The unitary implementation of a measured quantum groupoid action

BLAISE PASCAL

Michel Enock

The unitary implementation of a measured quantum groupoid action

Volume 17, n^o2 (2010), p. 233-302.

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The unitary implementation of a measured quantum groupoid action

Michel Enock

Abstract

Mimicking the von Neumann version of Kustermans and Vaes’ locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In a former article, the author had introduced the notions of actions, crossed-product, dual actions of a measured quantum groupoid; a biduality theorem for actions has been proved. This article continues that program: we prove the existence of a standard implementation for an action, and a biduality theorem for weights. We generalize this way results which were proved, for locally compact quantum groups by S. Vaes, and for measured groupoids by T. Yamanouchi.

L’implémentation unitaire d’une action de groupoïde quantique mesuré

Résumé

Frank Lesieur a introduit une notion de groupoïde quantique mesuré, dans le cadre des algèbres de von Neumann, en s’inspirant des groupes quantiques locale- ment compacts de Kustermans et Vaes (dans la version de cette construction faite dans le cadre des algèbres de von Neumann). Dans un article précédent, l’auteur a introduit les notions d’action, de produit croisé, d’action duale d’un groupoïde quantique mesuré ; un théorème de bidulaité des actions a éte démontré. Cet article continue ce programme : nous démontrons l’existence d’une implémentation stan- dard d’une action, et un théorème de bidulaité pour les poids. Sont ainsi généralisés des résultats qui avaient été démontrés par S. Vaes pour les groupes quantiques localement compacts, et par T. Yamanouchi pour les groupoïdes mesurés.

1. Introduction

In two articles ([39], [40]), J.-M. Vallin has introduced two notions (pseudo- multiplicative unitary, Hopf-bimodule), in order to generalize, up to the

Keywords:Measured quantum groupoids, actions, biduality theorems.

Math. classification:46L55, 46L89.

groupoid case, the classical notions of multiplicative unitary [1] and of Hopf-von Neumann algebras [21] which were introduced to describe and explain duality of groups, and leaded to appropriate notions of quantum groups ([21], [44], [45], [1], [28], [46], [24], [25], [27]).

In another article [22], J.-M. Vallin and the author have constructed, from a depth 2 inclusion of von Neumann algebrasM0 ⊂M1, with an operator- valued weight T1 verifying a regularity condition, a pseudo-multiplicative unitary, which leaded to two structures of Hopf bimodules, dual to each other. Moreover, we have then constructed an action of one of these struc- tures on the algebra M1 such that M0 is the fixed point subalgebra, the algebra M2 given by the basic construction being then isomorphic to the crossed-product. We construct on M2 an action of the other structure, which can be considered as the dual action.

If the inclusion M0 ⊂M1 is irreducible, we recovered quantum groups, as proved and studied in former papers ([19], [13]).

Therefore, this construction leads to a notion of "quantum groupoid", and a construction of a duality within "quantum groupoids".

In a finite-dimensional setting, this construction can be mostly sim- plified, and is studied in [29], [7], [6], [34],[41], [42], and examples are described. In [30], the link between these "finite quantum groupoids" and depth 2 inclusions of II1 factors is given.

F. Lesieur, in [26], starting from a Hopf-bimodule, as introduced in [39], when there exist a left-invariant operator-valued weight, and a right- invariant operator-valued weight, mimicking in that wider setting the tech- nics of Kustermans and Vaes ([24], [25]), obtained a pseudo-multiplicative unitary, which, as in quantum group theory, "contains" all the information about the object (the von Neumann algebra, the coproduct) and allows to construct important data (an antipod, a co-inverse, etc.) Lesieur gave the name of "measured quantum groupoids" to these objects. A new set of axioms for these had been given in an appendix of [16]. In [14] had been shown that, with suitable conditions, the objects constructed from [22] are

"measured quantum groupoids" in the sense of Lesieur.

In [16] have been developped the notions of action (already introduced in [22]), crossed-product, etc, following what had been done for locally

compact quantum groups in ([12], [20], [36]); a biduality theorem for ac- tions had been obtained in ([16], 11.6). Several points were left apart in [16], namely the generalization of Vaes’ theorem ([36], 4.4) on the stan- dard implementation of an action of a locally compact quantum group, which was the head light of [36], and a biduality theorem for weights, as obtained in [49], [51] (in fact, we were much more inspired by the shorter proof given in an appendix of [3]).

We solve here these two problems when there exists a normal semi-finite faithful operator-valued weight from the von Neumann algebra on which the measured quantum groupoid is acting, onto the copy of the basis of this measured quantum groupoid which is put inside this algebra. In fact, these results appear much more as a biduality theorem of operator-valued weights rather than a biduality theorem on weights, which seems quite natural in the spirit of measured quantum groupoids, where, for instance, left-invariant weight on a locally compact quantum group is replaced by a left-invariant operator-valued weight. The strategy for the proofs had been mostly inspired by [36] and [3].

This article is organized as follows:

In chapter 2, we recall very quickly all the notations and results needed in that article; we have tried to make these preliminaries as short as possible, and we emphazise that this article should be understood as the continua- tion of [16].

In chapter 3, we follow ([36], 4.1 to 4.4), and prove, for any dual action, the result on the standard implementation of an action.

Chapter 4 is rather technical; let G = (N, M, α, β,Γ, T, T⁰, ν) be a mea- sured quantum groupoid, and let b be an injective ∗-anti-homomorphism from N into a von Neumann algebra A; let us suppose that there exists a normal semi-finite faithful operator-valued weight Tfrom A ontob(N), and let us write ψ =ν^o◦b⁻¹ ◦T. Then, we can define on A_b∗_α

N

L(H) a weight ψ, which will generalize the tensor product ofψ andT r

∆b⁻¹ (when G is a locally compact quantum group, and therefore N =C).

In chapter 5, using this auxilliary weight introduced in chapter 4, and the particular case of the dual actions studied in chapter 3, we calculate the standard implementation of an action, whenever there exists a normal semi-finite faithful operator-valued weight from A ontob(N). This condi- tion is fulfilled trivially when the measured quantum groupoid is a locally

compact quantum group, or is a measured groupoid; therefore, we recover in both cases the results already obtained.

Chapter 6 is another technical chapter; we define conditions on a weight ψ defined on A which allow us to construct on A_b∗_α

N

L(H) a weight ψ_δ which generalize the tensor product ofψandT r_(δ

∆)b ⁻¹(whenGis a locally compact quantum group, and therefore N =C).

In chapter 7 we use both auxilliary weights constructions made in chapters 4 and 6; then, when there exists a normal semi-finite faithful operator- valued weight T from A onto b(N) such that ψ = ν^o ◦b⁻¹◦T, we can define a Radon-Nikodym derivative of the weight ψ with respect to the action, which will be a cocycle for this action. This condition is fulfilled trivially when the measured quantum groupoid is a locally compact quan- tum group, or is a measured groupoid, and, therefore, we recover in both cases the results already obtained.

2. Definitions and notations

This article is the continuation of [16]; preliminaries are to be found in [16], and we just recall herafter the following definitions and notations:

2.1. Spatial theory; relative tensor products of Hilbert spaces and fiber products of von Neumann algebras ([8], [32], [35], [22])

Let N a von Neumann algebra, ψ a normal semi-finite faithful weight on N; we shall denote by H_ψ, N_ψ, S_ψ, J_ψ, ∆_ψ... the canonical objects of the Tomita-Takesaki theory associated to the weight ψ; let α be a non degenerate faithful representation of N on a Hilbert space H; the set of ψ-bounded elements of the left-moduleαHis:

D(αH, ψ) ={ξ ∈ H;∃C <∞,kα(y)ξk ≤CkΛ_ψ(y)k,∀y∈N_ψ}. Then, for any ξ in D(_αH, ψ), there exists a bounded operator R^α,ψ(ξ) from H_ψ toH, defined, for ally in N_ψ by:

R^α,ψ(ξ)Λψ(y) =α(y)ξ

which intertwines the actions ofN.

If ξ,η are bounded vectors, the operator product hξ, ηi_α,ψ=R^α,ψ(η)^∗R^α,ψ(ξ)

belongs to π_ψ(N)⁰, which, thanks to Tomita-Takesaki theory, will be iden- tified to the opposite von Neumann algebra N^o.

If nowβ is a non degenerate faithful antirepresentation ofN on a Hilbert space K, the relative tensor product K_β⊗_α

ψ

H is the completion of the algebraic tensor product KD(_αH, ψ) by the scalar product defined, if ξ1,ξ2 are in K,η1,η2 are in D(αH, ψ), by the following formula:

(ξ1η1|ξ2η2) = (β(hη1, η2i_α,ψ)ξ1|ξ2). If ξ ∈ K,η ∈ D(_αH, ψ), we shall denote ξ_β⊗_α

ψ

η the image ofξη into K_β⊗_α

ψ

H, and, writingρ^β,α_η (ξ) =ξ_β⊗_α

ψ

η, we get a bounded linear operator from K into K_β⊗_α

ν

H, which is equal to1_K⊗_ψR^α,ψ(η).

Changing the weight ψwill give a canonical isomorphic Hilbert space, but the isomorphism will not exchange elementary tensors !

We shall denoteσ_ψ the relative flip, which is a unitary sendingK_β⊗_α

ψ

H onto H_α⊗_β

ψ^o

K, defined, for any ξ inD(K_β, ψ^o),η inD(αH, ψ), by:

σ_ψ(ξ_β⊗_α

ψ

η) =η_α⊗_β

ψ^o

ξ .

In x ∈β(N)⁰, y ∈α(N)⁰, it is possible to define an operator x_β⊗_α

ψ

y on K_β⊗_α

ψ

H, with natural values on the elementary tensors. As this operator does not depend upon the weight ψ, it will be denoted x_β⊗_α

N

y. We can define a relative flip ςN at the level of operators such that ςN(xβ⊗_α

N

y) = yα⊗_β

N^o

x. If P is a von Neumann algebra on H, with α(N) ⊂ P, and Q a von Neumann algebra on K, with β(N) ⊂Q, then we define the fiber

product Q_β∗_α

N

P as{x_β⊗_α

N

y, x∈Q⁰, y ∈P⁰}⁰, and we get that ςN(Qβ∗_α

N

P) =Pα∗_β

N^o

Q.

Moreover, this von Neumann algebra can be defined independantly of the Hilbert spaces on which P and Q are represented; if (i= 1,2), αi is a faithful non degenerate homomorphism from N intoP_i,β_i is a faithful non degenerate antihomomorphism from N into Qi, and Φ (resp. Ψ) an homomorphism fromP1 toP2 (resp. fromQ1 toQ2) such thatΦ◦α1 =α2

(resp.Ψ◦β1 =β2), then, it is possible to define an homomorphismΨ_β1∗_α1

N

Φ from Q1β1∗_α1

N

P1 intoQ2β2∗_α2

N

P2.

The operators θ^α,ψ(ξ, η) = R^α,ψ(ξ)R^α,ψ(η)^∗, for all ξ, η in D(_αH, ψ), generates a weakly dense ideal in α(N)⁰. Moreover, there exists a family (e_i)i∈I of vectors in D(_αH, ψ) such that the operators θ^α,ψ(e_i, e_i) are 2 by 2 orthogonal projections (θ^α,ψ(e_i, e_i) being then the projection on the closure of α(N)ei). Such a family is called an orthogonal (α, ψ)-basis of H.

2.2. Measured quantum groupoids ([26], [16])

A measured quantum groupoid is an octupletG= (N, M, α, β,Γ, T, T⁰, ν) such that ([16], 3.8):

(i) (N, M, α, β,Γ)is a Hopf-bimodule (as defined in [16], 3.1),

(ii)T is a left-invariant normal, semi-finite, faithful operator valued weight T fromM toα(N),

(iii) T⁰ is a right-invariant normal, semi-finite, faithful operator-valued weight T⁰ fromM toβ(N),

(iv)ν is normal semi-finite faitfull weight on N, which is relatively invari- ant with respect toT and T⁰.

We shall write Φ =ν◦α⁻¹◦T, andH=HΦ,J =JΦ, and, for alln∈N, βˆ(n) = J α(n^∗)J, α(n) =ˆ J β(n^∗)J. The weight Φ will be called the left- invariant weight on M.

Then, G can be equipped with a pseudo-multiplicative unitary W from H_β⊗_α

ν

H onto H_α⊗_βˆ

ν^o

H ([16], 3.6), a co-inverseR, a scaling groupτ_t, an antipodS, a modulusδ, a scaling operatorλ, a managing operatorP, and a canonical one-parameter groupγtof automorphisms on the basisN ([16],

3.8). Instead ofG, we shall mostly use (N, M, α, β,Γ, T, RT R, ν) which is another measured quantum groupoid, denotedG, which is equipped with the same data (W,R, . . . ) asG.

A dual measured quantum group G, which is denotedb (N,M , α,^c β,ˆ Γ,^b T ,^b R^bT^bR, ν),^b can be constructed, and we have G^bb =G.

Canonically associated to G, can be defined also the opposite measured quantum groupoid isG^o= (N^o, M, β, α, ς_NΓ, RT R, T, ν^o)and the commu- tant measured quantum groupoid G^c= (N^o, M⁰,β,ˆ α,ˆ Γ^c, T^c, R^cT^cR^c, ν^o);

we have (G^o)^o = (G^c)^c = G, Gc^o = (G)b ^c, Gc^c = (G)b ^o, and G^oc = G^co is canonically isomorphic to G ([16], 3.12).

The pseudo-multiplicative unitary of Gb (resp.G^o,G^c) will be denotedWc (resp. W^o,W^c). The left-invariant weight onGb (resp. G^o,G^c) will be de- noted Φ^b (resp. Φ^o,Φ^c).

Let_aH_bbe aN−N-bimodule, i.e. an Hilbert spaceHequipped with a nor- mal faithful non degenerate representationaofNonHand a normal faith- ful non degenerate anti-representation bonH, such that b(N)⊂a(N)⁰. A corepresentation ofGon_aH_b is a unitaryV fromH_a⊗_β

ν^o

HΦ ontoH_b⊗_α

ν HΦ, satisfying, for all n∈N:

V(b(n)_a⊗_β

N^o

1) = (1_b⊗_α

N

β(n))V V(1_a⊗_β

N^o

α(x)) = (a(n)_b⊗_α

N

1)V

such that, for any ξ ∈ D(_aH, ν) and η ∈ D(H_b, ν^o), the operator (ω_ξ,η∗ id)(V) belongs to M (then, it is possible to define (id∗θ)(V), for anyθ in M∗^α,β which is the linear set generated by the ωξ, withξ∈D(αH, ν)∩ D(H_β, ν^o)), and such that the applicationθ→(id∗θ)(V)fromM∗^α,β into L(H) is multiplicative ([16] 5.1, 5.5).

2.3. Action of a measured quantum groupoid ([16])

An action ([16], 6.1) of Gon a von Neumann algebraA is a couple (b,a), where:

(i) b is an injective∗-antihomomorphism from N intoA;

(ii) ais an injective ∗-homomorphism from AintoA_b∗_α

N

M; (iii) b and aare such that, for alln inN:

a(b(n)) = 1b⊗_α

N

β(n) (which allow us to define a_b∗_α

N

idfrom A_b∗_α

N

M intoA_b∗_α

N

M_β∗_α

N

M) and such that:

(a_b∗_α

N

id)a= (idb∗_α

N

Γ)a.

The set of invariants is defined as the sub von Neumann algebra:

A^a={x∈A∩b(N)⁰,a(x) =x_b⊗_α

N

1}.

If the von Neumann algebra acts on a Hilbert space H, and if there exists a representationaofN onHsuch thatb(N)⊂A⊂a(N)⁰, a corepresenta- tionV ofGon the bimodule_aH_bwill be called an implementation ofaif we havea(x) =V(x_a⊗_b

N^o

1)V^∗, for allx∈A([16], 6.6); we shall look at the fol- lowing more precise situation: let ψis a normal semi-finite faithful weight on A, andV an implementation ofaon _a(H_ψ)_b (with a(n) =J_ψb(n^∗)J_ψ), such that:

V^∗ = (Jψ α⊗_β

ν^o

J

bΦ)V(Jψ b⊗_α

ν J

Φb).

Such an implementation had been constructed ([16] 8.8) in the particular case when the weight ψ is called δ-invariant, which means that, for all η ∈ D(_αHΦ, ν)∩ D(δ^1/2), such that δ^1/2η belongs to D((HΦ)_β, ν^o), and for all x∈N_ψ, we have:

ψ((id_b∗_α

N

ωη)a(x^∗x)) =kΛ_ψ(x)a⊗_β

ν^o

δ^1/2ηk² and bears the density property, which means that the subset

D((H_ψ)_b, ν^o)∩D(_aH_ψ, ν)

is dense inHψ. This standard implementation is then given by the formula ([16], 8.4):

V_ψ(Λ_ψ(x)a⊗_β

ν^o

δ^1/2η) =^X

i

Λ_ψ((id_b∗_α

N

ωη,ei)a(x))_b⊗_α

ν ei

for all x ∈ N_ψ, η ∈ D(_αH, ν) ∩ D(δ^1/2) such that δ^1/2η belongs to D(Hβ, ν^o),(ei)i∈Iany orthonormal(α, ν)-basis ofH. Moreover ([16], 8.9),

it is possible to define one parameter groups of unitaries∆^it_{ψ a}⊗_β

N^o

δ^−it∆^−it bΦ

and ∆^it_{ψ b}⊗_α

N

δ^−it∆^−it

Φb , with natural values on elementary tensors, and we have:

Vψ(∆^it_{ψ a}⊗_β

N^o

δ^−it∆^−it

bΦ ) = (∆^it_{ψ b}⊗_α

N

δ^−it∆^−it

Φb )Vψ

and, therefore, for any x inA,t inR, we have:

a(σ_t^ψ(x)) = (∆^it_{ψ b}⊗_α

N

δ^−it∆^−it

Φb )a(x)(∆^−it_ψ b⊗_α

N

δ^it∆^it

Φb). 2.4. Crossed-product ([16])

The crossed-product of A by G via the action a is the von Neumann algebra generated bya(A)and1_b⊗_α

N

Mc⁰ ([16], 9.1) and is denotedAoaG;

then there exists ([16], 9.3) an action (1_b⊗_α

N

α,ˆ ˜a) of(G)b ^c onAoaG.

The biduality theorem ([16], 11.6) says that the bicrossed-product (Aoa

G) o˜a Gb^c is canonically isomorphic to Ab∗_α

N

L(H); more precisely, this isomorphism is given by:

Θ(a_b∗_α

N

id)(Ab∗_α

N

L(H)) = (AoaG)o˜aGb^c where Θ is the spatial isomorphism between L(H _b⊗_α

ν H β⊗_α

ν

H) and L(H_b⊗_α

ν Hαˆ⊗_β

ν^o

H) implemented by1_Hb⊗_α

ν σνW^oσν; the biduality theo- rem says also that this isomorphism sends the action(1_b⊗_α

N

β,ˆ a) ofGon Ab∗_α

N

L(H), defined, for any X ∈Ab∗_α

N

L(H), by:

a(X) = (1_b⊗_α

N

σ_ν^oW σ_ν^o)(id_b∗_α

N

ς_N)(a_b∗_α

N

id)(X)(1_b⊗_α

N

σ_ν^oW σ_ν^o)^∗ on the bidual action (ofG^co) on(AoaG)o˜aGb^o.

There exists a normal faithful semi-finite operator-valued weight T˜a from A oa G onto a(A); therefore, starting with a normal semi-finite weight ψ on A, we can construct a dual weight ψ˜ on Aoa G by the formula ψ˜ = ψ◦ a⁻¹ ◦T˜a ([16] 13.2). These dual weights are exactly the δˆ⁻¹- invariant weights on AoaGbearing the density property ([16] 13.3).

Moreover ([16] 13.3), the linear set generated by all the elements (1_b⊗_α

N

a)a(x), for all x∈N_ψ,a∈N

bΦ^c∩N_Tˆc, is a core for Λψ˜, and it is possible to identify the GNS representation of AoaGassociated to the weight ψ˜ with the natural representation on H_{ψ b}⊗_α

ν HΦ by writing:

Λ_ψ(x)_b⊗_α

ν

Λ

bΦ^c(a) = Λψ˜[(1_b⊗_α

N

a)a(x)]

which leads to the identification of Hψ˜ with Hψ b⊗_α

ν H. Moreover, using that identification, the linear set generated by the elements of the form a(y^∗)(Λ_ψ(x)_b⊗_α

ν

Λ

bΦ^c(a)), forx, y inN_ψ, andainN

Φb^c∩N_Tˆc∩N^∗ bΦ^c∩N^∗_ˆ

T^c

is a core for Sψ˜, and we have:

Sψ˜a(y^∗)(Λ_ψ(x)_b⊗_α

ν

Λ

Φb^c(a)) =a(x^∗)(Λ_ψ(y)_b⊗_α

ν

Λ

bΦ^c(a^∗)). Then, the unitaryU_ψ^a =Jψ˜(J_{ψ a}⊗_β

N^o

J

bΦ)fromH_{ψ a}⊗_β

ν^o

HΦ onto H_{ψ b}⊗_α

ν HΦ

satisfies:

U_ψ^a(J_{ψ b}⊗_α

N

J

bΦ) = (J_{ψ b}⊗_α

N

J

Φb)(U_ψ^a)^∗ and we have ([16] 13.4):

(i) for all y∈A:

a(y) =U_ψ^a(y_a⊗_β

N^o

1)(U_ψ^a)^∗, (ii) for all b∈M:

(1_b⊗_α

N

JΦbJΦ)U_ψ^a =U_ψ^a(1_a⊗_β

N^o

JΦbJΦ), (iii) for all n∈N:

U_ψ^a(b(n)a⊗_β

N^o

1) = (1_b⊗_α

N

β(n))U_ψ^a U_ψ^a(1_a⊗_β

N^o

α(n)) = (a(n)_b⊗_α

N

1)U_ψ^a.

Therefore, we see that this unitaryU_ψ^a"implements"a, but we do not know whether it is a corepresentation. If it is, we shall say that it is a standard implementation of a.

We can define the bidual weight ψ˜˜ on (AoaG)o˜a Gb^o, and the weight

˜˜

ψ◦Θ◦(a_b∗_α

N

id) onAb∗_α

N

L(H), that we shall denoteψafor simplification (or ψ if there is no ambiguity about the action). Then we get ([16], 13.6) that the spatial derivative _dψ^dψo is equal to the modulus operator∆ψ˜. There

exists a normal semi-finite faithful operator-valued weight Ta from A_b∗_α

N

L(H) onto AoaGsuch thatψa= ˜ψ◦Ta.

Using twice ([35] 4.22(ii)), we obtain, for any x ∈ A and t ∈ R, that σ^ψ_t^a(a(x)) =a(σ_t^ψ(x)); and ifψ1andψ2are two normal semi-finite faithful weights onA, we get:

(Dψ_1a:Dψ_2a)_t= (Dψ˜1 :Dψ˜2)_t=a((Dψ1 :Dψ2)_t). 2.5. Examples of measured quantum groupoids Examples of measured quantum groupoids are the following:

(i) Locally compact quantum groups, as defined and studied by J. Kuster- mans and S. Vaes ([24], [25], [36]); these are, trivially, the measured quan- tum groupoids with the basis N =C.

(ii) Measured groupoids, equipped with a left Haar system and a quasi- invariant measure on the set of units, as studied mostly by T. Yamanouchi ([47], [48], [49], [51]); it was proved in [17] that these measured quantum groupoids are exactly those whose underlying von Neumann algebra is abelian.

(iii) The finite dimensional case had been studied by D. Nikshych and L. Vainermann ([29], [30], [31]), J.-M. Vallin ([41], [42]) and M.-C. David ([10]); in that case, non trivial examples are given, for instance Temperley- Lieb algebras ([31], [10]), which had appeared in subfactor theory ([23]).

(iv) Continuous fields of (C^∗-version of) locally compact quantum groups, as studied by E. Blanchard in ([4], [5]); it was proved in [17] that these measured quantum groupoids are exactly those whose basis is central in the underlying von Neumann algebras of both the measured quantum groupoid and its dual.

(v) In [11], K. De Commer proved that, in the case of a monoidal equiv- alence between two locally compact quantum groups (which means that these two locally compact quantum group have commuting ergodic and integrable actions on the same von Neumann algebra), it is possible to construct a measurable quantum groupoid of basis C² which contains all the data. Moreover, this construction was useful to prove new results on locally compact quantum groups, namely on the deformation of a locally compact quantum group by a unitary2-cocycle; he proved that these mea- sured quantum groupoids are exactly those whose basis C² is central in the underlying von Neumann algebra of the measured quantum groupoid,

but not in the underlying von Neumann algebra of the dual measured quantum groupoid.

(vi) Starting from a depth 2 inclusionM0 ⊂M1of von Neumann algebras, equipped with an operator-valued weightT1 from M1 ontoM0, satisfying appropriate conditions, such that there exists a normal semi-finite faith- ful weight χ on the first relative commutant M₀⁰ ∩M1, invariant under the modular automorphism groupσ^T_t¹, it has been proved ([22], [14]) that it is possible to put on the second relative commutant M₀⁰ ∩M2 (where M0 ⊂ M1 ⊂ M2 ⊂ M3 ⊂ · · · is Jones’ tower associated to the inclu- sion M0 ⊂ M1) a canonical structure of a measured quantum groupoid;

moreover, its dual is given then by the same construction associated to the inclusion M1 ⊂ M2, and this dual measured quantum groupoid acts canonically on the von Neumann algebra M1, in such a way that M0 is equal to the subalgebra of invariants, and the inclusion M1 ⊂ M2 is iso- morphic to the inclusion ofM1 into its crossed-product. This gives a "geo- metrical" construction of measured quantum groupoids; in another article in preparation ([18]), in which is used the biduality theorem for weights proved in 7.3, had been proved that any measured quantum groupoid has an outer action on some von Neumann algebra, and can be, therefore, obtained by this "geometrical construction". The same result for locally compact quantum groups relies upon [37] and the corresponding result for measured quantum groupoids had been pointed out in [16].

(vii) In [38] and [2] was given a specific procedure for constructing locally compact quantum groups, starting from a locally compact groupG, whose almost all elements belong to the product G1G2 (where G1 and G2 are closed subgroups of G whose intersection is reduced to the unit element of G); such (G1, G2) is called a "matched pair" of locally compact groups (more precisely, in [38], the set G1G2 is required to be open, and it is not the case in [2]).Then, G1 acts naturally onL^∞(G2) (and vice versa), and the two crossed-products obtained bear the structure of two locally compact quantum groups in duality. In [43], J.-M. Vallin generalizes this constructions up to groupoids, and, then, obtains examples of measured quantum groupoids; more specific examples are then given by the action of a matched pair of groups on a locally compact space, and also more exotic examples.

3. The standard implementation of an action: the case of a dual action

In this chapter, following [36], we prove that the unitaryU_ψ^a introduced in 2.4 is a standard implementation of a, for all normal semi-finite faithful weight ψ on A, whenever a is a dual action (3.4). For this purpose, we prove first that, if for some weight ψ1, the unitary U_ψ^a

1 is a standard implementation, then, for any weight ψ,U_ψ^a is a standard implementation (3.1). Second (3.2), we prove, for a δ-invariant weight ψ, that U_ψ^a is equal to the implementation V_ψ constructed in ([16] 8.8) and recalled in 2.3.

Thanks to ([16] 13.3), recalled in 2.4, we then get the result.

3.1. Proposition

Let G be a measured quantum groupoid, and (b,a) an action of G on a von Neumann algebra A; let ψ1 andψ2 be two normal faithful semi-finite weights on A and U_ψ^a₁ and U_ψ^a₂ the two unitaries constructed in 2.4; let u be the unitary fromH_ψ1 onto H_ψ2 intertwining the representations π_ψ1 and π_ψ2; then:

(i) The unitary u _b⊗_α

N

1 intertwines the representations of A oa G on H_{ψ1 b}⊗_α

ν HΦ and on H_{ψ2 b}⊗_α

ν HΦ; moreover, we have:

(ub⊗_α

N

1)U_ψ^a₁ =U_ψ^a₂(ua1⊗_β

N^o

1) where a1(n) =J_ψ1π_ψ1(b(n^∗))J_ψ1, for alln∈N.

(ii) If U_ψ^a₁ is a corepresentation of GonHψ1, then U_ψ^a₂ is a corepresenta- tion of Gon H_ψ2.

(iii) If U_ψ^a

1 is a standard implementation of a, then U_ψ^a

2 is a standard implementation of a.

Proof. Let us write J2,1 the relative modular conjugation, which is an antilinear surjective isometry from H_ψ1 onto H_ψ2. Then we have u = J_2,1J_ψ1 = J_ψ2J_2,1, by ([33] 3.16). Moreover, let us define, for x ∈A, and t ∈ R, σ_t^2,1(x) = [Dψ2 : Dψ1]_tσ^ψ_t¹(x); then, by ([33], 3.15), for x ∈ N_ψ1, y ∈D(σ^2,1_−i/2),xy^∗ belongs to N_ψ2 and:

Λ_ψ2(xy^∗) =J2,1π_ψ1(σ_−i/2^2,1 (y))J_ψ1Λ_ψ1(x).

Therefore, ifa∈N

Φb^c,(1_b⊗_α

N

a)a(xy^∗)belongs toNψ˜2, and, we have, where V_i (i= (1,2)) denotes the unitary fromH_ψib⊗_α

ν HΦ onto Hψ˜i defined in 2.4:

Λψ˜2[(1_b⊗_α

N

a)a(xy^∗)] = V2(Λ_ψ2(xy^∗)_b⊗_α

ν

Λ bΦ^c(a))

= V2J2,1πψ1(σ^2,1_−i/2(y))Jψ1Λ_ψ1(x)b⊗_α

ν Λ bΦ^c(a)) which is equal to:

V2(J_2,1π_ψ1(σ^2,1_−i/2(y))J_{ψ1 b}⊗_α

N

1)V₁^∗Λ_ψ˜

1[(1_b⊗_α

N

a)a(x)]

and, as the linear set generated by the elements of the form (1_b⊗_α

N

a)a(x) is a core for Λψ˜1, we get, for any z ∈ Nψ˜1, that za(y^∗) belongs to Nψ˜2, and that:

Λψ˜2(za(y^∗)) =V2(J2,1πψ1(σ_−i/2^2,1 (y))Jψ1 b⊗_α

N

1)V₁^∗Λψ˜1(z).

Let us denote by J˜_2,1 the relative modular conjugation constructed from the weights ψ˜1 and ψ˜2, and σ˜_t^2,1 the one-parameter group of isometries of AoaG constructed from these two weights by the formula, for any X ∈AoaG:

σ˜_t^2,1(X) = [Dψ˜2:Dψ˜1]_tσ_t^ψ˜1(X).

Using ([33], 3.15) applied to these two weights, we get that a(y) belongs toD(˜σ_−i/2^2,1 ) and that:

J˜_2,1πψ˜1(˜σ^2,1_−i/2(a(y)))Jψ˜1 =V2(J_2,1π_ψ1(σ^2,1_−i/2(y))J_{ψ1 b}⊗_α

N

1)V₁^∗. We easily get thatσ˜^2,1_t (a(y)) =a(σ_t^2,1(y))and, therefore, we have:

πψ˜1(a(σ_−i/2^2,1 (y)) = ˜J2,1

∗V2(J2,1πψ1(σ^2,1_−i/2(y))Jψ1 b⊗_α

N

1)V₁^∗Jψ˜1. As we have, using 2.4:

(J_ψ1b⊗_α

N

J

bΦ)V₁^∗Jψ˜1 =U_ψ^a₁V₁^∗ we get:

πψ˜1(a(σ_−i/2^2,1 (y)) = ˜J_2,1^∗V2(J_2,1a1⊗_β

N^o

J

bΦ)(π_ψ1(σ^2,1_−i/2(y))_a1⊗_β

N^o

1)U_ψ^a₁V₁^∗

and, therefore, using 2.4:

J˜2,1

∗V2(J2,1a1⊗_β

N^o

J

bΦ)(π_ψ1(σ_−i/2^2,1 (y))a1⊗_β

N^o

1) = πψ˜1(a(σ^2,1_−i/2(y))V1(U_ψ^a₁)^∗

= V1a(σ_−i/2^2,1 (y))(U_ψ^a₁)^∗ which, using 2.4, is equal to:

V1U_ψ^a₁(π_ψ1(σ_−i/2^2,1 (y))_a1⊗_β

N^o

1).

By density, we get:

U_ψ^a₁ =V₁^∗J˜_2,1^∗V2(J_2,1a1⊗_β

N^o

J bΦ) and, therefore, using 2.4 again:

1_Hψ

1 b⊗_α

N

1_HΦ = V₁^∗J˜2,1

∗V2(J2,1a1⊗_β

N^o

J

Φb)(Jψ1 b⊗_α

N

J

Φb)V₁^∗Jψ˜1V1

= V₁^∗J˜_2,1^∗V2(u_b⊗_α

N

1)V₁^∗Jψ˜1V1

which implies that:

1_H˜

ψ1 b⊗_α

N

1_HΦ = ˜J_2,1^∗V2(u_b⊗_α

N

1)V₁^∗Jψ˜1

and:

V2(ub⊗_α

N

1)V₁^∗ = ˜J2,1Jψ˜1. ButJ˜2,1Jψ˜1 =Jψ˜2

J˜2,1is the unitary fromHψ˜1 ontoHψ˜2 which intertwines πψ˜1 andπψ˜2; from which we get the first result.