Note on bi-exactness for creation operators on Fock spaces
aa r X i v : . [ m a t h . OA ] J a n Note on bi-exactness for creation operators onFock spaces
Kei Hasegawa Yusuke Isono ∗ Tomohiro Kanda
Abstract
In this note, we introduce and study a notion of bi-exactness for creation operatorsacting on full, symmetric and anti-symmetric Fock spaces. This is a generalization ofour previous work, in which we studied the case of anti-symmetric Fock spaces. As aresult, we obtain new examples of solid actions as well as new proofs for some knownsolid actions. We also study free wreath product groups in the same context.
This article is a complementary note of our previous work [HIK20], in which we studiedsome boundary amenability phenomenon for creation operators on anti-symmetric Fockspaces. Our work was inspired by Ozawa’s work on Bernoulli actions and wreath productgroups [Oz04, BO08], which we briefly explain here.Let Λ be an amenable group and Γ an exact group. Let Λ ≀ Γ = [ L Γ Λ] ⋊ Γ be thewreath product group. Ozawa constructed a C ∗ -subalgebra C ( X ) ⊂ ℓ ∞ ( L Γ Λ) such that,with the quotient C ( ∂X ) := C ( X ) /c ( L Γ Λ), (i) the left and right translation of L Γ Λ on C ( ∂X ) is trivial, and (ii) the left and right translation of Γ × Γ on C ( ∂X ) is topologicallyamenable . This amenability implies that the following map C ( X ) ⊗ alg C ∗ λ (Λ ≀ Γ) ⊗ alg C ∗ ρ (Λ ≀ Γ) → B ( ℓ (Λ ≀ Γ))arising from inclusions is a minimal norm bounded ∗ -homomorphism up to some relative compact operators. Note that if there is no such compact operators, this boundedness isequivalent to amenability of Λ ≀ Γ. Ozawa used the boundedness to deduce some rigidity of L (Λ ≀ Γ). This is the idea in [Oz04] and a general framework is given in [BO08, Section15].In [HIK20, Section 3], we have developed a way of applying these techniques to creationoperators acting on anti-symmetric Fock spaces. In this note, we introduce a more generalframework that covers • (Subsection 3.1) wreath product groups and Z ⋊ SL(2 , Z ) [Oz04, BO08, Oz08]; • (Section 4) creation operators on full, symmetric, and anti-symmetric Fock spaces; • (Section 5) free wreath product groups.We will obtain some boundary amenability phenomenon for all these examples. Thisframework unifies Ozawa’s works and our previous one, hence it is useful to understandhow they are related. ∗ Research Institute for Mathematical Sciences, Kyoto University, 606-8502, Kyoto, JapanE-mail: [email protected]
YI is supported by JSPS KAKENHI Grant Number 20K14324.
1s in [HIK20], our boundary amenability has an application to rigidity of associatedvon Neumann algebras. Recall that a discrete group action Γ y M on a diffuse vou Neu-mann algebra is solid if for any diffuse von Neumann subalgebra A ⊂ M , which is a rangeof a faithful normal conditional expectation, the relative commutant A ′ ∩ M is amenable(see [HIK20, Appendix]). In Section 6, we will show that, under certain assumptions, thegroup action Γ y X on a set X gives rise to solid actions of Γ on associated von Neumannalgebras. Most of them were already obtained in Popa’s deformation/rigidity theory , butsome of them give new examples, see Remark 6.3. In this note, we focus on the case ofstate preserving actions, so technical difficulties discussed in [HIK20, Section 4] do notappear.
Contents c -functions and compact operators . . . . . . . . . . . . . . . . . . 22.2 Bi-exactness for groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Standard representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Fock spaces and associated von Neumann algebras . . . . . . . . . . . . . . 5 c -functions and compact operators Let X be a set. We recall generalized c -functions and compact operators. Definition 2.1.
Let X be a family of subsets in X satisfying E, F ∈ X ⇒ E ∪ F ∈ X . We say that a subset A ⊂ X is small relative to X (and write small / X ) if there is E ∈ X such that A ⊂ E . We use the following notations.2 For any net ( x λ ) λ in X , we write X ∋ x λ → ∞ / X if for any E ∈ X , there is λ suchthat x λ E for all λ ≥ λ . • We denote by c ( X ) ⊂ ℓ ∞ ( X ) the C ∗ -algebra generated by functions whose supportsare small relative to X .We note that if X is the family of all finite subsets, then c ( X ) = c ( X ). The followinglemma is straightforward. Lemma 2.2.
The following statements hold true.1. Let c alg0 ( X ) ⊂ ℓ ∞ ( X ) be the set of all functions whose supports are small relativeto X . Then it is a dense ∗ -subalgebra in c ( X ) . In particular c ( X ) ≤ ℓ ∞ ( X ) is aclosed ideal.2. For any f ∈ ℓ ∞ ( X ) , the following conditions are equivalent: (a) f ∈ c ( X ) ; (b) { x ∈ X | | f ( x ) | > ε } is small / X for any ε > ; (c) for any net ( x λ ) λ in X , x λ → ∞ / X implies lim λ →∞ f ( x λ ) = 0 (we write it as lim x →∞ / X f ( x ) = 0 ).3. Let Y ⊂ X be a non-empty subset. Define Y := { E ∩ Y | E ∈ X } (so that E, F ∈ Y ⇒ E ∪ F ∈ Y ). We have • for any net { y i } i in Y , y i → ∞ / Y ⇔ y i → ∞ / X ; • c ( Y ) = c ( X ) ∩ ℓ ∞ ( Y ) .4. Let Γ y π X be any action of a discrete group Γ which globally preserves Y such that π g ( E ) ∈ X , for all E ∈ X , g ∈ Γ . In this case, the natural action Γ y ℓ ∞ ( Y ) induces an action on c ( Y ) . Using c ( X ), define the relative compact operators and its multiplier algebra by K ( X ) := c ( X ) B ( ℓ ( X )) c ( X ) norm M( K ( X )) := the multiplier algebra of K ( X ) . Note that K ( X ) is the minimum hereditary C ∗ -algebra which contains c ( X ). If there is adiscrete group action Γ y π X , by using the natural unitary representation Γ y U π ℓ ( X ),Ad( U πg ) globally preserves c ( X ) and K ( X ), so that U πg ∈ M( K ( X )).We record the following elementary lemma. Lemma 2.3.
For any a ∈ B ( ℓ ( X )) , a ∈ M( K ( X )) ⇔ [ a, f ] ∈ K ( X ) , for all f ∈ c ( X ) . .2 Bi-exactness for groups We refer the reader to [BO08, Section 2] for nuclearlity, [BO08, Subsection 4.4] foramenability of actions, [BO08, Section 5] for exactness, and [BO08, Section 15] for bi-exactness of groups.Let Γ y X be an action of a discrete group on a compact Hausdorff space. Considerthe associated action Γ y C ( X ). We say that it is (topologically) amenable if the reducedand full crossed products coincide: C ( X ) ⋊ red Γ = C ( X ) ⋊ full Γ. More generally for anyaction Γ y A on a unital C ∗ -algebra A , the action is amenable if the restriction to thecenter of A is amenable. In that case, we also have A ⋊ red Γ = A ⋊ full Γ.Let Γ be a discrete group and G a family of subsets in Γ such that sEt, E ∪ F ∈ G for all s, t ∈ Γ , E, F ∈ G . We can define c ( G ) as in the previous subsection, which admits the left and right transla-tion action Γ × Γ y c ( G ). We say that Γ is bi-exact relative to G (and write bi-exact / G ) ifthe left and right translation action Γ × Γ y ℓ ∞ (Γ) /c ( G ) is amenable. When G consistsof all finite subsets in Γ, this means amenability of Γ × Γ y ℓ ∞ (Γ) /c (Γ). Recall that Γis exact if and only if the left translation Γ y ℓ ∞ (Γ) /c (Γ) is amenable [BO08, Theorem5.1.7], hence the terminology bi -exact makes sense.Assume that there is a family G of subgroups in Γ which generates G in the followingsense: G = { [ finite s Λ t | s, t ∈ Γ , Λ ∈ G } . In this case, c ( G ) coincides with c (Γ; G ) given in [BO08, Subsection 15.1] and henceour definition of bi-exactness is a generalization of Ozawa’s one, see [BO08, Proposition15.2.3]. The proof of 3 ⇒ / G , thenthe algebraic ∗ -homomorphism ν : C ∗ λ (Γ) ⊗ alg C ∗ ρ (Γ) → M( K ( G )) / K ( G ); a ⊗ b ab + K ( G )is bounded with respect to the minimal tensor norm (say, min-bounded ). Further if Γ iscountable, it has a ucp lift θ : C ∗ λ (Γ) ⊗ alg C ∗ ρ (Γ) → M( K ( G )) in the sense that θ ( a ⊗ b ) − ab ∈ K ( G ) for all a ⊗ b ∈ C ∗ λ (Γ) ⊗ min C ∗ ρ (Γ). This boundedness is called condition (AO) [Oz04].In this article, we study condition (AO) for several concrete examples.Recall that a ucp map π : A → B between unital C ∗ -algebras is nuclear if and only iffor any C ∗ -algebra C , the map π ⊗ id C : A ⊗ max C → B ⊗ max C factors through A ⊗ min C .We will need the following facts. Proposition 2.4.
Let π : A → B be a unital ∗ -homomorphism between unital C ∗ -algebras A and B . The following assertions hold true.1. Assume that a discrete group Γ acts on A and B . If π is Γ -equivariant, nuclear, and Γ y A is amenable, then the natural extension π : A ⋊ red Γ = A ⋊ full Γ → B ⋊ full Γ is nuclear.2. If π is nuclear and A is exact, then the induced map A/ ker π → B is nuclear.3. Let J ⊂ A and I ⊂ B be ideals such that π ( I ) ⊂ J . Consider the following diagram / / J / / A / / A/J / / / / I / / ϕ = π | I O O B / / π O O B/I ψ O O / / here ψ is the induced map. Assume that there is an approximate unit ( a i ) i in I such that ( ϕ ( a i )) i is an approximate unit for J . If ϕ and ψ are nuclear and B isexact, then π is nuclear. Proof.
1. This follows by the characterization of nuclearlity above.2. Observe first that A is locally reflexive (e.g. [BO08, Corollary 9.4.1]). Let C be anyC ∗ -algebra and consider π ⊗ id C : A ⊗ min C → B ⊗ max C . With J := ker π and by [BO08,Corollary 9.1.5], it induces A/J ⊗ min C ≃ A ⊗ min CJ ⊗ min C → B ⊗ max C. This shows that
A/J ⊗ max C → B ⊗ max C factors the minimal tensor product.3. Observe that ϕ ∗∗ , ψ ∗∗ are weakly nuclear, since ϕ, ψ are nuclear and I, B/I areexact (hence locally reflexive). Consider the map I ∗∗ ⊕ ( B/I ) ∗∗ ≃ B ∗∗ → π ∗∗ A ∗∗ ≃ J ∗∗ ⊕ ( A/J ) ∗∗ Then π ∗∗ restricts to ϕ ∗∗ : I ∗∗ → J ∗∗ , which is unital by the assumption on approximateunits. Then π ∗∗ restricts to ( B/I ) ∗∗ → ( A/J ) ∗∗ , which coincides with ψ ∗∗ by construction.We conclude that π ∗∗ is weakly nuclear. This implies π is nuclear. Note that we candirectly prove it by diagram chasing (without von Neumann algebras). For Tomita–Takesaki’s modular theory, we refer the reader to [Ta03]. For a von Neu-mann algebra M with a faithful normal state ϕ , we denote by ∆ ϕ and J ϕ the modularoperator and the modular conjugation . When ϕ is tracial, ∆ ϕ is trivial and J ϕ is given bythe involution a a ∗ . The GNS representation L ( M ) := L ( M, ϕ ) is called the standardrepresentation .Let α : Γ y M be an action of a discrete group Γ and U g ∈ U ( L ( M )) the standardimplementation of α g for each g ∈ Γ. We use the standard representation L ( M ) ⊗ ℓ (Γ)of M ⋊ Γ given by: with the J -map J M on L ( M ),Left action M ∋ a a ⊗ Γ ; Γ ∋ g U g ⊗ λ g ;Right action J M M J M ∋ a π r ( a ) = X h ∈ Γ U h aU − h ⊗ e h,h ; Γ ∋ g ⊗ ρ g ; J -map J = X h ∈ Γ U h J M ⊗ e h,h − . Here { e g,h } g,h ∈ Γ is the matrix unit in B ( ℓ (Γ)). In the case that M ⋊ Γ = L (∆ ⋊ Γ) (agroup von Neumann algebra of a semidirect product group ∆ ⋊ Γ), Ad( U g ⊗ λ g ρ h ) for g, h ∈ Γ restricts to an automorphism on ℓ ∞ (∆ ⋊ Γ), which corresponds to translations s g − sh on ∆ ⋊ Γ. Similarly, Ad( J ) corresponds to the inverse map on ∆ ⋊ Γ. Let q ∈ { , , − } and let H be a Hilbert space. For each n ∈ N , we denote H ⊗ alg n bythe n -algebraic tensor product. Define a (possibly degenerate) inner product on H ⊗ alg n by h ξ ⊗ · · · ⊗ ξ n , η ⊗ · · · ⊗ η n i = X σ ∈ S n q i ( σ ) h ξ σ (1) ⊗ · · · ⊗ ξ σ ( n ) , η ⊗ · · · ⊗ η n i , ξ , . . . , ξ n , η , . . . , η n ∈ H , where i ( σ ) is the number of inversions. We define the q -Fockspace F q ( H ) by separation and completion of the algebraic Fock space F alg ( H ) = C Ω ⊕ M n ≥ H ⊗ alg n . When q = 0 ,
1, and − F q ( H ) is called the full , symmetric , and anti-symmetric Fockspace, respectively. For ξ ∈ H , define the left and right creation operator by ℓ ( ξ )( ξ ⊗ · · · ⊗ ξ n ) := ξ ⊗ ξ ⊗ · · · ⊗ ξ n ; r ( ξ )( ξ ⊗ · · · ⊗ ξ n ) := ξ ⊗ · · · ⊗ ξ n ⊗ ξ for all n ∈ N and ξ , . . . , ξ n ∈ H . They satisfy the q -commutation relation ℓ ( ξ ) ℓ ( η ) ∗ − qℓ ( η ) ∗ ℓ ( ξ ) = h ξ, η i , for all ξ, η ∈ H. Note that for any 0 = ξ ∈ H , ℓ ( ξ ) is bounded if q ∈ { , − } , and is unbounded if q = 1.We next introduce associated von Neumann algebras. For this, consider a real Hilbertspace H R and extend it to a complex Hilbert space by H := H R ⊗ R C . We identify H R ⊗ R H R . Let I be the involution for H R ⊂ H , that is, I ( ξ ⊗ λ ) = ξ ⊗ λ for ξ ⊗ λ ∈ H R ⊗ R C . Note that H R = { ξ ∈ H | Iξ = ξ } .Assume first q = 1. Then putting W ( ξ ) := ℓ ( ξ ) + ℓ ( ξ ) ∗ for ξ , we can define the Gaussian algebra by Γ q ( H R ) := W ∗ { e i W ( ξ ) | ξ ∈ H R } . Then Ω is a cyclic separating vector for Γ q ( H R ). In this case, Γ q ( H R ) is commutativeand the vacuum state h · Ω , Ω i corresponds the the one arising from (a product of) theGaussian measure. See [Bo14, Subsection II.1] for these facts.Assume next q ∈ { , − } . Let U : R → O ( H R ) be any strongly continuous representa-tion of R on H R . We extend U on H by U t ⊗ id C for t ∈ R and use the same notation U t .By Stone’s theorem, take the infinitesimal generator A satisfying U t = A i t for all t ∈ R .Consider an embedding H → H ; ξ
7→ √ √ A − ξ =: b ξ. We define the associated von Neumann algebra byΓ q ( H R , U ) := W ∗ { W ( b ξ ) | ξ ∈ H } , where W ( b ξ ) := ℓ ( b ξ ) + ℓ ∗ ( c Iξ ). Then Ω is cyclic and separating, and the vacuum state h · Ω , Ω i defines a faithful normal state on Γ q ( H R , U ). When q = 0, Γ q ( H R , U ) is the freeAraki–Woods factor introduced in [Sh97]. When q = − U is almostperiodic, Γ q ( H R , U ) is the classical Araki–Woods factor , see [HIK20, Section 4]. For themodular theory of these von Neumann algebras, we refer the reader to [Hi02, Lemma 1.4],which treat the case q ∈ ( − ,
1) but the same proof works for q = − q ∈ { , , − } , the modular conju-gation J on F q ( H ) is given by J ( ξ ⊗ · · · ⊗ ξ n ) = ( Iξ n ) ⊗ · · · ⊗ ( Iξ )for all n ∈ N and ξ . . . , ξ n ∈ H . 6 Bi-exactness for groups acting on semigroups
We first review Ozawa’s work on bi-exactness for some semidirect product groups.
Example 3.1.
Let Λ , Γ be (nontrivial) countable discrete groups and consider the wreathproduct Λ ≀ Γ = M Γ Λ ! ⋊ Γ . Set G := { S finite s Γ t | s, t ∈ Λ ≀ Γ } . Then it is proved in [Oz04] [BO08, Proposition 15.3.6and Corollary 15.3.9] that, if Γ is exact and Λ is amenable, then Γ is bi-exact / G . Thismeans the left-right translation Γ × Γ y ℓ ∞ (Γ) /c ( G ) is amenable. When Γ is bi-exact,then Λ ≀ Γ is also bi-exact.
Example 3.2.
Let Γ := SL(2 , Z ) and Λ := Z and consider Λ ⋊ Γ arising from the naturalaction Γ y Λ. Set G := { S finite s Γ t | s, t ∈ Λ ⋊ Γ } . Then Γ is bi-exact / G . Since Γ isbi-exact, the group Λ ⋊ Γ is bi-exact. See [Oz08].In both examples, semidirect product groups Λ ⋊ Γ with families G := { S finite s Γ t | s, t ∈ Λ ⋊ Γ } of subsets in Λ ⋊ Γ are considered. Here we briefly recall Ozawa’s proofs.Later we give detailed proofs in our more general setting.First, we consider ℓ ∞ ( G ) := { f ∈ ℓ ∞ (Λ ⋊ Γ) | f ( s · ) − f, f ( · s ) − f ∈ c ( G ) for all s ∈ Λ } . Then the left-right actions of Λ on ℓ ∞ ( ∂ G ) := ℓ ∞ ( G ) /c ( G ) is trivial, hence it can beregarded as a boundary for Λ-actions. Next we prove that the Γ × Γ-action on ℓ ∞ ( ∂ G )induced by the left-right translation of Γ is amenable. Finally since ℓ ∞ ( ∂ G ) commuteswith C λ (Λ) and C ρ (Λ), we have a Γ × Γ-equivariant map ℓ ∞ ( ∂ G ) ⊗ min C λ (Λ) ⊗ min C ρ (Λ) → M( K ( G )) / K ( G ) . By the amenability of the action, we can extend this map to the reduced crossed product.By restriction, we get that C λ (Λ ⋊ Γ) ⊗ min C ρ (Λ ⋊ Γ) → M( K ( G )) / K ( G )is min-bounded, hence Λ ⋊ Γ is bi-exact / G . Note that the following algebra was also usedto prove the amenability: C ( G ) := { f ∈ ℓ ∞ ( G ) | f ( · h ) − f ∈ c ( G ) for all h ∈ Γ } . Note that in the above argument, we need the following objects. • We denote the inverse map s s − in Λ ⋊ Γ by I . We have an anti-linear isometry J on ℓ (Λ ⋊ Γ) given by
J δ s = δ s − for all s ∈ Λ ⋊ Γ. This restricts to the one J Λ on ℓ (Λ). • If we write the action Γ y Λ by π , it naturally extends to a unitary representation U π : Γ y ℓ (Λ). • Left and right regular representations λ and ρ of Λ satisfy λ g δ s = δ gs and ρ g δ s = δ sg − for all s, g ∈ Λ. • For all g, h ∈ Γ, s, t ∈ Λ, we have that (inside B ( ℓ (Λ))) U πg λ s U π ∗ g = λ π g ( s ) , U πg ρ ( s ) U π ∗ g = ρ π g ( s ) , J Λ λ s J Λ = ρ s . We thus have
I, J Λ , U πg , λ, ρ in this setting. Our aim of this section is to generalize theabove framework to the ones for groups Γ acting on semigroups S rather than Λ.7 .2 Basic setting Let S be a semigoup with unit 1 S . We always assume the following two conditions: • (cancellative) for any x, y, s ∈ S , sx = sy or xs = ys implies x = y ; • (involution) there is a bijection I : S → S such that I = id S and I ( st ) = I ( t ) I ( s ) ( s, t ∈ S ) . They are trivially satisfied when S is a group. Let Γ be a discrete group and consideran action Γ y π S such that each π g for g ∈ Γ preserves the structure of the involutivesemigroup: π g ( ab ) = π g ( a ) π g ( b ) for a, b ∈ S and Iπ g = π g I. Consider the semidirect product S ⋊ Γ and we use the notation ( s, g ) = sg = gπ − g ( s )for s ∈ S and g ∈ Γ as an element in S ⋊ Γ. Observe that S ⋊ Γ is also a cancellativesemigroup with involution given by I ( s, g ) := ( π − g ( Is ) , g − ) = g − ( Is ) . Since S ⋊ Γ = S × Γ as a set, we have a natural inclusion ℓ ∞ ( S ⋊ Γ) = ℓ ∞ ( S ) ⊗ ℓ ∞ (Γ) ⊂ B ( ℓ ( S ) ⊗ ℓ (Γ)) . In this section, we always keep the following assumptions. They are objects in B ( ℓ ( S ))which correspond to structures of S ⋊ Γ up to multiples of T (or C ). Below we use thenotation a = C b if there is λ ∈ C \ { } such that a = λb, and a = T b if a = λb for λ ∈ T . Assumption 3.3.
Let Γ y π S be as above. Assume the following conditions. • (Involution) There is an anti-linear isometry J S on ℓ ( S ) which satisfies J S δ x = T δ Ix for all x ∈ S. • (Unitary representation) There is a unitary representation U π : Γ y ℓ ( S ) such that U πg δ x = T δ π g ( x ) and U πg J S = J S U πg for all g ∈ Γ , x ∈ S. • (Creation operators) For any s ∈ S , there are ℓ ( s ) , r ( s ) ∈ B ( ℓ ( S )) such that for any x ∈ X , ℓ ( s ) δ x = C s,x δ sx and r ( s ) δ x = C rs,x δ xs for some C s,x , C rs,x ∈ C , where C s,x , C rs,x are possibly zero. We further assume ℓ ( s ) ℓ ( t ) = C ℓ ( st ) and r ( s ) r ( t ) = C r ( ts ) for all s, t ∈ S. • (Covariance conditions) For all g, h ∈ Γ, s, t ∈ S , we have U πg ℓ ( s ) U π ∗ g = ℓ ( π g ( s )) , U πg r ( s ) U π ∗ g = r ( π g ( s )); J S ℓ ( s ) J S = r ( Is ) , J S r ( s ) J S = ℓ ( Is ) . (Relativity) There is a family G of subsets in S ⋊ Γ such that E ∪ F ∈ G , sEt ∈ G , and IE ∈ G , for all E, F ∈ G , s, t ∈ S ⋊ Γ . Assume Γ
6∈ G , so that c ( G ) ( ℓ ∞ ( S ⋊ Γ).Observe that examples from groups in the previous subsection trivially satisfy all theassumptions (regular representations of Λ correspond to creations). Our main examplesare ones from creations operators on Fock spaces, see Subsection 4.1.Using the family G , we can define relative algebras: c ( G ) := { f ∈ ℓ ∞ ( S ⋊ Γ) | lim x →∞ / G f ( x ) = 0 } ; K ( G ) := c ( G ) B ( ℓ ( S ⋊ Γ)) c ( G ) norm . We denote by M( K ( G )) the multiplier algebra of K ( G ). Consider the following represen-tations, Left action π ℓ : B ( ℓ ( S )) ∋ a a ⊗ Γ ; Γ ∋ g U g ⊗ λ g ;Right action π r : B ( ℓ ( S )) ∋ a X h ∈ Γ U πh aU π ∗ h ⊗ e h,h ; Γ ∋ g ⊗ ρ g ; J -map J = X h ∈ Γ U πh J S ⊗ e h,h − . Note that this coincides with the one in Subsection 2.3 when S is a group. Remark 3.4.
We assumed that ℓ ( s ) , r ( s ) are all bounded operators. In the case ofsymmetric Fock spaces, however, creation operators are all unbounded . We will prove somebi-exactness results for this case, so we sometimes consider below the case that ℓ ( x ) , r ( x )are closed operators. This exactly means that if we regard constants C s,x , C rs,x as functionson x ∈ S denoted by C s, · and C rs, · , then they are (possibly) unbounded functions.We also note that, if a = ℓ ( s ) for s ∈ S is a closed operator, then π ℓ ( a ) = a ⊗ π r ( a ) = J ( J S aJ S ⊗ J are again closed, hence π ℓ , π r are defined on closed operators on ℓ ( S ). In particular π ℓ ( C s, · ) and π r ( C rs, · ) define closed operators.Before proceeding, we see two elementary lemmas. We omit most proofs. Lemma 3.5.
The following conditions hold true.1. For all g ∈ Γ and a ∈ B ( ℓ ( S )) , we have J ( U πg ⊗ λ g ) J = 1 ⊗ ρ g , π r ( a ) = J ( J S aJ S ⊗ J. Ad( U g ⊗ λ g ) , Ad(1 ⊗ ρ g ) for g ∈ Γ , and Ad J restrict to automorphisms on ℓ ∞ ( S ⋊ Γ) ,which correspond to left, right translations by g and the involution I . More precisely, Ad( U πg ⊗ λ g ρ h ) f = f ( g − · h ) , for f ∈ ℓ ∞ ( S ⋊ Γ);Ad( J ) f = f ◦ I, for f ∈ ℓ ∞ ( S ⋊ Γ) . It follows that • Ad( U πg ⊗ λ g ρ h ) and Ad( J ) globally preserve c ( G ) and K ( G ) ; • for any (possibly unbounded) map f : S → C , π ℓ ( f ) = f ⊗ , π r ( f ) = ( f ◦ I S ⊗ ◦ I can be regarded as maps on S ⋊ Γ . emma 3.6. Fix s ∈ S and assume that ℓ ( s ) , r ( s ) are possibly unbounded. The followingconditions hold true.1. For any map f ∈ ℓ ∞ ( S ⋊ Γ) , we have f π ℓ ( ℓ ( s )) = π ℓ ( ℓ ( s )) f ( s · ) , π ℓ ( ℓ ( s )) f = f ( s − · )1 s ( S ⋊ Γ) π ℓ ( ℓ ( s )); f π r ( r ( s )) = π r ( r ( s )) f ( · s ) , π r ( r ( s )) f = f ( · s − )1 ( S ⋊ Γ) s π r ( r ( s )) . where X means the characteristic function on a subset X ⊂ S ⋊ Γ .2. For any f ∈ ℓ ∞ ( S ⋊ Γ) , the following implications hold: π ℓ ( C s, · )( f − f ( s · )) ∈ c ( G ) ⇒ [ π ℓ ( ℓ ( s )) , f ] ∈ K ( G ); π r ( C rs, · )( f − f ( · s )) ∈ c ( G ) ⇒ [ π r ( r ( s )) , f ] ∈ K ( G ) . In particular π ℓ ( ℓ ( s )) , π r ( r ( s )) ∈ M( K ( G )) if ℓ ( s ) , r ( s ) are bounded. Proof.
1. Apply them to δ x ⊗ δ s for x ∈ S and g ∈ Γ.2. We see the case π ℓ . Observe that ℓ ( s ) C − s, · is a well defined isometry. Then item 1implies [ π ℓ ( ℓ ( s )) , f ] = π ℓ ( ℓ ( s ))( f − f ( s · ) = π ℓ ( ℓ ( s ) C − s, · ) π ℓ ( C s, · )( f − f ( s · )) . Since π ℓ ( ℓ ( s ) C − s, · ) is an isometry, if π ℓ ( C s, · )( f − f ( s · )) ∈ c ( G ), we have [ π ℓ ( ℓ ( s )) , f ] ∈ B ( ℓ ( S ⋊ Γ)) c ( G ). A similar argument works for [ π ℓ ( ℓ ( s )) , f ] = ( f ( s − · )1 s ( S ⋊ Γ) − f ) π ℓ ( ℓ ( s )),hence we get the conclusion. The last statement follows by Lemma 2.3. Boundary C ∗ -algebras and associated actions Keep Assumption 3.3 and define C ∗ -algebras by ℓ ∞ ( G ) := { f ∈ ℓ ∞ ( S ⋊ Γ) | π ℓ ( C s, · )( f − f ( s · )) , π r ( C rs, · )( f − f ( · s )) ∈ c ( G ) for all s ∈ S } ; C ( G ) := { f ∈ ℓ ∞ ( G ) | f − f ( · h ) ∈ c ( G ) for all h ∈ Γ } . Here ℓ ∞ ( G ) should be regarded as a boundary for left-right creations (and C ( G ) for theright Γ-action). Since they contain c ( G ), we can define the following boundary C ∗ -algebras ℓ ∞ ( ∂ G ) := ℓ ∞ ( G ) /c ( G ) , C ( ∂ G ) := C ( G ) /c ( G ) . Lemma 3.7.
The map
Ad( J ) globally preserves ℓ ∞ ( G ) . For any g ∈ Γ , Ad( U πg ⊗ λ g ) and Ad(1 ⊗ ρ g ) globally preserves ℓ ∞ ( G ) and C ( G ) . Proof.
Let f ∈ ℓ ∞ ( G ) and put F := J f J = f ◦ I . Then π r ( C s,I · )( F − F ( · ( Is ))) = J π ℓ ( C s, · )( f − f ( s · )) J ∈ J c ( G ) J = c ( G ) . Since C s,I · = C rIs, · (which comes from ℓ ( s ) = J S r ( Is ) J S ), by putting Is =: t , we have π r ( C rt, · )( F − F ( · t )) ∈ c ( G ) . This means F satisfies the condition for π r . Similarly F satisfies the desired condition for π ℓ , hence F ∈ ℓ ∞ ( G ). For group actions of Γ, we can use a similar argument (use theequation U πg ℓ ( s ) U π ∗ g = ℓ ( π g ( s ))).By the lemma, we get group actions arising from adjoint maps: for g, h ∈ Γ,Ad( U πg ⊗ λ g ρ h ) : Γ × Γ y ℓ ∞ ( ∂ G );Ad( U πg ⊗ λ g ) : Γ y C ( ∂ G ) . By Lemma 3.5, they coincide with the ones arising from translation actions on S ⋊ Γ.10 .3 Bi-exactness and condition (AO)
We now define bi-exactness in our setting. We keep S, Γ , G in assumption 3.3. Definition 3.8.
We say that Γ y π S is bi-exact relative to G (or bi-exact/ G in short) ifthe Γ-action Γ × Γ y ℓ ∞ ( ∂ G ) arising from the left and right translations is (topologically)amenable. Remark 3.9.
As we will explain at the last part in this subsection, for certain examples,we should exchange S with a subset S ⊂ S .Note that this notion obviously depends on other objects such as J S , U π , ℓ, r . Notethat Γ is automatically exact since it has a (topologically) amenable action.Here is a sufficient condition of bi-exactness. We will indeed check this condition later. Lemma 3.10.
If the left translation Γ y C ( ∂ G ) is amenable, then Γ y S is bi-exact/ G . Proof.
Consider the C ∗ -algebra J C ( G ) J = { f ∈ ℓ ∞ ( G ) | f − f ( g · ) ∈ c ( G ) for all g ∈ Γ } . This is a boundary for the left Γ-action. Since Ad( J ) globally preserves c ( G ), it inducesan anti-linear ∗ -isomorphism on M( K ( G )), which we again denote by Ad( J ). Then itfollows that Ad( J )( C ( ∂ G )) = J C ( G ) J/c ( G ) =: J C ( ∂ G ) J. By applying Ad( J ) to Γ y C ( ∂ G ), the right translation Γ y J C ( ∂ G ) J is also amenable.Observe that there are inclusions C ( ∂ G ) , J C ( ∂ G ) J ⊂ ℓ ∞ ( ∂ G ) . Then the left translation on ℓ ∞ ( ∂ G ) restricts to an amenable action on C ( ∂ G ), so thecrossed product ℓ ∞ ( ∂ G ) ⋊ red Γ is nuclear. Next consider the right translation on ℓ ∞ ( ∂ G ). Itcommutes with the left translation and is amenable on J C ( ∂ G ) J . Since the left translationon J C ( ∂ G ) J is trivial, J C ( ∂ G ) J is contained in the center of ℓ ∞ ( ∂ G ) ⋊ red Γ. This impliesthat [ ℓ ∞ ( ∂ G ) ⋊ red Γ] ⋊ red Γ = ℓ ∞ ( ∂ G ) ⋊ red (Γ × Γ)is nuclear. This means that the Γ × Γ-action is topologically amenable.We next study relationship between bi-exactness and condition (AO). Define C := C ∗ { ℓ ( s ) | s ∈ S } , C r := C ∗ { r ( s ) | s ∈ S } = J S C J S ⊂ B ( ℓ ( S )) . By the covariance conditions, Ad U π defines actions on C , C r . We can consider the left andright representations π ℓ , π r for crossed products of these actions, hence π ℓ ( C ⋊ red Γ) , π r ( C r ⋊ red Γ) ⊂ B ( ℓ ( S ⋊ Γ)) , Note that π r ( C r ⋊ red Γ) =
J π ℓ ( C ⋊ red Γ) J . The next lemma is straightforward, see Lemmas3.5 and 3.6. Lemma 3.11.
We have π ℓ ( C ⋊ red Γ) , π r ( C r ⋊ red Γ) ⊂ M( K ( G )) . B ⊂ C be a unital C ∗ -subalgebra such that the Γ-action on C globally preserves B and that [ B, B r ] = 0, where B r := J S BJ S . For example, we will choose B = C ∗ { W ( ξ ) | ξ ∈ H } in the Fock space case. By the previous lemma, one can define an algebraic ∗ -homomorphism ν : ( B ⋊ red Γ) ⊗ alg ( B r ⋊ red Γ) → M( K ( G )) / K ( G ); a ⊗ b π ℓ ( a ) π r ( b ) + K ( G ) . Since ℓ ∞ ( G ) ⊂ M( K ( G )) and c ( G ) ⊂ K ( G ), there is a ∗ -homomorphism π G : ℓ ∞ ( ∂ G ) → M( K ( G )) / K ( G ) . Then by the definition of ℓ ∞ ( G ), we have an algebraic ∗ -homomorphism ν G : ℓ ∞ ( ∂ G ) ⊗ alg B ⊗ alg B r → M( K ( G )) / K ( G ); ν G = π G ⊗ π ℓ ⊗ π r . The next proposition explains how to use our bi-exactness. We will call the boundednessof ν below condition (AO) relative to K ( G ). Proposition 3.12.
Keep the setting and assume that • ν G is min-bounded and nuclear (e.g. B is nuclear); • Γ y S is bi-exact / G .Then ν is min-bounded and nuclear. Further if B ⋊ red Γ is separable, then ν has a ucplift, that is, there is a ucp map θ : ( B ⋊ red Γ) ⊗ min ( B r ⋊ red Γ) → M( K ( G )) such that θ ( a ⊗ b ) − π ℓ ( a ) π r ( b ) ∈ K ( G ) for all a ∈ B ⋊ red Γ and b ∈ B r ⋊ red Γ . Proof.
Put e Γ := Γ × Γ. Consider e Γ-actions on ℓ ∞ ( ∂ G ) and B ⊗ min B r by the left and righttranslations and by Ad( U πg ⊗ U πh ) for ( g, h ) ∈ e Γ respectively. By assumption, the tensorproduct e Γ-action on ℓ ∞ ( ∂ G ) ⊗ min B ⊗ min B r is amenable. Observe that ν G is e Γ-equivariantfrom this action to the one given by Ad( U πg ⊗ λ g ρ h ) for ( g, h ) ∈ e Γ, hence the followingmap is bounded and is nuclear, see item 1 in Proposition 2.4,[ ℓ ∞ ( ∂ G ) ⊗ min B ⊗ min B r ] ⋊ red e Γ → M( K ( G )) / K ( G ) . The domain contains[ C ⊗ min B ⊗ min B r ] ⋊ red e Γ ≃ ( B ⋊ red Γ) ⊗ min ( B r ⋊ red Γ)and the restriction to this algebra coincides with ν . This is the conclusion. If B ⋊ red Γ isseparable, we can use Choi–Effros’s lifting theorem (e.g. [BO08, Theorem C.3]).
A modification for anti-symmetric Fock spaces
Assumption 3.3 contains the case that ℓ ( s ) = 0 for some s ∈ S . This indeed happensin the case of anti-symmetric Fock space. In such a case, we should ignore non-importantpart in ℓ ( S ). To do this, we keep Assumption 3.3 and consider S := { s ∈ S | ℓ ( s ) = 0 or ℓ ( Is ) = 0 } = { s ∈ S | C s, · = 0 or C Is, · = 0 } . Observe that by the covariance conditions, π and I restrict to bijective maps on S andthat S can be defined by right creations. Write S ⋊ Γ := S × Γ and we denote by12 S ∈ B ( ℓ ( S )) the orthogonal projection onto ℓ ( S ). We use the following items in B ( ℓ ( S )): for g ∈ Γ and s ∈ S , U πg P S , J S P S , ℓ ( s ) P S and r ( s ) P S . With abuse of notations, we again denote by U πg , J S , ℓ ( s ) , r ( s ) respectively. As we haveseen in Lemma 2.2, G := G ∩ ( S ⋊ Γ) satisfies c ( G ) = c ( G ) ∩ ℓ ∞ ( S ⋊ Γ) = c ( G )( P S ⊗ , K ( G ) = ( P S ⊗ K ( G )( P S ⊗ . Then it is straightforward to check that all lemmas in Subsection 3.2 hold for S ⋊ Γ (onecan apply the compression map by P S ⊗ × Γ y ℓ ∞ ( ∂ G ) and Γ y C ( ∂ G ) . In this case, we say that Γ y π S is bi-exact relative to G if this Γ × Γ-action is amenable.This is weaker than bi-exactness relative to G , but is enough to prove Proposition 3.12.More precisely if we exchange S ⋊ Γ and G in Proposition 3.12 with S ⋊ Γ and G , thesame proof still works.Thus we can use all the results in Subsections 3.2 and 3.3 (including Lemma 3.13below) if we exchange S with S which supports all creation operators. An intersection lemma
We keep the setting from Proposition 3.12. If there is another family F of subsets in S ⋊ Γ satisfying the relativity condition in Assumption 3.3, then
G ∩ F := { G ∩ F | G ∈ G , F ∈ F } also satisfy the same condition. It is then natural to ask when we get condition (AO)relative to K ( G ∩ F ), using information of G and F . It was discussed in [BO08, Proposition15.2.7] for the group case.In our setting, we need to assume that F is of the following form: assume that thereis a family E of subsets in Γ such that E − , gEh ∈ E , E ∪ F for all E ∈ E , F and g, h ∈ Γ . Then we put F := { SE | E ∈ E} . Since SE = ES for E ∈ E , it is easy to check therelativity condition in Assumption 3.3 for F . In this notation, if Γ is bi-exact / E , thenΓ y S is bi-exact / F . To see this, we have only to see 1 ⊗ ℓ ∞ (Γ) ⊂ ℓ ∞ ( G ), which inducesa Γ × Γ-equivariant map ℓ ∞ (Γ) /c ( E ) → ℓ ∞ ( ∂ G ) . For such G and F , we prepare the following notation: • K G := K ( G ) and M G := M( K G ) (similarly for F and G ∩ F ); • K := K G ∩ K F (= K G∩F ) and M := C ∗ { π ℓ ( C ) , π r ( C r ) , ℓ ∞ ( S ⋊ Γ) } .With abuse of notation, we omit intersections in quotients such asM / K := M / ( K ∩ M) . As in Proposition 3.12, we consider ∗ -homomorphisms (with smaller ranges) ν G : ℓ ∞ ( ∂ G ) ⊗ alg B ⊗ alg B r → M / K G , ν G = π G ⊗ π ℓ ⊗ π r ; ν F : ℓ ∞ ( ∂ F ) ⊗ alg B ⊗ alg B r → M / K F , ν F = π F ⊗ π ℓ ⊗ π r . In this notation, we prove the following lemma. Note that this also holds for S ⊂ S inthe above sense. 13 emma 3.13. Keep the setting and assume that • π ℓ ⊗ π r : B ⊗ alg B r → M / K is min-bounded; • ν G , ν F are min-bounded and nuclear; • Γ y S is bi-exact / G and Γ is bi-exact / E .Then B ⋊ red Γ satisfies condition (AO) relative to K G∩F and the associated bounded mapis nuclear. It has a ucp lift if B ⋊ red Γ is separable. Proof.
We put e Γ := Γ × Γ and A := C ∗ { π ℓ ( B ) , π r ( B r ) , ℓ ∞ ( G ) } , B := C ∗ { π ℓ ( B ) , π r ( B r ) , ⊗ ℓ ∞ (Γ) } . Observe that
B ⊂ A ⊂
M and e Γ-actions on B , A , M are defined by Ad( U πg ⊗ λ g ρ h ) for( g, h ) ∈ e Γ, which induces ones on quotients such as e Γ y M / K . Claim.
Assume that the ∗ -homomorphism θ : [ B / K ] ⋊ alg e Γ → [M / K ] ⋊ alg e Γ arising from the inclusion B ⊂ M is bounded from the reduced norm to the full norm andis nuclear. Then the conclusion follows. Proof.
We consider the following maps:( B ⋊ red Γ) ⊗ min ( B r ⋊ red Γ) ≃ [ B ⊗ min B r ] ⋊ red e Γ → [ B / K ] ⋊ red e Γ → θ [M / K ] ⋊ full e Γ → M G∩F / K . Here the first map is from B ⊗ min B r → B / K (in the assumption), and the last one isfrom the inclusion M ⊂ M G∩F with e Γ ∋ ( g, h ) U πg ⊗ λ g ρ h . Then since θ is nuclear, theconclusion follows.To study θ in the claim, we consider the following two exact sequences:0 / / [( K F ∩ M) / K ] ⋊ full e Γ / / [M / K ] ⋊ full e Γ / / [M / K F ] ⋊ full e Γ / / / / [( K F ∩ A ) / K ] ⋊ red e Γ / / ϕ O O [ A / K ] ⋊ red e Γ / / π O O [ A / K F ] ⋊ red e Γ ψ O O / / , which arise from surjective e Γ-equivariant maps M / K → M / K F and A / K → A / K F . Here ϕ, π, ψ arise from inclusions and defined only at the algebraic level. Note that the bottomline is exact since e Γ is exact. The first step of the proof is to show the following claim.
Claim.
The maps ϕ, π, ψ are bounded.
Proof.
Observe that π F (( ℓ ∞ (Γ) /c ( E )) is contained in the center of A / K F . Then e Γ y A / K F is amenable since it restricts to the amenable action e Γ y π F (( ℓ ∞ (Γ) /c ( E )). Weget that [ A / K F ] ⋊ red e Γ = [ A/ K F ] ⋊ full e Γ, so that ψ is bounded.We next see ϕ . If we put I F := K F ∩ A , I G := K G ∩ A , then there is an inclusion as aclosed ideal ( K F ∩ A ) / K = I F / ( I G ∩ I F ) ≃ ( I G + I F ) / I G ⊂ A / I G , e Γ-actions. Then e Γ y A / I G is amenable since it restricts to the amenableaction e Γ y π G ( ℓ ∞ ( ∂ G )) contained in the center. We get [ A / I G ] ⋊ red e Γ = [ A / I G ] ⋊ full e Γ,hence its restriction to the closed ideal gives [( K F ∩ A ) / K ] ⋊ red e Γ = [( K F ∩ A ) / K ] ⋊ full e Γ.This implies the boundedness of ϕ .We finally see π . Since we have already proved that the reduced and the full crossedproduct coincide on domains on ψ, ϕ , by the 5 lemma, the same holds for the domain of π . Thus π is also bounded.We next see some nuclearlity of ϕ and ψ . Claim.
The maps ϕ and the restriction of ψ to [ B / K F ] ⋊ red e Γ are nuclear. Proof.
Observe first that by assumption and item 2 in Proposition 2.4, inclusions A / K G = Im( ν G ) ⊂ M / K G and ( B / K F ⊂ ) Im( ν F ) ⊂ M / K F are nuclear. Since e Γ y A / K G , B / K F are amenable, by item 1 in Proposition 2.4,[ A / K G ] ⋊ red e Γ → [M / K G ] ⋊ full e Γ and [ B / K F ] ⋊ red e Γ → [M / K F ] ⋊ full e Γare nuclear. The second map is the restriction of ψ . We restricts the first map to [( K F ∩A ) / K ] ⋊ red e Γ as in the proof of the last claim. Then the same reasoning shows that therange [M / K G ] ⋊ full e Γ contains [( K F ∩ M) / K ] ⋊ full e Γ as a closed ideal. Then the restrictionto these ideals is again nuclear and it coincides with ϕ .We now deduce our conclusion as follows. Claim.
The restriction of π to [ B / K ] ⋊ red e Γ is nuclear. Hence the conclusion follows bythe first claim. Proof.
We restrict ϕ, π, ψ to0 / / [( K F ∩ B ) / K ] ⋊ red e Γ / / [ B / K ] ⋊ red e Γ / / [ B / K F ] ⋊ red e Γ / / . We will apply item 3 in Proposition 2.4. For this, since ϕ, ψ are nuclear by the lastclaim and since [ B / K ] ⋊ red e Γ is exact (because B / K is an image of the exact algebra ℓ ∞ (Γ) ⊗ min B ⊗ min B r ), we have only to check the assumption on approximate units.To see this, since ϕ arises from the inclusion K F ∩ B ⊂ K F ∩ M, it suffices to show thatthere is an approximate unit for K F ∩ B which is also one for K F ∩ M. Then we can use(1 SE ) E ∈E , which is an approximate unit for K F and is contained in K F ∩ B . Throughout this section, we keep the following setting and notation.Let Γ y π X be an action of a discrete group Γ on a set X . Consider the associatedunitary representation π : Γ y ℓ ( X ); π g δ x = δ π g ( x ) . For q ∈ { , , − } , define the q -Fock space F q := F q ( ℓ ( X )) = C Ω ⊕ M n ≥ ℓ ( X ) ⊗ nq , ℓ ( X ) ⊗ nq is the completion of ℓ ( X ) ⊗ alg n by the q -inner product. For ξ ∈ ℓ ( X ), wedenote by ℓ ( ξ ) and r ( ξ ) creation operators. We put ℓ ( x ) := ℓ ( δ x ) and r ( x ) := r ( δ x ) for x ∈ X , and C ℓ := C ∗ { ℓ ( x ) | x ∈ X } , C r := C ∗ { r ( x ) | x ∈ X } (if q = 1); C ℓ := C ∗ { e i( ℓ ( x )+ ℓ ( x ) ∗ ) | x ∈ X } = C r (if q = 1) . Let U π : Γ → U ( F q ) be the associated unitary representation on F q given by U πg = 1 Ω ⊕ M n ≥ π ⊗ ng , g ∈ Γ . Note that Ad U π gives Γ-actions on C ℓ and C r . Finally assume that there is a bijection I : X → X such that I = id X and Iπ g = π g I for all g ∈ Γ. Define an anti-linear isometry I : ℓ ( X ) → ℓ ( X ); aδ x aδ Ix for a ∈ C , x ∈ X. We extend it on F q (as an anti-linear isometry) by J q ( ξ ⊗ · · · ⊗ ξ n ) = Iξ n ⊗ · · · ⊗ Iξ and J q Ω = Ω. We have J q = id. This J q should be understood as the modular con-jugation arising from von Neumann algebras given in Subsection 2.4. Examples of suchvon Neumann algebras will be given in Subsection 6.1 (the bijectioin I : X → X is indeedimportant).To study bi-exactness on Fock spaces, we introduce semigroup structures. More pre-cisely, we will define a semigroup S satisfying F q ⊗ ℓ (Γ) = ℓ ( S ⋊ Γ)and Assumption 3.3.
The case of full Fock spaces
We consider the case F =: F full . Each ℓ ( X ) ⊗ n in F full has an identification ℓ ( X ) ⊗ n = ℓ ( X n ); δ x ⊗ · · · ⊗ δ x n = δ ( x ,...,x n ) . So by using the set X full := { ⋆ } ⊔ G n ≥ X n , where { ⋆ } is the singleton, we have ℓ ( X full ) = F full with C δ ⋆ = C Ω. This X full has asemigroup structure by xy := ( x , . . . , x n , y , . . . , y m ) , for x = ( x , . . . , x n ) , y = ( y , . . . , y m ) ∈ X full . Note that X full = ∗ X N and ⋆ is the unit. We define creation operators for x ∈ X full by ℓ ( x ) := ℓ ( x ) · · · ℓ ( x n ) , r ( x ) := r ( x n ) · · · r ( x ) . They satisfy ℓ ( x ) δ y = δ xy , r ( x ) δ y = δ yx , x, y ∈ X full . We extend π : Γ y X on X full by π g ( x , . . . , x n ) := ( π g ( x ) , . . . , π g ( x )) . U πg on F full satisfies δ ( x ,...,x n ) = δ x ⊗ · · · ⊗ δ x n U πg δ π g ( x ) ⊗ · · · ⊗ δ π g ( x n ) = δ ( π g ( x ) ,...,π g ( x )) . We extend I on X full by I : X full → X full ; ( x , . . . , x n ) ( Ix n , . . . , Ix ) . It holds that Iπ g = π g I and I ( xy ) = I ( y ) I ( x ) for all g ∈ Γ and x, y ∈ X full . As inAssumption 3.3, we have an associated map J full := J X full on F q = ℓ ( X full ) and it holdsthat J full = J q . They satisfy all the conditions in Assumption 3.3. The case of symmetric Fock spaces
We consider the case F sym := F ( ℓ ( X )). Recall that for ξ , . . . , ξ n , η , . . . , η n ∈ ℓ ( X ), h ξ ⊗ · · · ⊗ ξ n , η ⊗ · · · ⊗ η m i = δ n,m X σ ∈ S n h ξ σ (1) ⊗ · · · ⊗ ξ σ ( n ) , η ⊗ · · · ⊗ η n i . Let ℓ ( X ) ⊗ n sym denote the completion of ℓ ( X ) ⊗ alg n and we have F sym = C Ω ⊕ M n ≥ ℓ ( X ) ⊗ n sym . For each n ≥
1, consider the natural action of the symmetric group S n on X n by permu-tations of indices. Put Y n := X n / S n and define X sym := { ⋆ } ⊔ ∞ G n =1 Y n , which has a natural surjection Q : X full → X sym . We write [ x ] := Q ( x ) for x ∈ X full . Wemake identifications for all n ∈ N by ℓ ( Y n ) ∋ δ [ x ] C [ x ] ( δ x ⊗ · · · ⊗ δ x n ) ∈ ℓ ( X ) ⊗ n sym , where C [ x ] > C [ x ] as follows. For any [ x , . . . , x n ] ∈ Y n , up to permutation, it has a form δ x ⊗ · · · ⊗ δ x n = δ ⊗ k z ⊗ · · · ⊗ δ ⊗ k m z m , where all z , . . . , z m ∈ X are distinct (hence k + · · · + k m = n ). It holds that k δ x ⊗ · · · ⊗ δ x n k = X σ ∈ S n h δ x σ (1) ⊗ · · · δ x σ ( n ) , δ x ⊗ · · · ⊗ δ x n i = k ! k ! · · · k m ! . We get C [ x ] = √ k ! ··· k m ! . We thus have the identification ℓ ( X sym ) = F sym .We consider a semigroup structure on X sym by[ x ][ y ] = [ x , . . . , x n , y , . . . , y m ] , for x = ( x , . . . , x n ) , y = ( y , . . . , y m ) ∈ X full . This means X sym = L X N and ⋆ is the unit. For any [ x ] = [ x , . . . , x n ] ∈ Y n , define ℓ ([ x ]) := ℓ ( x ) · · · ℓ ( x n ) , r ([ x ]) := r ( x n ) · · · r ( x )17actually ℓ ([ x ]) = r ([ x ])). By Lemma 4.1 below, there are C x,y for x, y ∈ X sym such that ℓ ( x ) δ y = C x,y δ xy = r ( x ) δ y . We can extend π and I on X sym in the same way as for X full , so that U πg : ℓ ( X sym ) → ℓ ( X sym ); δ [ x ,...,x n ] δ [ π g ( x ) ,...,π g ( x n )] ( J sym :=) J X sym : ℓ ( X sym ) → ℓ ( X sym ); δ [ x ,...,x n ] δ [ Ix n ,...,Ix ] are defined. They satisfy conditions in Assumption 3.3. Lemma 4.1.
For any x ∈ X = Y and [ y ] ∈ Y n , there is a constant C x, [ y ] such that ℓ ( x ) δ [ y ] = C x, [ y ] δ [( x,y )] , ≤ C x, [ y ] ≤ √ n + 1 . For any x, y ∈ X sym , there is a constant ≤ C x,y such that ℓ ( x ) δ y = C x,y δ xy = C x,y δ yx = r ( x ) δ y . Proof.
We have only to prove the first half of the statement. By the above notation, weidentify δ [ y ] as δ ⊗ k z ⊗ · · · ⊗ δ k m z m . Then ℓ ( x ) δ [ y ] = ( k ! · · · k m !) − / δ x ⊗ δ ⊗ k z ⊗ · · · ⊗ δ k m z m = k δ x ⊗ δ ⊗ k z ⊗ · · · ⊗ δ k m z m k ( k ! · · · k m !) / δ [( x,y )] =: C x, [ y ] δ [( x,y )] . If there is i such that x = z i , then it is straightforward to compute that C x, [ y ] = k i + 1. Ifthere are no such i , then C x, [ y ] = 1. We get the inequality. The case of anti-symmetric Fock spaces
We consider the case F anti := F − ( ℓ ( X )). Recall that for ξ , . . . , ξ n , η , . . . , η n ∈ ℓ ( X ), h ξ ⊗ · · · ⊗ ξ n , η ⊗ · · · ⊗ η m i = δ n,m X σ ∈ S n ( − i ( σ ) h ξ σ (1) ⊗ · · · ⊗ ξ σ ( n ) , η ⊗ · · · ⊗ η n i , where i ( σ ) is the number of inversions. Let ℓ ( X ) ⊗ n anti denote the completion of ℓ ( X ) ⊗ alg n and we have F anti = C Ω ⊕ M n ≥ ℓ ( X ) ⊗ n anti . For each n ∈ N , define Z n := { [ x ] = [ x , . . . , x n ] ∈ Y n | x i = x j for all i = j } and put X anti := { ⋆ } ⊔ ∞ G n =1 Z n ⊂ X sym . We would like to make an identification ℓ ( Z n ) ∋ δ [ x ] = δ [ x ,...,x n ] δ x ⊗ · · · ⊗ δ x n ∈ ℓ ( X ) ⊗ n anti .
18t is however not well defined, since it depends on the order of x , . . . , x n . So we fix asection s : X anti → X sym such that s ( Z n ) ⊂ X n , and then the assignment ℓ ( Z n ) ∋ δ [ x ] δ x ⊗ · · · ⊗ δ x n ∈ ℓ ( X ) ⊗ n anti , if s ([ x ]) = ( x , . . . , x n )defines a well defined unitary. We get an identification ℓ ( X anti ) = F anti , which obviouslydepends on the choice of the section. Observe that for any z ∈ Z n and any its representative x = ( x , . . . , x n ), there is C x ∈ { , − } such that δ z = C x ( δ x ⊗ · · · ⊗ δ x n ) . Note that C σ ( x ) = sgn( σ ) C x for any σ ∈ S n . Since X anti ⊂ X sym , there is a naturalinclusion ℓ ( X anti ) ⊂ ℓ ( X sym ) and we denote by P X anti the orthogonal projection onto ℓ ( X anti ). We will use the semigroup structure of X sym and make creation operators whichare supported on X anti , as explained in the last part in Subsection 3.3.For the semigroup structure on X anti , we use the one from X sym . Then X anti is not asubsemigroup but we can use creation operators as follows. First if x ∈ X , then we have ℓ ( x ) , r ( x ) on F anti , so we can induce them on ℓ ( X anti ). By Lemma 4.2 below, there are C x,z , C rx,z ∈ { , , − } for z ∈ X anti such that ℓ ( x ) δ z = C x,z δ xz , r ( x ) δ z = C rx,z δ zx . Note that C x,z = 0 if and only if xz X anti . For any z ∈ X sym \ X anti , we put ℓ ( z ) = r ( z ) =0. For any [ x ] ∈ Z n ⊂ X sym with s ([ x ]) = x = ( x , . . . , x n ) by the section s , we define ℓ ([ x ]) = ℓ ( x ) · · · ℓ ( x n ) , r ([ x ]) = r ( x n ) · · · r ( x ) . Observe that for any other representative y ∈ X n with [ x ] = [ y ], we have ℓ ([ x ]) = C y ℓ ( y ) · · · ℓ ( y n ). This implies that for any z , z ∈ X anti , ℓ ( z ) ℓ ( z ) = T ℓ z ,z ℓ ( z z ) (pos-sibly 0 = T ℓ ( s ) , r ( s )for s ∈ X sym as operators on ℓ ( X sym ), which are supported on ℓ ( X anti ). We thus get afamily of creation operators.We next consider the unitary representation and the anti-linear map. Observe that theinvolution I and the action π : Γ y X sym naturally restrict to ones on X anti . Then J and U π on ℓ ( X sym ) given in the symmetric case naturally restrict to ones on ℓ ( X anti ), so we usethem as our objects for X anti (hence we write J − := J ). We have to show that they satisfyconditions in Assumption 3.3. For any g ∈ Γ and [ x ] ∈ Z n with x = ( x , . . . , x n ) ∈ X n , δ [ x ,...,x n ] = C x ( δ x ⊗ · · · ⊗ δ x n ) U πg C x ( δ π g ( x ) ⊗ · · · ⊗ δ π g ( x n ) ) = C x C − π g ( x ) δ [ π g ( x ) ,...,π g ( x n )] ; δ [ x ,...,x n ] = C x ( δ x ⊗ · · · ⊗ δ x n ) J − C x ( δ Ix n ⊗ · · · ⊗ δ Ix ) = C x C − I ( x ) δ [ Ix n ,...,Ix ] . This means that for any g ∈ Γ and [ x ] ∈ X anti ⊂ X sym , U πg δ [ x ] = T δ π g ([ x ]) and J − δ [ x ] = T δ I ([ x ]) . For any [ x ] ∈ X sym \ X anti , we trivially have U πg δ [ x ] = δ π g ([ x ]) and J − δ [ x ] = δ I ([ x ]) . Thusputting J X sym := J − in this setting, they satisfy conditions in Assumption 3.3 for X sym which are supported on X anti ⊂ X sym in the sense that X anti = { z ∈ X sym | ℓ ( z ) = 0 or ℓ ( Iz ) = 0 } = { z ∈ X sym | ℓ ( z ) = 0 } . We will use this semigroup structure for X anti .19 emma 4.2. For any x ∈ X = Z and [ z ] ∈ X anti , there are C x, [ z ] , C rx, [ z ] ∈ { , , − } suchthat ℓ ( x ) δ [ z ] = C x, [ z ] δ [( x,z )] , r ( x ) δ [ z ] = C rx, [ z ] δ [( z,x )] . For any x, y ∈ X anti , there are C x,y , C rx,y ∈ { , , − } such that ℓ ( x ) δ y = C x,y δ xy , r ( x ) δ y = C rx,y δ yx . For x, y ∈ X anti , we have that C x,y = 0 (or C rx,y = 0 ) if and only xy X anti . Proof.
We have only to prove the first part of the statement. Take any representative z = ( z , . . . , z n ) of [ z ]. If there is i such that x = z i , then ℓ ( x ) δ [ z ] = 0, so we can put C x, [ z ] = 0. If there is no such i , then ( x, z ) defines an element in X anti , hence ℓ ( x ) δ [ z ] = ℓ ( x ) C z ( δ z ⊗ · · · ⊗ δ z n )= C z ( δ x ⊗ δ z ⊗ · · · ⊗ δ z n )= C z C − xz δ [( x,z )] . Then we can put C x, [ z ] := C z C − xz (which is well defined by C σ ( z ) = sgn( σ ) C z ). Family G for relativity We keep the action Γ y π X and associated semigroups X full , X sym and the subset X anti ⊂ X sym . We first define a family of subsets in X sym ⋊ Γ by G sym := ( [ finite s Γ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, t ∈ X sym ⋊ Γ ) = ( [ finite s Γ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s, t ∈ X sym ) . We then define G full := { Q − ( E ) ⊂ X full ⋊ Γ | E ∈ G sym } , where Q : X full ⋊ Γ → X sym ⋊ Γ is the canonical surjection Q ( x, g ) = ([ x ] , g ), and G anti := { E ∩ X anti | E ∈ G sym } . Observe that G sym and G full satisfy the relativity condition in Assumption 3.3.Recall the following lemma, see [HIK20, Lemma 3.6]. Lemma 4.3.
Assume that Γ is countable and that Γ y π X has finite stabilizers andfinitely many orbits. Then there is a function | · | X : Γ / Λ → R ≥ and a proper symmetriclength function | · | Γ / Λ : Γ / Λ → R ≥ , where Λ := T i Λ i ≤ Γ , such that (i) | g · x | X ≤ | g Λ | Γ / Λ + | x | X ; (ii) { x ∈ X | | x | X ≤ R } is finite for all R > . From now on we assume that the assumption in Lemma 4.3 is satisfied. Then using thefunctions | · | X and | · | Γ (:= | · | Γ / Λ ), we introduce the following functions on our semigroups.For any x = ( x , . . . , x n ) ∈ X n and g ∈ Γ, define | ( x, g ) | := n, | ( x, g ) | := n X i =1 min {| x i | X , | π − g ( x i ) | X } , and | ( ⋆, g ) | = | ( ⋆, g ) | = 0. Since they do not depend on the order of x , we can regardthem as functions on Y n × Γ by | ([ x ] , g ) | k := | ( x, g ) | k for k = 0 ,
1. We define functions onsemigroups by, for any ( x, g ) ∈ X full ⋊ Γ, | ( x, g ) | full := | ( x, g ) | + | ( x, g ) | , | ([ x ] , g ) | sym := | ([ x ] , g ) | + | ([ x ] , g ) | . Put | ([ x ] , g ) | anti := | ([ x ] , g ) | sym for ([ x ] , g ) ∈ X anti ⋊ Γ. Note that | ([ x ] , g ) | sym = | ( x, g ) | full for all ( x, g ) ∈ X full ⋊ Γ. 20 emma 4.4.
Let ∗ be full, sym, or anti . Then for any net ( z λ ) λ in X ∗ ⋊ Γ , we have z λ → ∞ / G ∗ ⇔ lim λ | z λ | ∗ = ∞ . Proof.
By the definition of G ∗ and | · | ∗ , we have only to prove this lemma for the casethat ∗ is sym.( ⇒ ) Fix R > x ] , g ) ∈ X sym ⋊ Γ satisfying | ([ x ] , g ) | sym ≤ R is small / G sym . Fix such ([ x ] , g ) and write x = ( x , . . . , x n ). Note that n = | ( x, g ) | ≤ R .Observe that min {| x i | X , | π − g ( x i ) | X ≤ | ( x, g ) | ≤ R, so for each i , | x i | X or | π − g ( x i ) | X is smaller than R . Since we can change the order of x in Y n , we may assume that | x i | X ≤ R ( i = 1 , . . . , k ) , | π − g ( x j ) | X ≤ R ( j = k + 1 , . . . , n ) . As an element in X full ⋊ Γ, we have( x, g ) = x · · · x n g = x · · · x k g π − g ( x k +1 ) · · · π − g ( x n ) ∈ B R ( X ) k Γ B R ( X ) n − k , where B R ( X ) = { a ∈ X | | a | ≤ R } , which is a finite set. This implies that any ([ x ] , g )with | ([ x ] , g ) | sym ≤ R is contained in [ n ≤ R n [ k =0 Q ( B R ( X ) k Γ B R ( X ) n − k ) , where Q is the canonical surjection onto X sym ⋊ Γ. This is small relative to G sym .( ⇐ ) Fix any A ⊂ X sym ⋊ Γ which is small / G sym . We have to prove sup a ∈ A | a | sym < ∞ .Since A is contained in [ finite s Γ t, s, t ∈ X sym , we may assume A = s Γ t for some s, t ∈ X sym . Write s = [ x, . . . , x n ] , t = [ y , . . . , y m ] ∈ X sym . Fix any g ∈ Γ and write sgt = ([ x , . . . , x n , π g ( y ) , . . . , π g ( y m )] , g ). We have | sgt | sym = n + m + n X i =1 min {| x i | X , | π − g ( x i ) | X } + m X j =1 min {| π g ( y j ) | X , | π − g ( π g ( y j )) | X }≤ n + m + n X i =1 | x i | X + m X j =1 | y j | X =: M. Since M does not depend on g ∈ Γ, we get sup a ∈ A | a | sym ≤ M . We use the framework given in Subsection 4.1, such as X full , I, G full , U πg , ℓ ( x ) , r ( x ) sat-isfying Assumption 3.3. Our goal is to prove the following theorem. Theorem 4.5.
Assume that Γ is exact and Γ y π X has finite stabilizers and finitely manyorbits. Then the left translation Γ y C ( ∂ G full ) is amenable. In particular Γ y π X full isbi-exact/ G full .
21e define a map ω : X full ⋊ Γ → ℓ ( X ) + by ω ( x, g ) := n X i =1 ( | ( x, g ) | + min {| x i | X , | π − g ( x i ) | X } ) δ x i for ( x, g ) ∈ X full ⋊ Γ , and ω ( ⋆, g ) is any nonzero element. Observe that k ω ( x, g ) k = n X i =1 | ( x, g ) | + min {| x i | X , | π − g ( x i ) | X } = | ( x, g ) | + | ( x, g ) | . Up to normalization, define µ : X full ⋊ Γ → Prob( X ); µ ( x, g ) := ω ( x, g ) k ω ( x, g ) k , ( x, g ) ∈ X full ⋊ Γ . We can induce a ucp map by µ ∗ : ℓ ∞ ( X ) → ℓ ∞ ( X full ⋊ Γ); f [ X full ⋊ Γ ∋ ( x, g )
7→ h f, µ ( x, g ) i ] . Lemma 4.6.
The following statements hold true.1. (Equivariance) For any g, h ∈ Γ and ϕ ∈ ℓ ∞ ( X ) , µ ∗ ( g · ϕ ) − µ ∗ ( ϕ )( g − · h ) ∈ c ( G full ) .
2. (Commutativity) For any x, y ∈ X full and ϕ ∈ ℓ ∞ ( X ) , µ ∗ ( ϕ ) − µ ∗ ( ϕ )( x · y ) ∈ c ( G full ) . In particular, µ ∗ ( ℓ ∞ ( X )) ⊂ C ( G full ) and we have a Γ -equivariant ucp map q ◦ µ ∗ : ℓ ∞ ( X ) → C ( ∂ G full ) , where q : C ( G full ) → C ( ∂ G full ) is the quotient map. Proof.
1. For any z ∈ X full ⋊ Γ, it is easy to compute that µ ∗ ( g · ϕ )( z ) − µ ∗ ( ϕ )( g − zh ) = h ϕ, [ g − · µ ( z ) − µ ( g − zh )] i . So we have only to prove that for any g, h ∈ Γ,lim X full ⋊ Γ ∋ z →∞ / G full k µ ( gzh ) − g · µ ( z ) k = 0 . Claim.
For any g, h ∈ Γ and z ∈ X full ⋊ Γ , k g · ω ( z ) − ω ( gz ) k ≤ | z | | g | Γ , k ω ( z ) − ω ( zh ) k ≤ | z | | h | Γ . Proof.
If we write z = ( a, t ), a = ( a , . . . , a n ), then it is easy to compute that g · ω ( z ) = ω ( a, t )( π − g · ) = n X i =1 ( n + min {| a i | X , | π − t ( a i ) |} ) δ π g ( a i ) ; ω ( gzh ) = ω (( π g ( a ) , gth )) = n X i =1 ( n + min {| π g ( a i ) | X , | π − th ( a i ) |} ) δ π g ( a i ) , k g · ω ( z ) − ω ( gzh ) k = n X i =1 (cid:12)(cid:12) min {| a i | X , | π − t ( a i ) |} − min {| π g ( a i ) | X , | π − th ( a i ) | (cid:12)(cid:12) . Using the inequality | min { a, c } − min { b, c }| ≤ | a − b | , for a, b, c ≥ , for the case h = e , we have k g · ω ( z ) − ω ( gz ) k ≤ n X i =1 || a i | X − | π g ( a i ) | X | ≤ n | g | Γ = | ( a, t ) | | g | Γ . For the case g = e , we have k ω ( z ) − ω ( zh ) k ≤ n X i =1 (cid:12)(cid:12) | π − t ( a i ) | − | π − th ( a i ) | (cid:12)(cid:12) ≤ n | h − | Γ = | ( a, t ) | | h | Γ . We compute that k g · µ ( z ) − µ ( gzh ) k ≤ (cid:13)(cid:13)(cid:13)(cid:13) g · ω ( z ) k ω ( z ) k − ω ( gzh ) k ω ( z ) k (cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) ω ( gzh ) k ω ( z ) k − ω ( gzh ) k ω ( gzh ) k (cid:13)(cid:13)(cid:13)(cid:13) ≤ k ω ( z ) k k g · ω ( z ) − ω ( gzh ) k + (cid:12)(cid:12)(cid:12)(cid:12) k ω ( gzh ) k k ω ( z ) k − (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ω ( z ) k k g · ω ( z ) − ω ( gzh ) k . By the claim, we get k g · µ ( z ) − µ ( gzh ) k ≤ | g | Γ + | h | Γ ) | z | | z | + | z | . Now we consider z → ∞ / G full , which is equivalent to | z | + | z | → ∞ by Lemma 4.4. It isstraightforward to prove that the right hand side of this inequality converges to 0. Hencewe finish the proof of item 1.2. As in the proof of item 1, we have only to show thatlim X full ⋊ Γ ∋ z →∞ / G full k µ ( xzy ) − µ ( z ) k = 0 . To see this, we may assume x, y ∈ X . Claim.
For any x, y ∈ X , k ω ( xz ) − ω ( z ) k ≤ | z | + 1 + | x | X , k ω ( zy ) − ω ( z ) k ≤ | z | + 1 + | y | X . Proof.
Write z = ( a, t ) and a = ( a , . . . , a n ). Using ω ( xz ) = ω (( xa, t )) = ( n + 1 + min {| x | X , | π − t ( x ) | X } ) δ x + n X i =1 ( n + 1 + min {| a i | X , | π − t ( a i ) | X } ) δ a i , it is easy to see that k ω ( xz ) − ω ( z ) k = | n + 1 + min {| x | X , | π − t ( x ) | X }| + | n | ≤ | z | + 1 + | x | X . The same computation works for y (or apply I ).23y the claim, a computation similar to item 1 shows k µ ( z ) − µ ( xzy ) k ≤ | z | + | z | (4 | z | + 4 + | x | X + | y | X ) . It is straightforward to see that the right hand side converges to 0 as z → ∞ / G full .The last statement is trivial by the definition of C ( G full ). Proof of Theorem 4.5.
We have a Γ-equivariant ucp map q ◦ µ ∗ : ℓ ∞ ( X ) → C ( ∂ G full ). SinceΓ is exact and since Γ y π X has finite stabilizers and finitely many orbits, Γ y ℓ ∞ ( X )is amenable. Then by the ucp map, we get that Γ y C ( ∂ G full ) is amenable (e.g. [BO08,Exercise 15.2.2]). As in the case of full Fock spaces, we use the framework given in Subsection 4.1. Recallthat we use algebras C := C ∗ { e i W ( x ) | x ∈ X } , W ( x ) := ℓ ( x ) + ℓ ( x ) ∗ and C r := J sym C J sym = C . Observe that Ad( U πg ) for g ∈ Γ defines a Γ-action e i W ( x ) e i W ( π g ( x )) . We define C ∗ -algebras by ℓ ∞ ( G sym ) := { f ∈ ℓ ∞ ( X sym ⋊ Γ) | [ π ℓ ( a ) , f ] , [ π r ( a ) , f ] ∈ K ( G sym ) for all a ∈ C} ; C ( G sym ) := { f ∈ ℓ ∞ ( G sym ) | f − f ( · h ) ∈ c ( G sym ) for all h ∈ Γ } . We will show that they contain c ( G sym ), hence we can define boundary C ∗ -algebrasequipped with actionsAd( U πg ⊗ λ g ρ h ) : Γ × Γ y ℓ ∞ ( ∂ G sym ) := ℓ ∞ ( G sym ) /c ( G sym );Ad( U πg ⊗ λ g ) : Γ y C ( ∂ G sym ) := C ( G sym ) /c ( G sym ) . Our goal is to prove the following theorem.
Theorem 4.7.
Assume that Γ is exact and Γ y π X has finite stabilizers and finitely manyorbits. Then the action Γ y C ( ∂ G sym ) is amenable. In particular, Γ × Γ y ℓ ∞ ( ∂ G sym ) isamenable. Let ω : X full ⋊ Γ → ℓ ( X ) + be given in Subsection 4.2. Observe that it induces maps ω sym : X sym ⋊ Γ → ℓ ( X ) + ; ω sym ([ x ] , g ) := ω ( x, g ); µ sym : X sym ⋊ Γ → Prob( X ); µ sym ([ x ] , g ) := ω sym ([ x ] , g ) k ω sym ([ x ] , g ) k = µ ( x, g ) . Define µ ∗ sym : ℓ ∞ ( X ) → ℓ ∞ ( X sym ⋊ Γ); f [ X sym ⋊ Γ ∋ ([ x ] , g )
7→ h f, µ sym ([ x ] , g ) i ] . Lemma 4.8.
The following statements hold true.1. (Equivariance) For any g, h ∈ Γ and ϕ ∈ ℓ ∞ ( X ) , µ ∗ sym ( g · ϕ ) − µ ∗ sym ( ϕ )( g − · h ) ∈ c ( G sym ) . . (Commutativity) For any x, y ∈ X and ϕ ∈ ℓ ∞ ( X ) , π ℓ ( C x, · )( µ ∗ sym ( ϕ ) − µ ∗ sym ( ϕ )( x · )) ∈ c ( G sym ); π r ( C ry, · )( µ ∗ sym ( ϕ ) − µ ∗ sym ( ϕ )( · y )) ∈ c ( G sym ) . Proof.
1. As in the proof of Lemma 4.6, we have only to provelim X sym ⋊ Γ ∋ z →∞ / G sym k µ sym ( gzh ) − g · µ sym ( z ) k = 0 . Let Q : X full ⋊ Γ → X sym ⋊ Γ be the canonical surjection. For any z ∈ X full ⋊ Γ, we have µ sym ( gQ ( z ) h ) = µ ( gzh ) , g · µ sym ( Q ( z )) = g · µ ( z ) . Since z → ∞ / G full and Q ( z ) → ∞ / G sym are equivalent by Lemma 4.4, we getlim Q ( z ) →∞ / G sym k µ sym ( gQ ( z ) h ) − g · µ sym ( Q ( z )) k = lim z →∞ / G full k µ ( gzh ) − g · µ ( z ) k = 0 , where the last equality is by Lemma 4.6.2. As in the proof of Lemma 4.6 and by using the inequality in Lemma 4.1, we have | π ℓ ( C x, · )( z )( µ ∗ sym ( ϕ )( z ) − µ ∗ sym ( ϕ )( xz )) | ≤ k ϕ k ∞ p | z | + 1 k µ sym ( z ) − µ sym ( xz ) k . As in the proof of item 1, the right hand side coincides with the term by µ , hence by theproof of item 2 in Lemma 4.6, we have (up to exchanging z with Q ( z )), p | Q ( z ) | + 1 k µ sym ( Q ( z )) − µ sym ( Q ( xz )) k = p | z | + 1 k µ ( z ) − µ ( xz ) k ≤ p | z | + 1 | z | + | z | (4 | z | + 4 + | x | X ) . It is straightforward to show that the last term converges to 0 as z → ∞ / G full , which isequivalent to Q ( z ) → ∞ / G sym . The same argument works for y .We next transfer the commutativity condition in the previous lemma to e i W ( x ) . Lemma 4.9.
The following statements hold true.1. For any x ∈ X and t ∈ R , π ℓ ( e i tW ( x ) ) , π r ( e i tW ( x ) ) ∈ M( K ( G sym )) .
2. For any x ∈ X and f ∈ c ( G sym ) , the maps R ∋ t π ℓ ( e i tW ( x ) ) f, π r ( e i tW ( x ) ) f ∈ K ( G sym ) are norm continuous.3. For any x ∈ X and b ∈ ℓ ∞ ( X sym ⋊ Γ) , if [ π ℓ ( W ( x )) , b ] ∈ K ( G sym ) , then π ℓ ( e i W ( x ) ) bπ ℓ ( e − i W ( x ) ) − b ∈ K ( G sym ) . In particular we can put b = µ ∗ sym ( ϕ ) for ϕ ∈ ℓ ∞ ( X ) . Proof.
1. Up to Ad( J ), we have only to see π ℓ . We will prove π ℓ ( e i tW ( x ) )1 A ∈ K ( G sym )for any A ⊂ X sym ⋊ Γ which is small / G sym and any 0 < | t | < /
2. Fix such A and t .25 laim. There is B ⊂ X sym ⋊ Γ which is small / G sym such that ( W ( x ) n ⊗ A = 1 B ( W ( x ) n ⊗ A . In particular they are contained in K ( G sym ) . Proof.
We may assume n = 1. For simplicity, assume A = s Γ t for some s, t ∈ X sym . Therange of ( ℓ ( x ) ⊗ A has a basis δ xsgt , while a basis from ( ℓ ( x ) ∗ ⊗ A is by subwordsfrom sgt (for which the letter x is removed). By counting these words and since Γ y π X has finite stabilizers, it is easy to find the desired B . Claim.
The sum P n ≥ n ! ((i tW ( x )) n ⊗ A has absolute convergence in norm. Proof.
For m ∈ N , let P ≤ m be the orthogonal projection onto the subspace spanned byvectors having tensor length less than m in F sym . Observe that k W ( x ) P ≤ m k ∞ ≤ √ m + 1for all m ∈ N . Since A is small / G sym , there is a large m ∈ N such that 1 A ≤ P ≤ m ⊗ Γ . Itholds that k ( W ( x ) n ⊗ A k ∞ ≤ k W ( x ) n P ≤ m k ∞ = k W ( x ) n − P ≤ m +1 W ( x ) P ≤ m k ∞ ≤ k W ( x ) P ≤ m + n − k ∞ · · · k W ( x ) P ≤ m +1 k ∞ k W ( x ) P ≤ m k ∞ ≤ (2 √ m + n ) · · · (2 √ m + 2)(2 √ m + 1) , so that X n ≥ t n n ! k ( W ( x )) n ⊗ A k ∞ ≤ X n ≥ (2 t ) n ( √ m + 1)( √ m + 2) · · · ( √ m + n ) n ! . Since 2 t <
1, it is easy to get the conclusion.Using these two claims, we get e i tW ( x ) A = P n ≥ n ! ((i tW ( x )) n ⊗ A ∈ K ( G sym ). Thisshows e i tW ( x ) ∈ M( K ( G sym )) for all | t | < /
2, hence for all t ∈ R .2. Up to Ad( J ), we have only to see π ℓ . Put F ( t ) := π ℓ ( e i tW ( x ) ) − t → k F ( t ) f k ∞ → f ∈ c ( G sym ). By approximating f , we may assumethat f = 1 A , A = s Γ t for some s, t ∈ X sym . Write s = [ a , . . . , a n ] and t = [ b , . . . , b m ].Since s Γ t = { sgt = ( sπ g ( t ) , g ) | g ∈ Γ } , we have 1 A = P g ∈ Γ { sπ g ( t ) } ⊗ { g } . It holds that k F ( t )1 A k ∞ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X g ∈ Γ [( e i tW ( x ) − { sπ g ( t ) } ] ⊗ e g,g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ = sup g ∈ Γ k ( e i tW ( x ) − { sπ g ( t ) } k ∞ = sup g ∈ Γ k ( e i tW ( x ) − δ sπ g ( t ) k F sym ≤ sup g ∈ Γ k ( e i tW ( x ) − ℓ ( π g ( t )) δ s k F sym = sup g ∈ Γ k ℓ ( π g ( t ))( e i tW ( x ) − δ s k F sym . We have to show that the last term converges to 0 as t →
0. To see this, observe that,since Γ y π X has finite stabilizers, there is a finite set E ⊂ Γ such that π g ( b i )
6∈ { x, a , . . . , a n } , for all i = 1 , . . . , m and g E. g E , since we know the exact value of C π g ( b ) , · by (the proof of) Lemma4.1 and since ( e i tW ( x ) − δ a is contained in the space spanned by vectors indexed by { x, a , . . . , a n } , we have k ℓ ( π g ( b ))( e i tW ( x ) − δ a k F sym ≤ √ · · · √ m k ( e i tW ( x ) − δ a k F sym for all g E. The last term goes to 0 as t →
0. Since E is finite, we get the conclusion.3. Up to Ad( J ), we have only to see π ℓ . Put a := W ( x ) ⊗ π ℓ ( e i W ( x ) ) = e i a .Consider the map F : R → K ( G sym ); F ( t ) := e i ta [ b, a ] e − i ta . Since e i ta ∈ M( K ( G sym )) by item 1 and [ b, a ] ∈ K ( G sym ) by assumption, F ( t ) is indeedcontained in K ( G sym ). By item 2, it is norm continuous.Let F alg ⊂ F sym be the image of the algebraic Fock space and fix ξ, η ∈ F alg ⊗ alg ℓ (Γ).Observe that by Stone’s theorem, the function f : R ∋ t
7→ h e i ta ξ, η i ∈ C is differentiableand f ′ ( t ) = h i ae i ta ξ, η i . Hence the function g ( t ) := h e i ta be − i ta ξ, η i = h be − i ta ξ, e − i ta η i is also differentiable and g ′ ( t ) = i h F ( t ) ξ, η i . For any s >
0, we have Z s i h F ( t ) ξ, η i dt = Z s g ′ ( t ) dt = g ( s ) − g (0) = h ( e i sa be − i sa − b ) ξ, η i . Since F ( t ) is norm continuous, the integral R s F ( t ) dt ∈ K ( G sym ) is defined, so that e i sa be − i sa − b = i Z s F ( t ) dt ∈ K ( G sym )for any s >
0. This is the conclusion. The last statement follows by Lemma 4.8 and3.6.
Proof of Theorem 4.7.
Thanks to the previous two lemmas, we have c ( G sym ) ⊂ ℓ ∞ ( G sym )and µ ∗ sym ( ℓ ∞ ( X )) ⊂ C ( G sym ). In particular, there is a Γ-equivariant ucp map q ◦ µ ∗ sym : ℓ ∞ ( X ) → C ( ∂ G sym ) , where q : C ( G sym ) → C ( ∂ G sym ) is the quotient map. We then follow the proof of Theorem4.5 and Lemma 3.10. We keep the framework given in Subsection 4.1. We prove the following Theorem.Here we are using the framework explained in Subsection 3.3 for the subset X anti ⊂ X sym . Theorem 4.10.
Assume that Γ is exact and Γ y π X has finite stabilizers and finitelymany orbits. Then the left translation Γ y C ( ∂ G anti ) is amenable. In particular Γ y π X anti is bi-exact/ G anti . Since X anti ⋊ Γ = X anti × Γ ⊂ X sym ⋊ Γ, by restriction, we can define µ anti = µ sym | X anti ⋊ Γ : X anti ⋊ Γ → Prob( X ); µ ∗ anti : ℓ ∞ ( X ) → ℓ ∞ ( X anti ⋊ Γ) . To prove Theorem 4.10, as in the proof of Theorem 4.5, we have only to show the followinglemma. 27 emma 4.11.
The following conditions hold true.1. (Equivariance) For any g, h ∈ Γ and ϕ ∈ ℓ ∞ ( X ) , µ ∗ anti ( g · ϕ ) − µ ∗ anti ( ϕ )( g − · h ) ∈ c ( G anti ) .
2. (Commutativity) For any x, y ∈ X and ϕ ∈ ℓ ∞ ( X ) , π ℓ ( C x, · )( µ ∗ anti ( ϕ ) − µ ∗ anti ( ϕ )( x · )) ∈ c ( G anti ); π r ( C ry, · )( µ ∗ anti ( ϕ ) − µ ∗ anti ( ϕ )( · y )) ∈ c ( G anti ) . In particular, µ ∗ anti ( ℓ ∞ ( X )) ⊂ C ( G anti ) and we have a Γ -equivariant ucp map q ◦ µ ∗ : ℓ ∞ ( X ) → C ( ∂ G anti ) , where q : C ( G anti ) → C ( ∂ G anti ) is the quotient map. Proof.
1. This is a special case of item 1 in Lemma 4.8.2. We see only the case for x ∈ X . As in the proof of Lemma 4.6, we have for z = ( a, g ) ∈ X anti ⋊ Γ, | π ℓ ( C x, · )( z )( µ ∗ anti ( ϕ )( z ) − µ ∗ anti ( ϕ )( xz )) | ≤ k ϕ k ∞ | C x,a |k µ anti ( z ) − µ anti ( xz ) k . Hence we have only to show lim ( X anti ⋊ Γ) x ∋ z →∞ / G anti k µ anti ( xz ) − µ anti ( z ) k = 0, where( X anti ⋊ Γ) x is the set of all z = ( a, g ) ∈ X anti ⋊ Γ such that C x,a = 0 (which means xz ∈ X anti ⋊ Γ). As in the proof of Lemma 4.8, this convergence follows by (the proof of)Lemma 4.6.
Remark 4.12.
We have proved the bi-exactness of Γ y π X anti . We are interested in cre-ation operators on F anti and we will use the bi-exactness via the unitary F anti ≃ ℓ ( X anti ),which is given by a fixed section in Subsection 4.1. In this respect, it is worth mentioningthat the associated C ∗ -algebras on F anti ⊗ ℓ (Γ) do not depend on the choice of the section.Indeed, it is easy to see that the inclusion ℓ ∞ ( X anti ) ⊂ B ( F anti ) does not depend on thesection (actually this coincides with the one given in [HIK20, Proposition 3.1]). Hencepositions of c ( G anti ) ⊂ ℓ ∞ ( X anti ⋊ Γ) , K ( G anti )in B ( F anti ⊗ ℓ (Γ)) are unique. Also U πg for g ∈ Γ, ℓ ( x ) , r ( x ) for x ∈ X , and C ℓ , C r are defined originally on F anti , hence they do not depend on the section. Finally theoperators ℓ ([ x ]) , r ([ x ]) for x ∈ X anti depend on the section, but the dependence is onlyits sign, hence the algebra C ( G anti ) does not depend on the section. Thus we can regard c ( G anti ) ⊂ C ( G anti ) as subalgebras in B ( F anti ⊗ ℓ (Γ)), and we have the amenability of theaction on the quotient algebra C ( ∂ G anti ).We finally note that the boundary constructed in [HIK20, Subsection 3.2] is differentfor ours. In [HIK20], we considered a quotient by K , which contains our compact K ( G anti ).In this sense, our result is better and the associated condition (AO) is a stronger resultsince it uses the quotient algebra by a smaller compact operators. We note that theboundary in [HIK20] is inspired by the construction in [Oz04], while ours is by the one in[Oz08]. 28 Other examples
To study free wreath product groups, we first see infinite free product groups. Theproof is a straightforward adaptation of known techniques.
Proposition 5.1.
Let I be a countable set and Γ i for i ∈ I countable discrete groups. Wedenote by Γ = ∗ i ∈ I Γ i the free product group. Then Γ is bi-exact if and only if so are all Γ i ’s. Proof.
The only if part is trivial, so we assume Γ i is bi-exact for all i ∈ I . For each i ∈ I ,put C (Γ i ) = { f ∈ ℓ ∞ (Γ i ) | f ( · h ) − f ∈ c (Γ i ) } and then Γ i y C (Γ i ) is amenable. In particular A i := C ∗ { C (Γ i ) , C ∗ λ (Γ i ) } ≃ C (Γ i ) ⋊ Γ isnuclear and contains all compact operators. By the proof of [Oz04, Lemma 2.4], ∗ i ∈ F A i is nuclear for all finite subsets F ⊂ I , hence A := ∗ i ∈ I A i is nuclear. Then define thefollowing ∗ -homomorphism ν : A ⊗ min J A J → B ( ℓ (Γ)) / K ( ℓ (Γ)); x ⊗ J yJ xJ yJ + K ( ℓ (Γ)) . (It is well defined, since [ A , J A J ] ⊂ K ( ℓ (Γ)).) Since A⊗ min J A J is nuclear, the restrictionof ν on C ∗ λ (Γ) ⊗ min C ρ (Γ) is nuclear. By the lifting theorem, ν has a ucp lift, hence Γ isbi-exact by [BO08, Lemma 15.1.4]. Let Γ , ∆ be countable discrete groups and Γ y I an action on a countable set. Definewreath product ∆ ≀ I Γ := ∆ I ⋊ Γ , where ∆ I := M I ∆;free wreath product ∆ ≀ free I Γ := ∆ free I ⋊ Γ , where ∆ free I := ∗ I ∆ . Here both Γ-actions are given by coordinate shifts along Γ y I . In this subsection, westudy some boundary amenability for Γ y ∆ free I given in Theorem 5.4. We will thenobtain the following characterization of bi-exactness for free wreath product groups. Theproof uses also Proposition 6.1 which will be proved in Subsection 6.2. Theorem 5.2.
Assume that Γ y I has finite stabilizers and finitely many orbits. Then ∆ ≀ free I Γ is bi-exact if and only if so are ∆ and Γ . Proof.
The only if part is trivial. By Theorem 5.4 and Proposition 6.1, if Γ , ∆ are bi-exact, L (∆ ≀ free I Γ) with its group C ∗ -algebra satisfies condition (AO) with a ucp lift. By[BO08, Lemma 15.1.4], ∆ ≀ free I Γ is bi-exact.This theorem should be compared to the following Ozawa’s theorem.
Theorem 5.3 ([Oz04, BO08]) . Assume that Γ y I has finite stabilizers and finitely manyorbits. Then ∆ ≀ I Γ is bi-exact if and only if ∆ is amenable and Γ is bi-exact. We start the proof of Theorem 5.4 below. We denote by ∆ i the i -th copy of ∆ in ∆ free I .Any non trivial y ∈ ∆ free I has a unique word decomposition y = y i · · · y i n , y i k ∈ ∆ i k , i = i , . . . , i n − = i n . y ) := { i , . . . , i n } ⊂ I . For any finite subsets E ⊂ ∆, F ⊂ I and n ∈ N , wedefine A ( E, F, n ) ⊂ ∆ free I ⋊ Γ in the following way. For any y ∈ ∆ free I and g ∈ Γ, ( y, g ) isin A ( E, F, n ) if y = e or the word decomposition y = y i · · · y i m satisfies(a) m ≤ n (b) y i k ∈ E for all i k (c) supp( y ) ⊂ F ∪ gF. It is then straightforward to check the following conditions: for any finite subsets
E, E ′ ⊂ ∆, F, F ′ ⊂ I , n, m ∈ N , x ∈ ∆ j , and g ∈ Γ, • A ( E, F, n ) ∪ A ( E ′ , F ′ , m ) ⊂ A ( E ∪ E ′ , F ∪ F ′ , max { n, m } ); • A ( E, F, n ) − = A ( E − , F, n ); • xA ( E, F, n ) ⊂ A ( E ∪ { x } ∪ xE, F ∪ { j } , n + 1), A ( E, F, n ) x ⊂ A ( E ∪ { x } ∪ Ex, F ∪ { j } , n + 1); • gA ( E, F, n ) ⊂ A ( E, F ∪ g − F, n ) , A ( E, F, n ) g ⊂ A ( E, F ∪ gF, n ).Hence if we define G free := { A ⊂ ∆ free I ⋊ Γ | A ⊂ A ( E, F, n ) for some finite subsets
E, F and n ∈ N } , then it is globally preserved by finite unions, group inverse, and left-right translations.We can consider the relativity condition in ℓ ∞ (∆ free I ⋊ Γ) given by G free . Note that G free isnot generated by subgroups in ∆ free I ⋊ Γ, hence it is not covered by Ozawa’s argument in[BO08]. Define C ( G free ) := { f ∈ ℓ ∞ (∆ free I ⋊ Γ) | f ( x · h ) − f ∈ c ( G free ) for all x ∈ ∆ free I , h ∈ ∆ free I ⋊ Γ } and C ( ∂ G free ) := C ( G free ) /c ( G free ). It admits the Γ-action arising from the left translation.Our goal in this subsection is to show the following theorem. Theorem 5.4.
Assume that Γ and exact and Γ y I has finite stabilizers and finitelymany orbits. Then the Γ -action Γ y C ( ∂ G free ) is amenable. In particular Γ y ∆ free I isbi-exact / G free . From now on, assume that Γ y I has finite stabilizers and finitely many orbits. ByLemma 4.3, take proper length functions | · | Γ (= | · | Γ / Λ ) in the statement) and | · | I .Also take | · | ∆ for ∆. Then we define a length in ∆ free I ⋊ Γ by, for any y ∈ ∆ free I withdecomposition y = y i · · · y i n and g ∈ Γ, | ( y, g ) | free := X i ∈ supp( y ) min {| i | I , | g − · i | I } + n X k =1 | y i k | ∆ . Put | ( e, g ) | free = 0. The next lemma is useful. Lemma 5.5.
The following conditions hold true.1. For any net ( z λ ) λ in ∆ free I ⋊ Γ , z λ → ∞ / G free ⇔ | z λ | free → ∞ . lim ( y,g ) →∞ / G free | supp( y ) || ( y,g ) | free = 0 . roof.
1. ( ⇒ ) Take any N ∈ N and we show that { z ∈ ∆ free I ⋊ Γ | | z | free ≤ N } issmall / G free . Take any such z = ( y, g ) and write y = y i · · · y i n . Then since n X k =1 | y i k | ∆ ≤ | ( y, g ) | free ≤ N, and since 1 ≤ | y i k | ∆ for all i k by construction, it follows that (a) n ≤ N and (b) y i k ∈ B N (∆) for all i k , where B N (∆) is the N -ball in ∆ with respect to | · | ∆ . For any i ∈ supp( y ), since min {| i | I , | g − · i | I } ≤ | z | free ≤ N, it follows that (c) i ∈ B N ( I ) ∪ gB N ( I ), where B N ( I ) is the N -ball in I . Thus z iscontained in A ( E, F, N ) for E = B N (∆) and F = B N ( I ), both of which are finite sets byproperness.( ⇐ ) Fix any A ( E, F, n ) and we show sup z ∈ A ( E,F,n ) | z | free < ∞ . If yg = y i · · · y i m g ∈ A ( E, F, n ), then | ( y, g ) | free = X i ∈ supp( y ) min {| i | I , | g − · i | I } + m X k =1 | y i k | ∆ ≤ X i ∈ supp( y ) sup j ∈ F | j | I + m X k =1 sup x ∈ E | x | ∆ ≤ | F | sup j ∈ F | j | I + n sup x ∈ E | x | ∆ < ∞ .
2. We prove a claim.
Claim.
For any ( y, g ) ∈ ∆ free I ⋊ Γ and C > , | ( y, g ) | free ≤ C | supp( y ) | ⇒ | ( y, g ) | free ≤ C | B C ( I ) | . Proof.
We can assume | ( y, g ) | free = 0. We have only to show | supp( y ) | ≤ | B C ( I ) | , sosuppose | supp( y ) | > | B C ( I ) | . Put B g := B C ( I ) ∪ gB C ( I ) and observe that | B g | ≤ | B C ( I ) | < | supp( y ) | . Then since supp( y ) \ B g = ∅ and since min {| i | I , | g − · i | I } > C for all i B g , it followsthat C | supp( y ) | ≥ | ( y, g ) | free ≥ X i ∈ supp( y ) \ B g min {| i | I , | g − · i | I } > C | supp( y ) \ B g | . We get | supp( y ) | > | supp( y ) \ B g | . This implies | supp( y ) | = | supp( y ) \ B g | + | B g ∩ supp( y ) | < | supp( y ) | + 12 | supp( y ) | = | supp( y ) | . This is a contradiction.Suppose by contradiction that there is a net z λ = ( y λ , g λ ) such that z λ → ∞ / G free and | supp( y λ ) || z λ | −
0. Up to a subnet, we can assume there is δ > | supp( y λ ) || z λ | free ≥ δ for all λ. Then the claim for the case C := δ − shows that | z λ | free ≤ C | B C ( I ) | for all λ . This is acontradiction, since | z λ | free → ∞ by item 1.31e define a map ω : ∆ full I ⋊ Γ → ℓ ( I ) + ; ω ( y, g ) = m ( y, g ) + a ( y ) , where for y = y i · · · y i n ∈ ∆ free I and g ∈ Γ, m ( y, g ) = X i ∈ supp( y ) min {| i | I , | g − · i | I } δ i and a ( y ) = n X k =1 | y ( i k ) | ∆ δ i k , and m ( e, g ) = 0 = a ( e ). We note that • k ω ( y, g ) k = k m ( y, g ) k + k a ( y ) k = | ( y, g ) | free ; • a ( y ) = P nk =1 a ( y i k ) and k a ( y ) k = P nk =1 k a ( y i k ) k for y = y i · · · y i n ; • ω ( y − , g ) = ω ( y, g ).Up to normalization, we define µ : ∆ full I ⋊ Γ → Prob( I ); µ ( z ) = ω ( z ) k ω ( z ) k . Lemma 5.6.
The following conditions hold true.1. For any z = ( y, h ) ∈ ∆ free I ⋊ Γ and g ∈ Γ , k g · ω ( z ) − ω ( gz ) k ≤ | g | Γ | supp( y ) | , k ω ( z ) − ω ( zg ) k ≤ | g | Γ | supp( y ) | .
2. For any z ∈ ∆ I ⋊ Γ and x ∈ ∆ j for j ∈ I , k ω ( xz ) − ω ( z ) k ≤ k ω ( x ) k , k ω ( zx ) − ω ( z ) k ≤ k ω ( x ) k . Proof.
For g ∈ Γ and y ∈ ∆ free I , we denote by π g ( y ) the associated action. If y = y i · · · y i n ,then π g ( y ) = π g ( y i ) · · · π g ( y i n ) and each π g ( y i k ) is y i k as an element in ∆ and is containedin ∆ g · i k .1. By definition, we have ω ( gz ) = X i ∈ supp( y ) min {| g · i | I , | ( gh ) − · g · i | I } δ g · i + n X k =1 | y i k | ∆ δ g · i k ; g · ω ( z ) = X i ∈ supp( y ) min {| i | I , | h − · i | I } δ g · i + n X k =1 | y i k | ∆ δ g · i k . It follows that k g · ω ( z ) − ω ( gz ) k = X i ∈ supp( y ) | min {| i | I , | h − · i | I } − min {| g · i | I , | h − · i | I }|≤ X i ∈ supp( y ) || i | I − | g · i | I | ≤ | supp( y ) || g | Γ . The second inequality follows similarly.2. We see a claim.
Claim.
For z = ( y, g ) ∈ ∆ free I ⋊ Γ and x ∈ ∆ j , it holds that k m ( xy, g ) − m ( y, g ) k ≤ k m ( x, g ) k , k a ( xy ) − a ( y ) k ≤ k a ( x ) k . roof. Write y = y i · · · y i n and xz = xy i · · · y i n g . Observe that • j = i ⇒ a ( xy ) = a ( x ) + a ( y ); • j = i ⇒ a ( xy ) = a ( xy i ) + a ( y i · · · y i n ).If j = i , then k a ( xy ) − a ( y ) k = k a ( x ) k . If j = i , then k a ( xy ) − a ( y ) k = k a ( xy i ) + a ( y i · · · y i n ) − a ( y ) k = k a ( xy i ) − a ( y i ) k = || xy i | ∆ − | y i | ∆ || ≤ | x | ∆ = k a ( x ) k . Observe next that • j supp( y ) ⇒ m ( xy, g ) = m ( x, g ) + m ( y, g ); • j ∈ supp( y ) ⇒ m ( xy, g ) = m ( y, g ).If j supp( y ), then k m ( xy, g ) − m ( y, g ) k = k m ( x, g ) k . If j ∈ supp( y ), then k m ( xy, g ) − m ( y, g ) k = 0 ≤ k m ( x, g ) k .By the claim, we get k ω ( xy, g ) − ω ( y, g ) k ≤ k m ( xy, g ) − m ( y, g ) k + k a ( xy ) − a ( y ) k ≤ k m ( x, g ) k + k a ( x ) k = k ω ( x, g ) k . Since k ω ( x, g ) k ≤ k ω ( x ) k , we get the first inequality.We see the second one. It is straightforward to see that k ω ( π g ( x ) , g ) k ≤ k ω ( x, e ) k for any g ∈ Γ and x ∈ ∆ free I . Then since ω ( y − , g ) = ω ( y, g ), k ω ( y ( g · x ) , g ) − ω ( y, g ) k = k ω (( g · x − ) y − , g ) − ω ( y − , g ) k ≤ k ω (( g · x − ) , g ) k ≤ k ω ( x − , e ) k = k ω ( x, e ) k . This is the conclusion.Now following the same arguments as in the Fock spaces, we get the following lemma.Theorem 5.4 obviously follows by this lemma.
Lemma 5.7.
The following statements hold true.1. (Equivariance) For any g, h ∈ Γ and ϕ ∈ ℓ ∞ ( I ) , µ ∗ ( g · ϕ ) − µ ∗ ( ϕ )( g − · h ) ∈ c ( G free ) .
2. (Commutativity) For any x, y ∈ ∆ free I and ϕ ∈ ℓ ∞ ( I ) , µ ∗ ( ϕ )( x · y ) − µ ∗ ( ϕ ) ∈ c ( G free ) . In particular, µ ∗ ( ℓ ∞ ( I )) ⊂ C ( G free ) and we have a Γ -equivariant ucp map q ◦ µ ∗ : ℓ ∞ ( I ) → C ( ∂ G free ) , where q : C ( G free ) → C ( ∂ G free ) is the quotient map. Application to rigidity of von Neumann algebras
We introduce concrete examples of crossed product B ⋊ red Γ ⊂ M ⋊ Γ, for which ourrigidity results are applied. We always assume that Γ y π X is an action of a countableexact group Γ on a set X , which has finite stabilizers and finitely many orbits. Consider ℓ ( X ) and associated Fock spaces F ∗ , where ∗ is full, sym, or anti. We have U πg and J ∗ acting on F ∗ . Gaussian algebras acting on F sym Assume I = id X in this case. As in Subsection 4.3, we put B := C ∗ { e i W ( x ) | x ∈ X } and M := B ′′ . We are interested in the inclusion B ⋊ red Γ ⊂ M ⋊ Γ. Note that Γ y M is the generalized Bernoulli action with diffuse base, arising from Γ y X . Observe that( F sym ⊗ ℓ (Γ) , J sym ) is the standard representation with the modular conjugation of M ⋊ Γ,and U πg is the standard implementation of the action Γ y M . (Free) Araki–Woods algebras acting on F full and F anti Let R → O ( H R ) be any strongly continuous representation on a real Hilbert space H R . Put H := H ⊗ R C and let I R be the involution for H R ⊂ H . Using the infinitesimalgenerator A , as in Subsection 2.4, consider Γ q ( H R , U ) for q = 0 , − j ( ξ ) = √ √ A − ξ, W ( j ( ξ )) := ℓ ( j ( ξ )) + ℓ ( j ( Iξ )) ∗ , ξ ∈ H ; B := C ∗ { W ( j ( ξ )) | ξ ∈ H } ⊂ W ∗ { W ( j ( ξ )) | ξ ∈ H } = Γ q ( H R , U ) =: M. It holds that L ( M ) = F ∗ , where ∗ is full or anti. Assume that there is an identification ℓ ( X ) = H such that • the involutions I from ℓ ( X ) and I R from H R ⊂ H coincide; • π g and U t on ℓ ( X ) = H commute for all g ∈ Γ and t ∈ R .In this assumption, Γ y π H induces a vacuum state preserving action Γ y M . Then itis straightforward to check that U πg and J ∗ arising from the structure of ℓ ( X ) coincidewith the standard implementation and the modular conjugation arising from that of H and ( H R , U t ).We give concrete examples satisfying the above assumptions. Let Γ y π I be anyaction of a countable group on a set I which has finite stabilizers and finitely many orbits.Fix λ ≥ H ( λ ) := ℓ ( I ) =: H ( λ − ) and H := H ( λ ) ⊕ H ( λ − ) = ℓ ( I ⊔ I ),where I = I = I . Put X := I ⊔ I and let i k : I → X be the k -th embedding for k = 1 ,
2. Extend π on X diagonally and define a bijection I on X by the flip, that is, Ii ( x ) = i ( x ) and Ii ( x ) = i ( x ) for all x ∈ I . We consider U : R y H by U t ξ = λ i t ξ for ξ ∈ H ( λ ) and U t ξ = λ − i t ξ for ξ ∈ H ( λ − ). Observe that we can regard U as an almostperiodic representation arising from H R ⊂ H . In this case, the involution I R for H R ⊂ H coincides with I for ℓ ( X ) and π g commutes with U t for all g ∈ Γ and t ∈ R . Hence theysatisfy the above assumptions. More generally we can take a finite direct sum of such H .34 ree wreath products Let ∆ be a countable bi-exact group. Putting I := X , consider the free wreath product∆ ≀ free I Γ = ∆ free I ⋊ Γ. We put B := C ∗ λ (∆ free I ) ⊂ L (∆ free I ) =: M and B ⋊ red Γ = C ∗ λ (∆ free I ⋊ Γ) ⊂ L (∆ free I ⋊ Γ) = M ⋊ Γ . The associated action Γ y ∆ free I and the group inverse naturally induce U πg and J free on ℓ (∆ free I ) in this setting. We also have the family G free and put X free := ∆ free I . Let B ⋊ red Γ ⊂ M ⋊ Γ be an inclusion given in the last subsection. We put B r := J ∗ BJ ∗ with Γ-action by Ad( U πg ) and consider B r ⋊ red Γ.Summarizing all our previous results, we obtain the following (relative) condition AOand solidity.
Proposition 6.1.
Keep the notation from Subsection 6.1. Then the following ∗ -homomorphismis min-bounded, nuclear, and has a ucp lift: ν : ( B ⋊ red Γ) ⊗ alg ( B r ⋊ red Γ) → M( K ( G ∗ )) / K ( G ∗ ); x ⊗ y π ℓ ( x ) π r ( y ) + K ( G ∗ ) . It Γ is bi-exact, we can replace K ( G ∗ ) with K ( L ( M ) ⊗ ℓ (Γ)) , hence M ⋊ Γ satisfiescondition (AO) with a ucp lift. Proof.
Let E be the family of all finite subsets in Γ and define F as in Lemma 3.13.Observe that K ( G ∗ ∩ F ) = K ( ℓ ( X ∗ ⋊ Γ )).We first consider the case for F anti . In this case, since Γ y π X anti is bi-exact / G anti byTheorem 4.10, B is nuclear, and since [ B, B r ] = 0, we can directly apply Proposition 3.12.For the last part of the statement, we can use Lemma 3.13 for F .We next consider the case for F full . We first claim that [ π ℓ ( a ) , π r ( b )] ⊂ K ( ℓ ( X full ⋊ Γ))for all a ∈ C ℓ and b ∈ C r . We may assume a = ℓ ( x ) and b = r ( y ) for some x, y ∈ X . Since[ ℓ ( x ) ∗ , r ( y )] = δ x,y P C Ω , where P C Ω is the projection onto C Ω, we have[ π ℓ ( ℓ ( x ) ∗ ) , π r ( r ( y ))] = X h ∈ Γ δ x,π h ( y ) P C Ω ⊗ e h,h =: f ∈ ℓ ∞ ( X full ⋊ Γ) . It satisfies f ( s, g ) = δ s,⋆ δ x,π g ( y ) , hence it is contained in c ( X full ⋊ Γ) (since π has finitestabilizers). The claim is proven. Then since C ℓ , C r are nuclear, ν G full and ν F in Proposition3.12 and Lemma 3.13 are min-bounded and nuclear (because they are defined at the levelof C ℓ , C r ). We get the conclusion.We consider the case for F sym . In this case, we can not apply Proposition 3.12 andLemma 3.13, but we can follow the same proofs in this setting by the amenability ofΓ y C ( ∂ G sym ) in Theorem 4.7. They are all straightforward, hence we omit it.We consider the case of free wreath products. Let A := ∗ i ∈I A i be a nuclear C ∗ -algebra containing B used in the proof of Proposition 5.1. Then since each A i is con-tained in C ∗ { C ∗ λ (∆ i ) , ℓ ∞ (∆ i ) } , C is contained in C ∗ { C ∗ λ (∆ free I ) , ℓ ∞ (∆ free I ) } . We claimthat [ π ℓ ( a ) , π r ( J bJ )] ∈ K ( ℓ (∆ free I ⋊ Γ)) for all a, b ∈ A . We may assume a ∈ A i and b ∈ A j for some i, j . Then[ π ℓ ( a ) , π r ( J bJ )] = X h ∈ Γ [ a, J α h ( b ) J ] ⊗ e h,h , where α h ( b ) is b contained in A h · j . As in the proof in Proposition 5.1, [ a, J α h ( b ) J ] = 0 if h · j = i and [ a, J α h ( b ) J ] is compact if h · j = i (which happens for finitely many h ∈ Γ).35e conclude [ π ℓ ( a ) , π r ( J bJ )] is compact and the claim is proven. Now we can define ν G full and ν F in Proposition 3.12 and Lemma 3.13 at the level of A and J A J , hence they aremin-bounded and nuclear. We get the conclusion. Theorem 6.2.
Keep the setting from Subsection 6.1. Then the action Γ y M is solid.Further M ⋊ Γ is solid if Γ is bi-exact. Proof.
Observe that K ( G ∗ ) is contained in K := { X g ∈ Γ x g ⊗ e g,g | x g ∈ K ( L ( M )) , sup g k x g k ∞ < ∞} . Then one can follow Ozawa’s proof [Oz04], see [HIK20, Theorem 4.7].
Remark 6.3.
We finally explain relationship between our Theorem 6.2 and known resultsin Popa’s deformation/rigidity theory. • The case of symmetric Fock spaces (i.e. actions on Gaussian algebras) was provedin [Bo12] in a more general setting. • The case of full symmetric Fock space (i.e. actions on free Araki–Woods factors) wasstudied in [HS09, Ho12, HT18]. When Γ is amenable, the solidity is easily deducedfrom [HT18, Theorem D]. When Γ is not amenable, the solidity result in Theorem6.2 is not discussed in these articles. • The case of anti-symmetric Fock space (i.e. actions on Araki–Woods factors) is notstudied, so Theorem 6.2 provides new examples. • For the case of free wreath product groups, the solidity is not known, so Theorem6.2 provides new examples.Thus Theorem 6.2 provides some new examples of solid actions and solid factors. Wehowever strongly believe that appropriate adaptations of known techniques in Popa’s de-formation/rigidity theory should be applied to above new examples. Therefore we do notemphasize that they are really new examples, but we do emphasize that our proofs involveboundary amenability which have independent interests.
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