have formulated Theorem 2.5 in the case of finite N it s alsopossible to obtain similar results in the semifinite case.We also point out the the automatic boundedness of id ⋊ T m : L p ( N ⋊ θ Γ) → L p ( N ⋊ θ Γ) for some actionson sufficiently large algebras N is very different, and a priori unrelated, to the problem of determining ifthe map T m : L p ( L Γ) → L p ( L Γ) is completely bounded. The analogous construction to what we haveused in Corollary 2.7 would be to try to embed arbitrarily large matrices M k ( C ) ֒ → M k (Γ) ⊗ L Γ ∼ = L Γ,when L Γ is McDuff, in a way that intertwines id M k ⊗ T m and T m , which is not doable for all m ∈ A (Γ),since m = δ g ∈ A (Γ) and T m is of rank 1, while id M k ⊗ T m would have larger rank.
3. Transference and amenable equivariant homomorphisms
Let M , N be von Neumann algebras. We have that a Hilbert space H is a W ∗ -correspondence iffthere are normal ∗ -homomorphisms r : N op → B ( H ) and ℓ : M → B ( H ), whose ranges commute.Such category of bimodules with bounded bimodular maps as their morphisms was introduced as asuitable substitute for groups representations in the context of von Neumann algebras, see [CJ85].More material on correspondences can be found on [AD95, Pop86] as well as [Con94, Appendix B] and[AP17, Chapter 13]. The homomorphisms of M into B ( H ) are usually called the right and left actionsrespectively and we will denote them simply by x · ξ · y = ℓ ( x ) r ( y ) ξ, for every x ∈ N , y ∈ M and ξ ∈ H . We will write H as N H M whenever we want to make thedependence of N and M explicit. Like in the context of representations, it is possible to define anatural notion of weak containment of correspondences N K M ≺ N H M as follows Definition 3.1 ([AP17, Definition 13.3.8]) . It is said that K is weakly contained in H , and denoted N K M ≺ N H M iff for every ε > and finite sets E ⊂ N and F ⊂ M , it holds that for every unitvector ξ , there are vectors { η , η , ..., η m } such that (cid:12)(cid:12)(cid:12) h ξ, x · ξ · y i − m X j =1 h η j , x · η j · y i (cid:12)(cid:12)(cid:12) < ε, for every x ∈ E , y ∈ F . Proposition 3.2.
Let N K M and N H M be correspondences. The following are equivalent (i) N K M ≺ N H M . (ii) There is a net of maps T α : K → H ⊗ ℓ such that (cid:13)(cid:13) T α ( x · ξ · y ) − x · T ( ξ ) · y (cid:13)(cid:13) → (cid:10) T α ( ξ ) , T α ( η ) (cid:11) → (cid:10) ξ, η (cid:11) . for every ξ, η ∈ H and x ∈ M and y ∈ N . (iii) There is a bimodular isometry T : N K M −→ N ( H ⊗ ℓ ) UM Let N be a semifinite von Neumann algebra with a n.s.f weight τ : M + → [0 , ∞ ]. Its trivial correspon-dence N L ( N ) N is given by the GNS representation associated to the tracial weight and the naturalleft and right actions. This correspondence plays the same role as the trivial representation of groups.We recall also that if K = L ( N ) is the proposition above, then we can remove the extra ℓ factor andtherefore N L ( N ) N ≺ N H N iff there is a bimodular isometry J from L ( N ) into H U . The followingfollows after taking a sequence of vectors ξ α ∈ Ball( H ) such that J ( ) = ( ξ α ) U α gives the following. Proposition 3.3.
Let N H M be a correspondence. The following are equivalent (i) N L ( N ) N ≺ N H N . (ii) There is a sequence of unit vectors ( ξ α ) α in H such that: • lim α (cid:13)(cid:13) x · ξ α − ξ α · x (cid:13)(cid:13) = 0 , for every x ∈ N . • lim α (cid:10) ξ α , x · ξ α (cid:11) = τ ( x ) , for every x ∈ N . Remark 3.4.
Recall that the second condition on Proposition 3.3.(ii) is superfluous if N is factor.Indeed, by the centrality of the vectors ξ α , we have that any weak- ∗ accumulation point ϕ : N → C of the states ϕ α ( x ) = (cid:10) ξ α , π ( x ) ξ α (cid:11) is a (non-necessarily normal) tracial state on N . But the uniqueness of the trace of N gives that τ = ϕ . GNS construction for completely positive maps.
Recall that given a normal and completelypositive map Φ :
N → M , where we are are assuming the algebra M to be semifinite and have afaithful tracial weight τ , there is an associated N - M -correspondence H (Φ). Let N τ ⊂ M be the densesubspace given by M ∩ L ( M ). Let N ⊗ alg N τ be the algebraic tensor product of N and N τ anddefine the positive sesquilinear form (cid:28) m X j =1 x ⊗ y , n X k =1 x ⊗ y (cid:29) Φ = τ (cid:26) m X j =1 n X k =1 y ∗ Φ( x ∗ x ) y (cid:27) , Quotienting out the nulspace N of the sesquilinear form above gives a pre-inner product and completingwith respect to the associated norm gives the Hilbert space H (Φ). The left and right actions are definedby extension of x · ( z ⊗ w ) · y = x z ⊗ w y, and trivial computations gives that both are bounded and normal representations.If H is a N - M -correspondence its contragradient ¯ H is the M - N correspondence given by the conjugateHilbert space ¯ H together with the actions x · ¯ ξ · y = y ∗ · ξ · x ∗ , ξ ∈ H , x ∈ M and y ∈ N . Connes’ tensor product.
The last notion that we need to recall in order to define what amenablecompletely positive maps is that of Connes’ tensor product. If H and K are N - M and M - Q cor-respondences respectively, for semifinite von Neumann algebras N , M and Q , the Connes’ tensorproduct is a third N - Q correspondence H ¯ ⊗ M K . Let H ◦ ⊂ H be the set of all left-bounded vectorsof H , ie vectors for which the operator x ξ · x extends to a bounded operator L ξ : L ( M ) → H .Given to left-bounded vectors ξ, η ∈ H ◦ their M -product is given by h ξ, η i M = L ∗ ξ L η . Observe that L ∗ ξ L η : L ( M ) → L ( M ) is right M -modular and therefore belongs to M ⊂ B ( L M ).We define H ¯ ⊗ M K as the space resulting from taking H ◦ ⊗ alg K with the sesquilinear positive formgiven by linear extension of h ξ ⊗ η , ξ ⊗ η i = (cid:10) η , h ξ , ξ i M η (cid:11) K . (3.1)Quotienting out the nulspace associated to that form and taking the metric closure gives the Connes’product. Slightly ambiguously, we will represent the class modulo the nulspace associated to ξ ⊗ η by ξ ⊗ η itself. It is easily seen that for every x ∈ M it holds that ξ ⊗ x · η − ξ · x ⊗ η = 0 , where ξ ∈ H ◦ and η ∈ K .The following notion of amenable correspondence has its origin in [Pop86] and [AD95]. It is also aparallel to the notion of amenable representation [Bek90]. Definition 3.5 ([AD95]) .(i)
A correspondence N H M is (left) amenable iff L ( N ) ≺ H ⊗ M H . (ii) A complete positive map
Φ :
N → M is left amenable iff its associated N - M correspondence H (Φ) is. The case in which the completely positive map above is a normal ∗ -homorphism between semifinitevon Neumann algebras π : N → M is quite illustrative and easy to describe. First start noticing that H ( π ) ∼ = L ( M ) as N - M -bimodules, where the left and right actions on L ( M ) are given by x · ξ · y = π ( x ) ξ y. The bimodular isomorphism W : H ( π ) → L ( M ) is just defined over N ⊗ alg M as n X j =1 x j ⊗ y j W n X j =1 π ( x j ) y j and a straightforward computation yields that the above map is isometric and bimodular, the surjec-tivity is immediate. Next, notice that H ( π ) ¯ ⊗ M H ( π ) ∼ = L ( M )is another isomorphism of N - N -bimodules, where the actions on the right hand side are given by x · ξ · y = π ( x ) ξπ ( y ). In order to construct that isomorphism just notice that L ( M ) ◦ = L ( M ) ∩ M and that, for left bounded vectors ξ, η ∈ L ( M ) ∩ M their M -valued inner product is h ξ, η i M = ξ ∗ η .Then (3.1) gives that ξ ⊗ η ξ η is a bimodular isomorphism. As a consequence of this discussion wehave that π is (left) amenable iff N L ( N ) N ≺ π [ N ] L ( M ) π [ N ] ,
19r equivalently iff there is a N -bimodular isometry L ( N ) ֒ → L ( M ) U , for some proper ultrafilter U .Observe that, in this particular case, we can choose the sequence of unit vectors ( ξ α ) α ⊂ L ( M ) thatsatisfy the properties of Proposition 3.3. (ii) to be in the positive cone L ( M ) + ⊂ L ( M ) withoutloss of generality. Indeed, let us notice that if ξ α is a centralizing sequence, meaning that ( ξ α ) U α ∈ L ( M ) U ∩ π [ N ] ′ , so is ξ ∗ α and therefore ξ α ξ ∗ α is again a centralizing sequence in L ( M ), i.e. it lays in L ( M ) U ∩ π [ N ] ′ . But by Lemma 2.2, we have that η α = | ξ ∗ α | = ( ξ α ξ ∗ α ) (3.2)is again a sequence of centralizing unit vectors in L ( M ) + . To see that they also satisfy the tracecondition, we just check that τ N ( x ) = lim α (cid:10) ξ α , π ( x ) ξ α (cid:11) = lim α τ M (cid:0) ξ α ξ ∗ α π ( x ) (cid:1) = lim α (cid:10) | ξ ∗ α | , π ( x ) | ξ ∗ α | (cid:11) = lim α (cid:10) η α , π ( x ) η α (cid:11) . The following proposition establishes a connection between the amenability of an embedding π : L Γ ֒ → L ∞ (Ω) ⋊ θ Γ and the transference results on Theorem A.
Theorem 3.6.
Let π : L G → L ∞ (Ω) ⋊ θ Γ be a normal ∗ -homomorphism. The following are equivalent (i) π is amenable. (ii) For every ≤ p < ∞ , there is an completely isometric map L p ( L Γ) J p −−−−−→ Y U L p (Ω ⋊ θ Γ) , that is also bimodular, i.e. J p ( x ϕ y ) = π ( x ) J p ( ϕ ) π ( y ) .Furthermore if, π is equivariant and the isometric and bimodular map L ( L Γ) ֒ → L (Ω ⋊ θ Γ) of point (i) intertwines T m and id ⋊ T m , for every m ∈ B (Γ) , the Fourier-Stieltjes algebra of Γ , so does everyother L p isometry J p in point (ii) . Proof (of Theorem 3.6).
For the proof of (i) = ⇒ (ii), we start by noticing that, since π is amenablethere is a sequence of vectors ( ξ α ) α ⊂ L (Ω ⋊ θ Γ) satisfying the properties of Proposition 3.3 (ii) ,that is, ξ α being asymptotically central and satisfying that the vector states ω ξ α ,ξ α ( x ) = h ξ α , x ξ α i converge on the image of π to the canonical trace of L Γ. But then, we can define the family of maps J αp : L p ( L Γ) → L p (Ω ⋊ θ Γ) as J αp ( x ) = u α | ξ α | p ι ( x ) | ξ α | p , where ξ α = u α | ξ α | is the polar decomposition of the vector ξ α ∈ L ∞ (Ω ⋊ θ Γ). Then, proceeding likein the proof of Theorem A we can obtain easily that (1) (cid:13)(cid:13) J αp : L p ( L Γ) → L p (Ω ⋊ θ Γ) (cid:13)(cid:13) ≤ . (2) lim α (cid:10) J αp ( x ) , J αp ( y ) (cid:11) = h x, y i for every x ∈ L p ( L Γ) and y ∈ L q ( L Γ) with p + q = 1.Therefore, the ultraproduct map J p = ( J αp ) U is isometric. The fact that ξ α is asymptotically centralin L (Ω ⋊ θ Γ) gives, by Lemma 2.2, that the sequences u α | ξ α | p and | ξ α | p are asymptotically centralin L p (Ω ⋊ θ Γ). This readily gives that the map J p is L Γ-bimodular in the sense that J p ( x ϕ y ) = π ( x ) J p ( ϕ ) π ( y ). Now, we have to show that, if J satisfies the intertwining identity J ◦ T m = (id ⋊ T m ) U ◦ J , (3.3)then, so does every other J p . Notice that, since every m ∈ B (Γ) is a combination of four positivetype functions, we can assume that m is of positive type without loss of generality. Similarly, by thecontinuity of the map J p over L p ( L Γ), we have that its is enough to prove the intertwining identity20ver a dense subset of L p ( L Γ). We will choose C [Γ] ⊂ L p ( L Γ). By linearity it is enough to prove that,for every h ∈ Γ, it holds that0 = (cid:13)(cid:13)(cid:0) J p ◦ T m (cid:1) ( x ) − (id ⋊ T m ) U ◦ J p ( x ) (cid:13)(cid:13) p = lim α →U (cid:13)(cid:13)(cid:13) u α | ξ α | p π (cid:0) T m x (cid:1) | ξ α | p − (id ⋊ T m ) (cid:8) u α | ξ α | p π ( x ) | ξ α | p (cid:9)(cid:13)(cid:13)(cid:13) p , where x = λ h . Using the fact that π ◦ T m = (id ⋊ T m ) ◦ π by the fact that π is equivariant and thethe fact that | ξ α | p ∈ L p ( M ) asymptotically centralizes π ( x ), we obtain that the above expression isequal to (cid:13)(cid:13)(cid:0) J p ◦ T m (cid:1) ( x ) − (id ⋊ T m ) ◦ J p ( x ) (cid:13)(cid:13) p = lim α →U (cid:13)(cid:13)(cid:13) m ( h ) u α | ξ α | p π ( λ h ) − (id ⋊ T m ) (cid:8) u α | ξ α | p π ( x ) (cid:9)(cid:13)(cid:13)(cid:13) p . The following modularity property of (id ⋊ T m )(id ⋊ T m )( ξ π ( λ h )) = (id ⋊ T ρ h m )( ξ ) π ( λ h ) , (3.4)where ρ h m ( g ) = m ( g h ). Therefore, we only have to verify the claim just for h = e and everypositive type m , since ρ h m is of positive type whenever m is. Similarly, by linearity, we can take m ( g ) = m ( e ) − m ( g ) and assume that m ( e ) = 1. gathering all the information together, the claimfollows by showing that lim α →U (cid:13)(cid:13)(cid:13) u α | ξ α | p − (id ⋊ T m ) (cid:8) u α | ξ α | p (cid:9)(cid:13)(cid:13)(cid:13) p = 0 , for every m of positive type with m ( e ) = 1. Since id ⋊ T m is a (completely) positive operator over L ∞ (Ω ⋊ θ Γ) we can apply the Theorem 2.1 on almost multiplicative maps to prove the result.Observe that the above proposition has to be read as an extrapolation argument, by which the isometricand bimodular map defined in the L -level L ( L Γ) J ֒ −−−−→ L (Ω ⋊ θ Γ)can be extended to the rest of the L p -scale, with 1 ≤ p < ∞ as a complete isometry preservingboth the L Γ-bimodular properties as well as the intertwining properties for the operators T m and(id ⋊ T m ) U . This is something that is rarely possible when working with general Hilbert spaces.Indeed, by Lamperti’s theorem [Lam58], see [Yea81] as well for a noncommutative analogue, there aremany L -isometries that to not extend to L p . Nevertheless, it seems that turning the L -space into abimodule gives you enough structure to have a natural candidate to the associated L p -space. Indeed,if N H N is a bimodule, a L p extrapolation would be any noncommutative L p -bimodule X , in the senseof [JS05], such that it will have a dense subset (its L -part) isomorphic as a bimodule to a dense subsetof H .Observe also that the map π : L Γ ֒ → L ∞ (Ω) ⋊ θ Γ intertwines T m and id ⋊ T m when π is equivariant,and thus of the form described in Theorem B.(i). We will next give a complete assessment of theamenable and equivariant ∗ -homomorphisms π : L Γ → L ∞ (Ω) ⋊ θ Γ as well as those amenable π suchthat their associated isometric map L ( L Γ) ֒ → L (Ω ⋊ θ Γ) U intertwine Fourier multipliers. Beforethat, we will say that given a L ∞ (Ω)-valued symbol ( m g ) g ∈ ℓ ∞ [ L ∞ (Ω)] = L ∞ (Γ × Ω) its associated L ∞ -valued Fourier multiplier its the operator T m acting on L ∞ (Ω) ⋊ θ Γ given by T m (cid:18) X g ∈ Γ f g ( ω ) ⋊ λ g (cid:19) = X g ∈ Γ m g ( ω ) f g ( ω ) ⋊ λ g . Observe that, like in the case of L ∞ (Ω)-valued Herz-Schur multipliers, there is a bit of ambiguity indenoting both the usual Fourier multipliers id ⋊ T m and the L ∞ (Ω)-valued ones by the T m .21 heorem 3.7. Let θ : Γ → Aut(Ω , µ ) be an action, κ : Γ → L ∞ (Ω; T ) a multiplicative -cocycle and π : L Γ → L ∞ (Ω) ⋊ θ Γ the associated ∗ -homomorphism given by π (cid:18) X g ∈ Γ f g ⋊ λ g (cid:19) = X g ∈ Γ f g κ g ⋊ λ g . Then, we have that (i) π is amenable iff there is a sequence of unit vectors ( ξ α ) α ⊂ L (Ω × Γ) such that • lim α (cid:13)(cid:13)(cid:0) θ h ⊗ Ad h (cid:1) ( ξ α ) − M ̟ h ( ξ α ) (cid:13)(cid:13) = 0 , for every h ∈ Γ , where ϕ h ∈ L ∞ (Γ × Ω) is given by ̟ h ( g, ω ) = κ h ( ω ) θ g ( κ h ) and M ̟ h is its associated pointwise multiplication operator. • Z Ω κ h (cid:18) X g ∈ Γ ξ α ( g, h ) θ h ( ξ α ( g )) (cid:19) dµ = δ { h = e } for every h ∈ Γ . (ii) π is amenable and the associated bimodular isometry L Γ L ( L Γ) L Γ −→ L Γ L (Ω ⋊ θ Γ) UL Γ intertwines T m and id ⋊ T m , for every m ∈ B (Γ) , iff Ω has a Γ -invariant mean. Proof.
Point (i) follows after a routine application of Proposition 3.3. Indeed, if π is amenable there isa sequence of unit vectors ( ξ α ) α ⊂ L (Ω ⋊ θ Γ) that is centralizing, or equivalently it is asymptoticallyinvariant by the action of Γ given by ξ π ( λ h ) ξ π ( λ h ) ∗ . Therefore, we have that( κ h ⋊ λ h ) (cid:18) X g ∈ Γ ξ α ( g ) ⋊ λ g (cid:19) ( θ h − ( κ h ) ⋊ λ h − )= X g ∈ Γ κ h θ h ( ξ α ( g )) θ hgh − ( κ h ) ⋊ λ h gh − = X g ∈ Γ κ h θ h ( ξ α ( h − g h )) θ g ( κ h ) ⋊ λ g = X g ∈ Γ ̟ h ( g, ω ) θ h ( ξ α ( h − g h ))( ω ) ⋊ λ g , where ξ α = P g ξ α ( g ) ⋊ λ g and each of the coefficients ξ α ( g ) is uniquely determined by E [ ξ α λ ∗ g ]. Usingthe Plancherel identity and the fact that µ is θ -invariant, we have that L (Ω ⋊ θ Γ) ∼ = L (Ω × Γ), wherethe isomorphism is given by sending ξ α ( g ) ⋊ λ g to ξ α ( g ) ⊗ δ g ∈ L (Ω) ⊗ ℓ (Γ). Therefore, the aboveexpression is at diminish distance of ξ α iff the net ( ξ α ) α ⊂ L (Ω × Γ) satisfies thatlim α (cid:13)(cid:13)(cid:0) θ h ⊗ Ad h (cid:1) ξ α − M ̟ h ξ α (cid:13)(cid:13) = 0 , or every h ∈ Γ. The second condition follows by imposing that τ ( ξ ∗ α π ( λ h ) ξ α ) = δ { h = e } .For point (ii) we need to see that if J α ( x ) = ξ α π ( x ) intertwines id ⋊ T m and T m , then without lossof generality we can assume that ξ α ∈ L (Ω) ⊗ C1 ⊂ L (Ω ⋊ Γ). Arguing like in the proof of 3.6 wehave that J ◦ T m = (id ⋊ T m ) ◦ J iff for every h ∈ Γ0 = lim α →U (cid:13)(cid:13)(cid:13) ξ α π (cid:0) T m λ h (cid:1) − (id ⋊ T m )( ξ α π ( λ h ) (cid:13)(cid:13)(cid:13) = lim α →U (cid:13)(cid:13)(cid:13) ξ α π (cid:0) T m λ h (cid:1) − (id ⋊ T m ) (cid:0) ξ α π ( λ h ) (cid:1)(cid:13)(cid:13)(cid:13) = lim α →U (cid:13)(cid:13)(cid:13) m ( h ) ξ α − (id ⋊ T ρ h m )( ξ α ) (cid:13)(cid:13)(cid:13) . We have used the fact that π in equivariant, the modularity property (3.4) and the fact that the L -norm is invariant by multiplication by the unitary π ( λ h ) = κ h ⋊ λ h . But now, taking m = δ e impliesthat the sequences ξ α and T δ e ( ξ α ) = ξ α ( e ) ⋊ λ e ∈ L (Ω) ⊂ L (Ω ⋊ θ Γ) induce the same element inthe ulptrapower and therefore we can assume without loss of generality that ξ α is equal to ξ α ( e ) ⋊ λ e .The condition (i) over vectors in L (Ω) ⊂ L (Ω ⋊ θ Γ) implies that the vectors are asymptotically Γinvariant and therefore Ω has a Γ-invariant mean.22 emark 3.8.
Several remarks are in order. The first is that in point (i) the second condition issuperfluous whenever Γ is i.c.c, see Remark 3.4. The next is that the theorem above has to beunderstood as a negative result stating that the transference technique used in Theorem A can not beapplied in more general context by changing the natural inclusion L Γ ⊂ L ∞ (Ω) ⋊ θ Γ by a more generalinjective, normal and equivariant ∗ -homomorphism π : L Γ ֒ → L ∞ (Ω) ⋊ θ Γ. Indeed, both Proposition3.6 and Theorem 3.7 above imply that such technique requires the existence of a Γ-invariant mean onΩ. Nevertheless, it is possible that, provided there are actions without a Γ-invariant mean such that thecondition (i) holds, that the map J may intertwine a subfamily of multiplies m . In particular, assumethat there is an infinite conjugacy class C ⊂
Γ such that all of the vectors ξ α on (i) are supported in L (Ω × C ) and assume that Γ ⊂ Γ is a proper subgroup generated by
C ⊂
Γ. Then, J intertwinesevery left Γ -invariant multiplier m ∈ B (Γ) and by the techniques on 3.6 J p intertwines the samefamily of multipliers at the same time. Assuming that Zimmer-amenable actions satisfying (i) couldbe found, that would gives families of multipliers for which the Schur and Fourier norms are equal,thus obtaining a positive solution of a weakening of Problem 2.4.
4. Transference and amenable equivariant cp maps
In this last section we will explore in which situation the completely positive and equivariant maps ofTheorem B.(ii) Φ : L Γ → L ∞ (Ω) ⋊ θ Γ given by extension of λ g Φ ϕ g ⋊ λ g , where ϕ g ( ω ) = h ξ, κ g ( ω ) ξ i is given by the matrix coefficients of a multiplicative 1-cocycle κ : Γ → L ∞ (Ω; U ( H )) for some Hilbert space H . We will also denote by π the normal ∗ -homomorphism L Γ π −−−−→ B ( H ) ¯ ⊗ (cid:0) L ∞ (Ω) ⋊ θ Γ (cid:1) ∼ = L ∞ (Ω; B ( H )) ⋊ θ ⊗ id Γgiven by linear extension of π ( λ g ) = κ g ( ω ) ⋊ λ g .We introduce the following amenability condition for multiplicative 1-cocycles. Definition 4.1.
Let θ : Γ → Aut(Ω , µ ) be an action and κ : Γ → L ∞ (Ω; U ( H )) a multiplicative -cocyle for some Hilbert space H . We will say that κ is amenable iff either of the following equivalentconditions hold (i) There is a (not necessarily normal) state m : L ∞ (Ω; B ( H )) → C such that m (cid:0) ( θ h ⊗ id) x (cid:1) = m (cid:0) Ad κ h − x (cid:1) , for every x ∈ L ∞ (Ω; B ( H )) and h ∈ Γ . (ii) There is a net of unit vectors ( ξ α ) α ⊂ L (Ω; S ( H )) , where S ( H ) are the Hilbert-Schmidtoperators on H , such that lim α (cid:13)(cid:13) Ad κ h ◦ ( θ h ⊗ id)( ξ α ) − ξ α (cid:13)(cid:13) = 0 . Remark 4.2.
Observe that, by doing the change of variable x Ad − κ h − x we obtain that the state m on point (i) is invariant under the action x ( θ h ⊗ id) ◦ Ad − κ h − x . But such action of Γ is givenprecisely by ( θ h ⊗ id) ◦ Ad − κ h − = Ad κ h ◦ ( θ h ⊗ id) . Let us denote by θ ⋊ κ : Γ → Aut( L ∞ (Ω; B ( H ))) the action above. Clearly, it satisfies that itsrestriction to L ∞ (Ω) recovers θ . Therefore, we can think of it as an extended action. Condition (i) isthus equivalent to the existence of a θ ⋊ κ -invariant state.23he equivalence between the two notions above follows easily. The fact that (ii) implies (i) follows byan application of Banach-Alaouglu theorem. Take the vector states ϕ α ( x ) = ω ξ α ,ξ α ( x ) = h ξ α , xξ α i ,we have that any weak- ∗ accumulation point is an invariant state m . For the reverse implication weneed to use Goldstine’s theorem. Indeed, given a state m ∈ Ball( L (Ω; S ( H )) ∗∗ ) we can approximateby states in ω α = L (Ω; S ( H )) in the weak- ∗ topology. But, taking square roots gives elements in ξ α ∈ L (Ω; S ). A routine application of Hanh-Banach theorem, analogous to that on [BO08, Theorem2.5.11] gives that the ξ α satisfy the condition in (ii).We can describe explicitly the correspondences H (Φ) and H (Φ) ¯ ⊗ L ∞ (Ω) ⋊ Γ H (Φ) as follows. Proposition 4.3.
Let
Φ : L Γ → L ∞ (Ω) ⋊ θ Γ be an equivariant cp map and let κ : Γ → L ∞ (Ω; U ( H )) be its associated multiplicative -cocycle. We have that (i) There is bimodular isomorphism L Γ H (Φ) L ∞ (Ω) ⋊ Γ ∼ = L Γ (cid:0) H ⊗ L (Ω ⋊ θ Γ) (cid:1) L ∞ (Ω) ⋊ Γ , where the left and right actions are given by λ h · ξ = π ( λ h ) ξξ · ( f ⋊ λ h ) = ξ ( ⊗ f ⋊ λ h ) respectively. (ii) There is a bimodular isomorphism L Γ (cid:0) H (Φ) ¯ ⊗ L ∞ (Ω) ⋊ Γ H (Φ) (cid:1) L Γ ∼ = L Γ (cid:0) S ( H ) ⊗ L (Ω ⋊ θ Γ) (cid:1) L Γ , where both the left and right actions are given by the image of π , i.e. λ h · ξ · λ g = π ( λ h ) ξ π ( λ g ) . We will just sketch the proof of the two isomorphisms above. For the first one we will use the fact thatthe GNS construction for cp maps that associates to them the correspondence H (Φ) is unique underisomorphisms, see [AP17, Chapter 13]. Indeed, if N and M are finite von Neumann algebras, givenany contractive and completely positive map Φ : N → M , we can associate a pair ( H (Φ) , ξ ), where N H (Φ) M is the correspondence described after Proposition 3.3 and ξ is the bi-cyclic left-boundedvector associated to the class of ⊗ . In this case we can recover Φ byΦ( x ) = L ∗ ξ x L ξ . We have that in the finite case such construction is unique up to isomorphisms. The subtlety in our case,in which M is just semifinite, is that ⊗ doesn’t belong to H (Φ) since τ M ( ) = ∞ . Nevertheless, itis possible to choose an increasing net of vectors ( ξ α ) α ⊂ M of finite trace, whose associated operators L ξ : L ( M ) → H are uniformly bounded and such that Φ( x ) is the strong limit of L ∗ ξ α x L ξ α . Indeed, itis enough to fix ξ α as the class of ⊗ p α , where p α is an increasing net of finite projections converging inthe strong topology to . It is again true that such property characterizes H (Φ) as a correspondence.Taking ξ α = ξ ⊗ p α , where ξ ∈ H satisfies that ϕ g = h ξ, κ g ξ i and p α = F α ⋊ λ e is an increasing netof finite projections where F α ⊂ Ω and µ ( F α ) < ∞ , gives the desired result. The second isomorphismfollows immediately. Theorem 4.4.
Let θ : Γ → Aut(Ω , µ ) be an action Φ : L Γ → L ∞ (Ω) ⋊ θ Γ an equivariant cp map and κ : Γ → L ∞ (Ω; U ( H )) its associated -cocycle. (i) Φ is (left) amenable iff there is a sequence of unit vectors ( ξ α ) α ⊂ S ( H ) ⊗ L (Ω × Γ) satisfyingthat • If Π : L (Ω; S ( H )) ⊗ ℓ (Γ) → L (Ω; S ( H )) ⊗ ℓ (Γ) is the operator defined by Π( ϕ ⊗ δ g ) = κ ∗ h ϕ θ g ( κ h ) ⊗ δ g , hen lim α (cid:13)(cid:13) (id S ⊗ θ h ⊗ Ad h )( ξ α ) − Π( ξ α ) (cid:13)(cid:13) = 0 , for every h ∈ Γ . • It holds that X g ∈ Γ Z Ω Tr (cid:8) ξ α ( h g ) ∗ κ h θ h ( ξ α ( k )) (cid:9) dµ = δ { h = e } , for every h ∈ Γ . (ii) Φ is (left) amenable and the associated L Γ -bimodular isometry L ( L Γ) J −−−−−→ Y U S ( H ) ⊗ L (Ω ⋊ θ Γ) intertwines T m and id S ⊗ id Ω ⋊ T m , for every m ∈ B (Γ) , iff κ is amenable in the sense ofDefinition 4.1. The proof of the theorem above is exactly like that of Theorem 3.7 land therefore we will omit it. Thefollowing corollary follows from point (ii) and the proof of Theorem 3.6.
Corollary 4.5.
Let θ : Γ → Aut(Ω , µ ) be an action Φ : L Γ → L ∞ (Ω) ⋊ θ Γ an equivariant cp map and κ : Γ → L ∞ (Ω; U ( H )) its associated -cocycle. If κ is amenable, then for every ≤ p < ∞ , there is acomplete isometry L p ( L Γ) J p −−−−−→ Y U S p ( H ) ⊗ p L p (Ω ⋊ θ Γ) , which is L Γ -bimodular and satisfies that L p ( L Γ) T m (cid:15) (cid:15) J p / / Y U S p ( H ) ⊗ p L p (Ω ⋊ θ Γ) (id ⊗ id ⋊ T m ) U (cid:15) (cid:15) L p ( L Γ) J p / / Y U S p ( H ) ⊗ p L p (Ω ⋊ θ Γ) . (4.1)Although the theorem above will give that Corollary 0.4 holds whenever the action θ admits anamenable 1-cocycle κ . Sadly, this doesn’t provide more exotic examples since restricting the mean m : L ∞ (Ω; B ( H )) → C to the unital von Neumann subalgebra L ∞ (Ω; C1 ) = L ∞ (Ω) yields a Γ-invariant mean on Ω. Similarly, in the case in which θ is Zimmer-amenable and κ is amenable, so is Γitself. Acknowledgement.
The author is thankful to Simeng Wang for a personal communication regardingthe mistake on [GP18] and for subsequent discussion on the present manuscript.
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Adri´an M. Gonz´alez-P´erez
University Clermont-Auvergne, LMBP3 Place Vasarely 63178 Aubi´ere, France [email protected]@[email protected]@gmail.com