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The Haagerup Property for Discrete Measured Groupoids

Claire Anantharaman-Delaroche

To cite this version:

Claire Anantharaman-Delaroche. The Haagerup Property for Discrete Measured Groupoids. Nord-

forsk Network Closing Conference, May 2012, Gjaargardur, Denmark. pp.1-30, �10.1007/978-3-642-

39459-1_1�. �hal-01138882�

The Haagerup property for discrete measured groupoids

Claire Anantharaman-Delaroche

Abstract We define the Haagerup property in the general context of countable groupoids equipped with a quasi-invariant measure. One of our objectives is to com- plete an article of Jolissaint devoted to the study of this property for probability measure preserving countable equivalence relations. Our second goal, concerning the general situation, is to provide a definition of this property in purely geometric terms, whereas this notion had been introduced by Ueda in terms of the associated inclusion of von Neumann algebras. Our equivalent definition makes obvious the fact that treeability implies the Haagerup property for such groupoids and that it is not compatible with Kazhdan property (T).

1 Introduction

Since the seminal paper of Haagerup [15], showing that free groups have the (now so-called) Haagerup property, or property (H), this notion plays an increasingly im- portant role in group theory (see the book [10]). A similar property (H) has been introduced for finite von Neumann algebras [12, 11] and it was proved in [11] that a countable groupΓ has property (H) if and only if its von Neumann algebraL(Γ) has property (H).

Later, given a von Neumann subalgebraAof a finite von Neumann algebraM, a property (H) forMrelative toAhas been considered [9, 25] and proved to be very useful. It is in particular one of the crucial ingredients used by Popa [25], to provide the first example of aII₁factor with trivial fundamental group.

A discrete (also called countable) measured groupoid (G,µ) with set of units X (see Sect. 2.1) gives rise to an inclusionA⊂M, whereA=L^∞(X,µ)andM= L(G,µ)is the von Neumann algebra of the groupoid. This inclusion is canonically equipped with a conditional expectationE_A:M→A. AlthoughM is not always a Laboratoire MAPMO - UMR6628, F´ed´eration Denis Poisson (FDP - FR2964), CNRS/Universit´e d’Orl´eans, B. P. 6759, F-45067 Orl´eans Cedex 2, e-mail: claire.anantharaman@univ-orleans.fr

1

finite von Neumann algebra, there is still a notion of property (H) relative toAand E_A(see [32]). However, to our knowledge, this property has not been translated in terms only involving(G,µ), as in the group case. A significant exception concerns the case where G=R is a countable equivalence relation on X, preserving the probability measureµ,i.e.a typeII1equivalence relation [18].

Our first goal is to extend the work of Jolissaint [18] in order to cover the general case of countable measured groupoids, and in particular the case of group actions leaving a probability measure quasi-invariant. Although it is not difficult to guess the right definition of property (H) for(G,µ)(see Definition 6.2), it is more intricate to prove the equivalence of this notion with the fact thatL(G,µ)has property (H) relative toL^∞(X,µ).

We begin in Sect. 2 by introducing the basic notions and notation relative to countable measured groupoids. In particular we discuss the Tomita-Takesaki theory for their von Neumann algebras. This is essentially a reformulation of the pioneering results of P. Hahn [16] in a way that fits better for our purpose. In Section 3 we discuss in detail several facts about the von Neumann algebra of the Jones’ basic construction for an inclusionA⊂Mof von Neumann algebras, assuming thatAis abelian. We also recall here the notion of relative property (H) in this setting.

In Sects. 4 and 5, we study the relations between positive definite functions on our groupoids and completely positive maps on the corresponding von Neumann algebras. These results are extensions of well known results for groups and of results obtained by Jolissaint in [18] for equivalence relations, but additional difficulties must be overcome. After this preliminary work, it is immediate (Sect. 6) to show the equivalence of our definition of property (H) for groupoids with the definition involving operator algebras (Theorem 6.1).

Our main motivation originates from the reading of Ueda’s paper [32] and con- cerns treeable groupoids. This notion was introduced by Adams for probability measure preserving countable equivalence relations [1]. Treeable groupoids may be viewed as the groupoid analogue of free groups. So a natural question, raised by C.C. Moore in his survey [22, p. 277] is whether a treeable equivalence relation must have the Haagerup property. In fact, this problem is solved in [32] using ope- rator algebras techniques. In Ueda’s paper, the notion of treeing is translated in an operator algebra framework regarding the inclusionL^∞(X,µ)⊂L(G,µ), and it is proved that this condition implies thatL(G,µ)has the Haagerup property relative toL^∞(X,µ).

Our approach is opposite. For us, it seems more natural to compare these two notions, treeability and property (H), purely at the level of the groupoid. Indeed, the definition of treeability is more nicely read at this level: roughly speaking, it means that there is a measurable way to endow each fibre of the groupoid with a structure of tree (see Definition 7.2). The direct proof that treeability implies property (H) is given in Sect. 7 (Theorem 7.3).

Using our previous work [6] on groupoids with property (T), we prove in Sect. 8 that, under an assumption of ergodicity, this property is incompatible with the Haagerup property (Theorem 8.2). As a consequence, we recover the result of Jolis- saint [18, Proposition 3.2] stating that ifΓ is a Kazhdan countable group which

acts ergodically on a Lebesgue space(X,µ)and leaves the probability measureµ invariant, then the orbit equivalence relation(RΓ,µ)has not the Haagerup prop- erty (Corollary 8.4). A fortiori,(RΓ,µ)is not treeable, a result due to Adams and Spatzier [2, Theorem 18] and recovered in a different way by Ueda.

This text is an excerpt from the survey [7] which is not intended to publication.

2 The von Neumann algebra of a measured groupoid 2.1 Preliminaries on countable measured groupoids

Our references for measured groupoids are [8, 16, 27]. Let us recall that agroupoid is a setGendowed with a product(γ,γ⁰)7→γ γ⁰defined of a subsetG⁽²⁾ofG×G, called the set ofcomposableelements, and with an inverse mapγ7→γ⁻¹, satisfying the natural properties expected for a product and an inverse such as associativity.

For everyγ∈G, we have that(γ⁻¹,γ)∈G⁽²⁾and if(γ,γ⁰)∈G⁽²⁾, thenγ⁻¹(γ γ⁰) = (γ⁻¹γ)γ⁰=γ⁰(and similarly(γ γ⁰)γ⁰⁻¹=γ(γ⁰γ⁰⁻¹) =γ). We setr(γ) =γ γ⁻¹,s(γ) = γ⁻¹γ andG⁽⁰⁾=r(G) =s(G). The mapsrandsare called respectively therange and thesourcemap. The pair(γ,γ⁰)is composable if and only ifs(γ) =r(γ⁰)and then we haver(γ γ⁰) =r(γ)ands(γ γ⁰) =s(γ⁰). The setG⁽⁰⁾is called theunit space ofG. Indeed, its elements are units in the sense thatγs(γ) =γandr(γ)γ=γ.

The fibres corresponding to r,s:G→G⁽⁰⁾ are denoted respectively by G^x= r⁻¹(x)andG_x=s⁻¹(x). Given a subsetAofG⁽⁰⁾, thereductionof GtoAis the groupoidG|_A=r⁻¹(A)∩s⁻¹(A). Two elementsx,yofG⁽⁰⁾are said to be equivalent if G^x∩G_y6=/0. We denote by RG this equivalence relation. Given A⊂G⁽⁰⁾, its saturation[A]is the sets(r⁻¹(A))of all elements inG⁽⁰⁾that are equivalent to some element ofA. WhenA= [A], we say thatAisinvariant.

ABorel groupoidis a groupoidGendowed with a standard Borel structure such that the range, source, inverse and product are Borel maps, whereG⁽²⁾has the Borel structure induced byG×GandG⁽⁰⁾has the Borel structure induced byG. We say thatGiscountable(ordiscrete) if the fibresG^x(or equivalentlyG_x) are countable.

In the sequel, we only consider such groupoids. Wealways denote by X the set G⁽⁰⁾of units of G. Abisection Sis a Borel subset ofGsuch that the restrictions ofr andstoSare injective. A useful fact, consequence of a theorem of Lusin-Novikov, states that, sincerandsare countable-to-one Borel maps between standard Borel spaces, there exists a countable partition of Ginto bisections (see [20, Theorem 18.10]).

Letµbe a probability measure on onX=G⁽⁰⁾. We define aσ-finite measureν onGby the formula

Z

G

Fdν= Z

X

{γ∈G:s(γ)=x}

∑

F(γ)

dµ(x) (2.1)

for every non-negative Borel functionFonG. The fact thatx7→∑{γ:s(γ)=x}F(γ)is Borel is proved as in [14, Theorem 2] by using a countable partition ofGby Borel subsets on wichsis injective.

We say thatµisquasi-invariantifνis equivalent to it imageν⁻¹underγ7→γ⁻¹. In other terms, for every bisectionS, one hasµ(s(S)) =0 if and only ifµ(r(S)) =0.

This notion only depends on the measure class ofµ. We setδ = ^dν_dν⁻¹. Whenever ν=ν⁻¹, we say thatµisinvariant.

Definition 2.1 Acountable (or discrete) measured groupoid¹(G,µ)is a countable Borel groupoidGwith a quasi-invariant probability measureµonX=G⁽⁰⁾.

In the rest of this paper,Gwill always be equipped with the correspondingσ- finite measureνdefined in (2.1).

Examples 2.2 (a) LetΓ yXbe a (right) action of a countable groupΓ on a stan- dard Borel spaceX, and assume that the action preserves the class of a probability measureµ. LetG=XoΓbe thesemi-direct product groupoid. We haver(x,t) =x ands(x,t) =xt. The product is given by the formula(x,s)(xs,t) = (x,st). Equipped with the quasi-invariant measureµ,(G,µ)is a countable measured groupoid. As a particular case, we find the groupG=Γ whenXis reduced to a point.

(b) Another important family of examples concernscountable measured equiva- lence relations. A countable Borel equivalence relationR⊂X×X on a standard Borel space X is a Borel subset ofX×X whose equivalence classes are finite or countable. It has an obvious structure of Borel groupoid withr(x,y) =x,s(x,y) =y and(x,y)(x,z) = (x,z). When equipped with a quasi-invariant probability measure µ, we say that(R,µ)is a countable measured equivalence relation. Here, quasi- invariance also means that for every Borel subsetA⊂X, we haveµ(A) =0 if and only if the measure of the saturations(r⁻¹(A))ofAis still 0.

The orbit equivalence relation associated with an actionΓ y(X,µ)is denoted (RΓ,µ).

A general groupoid is a combination of an equivalence relation and groups.

Indeed, let (G,µ) be a countable measured groupoid. Let c= (r,s)be the map γ7→(r(γ),s(γ))fromGintoX×X. The range ofcis the graphRGof the equiva- lence relation induced onX byG. Moreover(RG,µ)is a countable measured equi- valence relation. The kernel of the groupoid homomorphismcis the subgroupoid {γ∈G:s(γ) =r(γ)}. For everyx∈X, the subsetG(x) =s⁻¹(x)∩r⁻¹(x), endowed with the induced operations, is a group, called theisotropy groupatx. So the kernel of cis the bundle of groupsx7→G(x)overX, called theisotropy bundle, andG appears as an extension of the equivalence relationRGby this bundle of groups.

A reduction(G_|U,µ|_U)such thatUis conull inXis calledinessential. Since we are working in the setting of measured spaces, it will make no difference to replace (G,µ)by any of its inessential reductions.

1In [8], a countable measured groupoid is calledr-discrete. Another difference is that we have swapped here the definitions ofνandν⁻¹.

2.2 The von Neumann algebra of (G, µ)

If f :G→Cis a Borel function, we set kfk_I=max

(

x7→

∑

r(γ)=x

|f(γ)|

∞

,

x7→

∑

s(γ)=x

|f(γ)|

∞

) .

LetI(G)be the set of functions such thatkfk_I<+∞. It only depends on the measure class ofµ. We endowI(G)with the (associative) convolution product

(f∗g)(γ) =

∑

γ₁γ₂=γ

f(γ₁)g(γ₂) =

∑

s(γ)=s(γ2)

f(γ γ₂⁻¹)g(γ₂) =

∑

r(γ1)=r(γ)

f(γ₁)g(γ₁⁻¹γ).

and the involution f^∗(γ) =f(γ⁻¹).

We haveI(G)⊂L¹(G,ν)∩L^∞(G,ν)⊂L²(G,ν), withkfk₁≤ kfk_I when f ∈ I(G). Therefore k·k_I is a norm onI(G), where two functions which coincide ν- almost everywhere are identified. It is easily checked thatI(G)is complete for this norm. Moreover forf,g∈I(G)we havekf∗gk_I≤ kfk_Ikgk_I. Therefore(I(G),k·k_I) is a Banach∗-algebra.

This variant of the Banach algebraI(G)introduced by Hahn [16] has been con- sidered by Renault in [29, p. 50]. Its advantage is that it does not involve the Radon- Nikodym derivativeδ.

For f∈I(G)we define a bounded operatorL(f)onL²(G,ν)by (L(f)ξ)(γ) = (f∗ξ)(γ) =

∑

γ1γ2=γ

f(γ₁)ξ(γ₂). (2.2)

We havekL(f)k ≤ kfk_I,L(f)^∗=L(f^∗)andL(f)L(g) =L(f∗g). Hence,Lis a representation ofI(G), called theleft regular representation.

Definition 2.3 The von Neumann algebra of the countable measured groupoid (G,µ)is the von Neumann subalgebraL(G,µ)ofB(L²(G,ν))generated byL(I(G)).

It will also be denoted byMin the rest of the paper.

Note thatL²(G,ν)is a direct integral of Hilbert spaces : L²(G,ν) =

Z ⊕ X

`²(G_x)dµ(x).

We define onL²(G,ν)a structure of L^∞(X)-module by(fξ)(γ) = f ◦s(γ)ξ(γ), where f ∈L^∞(X)andξ ∈L²(G,ν). In factL^∞(X)is the algebra of diagonalizable operators with respect to the above disintegration ofL²(G,ν).

Obviously, the representationLcommutes with this action ofL^∞(X). It follows that the elements ofL(G,µ)are decomposable operators ([13, Theorem 1, p. 164]).

We have L(f) =^R_X^⊕L_x(f)dµ(x), whereL_x(f):`<sup>2</sup>(G<sub>x</sub>)→`²(G_x)is defined as in (2.2), but forξ ∈`²(G_x).

LetC_n={1/n≤δ ≤n}. Then (C_n)is an increasing sequence of measurable subsets ofGwith∪_nC_n=G(up to null sets). We denote byI_n(G)the set of elements inI(G)taking value 0 outsideC_nand we setI∞(G) =∪_nI_n(G). Obviously,I∞(G) is an involutive subalgebra of I(G). It is easily checked that I∞(G)is dense into L²(G,ν)and thatL(G,µ)is generated byL(I_∞(G)).

The von Neumann algebra L^∞(X)is isomorphic to a subalgebra of I_∞(G), by giving to f ∈L^∞(X)the value 0 outsideX⊂G. Note that, forξ∈L²(G,ν),

(L(f)ξ)(γ) = f◦r(γ)ξ(γ).

In this way,A=L^∞(X)appears as a von Neumann subalgebra ofM.

Obviously, the pairA⊂Monly depends on the measure class ofµ, up to unitary equivalence.

We viewI(G)as a subspace ofL²(G,ν). The characteristic function1_XofX⊂G is a norm one vector inL²(G,ν). Letϕbe the normal state onMdefined by

ϕ(T) =h1_X,T1_Xi_L2(G,ν).

For f∈I(G), we haveϕ(L(f)) =^R_Xf(x)dµ(x), and therefore, for f,g∈I(G), ϕ(L(f)^∗L(g)) =hf,gi_L2(G,ν). (2.3) Lemma 2.4 Let g be a Borel function on G such thatδ^−1/2g= f ∈I(G)(for in- stance g∈I_∞(G)). Thenξ 7→ξ∗g is a bounded operator on L²(G,ν). More pre- cisely, we have

kξ∗gk₂≤ kfk_Ikξk₂. Proof. Straightforward. ut

We setR(g)(ξ) =ξ∗g. We haveL(f)◦R(g) =R(g)◦L(f)for everyg∈I_∞(G) and f ∈I(G). We denote by R(G,µ) the von Neumann algebra generated by R(I_∞(G)).

Lemma 2.5 The vector1_X is cyclic and separating for L(G,µ), and thereforeϕ is a faithful state.

Proof. Immediate from the fact thatL(f)andR(g)commute for f,g∈I_∞(G), with L(f)1_X=f andR(g)1_X=g, and from the density ofI_∞(G)intoL²(G,ν). ut

The von Neumann algebra L(G,µ) is on standard form on L²(G,ν), canoni- cally identified withL²(M,ϕ)(see (2.3)). We identifyM with a dense subspace of L²(G,ν)by T 7→Tb=T(1_X). The modular conjugation J and the one-parameter modular group σ relative to the vector 1_X (andϕ) have been computed in [16].

With our notations, we have

∀ξ∈L²(G,ν), (Jξ)(γ) =δ(γ)^1/2ξ(γ⁻¹) and

∀T ∈L(G,µ), σ_t(T) =δ^itTδ^−it.

Here, fort ∈R, the functionδ^it acts onL²(G,ν)by pointwise multiplication and defines a unitary operator. Note that forf∈L(G,µ), we haveδ^itL(f)δ^−it=L(δ^itf).

In particular,σacts trivially onA. Therefore (see [31]), there exists a unique faithful conditional expectationEA:M→Asuch thatϕ=ϕ◦EA, and forT∈M, we have

E\_A(T) =e_A(Tb),

wheree_Ais the orthogonal projection fromL²(G,ν)ontoL²(X,µ). If we view the elements ofM as functions on G, thenE_A is the restriction map toX. The triple (M,A,E_A)only depends on the class ofµ, up to equivalence.

For f∈I(G)andξ∈L²(G,ν)we observe that

(JL(f)J)ξ=R(g)ξ =ξ∗g with g=δ^1/2f^∗. (2.4)

3 Basic facts on the module L

(M)

We consider, in an abstract setting, the situation we have met above. LetA⊂M be a pair of von Neumann algebras, where A=L^∞(X,µ) is abelian. We assume the existence of a normal faithful conditional expectationE_A:M→Aand we set ϕ=τµ◦EA, whereτµ is the state onAdefined by the probability measureµ. Recall thatMis on standard form on the Hilbert spaceL²(M,ϕ)of the Gelfand-Naimark- Segal construction associated withϕ. We viewL²(M,ϕ)as a leftM-module and a rightA-module. Identifying²M with a subspace ofL²(M,ϕ), we know thatE_A is the restriction toMof the orthogonal projectione_A:L²(M,ϕ)→L²(A,τµ).

For further use, we make the following observation

∀m∈M,∀a∈A, mab =Ja^∗Jmb=mac=ma.b (3.5) Indeed, ifSis the closure of the mapmb7→mc^∗and ifS=J∆^1/2=∆^−1/2Jis its polar decomposition, then everya∈Acommutes with∆ since it is invariant underσ^ϕ. Then (3.5) follows easily. Note that (2.4) gives a particular case of this remark.

3.1 The commutant hM, e

i of the right action

The algebra of all operators which commute with the right action ofAis the von Neumann algebra of the basic construction forA⊂M. It is denotedhM,e_Aisince it is generated byMande_A. The linear span of{m₁e_Am₂:m₁,m₂∈M}is a∗-subalgebra which is weak operator dense inhM,e_Ai. MoreoverhM,e_Aiis a semi-finite von Neu-

2When necessary, we shall writembthe elementm∈M, when viewed inL²(M,ϕ), in order to stress this fact.

mann algebra, carrying a canonical normal faithful semi-finite trace Trµ(depending on the choice ofµ), defined by

Tr_µ(m₁e_Am₂) = Z

X

E_A(m₂m₁)dµ=ϕ(m₂m₁).

(for these classical results, see [19], [24]). We shall give more information on this trace in Lemma 3.4 and its proof. We need some preliminaries.

Definition 3.1 A vectorξ ∈L²(M,ϕ)isA-boundedif there existsc>0 such that kξak₂≤cτ_µ(a^∗a)^1/2for everya∈A.

We denote byL²(M,ϕ)⁰, orL²(M,ϕ), the subspace ofA-bounded vectors. It contains M. We also recall the obvious fact that T 7→T(1_A) is an isomorphism from the spaceB(L²(A,τµ)_A,L²(M,ϕ)_A)of bounded (right)A-linear operatorsT: L²(A,τµ)→L²(M,ϕ)ontoL²(M,ϕ). Forξ ∈L²(M,ϕ), we denote byL_ξ the corresponding operator fromL²(A,τµ)intoL²(M,ϕ). In particular, form∈M, we haveL_m=m_|

L2(A,τµ). It is easy to see thatL²(M,ϕ)is stable under the actions of

hM,eAiandA, and thatL_Tξa=T◦L_ξ◦aforT∈ hM,eAi,ξ∈L²(M,ϕ),a∈A.

Forξ,η∈L²(M,ϕ), the operatorL^∗_ξLη∈B(L²(A,τµ))is inA, since it com- mutes withA. We sethξ,ηi_A=L^∗

ξL_η. In particular, we havehm₁,m₂i_A=E_A(m^∗₁m₂) form1,m2∈M. TheA-valued inner producthξ,ηi_A=L^∗_ξLηgives toL²(M,ϕ)the structure of a self-dual Hilbert rightA-module [23]. It is a normed space with respect to the normkξk_L2(M)=khξ,ξi_Ak^1/2_A . Note that

kξk²_L2(M)=τ_µ(hξ,ξi_A)≤ kξk²_L2(M).

On the algebraic tensor productL²(M,ϕ)L²(A)a positive hermitian form is defined by

hξ⊗f,η⊗gi= Z

X

f ghξ,ηi_Adµ.

The Hilbert spaceL²(M,ϕ)⊗_AL²(A) obtained by separation and completion is isomorphic toL²(M,ϕ)as a rightA-module byξ⊗f7→ξf. Moreover the von Neu- mann algebraB(L²(M,ϕ)_A)of boundedA-linear endomorphisms ofL²(M,ϕ)is isomorphic to hM,e_AibyT 7→T⊗1. We shall identify these two von Neumann algebras (see [23], [30] for details on these facts).

Definition 3.2 Anorthonormal basisof theA-moduleL²(M,ϕ)is a family(ξ_i)of elements ofL²(M,ϕ)such that∑iξiA=L²(M,ϕ)and

ξi,ξj

A=δi,jp_jfor alli,j, where thep_jare projections inA.

It is easily checked thatL_ξ

iL^∗

ξ_i is the orthogonal projection onξiA, and that these projections are mutually orthogonal with∑iL_ξ

iL^∗

ξ_i=1.

Using a generalization of the Gram-Schmidt orthonormalization process, one shows the existence of orthonormal bases (see [23]).

Lemma 3.3 Let(ξ_i)be an orthonormal basis of the A-module L²(M,ϕ). For every ξ ∈L²(M,ϕ), we have (weak* convergence)

hξ,ξi_A=

∑

i

hξ,ξii_Ahξ_i,ξi_A. (3.6) Proof. Indeed

hξ,ξi_A=L^∗_ξL_ξ =L^∗_ξ(

∑

i

L_ξiL_ξ^∗

i)L_ξ =

∑

i

(L^∗_ξL_ξi)(L^∗_ξ

iL_ξ) =

∑

i

hξ,ξ_ii_Ahξ_i,ξi_A. u t

Lemma 3.4 Let(ξ_i)_i∈Ibe an orthonormal basis of the A-module L²(M,ϕ).

(i) For every x∈ hM,e_Ai₊we have Tr_µ(x) =

∑

i

τµ(hξ_i,xξ_ii_A) =

∑

i

hξ_i,xξ_ii_L2(M). (3.7)

(ii) span

L_ξL^∗_η:ξ,η∈L²(M,ϕ) is contained in the ideal of definition ofTr_µand we have, forξ,η∈L²(M,ϕ),

Tr_µ(L_ξL^∗_η) =τµ(L^∗_ηL_ξ) =τµ(hη,ξi_A). (3.8) Proof. (i) The mapU:L²(M,ϕ) =⊕_iξiA→ ⊕_ip_iL²(A)defined byU(ξ_ia) = p_ia is an isomorphism which identifies L²(M,ϕ) to the submodule p(`<sup>2</sup>(I)⊗L<sup>2</sup>(A)) of `²(I)⊗L²(A), with p=⊕_ip_i. The canonical trace onhM,e_Aiis transfered to the restriction to p B(`²(I))⊗A

p of the trace Tr⊗τ_µ, defined on T = [T_i,j]∈ (B(`²(I)⊗A)₊by(Tr⊗τµ)(T) =∑iτµ(T_ii). It follows that

Tr_µ(x) =

∑

i

τµ((U xU^∗)_ii) =

∑

i

hξ_i,xξ_ii_L2(M)=

∑

i

τµ(hξ_i,xξ_ii_A).

(ii) Takingx=L_ξL^∗

ξ in (i), the equality Tr_µ(L_ξL^∗

ξ) =τ_µ(hξ,ξi_A)follows from Equations (3.6) and (3.7). Formula (3.8) is deduced by polarization. ut

3.2 Compact operators

In a semi-finite von Neumann algebraN, there is a natural notion of ideal of compact operators, namely the norm-closed idealI(N)generated by its finite projections (see [25, Sect. 1.3.2] or [26]).

ConcerningN=hM,e_Ai, there is another natural candidate for the space of com- pact operators. First, we observe that givenξ,η∈L²(M,ϕ), the operatorL_ξL^∗_η∈ hM,e_Aiplays the role of a rank one operator in ordinary Hilbert spaces: indeed, if α∈L²(M,ϕ), we have(L_ξL^∗_η)(α) =ξhη,αi_A. In particular, form1,m2∈M, we

note thatm₁e_Am₂is a “rank one operator” sincem₁e_Am₂=L_m1L^∗_m∗

2. We denote by K(hM,e_Ai)the norm closure intohM,e_Aiof

span

L_ξL_η^∗:ξ,η∈L²(M,ϕ) . It is a two-sided ideal ofhM,e_Ai.

For everyξ∈L²(M,ϕ), we haveL_ξe_A∈ hM,e_Ai. Since L_ξL^∗_η= (L_ξe_A)(L_ηe_A)^∗

we see that K(hM,e_Ai) is the norm closed two-sided ideal generated by e_A in hM,e_Ai. The projectione_Abeing finite (because Trµ(e_A) =1), we have

K(hM,e_Ai)⊂I(hM,e_Ai).

The subtle difference between K(hM,eAi)andI(hM,e_Ai) is studied in [25, Sect. 1.3.2]. We recall in particular that for everyT∈I(hM,e_Ai)and everyε>0, there is a projection p∈Asuch thatτµ(1−p)≤ε andT J pJ∈K(hM,e_Ai)(see [25, Proposition 1.3.3 (3)]).³

3.3 The relative Haagerup property

LetΦ be a unital completely positive map fromM intoMsuch thatE_A◦Φ=E_A. Then form∈M, we have

kΦ(m)k²₂=ϕ(Φ(m)^∗Φ(m))≤ϕ(Φ(m^∗m)) =ϕ(m^∗m) =kmk²₂.

It follows that Φ extends to a contraction Φb of L²(M,ϕ). Whenever Φ is A- bimodular,Φb commutes with the right action ofA (due to (3.5)) and so belongs tohM,e_Ai. It also commutes with the left action ofAand so belongs toA⁰∩ hM,e_Ai.

Definition 3.5 We say thatMhas theHaagerup property(or property(H))relative to A and E_Aif there exists a net(Φ_i)of unitalA-bimodular completely positive maps fromMtoMsuch that

(i) E_A◦Φi=E_Afor alli; (ii) cΦ_i∈K(hM,e_Ai)for alli;

(iii) lim_ikΦ_i(x)−xk₂=0 for everyx∈M.

This notion is due to Boca [9]. In [25], Popa uses a slightly different formulation.

Lemma 3.6 In the previous definition, we may equivalently assume that cΦi ∈ I(hM,e_Ai)for every i.

3In [25],K(hM,e_Ai)is denotedI0(hM,e_Ai.

Proof. This fact is explained in [25]. LetΦ be a unitalA-bimodular completely positive map fromMtoMsuch thatE_A◦Φ=E_AandΦb∈I(hM,e_Ai). As already said, by [25, Proposition 1.3.3 (3)], for every ε>0, there is a projection pinA withτµ(1−p)<εandΦbJ pJ∈K(hM,e_Ai). Thus we havepΦJ pJb ∈K(hM,eAi).

Moreover, this operator is associated with the completely positive map Φp:m∈ M7→Φ(pmp), since

(pΦbJ pJ)(m) =b pΦb(mp) =b pΦb(mp) =c pΦ\(mp) =Φ\(pmp).

Then,Φ⁰=Φp+ (1−p)E_Ais unital, satisfiesE_A◦Φ⁰=E_Aand still provides an ele- ment ofK(hM,e_Ai). This modification allows to prove that if Definition 3.5 holds withK(hM,e_Ai)replaced byI(hM,e_Ai), then the relative Haagerup property is

satisfied (see [25, Proposition 2.2 (1)]). ut

3.4 Back to L

(G, ν )

We apply the facts just reminded toM=L(G,µ), which is on standard form on L²(G,ν) =L²(M,ϕ). This Hilbert space is viewed as a rightA-module: forξ ∈ L²(G,µ)andf ∈A, the action is given byξf◦s.

It is easily seen that L²(M,ϕ) is the space of ξ ∈L²(G,ν) such that x7→

∑s(γ)=x|ξ(γ)|²is inL^∞(X). Moreover, forξ,η∈L²(M,ϕ)we have hξ,ηi_A=

∑

s(γ)=x

ξ(γ)η(γ).

For simplicity of notation, we shall often identify f ∈I(G)⊂L²(G,ν)with the operatorL(f).⁴For instance, for f,g∈I(G), the operatorL(f)◦L(g)is also written

f∗g, and forT∈B(L²(G,µ)), we writeT◦f instead ofT◦L(f).

LetS⊂Gbe a bisection. Its characteristic function1_Sis an element ofI(G)and a partial isometry inMsince

1^∗_S∗1_S=1_s(S), and 1_S∗1^∗_S=1_r(S).

LetG=tS_nbe a countable partition ofGinto Borel bisections. Another straight- forward computation shows that (1_Sn)_n is an orthonormal basis of the right A- moduleL²(M,ϕ).

By Lemma 3.4, forx∈ hM,e_Ai₊we have Tr_µ(x) =

∑

n

h1_Sn,x1_Sni_L2(M).

4The reader should not confuseL(f):L²(G,ν)→L²(G,ν)with its restrictionLf:L²(A,τ_µ)→ L²(G,ν).

In particular, wheneverxis the multiplication operatorm(f)by some bounded non- negative Borel function f, we get

Tr_µ(m(f)) = Z

G

fdν. (3.9)

4 From completely positive maps to positive definite functions

Recall that ifGis a countable group, andΦ:L(G)→L(G)is a completely positive map, thent7→F_Φ(t) =τ(Φ(u_t)u_t^∗)is a positive definite function onG, whereτ is the canonical trace onL(G)andu_t,t∈G, are the canonical unitaries inL(G).

We want to extend this classical fact to the groupoid case. This was achieved by Jolissaint [18] for countable probability measure preserving equivalence relations.

Let(G,µ)be a countable measured groupoid andM=L(G,µ). LetΦ:M→M be a normalA-bimodular unital completely positive map. LetG=tS_nbe a partition into Borel bisections. We defineF_Φ:G→Cby

F_Φ(γ) =E_A(Φ(1_Sn)◦1^∗_Sn)◦r(γ), (4.10) whereS_nis the bisection which containsγ.

That FΦ does not depend (up to null sets) on the choice of the partition is a consequence of the following lemma.

Lemma 4.1 Let S1and S2be two Borel bisections. Then E_A(Φ(1_S1)◦1^∗_S

1) =E_A(Φ(1_S2)◦1^∗_S

2)

almost everywhere on r(S₁∩S₂).

Proof. Denote byethe characteristic function ofr(S₁∩S2). Thene∗1_S1=e∗1_S2= 1_S1∩S2. Thus we have

eEA(Φ(1_S1)◦1^∗_S

1)e=EA Φ(e∗1_S1)◦(1^∗_S

1∗e)

=E_A Φ(e∗1_S2)◦(1^∗_S

2∗e) =eE_A(Φ(1_S2)◦1^∗_S

2)e.

u t We now want to show thatF_Φ is a positive definite function in the following sense. We shall need some preliminary facts.

Definition 4.2 A Borel functionF:G→Cis said to bepositive definite if there exists aµ-null subsetNofX=G⁽⁰⁾such that for everyx∈/N, and everyγ1, . . . ,γk∈ G^x, thek×kmatrix[F(γ_i⁻¹γj)]is non-negative.

Definition 4.3 We say that aBorel bisection S is admissibleif there exists a constant c>0 such that 1/c≤δ(γ)≤calmost everywhere onS.

In other terms, 1_S∈I∞(G) and so the convolution to the right by 1_S defines a bounded operatorR(1_S)onL²(M,ϕ), by (2.4).

Lemma 4.4 Let S be a Borel bisection and let T ∈M. We have1\_S◦T =1_S∗T .b Moreover, if S is admissible, we have\T◦1_S=Tb∗1_S.

Proof. First, we observe that\1_S◦T =1_S◦T(1_X) =1_S∗Tb.

On the other hand, given f ∈I(G), we haveL(f)(1b_S) = f∗1_S. So, if(f_n)is a sequence inI(G)such that lim_nL(f_n) =T in the strong operator topology, we have

\T◦1_S=T(1b_S) =lim

n L(f_n)(1b_S) =lim

n f_n∗1_S

inL²(G,ν). But, whenS is admissible, the convolution to the right by 1_S is the bounded operatorR(1_S). Noticing that lim_n

f_n−Tb

2=0, it follows that

\T◦1_S=lim

n f_n∗1_S=Tb∗1_S.

u t

Lemma 4.5 Let T∈M, and let S be an admissible bisection. Then 1_S(γ)E_A(T◦1_S)(s(γ)) =1_S(γ)E_A(1_S◦T)(r(γ)) for almost everyγ.

Proof. We have

(\T◦1_S)(x) =

∑

γ₁γ₂=x

Tb(γ₁)1_S(γ₂) =Tb(γ₂⁻¹)

wheneverx∈s(S), whereγ2is the unique element ofSwiths(γ₂) =x. Otherwise (T◦1_S)(x) =0.

On the other hand,

(\1_S◦T)(x) =Tb(γ₁⁻¹)

wheneverx∈r(S), whereγ₁is the unique element ofSwithr(γ₁) =x. Otherwise (\1_S◦T)(x) =0. Our statement follows immediately. ut Lemma 4.6 F_Φ is a positive definite function.

Proof. We assume thatF_Φ is defined by equation (4.10) through a partition under admissible bisections. We setS_{i j}=S⁻¹_i S_j=

γ⁻¹γ⁰:γ∈S_i,γ⁰∈S_j . Note that1^∗_S

i∗ 1_Sj =1_{Si j}. Morever, theS_{i j}are admissible bisections. We set

Zi jm=n

x∈r(S_{i j}∩Sm):E_A(Φ(1_{Si j})◦1^∗_S

i j)(x)6=E_A(Φ(1_Sm)◦1^∗_S

m)(x)o andZ=∪_i,j,mZ_{i jm}. It is a null set by Lemma 4.1.

By Lemma 4.5, for everyithere is a null setE_i⊂r(S_i)such that forγ∈S_iwith r(γ)∈/E_iand for every j, we have

E_A(Φ(1_{Si j})◦1^∗_S

j◦1_Si)(s(γ)) =E_A(1_Si◦Φ(1_{Si j})◦1^∗_S

j)(r(γ))

We setE=∪_iE_i. LetY be the saturation ofZ∪E. It is a null set, sinceµis quasi- invariant.

Let x∈/ Y, and γ1, . . . ,γk ∈G^x. Assume that γ_i⁻¹γj ∈S⁻¹_n

i S_nj ∩S_m. We have r(γ_i⁻¹γj) =s(γ_i)∈/Ysincer(γ_i) =x∈/Y. Therefore,

F_Φ(γ_i⁻¹γ_j) =E_A(Φ(1_{Snin j})◦1^∗_S

n j◦1_Sni)(s(γ_i)).

Butγi∈Sn_i withr(γ_i) =x∈/Y, so E_A(Φ(1_{Snin j})◦1^∗_S

n j◦1_Sni)(s(γ_i)) =E_A(1_Sni◦Φ(1_{Snin j})◦1^∗_S

n j)(r(γ_i)).

Givenλ₁, . . . ,λ_k∈C, we have

k i,

∑

λiλjF_Φ(γ_i⁻¹γj) =E_A ^k

i=1

∑

(λ_i1_Sni)◦Φ(1^∗_S

ni◦1_{Sn j})◦

k

∑

(λ_j1_{Sn j})^∗ (x)≥0.

u t Obviously, ifΦ is unital,FΦ takes value 1 almost everywhere onX.

Proposition 4.7 We now assume that Φ is unital, with EA◦Φ =EA and Φb ∈ K(hM,e_Ai). Then, for everyε>0, we have

ν({|F_Φ|>ε}<+∞.

Proof. Let(S_n)be a partition ofGinto Borel bisections. Givenε>0 we choose ξ1, . . . ,ξk,η1, . . . ,ηk∈L²(M,ϕ)such that

Φb−

k i=1

∑

L_ξ

iL^∗_η

i

≤ε/2.

We viewΦb−∑^ki=1L_ξiL^∗_η

i as an element ofB(L²(M,ϕ)_A)and we apply it to1_Sn ∈ L²(M,ϕ). Then

Φ(1_Sn)−

k i=1

∑

ξihη_i,1_Sni_{A L2(M)}

≤ε/2k1_Snk_L2(M)≤ε/2.

Using the Cauchy-Schwarz inequality hξ,ηi^∗_Ahξ,ηi_A≤ kξk²_L2(M)hη,ηi_A, we get

*

1_Sn,Φ(1_Sn)−

k

∑

i=1

ξ_ihη_i,1_Sni_A +

A

≤

Φ(1_Sn)−

k

∑

i=1

ξ_ihη_i,1_Sni_{A L2}

(M)

≤ε/2.

We have, for almost everyγ∈S_nandx=s(γ),

|F_Φ(γ)|=

EA(Φ(1_Sn)◦1^∗_S

n)(r(γ)) =

EA(1^∗_S

n◦Φ(1_Sn))(x) =

h1_Sn,Φ(1_Sn)i_A(x)

≤

*

1_Sn,Φ(1_Sn)−

k

∑

i=1

ξihη_i,1_Sni_A +

A

(x)

+

k

∑

i=1

1_Sn,ξihη_i,1_Sni_A

A(x) .

The first term is≤ε/2 for almost everyx∈s(S_n). As for the second term, we have, almost everywhere,

h1_Sn,ξ_ii_A(x)hη_i,1_Sni_A(x)

≤ kξ_ik_L2(M)

hη_i,1_Sni_A(x) .

Hence, we get

|F_Φ(γ)| ≤ε/2+

k i=1

∑

kξ_ik_L2(M)

hη_i,1_Sni_A(s(γ))

for almost everyγ∈S_n. We want to estimate

ν({|F_Φ|>ε}) =

∑

n

ν({γ∈S_n:|F_Φ(γ)|>ε}).

For almost everyγ∈Snsuch that|F_Φ(γ)|>ε, we see that

k

∑

i=1

kξ_ik_L2(M)

hη_i,1_Sni_A(s(γ)) >ε/2.

Therefore

ν({|F_Φ|>ε})≤

∑

n

ν

( γ∈S_n:

k i=1

∑

kξ_ik_L2(M)

hη_i,1_Sni_A(s(γ)) >ε/2

)

≤

∑

n

µ

(

x∈s(S_n):

k

∑

i=1

kξ_ik_L2(M)

hη_i,1_Sni_A(x) >ε/2

)

.

Now,

k

∑

i=1

kξ_ik_L2(M)

hη_i,1_Sni_A(x) ≤(

k

∑

i=1

kξ_ik²_L2(M))^1/2

k

∑

i=1

hη_i,1_Sni_A(x)

21/2

.

We setc=∑^ki=1kξ_ik²_L2(M))andf_n(x) =∑^ki=1

hη_i,1_Sni_A(x)

2. We have

∑

f_n(x) =

∑

n k

∑

i=1

hη_i,1_Sni_A(x)

2=

k

∑

i=1

∑

n

hη_i,1_Sni_A(x)

2

=

k

∑

i=1

hη_i,η_ii_A(x)≤

k

∑

i=1

kη_ik²_L2(M),

since, by Lemma 3.3 (or directly here), hη_i,ηii_A=

∑

k

ηi,1_Sk

A

1_Sk,ηi

A=

∑

k

ηi,1_Sk

A

2.

We setd=∑^ki=1kη_ik²_L2(M). We have

ν({|F_Φ|>ε})≤

∑

n

µ(

x∈s(S_n):c f_n(x)>(ε/2)² .

We setα=c⁻¹(ε/2)². Denote byi(x)the number of indicesnsuch that f_n(x)>α. Theni(x)≤N, whereN is the integer part ofd/α. We denote byP={P_n}the set of subsets ofNwhose cardinal is≤N. Then there is a partitionX=t_mB_minto Borel subsets such that

∀x∈B_m, P_m={n∈N: f_n(x)>α}.

We have

ν({|F_Φ|>ε})≤

∑

n,m

µ({x∈B_m∩s(S_n):f_n(x)>α})

≤

∑

m

∑

n

µ({x∈B_m∩s(S_n): f_n(x)>α})

≤

∑

m

∑

n∈P_m

µ({x∈B_m∩s(S_n):f_n(x)>α})≤

∑

m

Nµ(B_m) =N

u t

5 From positive type functions to completely positive maps

Again, we want to extend a well known result in the group case, namely that, given a positive definite functionFon a countable groupG, there is a normal completely positive mapΦ :L(G)→L(G), well defined by the formulaΦ(u_t) =F(t)u_t for everyt∈G.

We need some preliminaries. For the notion of representation used below, see for instance [6, Sect. 3.1].

Lemma 5.1 Let F be a positive definite function on(G,µ). There exists a repre- sentationπ of G on a measurable fieldK ={K(x)}_x∈Xof Hilbert spaces, and a

measurable sectionξ :x7→ξ(x)∈K(x)such that F(γ) =hξ◦r(γ),π(γ)ξ◦s(γ)i

almost everywhere, that is F is the coefficient of the representationπ, associated withξ.

Proof. This classical fact may be found in [28]. The proof is straightforward, and similar to the classical GNS construction in the case of groups. LetV(x)the space of finitely supported complex-valued functions onG^x, endowed with the semi-definite positive hermitian form

hf,gi_x=

∑

γ₁,γ₂∈G^x

f(γ₁)g(γ₂)F(γ₁⁻¹γ2).

We denote byK(x)the Hilbert space obtained by separation and completion of V(x), andπ(γ):K(s(γ))→K(r(γ))is defined by(π(γ)f)(γ₁) = f(γ⁻¹γ1). The Borel structure on the field{K(x)}_x∈X is provided by the Borel functions onG whose restriction to the fibresG^xare finitely supported. Finally,ξ is the characte-

ristic function ofX, viewed as a Borel section. ut

Now we assume thatF(x) =1 for almost everyx∈X, and thusξis a unit section.

We consider the measurable field

`²(G_x)⊗K(x) _x∈X. Note that

`<sup>2</sup>(G<sub>x</sub>)⊗K(x) =`²(G_x,K(x)).

Let f ∈`<sup>2</sup>(G<sub>x</sub>). We defineS<sub>x</sub>(f)∈`²(G_x,K(x))by Sx(f)(γ) =f(γ)π(γ)^∗ξ◦r(γ) forγ∈Gx. Then

∑

s(γ)=x

kS_x(f)(γ)k²_K(x)=kfk_`2(Gx).

The field(S_x)_x∈X of operators defines an isometry S:L²(G,ν)→

Z ⊕ X

`²(G_x,K(x))dµ(x), by

S(f)(γ) =f(γ)π(γ)^∗ξ◦r(γ).

Note that^R_X^⊕`²(G_x,K(x))dµ(x)is a rightA-module, by (ηa)_x=η_xa(x):γ∈G_x7→η(γ)a◦s(γ).

Of course,Scommutes with the right actions ofA. We also observe that, as a rightA- module,L²(M,ϕ)⊗_A^R_X^⊕K(x)dµ(x)and^R_X^⊕`²(G_x,K(x))dµ(x)are canonically isomorphic under the map

ζ⊗η7→ζ η◦s, ∀ζ ∈L²(M,ϕ),∀η∈ Z ⊕

X K(x)dµ(x),

where(ζ η◦s)_xis the functionγ∈G_x7→ζ(γ)η◦s(γ)in`<sup>2</sup>(G<sub>x</sub>,K(x)). It follows thatMacts on<sup>R</sup><sub>X</sub><sup>⊕</sup>`²(G_x,K(x))dµ(x)bym7→m⊗Id . In particular, for f ∈I(G), we see thatL(f)⊗Id , viewed as an operator on^R_X^⊕`²(G_x,K(x))dµ(x), is acting as

((L(f)⊗Id)η)(γ) =

∑

γ₁γ₂=γ

f(γ₁)η(γ₂)∈K(s(γ)).

Lemma 5.2 For f∈I(G), we have S^∗ L(f)⊗Id

S=L(F f).

Proof. A straightforward computation shows that forη∈^R_X^⊕`²(G_x,K(x))dµ(x), we have

(S^∗η)(γ) =hπ(γ)^∗ξ◦r(γ),η(γ)i_K(s(γ)). Moreover, givenh∈L²(G,ν), we have

(L(f)⊗Id)S h

(γ) =

∑

γ1γ2=γ

f(γ₁)(S h)(γ₂) =

∑

γ1γ2=γ

f(γ₁)h(γ₂)π(γ₂)^∗ξ◦r(γ₂).

Hence,

S^∗(L(f)⊗Id)S h (γ) =

*

π(γ)^∗ξ◦r(γ),

∑

γ₁γ₂=γ

f(γ₁)h(γ₂)π(γ₂)^∗ξ◦r(γ₂) +

=

*

ξ◦r(γ),

∑

γ₁γ₂=γ

f(γ₁)h(γ₂)π(γ₁)ξ◦r(γ₂) +

=

∑

γ1γ2=γ

f(γ₁)h(γ₂)hξ◦r(γ₁),π(γ₁)ξ◦r(γ₂)i

=

∑

γ₁γ₂=γ

f(γ₁)F(γ₁)h(γ₂) = (L(F f)h)(γ).

u t Proposition 5.3 Let F:G→Cbe a Borel positive type function on G such that F_|X =1. Then there exists a unique normal completely positive mapΦ from M into M such thatΦ(L(f)) =L(F f)for every f∈I(G). Morever,Φis A-bimodular, unital and E_A◦Φ=E_A.

Proof. The uniqueness is a consequence of the normality ofΦ, combined with the density ofL(I(G))intoM. With the notation of the previous lemma, form∈Mwe putΦ(m) =S^∗ m⊗Id

S. Obviously,Φ satisfies the required conditions. ut Remark 5.4 We keep the notation of the previous proposition. A straightforward computation shows thatF is the positive definite functionF_Φconstructed fromΦ. Proposition 5.5 Let F be a Borel positive definite function on G such that F_|X = 1. We assume that for everyε>0, we have ν({|F|>ε})<+∞. Let Φ be the