New York Journal of Mathematics New York J. Math.

New York J. Math. 11(2005)225–245.

Isomorphic groupoid C

-algebras associated with different Haar systems

M˘ ad˘ alina Roxana Buneci

Abstract. We shall consider a locally compact groupoid endowed with a Haar systemνand having proper orbit space. We shall associate to each appropriate cross sectionσ:G⁽⁰⁾→G^F ford_F:G^F →G⁽⁰⁾(whereF is a Borel subset of G⁽⁰⁾meeting each orbit exactly once) aC^∗-algebraM_σ^∗(G, ν). We shall prove that theC^∗-algebras associated with different Haar systems are∗-isomorphic.

Contents

1. Introduction 225

2. Basic definitions and notations 227

3. The decomposition of a Haar system over the principal groupoid 229 4. AC^∗-algebra associated to a locally compact groupoid 232

5. The case of locally transitive groupoids 236

6. The case of principal proper groupoids 241

References 244

1. Introduction

The reader is referred to Section 2 for the basic defintions and notations we shall use here.

TheC^∗-algebra of a locally compact groupoid was introduced by J. Renault in [9]. The construction extends the case of a group: the space of continuous functions with compact support on the groupoid is made into a∗-algebra and endowed with the smallestC^∗-norm making its representations continuous. In order to define the convolution on the groupoid one needs to assume the existence of a Haar system which is an analogue of Haar measure on a group. Unlike the case for groups, Haar systems need not be unique. A result of Paul Muhly, Jean Renault and Dana Williams establishes that theC^∗-algebras of Gassociated with two Haar systems are strongly Morita equivalent [4, Theorem 2.8, p. 10]. If the groupoidGis transitive

Received February 12, 2004, and in revised form on June 6, 2005.

Mathematics Subject Classification. 22A22, 43A22, 43A65, 46L99.

Key words and phrases. locally compact groupoid,C^∗-algebra,∗-isomorphism.

This work was partly supported by the MEC-CNCSIS grant At127/2004 and by the Postdoc- toral Training Program HPRN-CT-2002-0277.

ISSN 1076-9803/05

225

they have proved that the C^∗-algebra of G is isomorphic to C^∗(H)⊗ K(L²(μ)), whereH is the isotropy group G^u_u at any unitu∈G⁽⁰⁾,μis an essentially unique measure onG⁽⁰⁾,C^∗(H) denotes the groupC^∗-algebra ofH, andK(L²(μ)) denotes the compact operators onL²(μ) [4, Theorem 3.1, p. 16]. Therefore theC^∗-algebras of atransitive groupoidGassociated with two Haar systems are∗-isomorphic.

In [8] Arlan Ramsay and Martin E. Walter have associated to a locally compact groupoid Ga C^∗-algebra denoted M^∗(G, ν). They have considered the universal representationω ofC^∗(G, ν) — the usual C^∗-algebra associated to a Haar system ν = {ν^u, u ∈ G⁽⁰⁾} (constructed as in [9]). Since every cyclic representation of C^∗(G, ν) is the integrated form of a representation of G, it follows that ω can be also regarded as a representation of Bc(G), the space of compactly supported bounded Borel functions on G. Arlan Ramsay and Martin E. Walter have used the notation M^∗(G, ν) for the operator norm closure of ω(Bc(G)). Since ω is an

∗-isomorphism on C^∗(G, ν), we can regardC^∗(G, ν) as a subalgebra ofM^∗(G, ν).

Definition 1. A locally compact groupoidGis proper if the map (r, d) :G→G⁽⁰⁾×G⁽⁰⁾

is proper (i.e., the inverse image of each compact subset ofG⁽⁰⁾×G⁽⁰⁾ is compact) [1, Definition 2.1.9].

Throughout this paper we shall assume that G is a second countable locally compact groupoid for which the orbit space is Hausdorff and the map

(r, d) :G→R,(r, d)(x) = (r(x), d(x))

is open, whereRis endowed with the product topology induced fromG⁽⁰⁾×G⁽⁰⁾. ThereforeR will be a locally compact groupoid. The fact thatRis a closed subset of G⁽⁰⁾×G⁽⁰⁾ and that it is endowed with the product topology is equivalent to the factR is a proper groupoid.

Throughout this paper by a groupoid with proper orbit space we shall mean a groupoidGfor which the orbit space is Hausdorff and the map

(r, d) :G→R,(r, d)(x) = (r(x), d(x))

is open, whereRis endowed with the product topology induced fromG⁽⁰⁾×G⁽⁰⁾. Let us give an example of a groupoid with proper orbit space that is not a proper groupoid. First let us make some remarks. Any locally compact principal groupoid can be viewed as an equivalence relation on a locally compact space X having its graph E ⊂X ×X endowed with a locally compact topology compatible with the groupoid structure. This topology can be finer than the product topology induced fromX×X. E is proper if and only ifEis endowed with the product topology and E is closed in X×X. LetE ⊂X×X be a proper principal groupoid and let Γ be a locally compact group. ThenE ×Γ is a groupoid under the following operations:

(u, v, x)⁻¹= (v, u, x⁻¹) (u, v, x)(v, w, y) = (u, w, xy).

It is easy to see that E ×Γ is a groupoid with proper orbit space. If Γ is not a compact group, thenE ×Γ is not a proper groupoid.

We shall assume that the orbit space of the groupoidGis proper and we shall choose a Borel subsetFofG⁽⁰⁾meeting each orbit exactly once and such thatF∩[K]

has a compact closure for each compact subset K of G⁽⁰⁾. For each appropriate cross sectionσ:G⁽⁰⁾→G^F ford_F :G^F →G⁽⁰⁾,d_F(x) =d(x), we shall construct a C^∗-algebra M_σ^∗(G, ν) which can be viewed as a subalgebra of M^∗(G, ν). Given two Haar systemν₁={ν₁^u, u∈G⁽⁰⁾}andν₂={ν₂^u, u∈G⁽⁰⁾}onG, we shall prove that theC^∗-algebrasM_σ^∗(G, ν₁) andM_σ^∗(G, ν₂) are∗-isomorphic.

For a transitive (or more generally, a locally transitive) groupoid G we shall prove that theC^∗-algebrasC^∗(G, ν),M^∗(G, ν) andM_σ^∗(G, ν) coincide.

IfGis a locally transitive groupoid endowed with a Haar system{ν^u, u∈G⁽⁰⁾}, then it is the topological disjoint union of its transitivity components G|_[u], and C^∗(G, ν) is the direct sum of theC^∗(G|_[u], ν_[u]), whereν_[u]={ν^s, s∈[u]}. This is a consequence of [2, Theorem 1, p. 10].

For a principal proper groupoidG, we shall prove that C^∗(G, ν)⊂M_σ^∗(G, ν)⊂M^∗(G, ν).

Letπ:G⁽⁰⁾→G⁽⁰⁾/Gbe the quotient map and let ν_i=

ε_u×μ^π(u)_i , u∈G⁽⁰⁾

, i= 1,2

be two Haar systems on the principal proper groupoidG. We shall also prove that if the Hilbert bundles determined by the systems of measures{μ^u_i^˙}u_˙ have continuous bases in the sense of Definition 24, then ∗-isomorphism between M_σ^∗(G, ν₁) and M_σ^∗(G, ν₂) can be restricted to a∗-isomorphism betweenC^∗(G, ν₁) andC^∗(G, ν₂).

2. Basic definitions and notations

For establishing notation, we include some definitions that can be found in several places (e.g., [9], [5]). A groupoid is a setGendowed with a product map

(x, y)→xy [:G⁽²⁾→G]

whereG⁽²⁾ is a subset ofG×Gcalled the set of composable pairs, and an inverse map

x→x⁻¹ [:G→G]

such that the following conditions hold:

(1) If (x, y) ∈ G⁽²⁾ and (y, z) ∈ G⁽²⁾, then (xy, z) ∈ G⁽²⁾, (x, yz) ∈ G⁽²⁾ and (xy)z=x(yz).

(2) (x⁻¹)⁻¹=xfor allx∈G.

(3) For allx∈G, (x, x⁻¹)∈G⁽²⁾, and if (z, x)∈G⁽²⁾, then (zx)x⁻¹=z.

(4) For allx∈G, (x⁻¹, x)∈G⁽²⁾, and if (x, y)∈G⁽²⁾, thenx⁻¹(xy) =y.

The mapsranddonG, defined by the formulaer(x) =xx⁻¹andd(x) =x⁻¹x, are called the range and the source maps. It follows easily from the definition that they have a common image called the unit space ofG, which is denotedG⁽⁰⁾. Its elements are units in the sense that xd(x) = r(x)x = x. Units will usually be denoted by letters asu, v, wwhile arbitrary elements will be denoted byx, y, z. It is useful to note that a pair (x, y) lies inG⁽²⁾precisely whend(x) =r(y), and that the cancellation laws hold (e.g.,xy=xz iffy=z). The fibres of the range and the source maps are denoted G^u =r⁻¹({u}) and G_v =d⁻¹({v}), respectively. More generally, given the subsetsA,B⊂G⁽⁰⁾, we defineG^A=r⁻¹(A),G_B=d⁻¹(B) and G^A_B=r⁻¹(A)∩d⁻¹(B). The reduction ofGtoA⊂G⁽⁰⁾isG|A=G^A_A. The relation

u∼viffG^u_v =φis an equivalence relation onG⁽⁰⁾. Its equivalence classes are called orbits and the orbit of a unit uis denoted [u]. A groupoid is called transitive iff it has a single orbit. The quotient space for this equivalence relation is called the orbit space ofGand denotedG⁽⁰⁾/G. We denote byπ:G⁽⁰⁾→G⁽⁰⁾/G, π(u) = ˙u the quotient map. A subset of G⁽⁰⁾ is said saturated if it contains the orbits of its elements. For any subsetAofG⁽⁰⁾, we denote by [A] the union of the orbits [u] for allu∈A.

A topological groupoid consists of a groupoidGand a topology compatible with the groupoid structure. This means that:

(1) x→x⁻¹ [:G→G] is continuous.

(2) (x, y)→xy [:G⁽²⁾→G] is continuous whereG⁽²⁾has the induced topology fromG×G.

We are exclusively concerned with topological groupoids which are second count- able, locally compact Hausdorff. It was shown in [7] that measured groupoids may be assume to have locally compact topologies, with no loss in generality.

If X is a locally compact space, C_c(X) denotes the space of complex-valuated continuous functions with compact support. The Borel sets of a topological space are taken to be theσ-algebra generated by the open sets. The space of compactly supported bounded Borel functions onX is denoted byBc(X).

For a locally compact groupoidG, we denote by G={x∈G: r(x) =d(x)} the isotropy group bundle ofG. It is closed inG.

Let G be a locally compact second countable groupoid equipped with a Haar system, i.e., a family of positive Radon measures onG,{ν^u, u∈G⁽⁰⁾}, such that:

(1) For allu∈G⁽⁰⁾, supp(ν^u) = G^u. (2) For allf ∈C_c(G),

u→

f(x)dν^u(x) [:G⁽⁰⁾→C] is continuous.

(3) For allf ∈C_c(G) and allx∈G,

f(y)dν^r(x)(y) =

f(xy)dν^d(x)(y).

As a consequence of the existence of continuous Haar systems,r, d:G→G⁽⁰⁾are open maps ([11]). Therefore, in this paper we shall always assume thatr:G→G⁽⁰⁾ is an open map

Ifμis a measure onG⁽⁰⁾, then the measureν =

ν^udμ(u), defined by

f(y)dν(y) = f(y)dν^u(y)

dμ(u), f ≥0 Borel

is called the measure on G induced by μ. The image of ν by the inverse map x→x⁻¹is denoted ν⁻¹. μis said to be quasi-invariant if its induced measureν is equivalent to its inverse,ν⁻¹. A measure belongings to the class of a quasi-invariant measure is also quasi-invariant. We say that the class is invariant.

If μ is a quasi-invariant measure on G⁽⁰⁾ and ν is the measure induced on G, then the Radon–Nikodym derivative Δ = _dν^dν₋₁ is called the modular function ofμ.

In order to define theC^∗-algebra of a groupoidG, the spaceC_c(G) of continuous functions with compact support onG, endowed with the inductive limit topology, is made into a topological ∗-algebra and is given the smallest C^∗-norm making its representations continuous. In somewhat more detail, for f, g ∈ C_c(G) the convolution is defined by:

f ∗g(x) =

f(xy)g(y⁻¹)dν^d(x)(y) and the involution by

f^∗(x) =f(x⁻¹).

Under these operations,C_c(G) becomes a topological∗-algebra.

A representation ofC_c(G) is a∗-homomorphism fromC_c(G) intoB(H), for some Hilbert spaceH, that is continuous with respect to the inductive limit topology on C_c(G) and the weak operator topology on B(H). The full C^∗-algebra C^∗(G) is defined as the completion of the involutive algebra C_c(G) with respect to the full C^∗-norm

f= supL(f)

whereLruns over all nondegenerate representations ofC_c(G) which are continuous for the inductive limit topology.

Every representation (μ, G⁽⁰⁾∗ H, L) [5, Definition 3.20/p. 68] of Gcan be in- tegrated into a representation, still denoted byL, of C_c(G). The relation between the two representation is:

L(f)ξ₁, ξ₂=

f(x)L(x)ξ₁(d(x)), ξ₂(r(x))Δ⁻¹²(x)dν^u(x)dμ(u) wheref ∈C_c(G),ξ₁, ξ₂∈_⊕

G⁽⁰⁾H(u)dμ(u).

Conversely, every nondegenerate ∗-representation of C_c(G) is obtained in this fashion (see [9] or [5]).

3. The decomposition of a Haar system over the principal groupoid

First we present some results on the structure of the Haar systems, as developed by J. Renault in Section 1 of [10] and also by A. Ramsay and M.E. Walter in Section 2 of [8].

In Section 1 of [10] Jean Renault constructs a Borel Haar system forG. One way to do this is to choose a function F₀ continuous with conditionally compact support which is nonnegative and equal to 1 at each u ∈ G⁽⁰⁾. Then for each u∈G⁽⁰⁾choose a left Haar measureβ_u^uonG^u_uso the integral ofF₀with respect to β_u^u is 1.

Renault defines β_v^u=xβ_v^v ifx∈G^u_v (wherexβ_v^v(f) =

f(xy)dβ_v^v(y) as usual).

Ifzis another element inG^u_v, thenx⁻¹z∈G^v_v, and sinceβ^v_v is a left Haar measure onG^v_v, it follows thatβ^u_v is independent of the choice ofx. IfKis a compact subset ofG, then sup

u,v

β^u_v(K)<∞. Renault also defines a 1-cocycleδ onGsuch that for everyu∈G⁽⁰⁾,δ|G^u_u is the modular function forβ_u^u. δandδ⁻¹= 1/δ are bounded on compact sets inG.

Let

R= (r, d)(G) ={(r(x), d(x)), x∈G}

be the graph of the equivalence relation induced onG⁽⁰⁾. ThisRis the image ofG under the homomorphism (r, d), so it is aσ-compact groupoid. With this apparatus in place, Renault describes a decomposition of the Haar system{ν^u, u∈G⁽⁰⁾} for G over the equivalence relation R (the principal groupoid associated to G). He proves that there is a unique Borel Haar systemαforRwith the property that

ν^u=

β_t^sdα^u(s, t) for allu∈G⁽⁰⁾. In Section 2 of [8] A. Ramsay and M.E. Walter prove that

sup

u

α^u((r, d)(K))<∞, for all compactK⊂G

For eachu∈G⁽⁰⁾ the measureα^u is concentrated on{u} ×[u]. Therefore there is a measure μ^u concentrated on [u] such thatα^u =ε_u×μ^u, whereε_u is the unit point mass at u. Since{α^u, u ∈G⁽⁰⁾} is a Haar system, we have μ^u =μ^v for all (u, v)∈R, and the function

u→

f(s)μ^u(s)

is Borel for allf ≥0 Borel onG⁽⁰⁾. For eachuthe measure μ^u is quasi-invariant (see Section 2 of [8]). Thereforeμ^u is equivalent tod_∗(v^u) [6, Lemma 4.5/p. 277].

If η is a quasi-invariant measure for{ν^u, u∈G⁽⁰⁾}, thenη is a quasi-invariant measure for {α^u, u ∈ G⁽⁰⁾}. Also if Δ_R is the modular function associated to {α^u, u ∈ G⁽⁰⁾} and η, then Δ = δΔ_R◦(r, d) can serve as the modular function associated to{ν^u, u∈G⁽⁰⁾} andη.

Sinceμ^u=μ^v for all (u, v)∈R, the system of measures{μ^u}u may be indexed by the elements of the orbit spaceG⁽⁰⁾/G.

Definition 2. We shall call the pair of systems of measures ({β_v^u}_(u,v)∈R,{μ^u˙}_u∈G˙ (0)/G)

(described above) the decomposition of the Haar system {ν^u, u ∈ G⁽⁰⁾} over the principal groupoid associated toG. Also we shall callδthe 1-cocycle associated to the decomposition.

Remark 3. Let us note that up to trivial changes in normalization, the system of measures {β^u_v} and the 1-cocycle in the preceding definition are unique. They do not depend on the Haar system, but only on the continuous functionF₀.

Lemma 4. Let G be a locally compact second countable groupoid such that the bundle mapr|G of G is open. Let {ν^u, u∈G⁽⁰⁾} be a Haar system on Gand let ({β_v^u},{μ^u˙})be its decomposition over the principal groupoid associated toG. Then for eachf ∈C_c(G)the function

x→

f(y)dβ_d(x)^r(x)(y) is continuous on G.

Proof. By Lemma 1.3/p. 6 of [10], for eachf ∈C_c(G) the function u→

f(y)dβ_u^u(y) is continuous.

Let x∈ Gand (x_i)_i be a sequence in G converging to x. Let f ∈ C_c(G) and letg be a continuous extension onGof y→f(xy) [:G^d(x)→C]. LetK be the compact set

({x, x_i, i= 1,2, . . .}⁻¹supp(f)∪supp(g))∩r⁻¹({d(x), d(x_i), i= 1,2, . . .}).

We have

f(y)dβ_d(x)^r(x)(y)−

f(y)dβ_d(x^r(xi)

i)(y) =

f(xy)dβ_d(x)^d(x)(y)−

f(x_iy)dβ_d(x^d(xi)

i)(y)

=

g(y)dβ_d(x)^d(x)(y)−

f(x_iy)dβ_d(x^d(xi)

i)(y)

≤

g(y)dβ_d(x)^d(x)(y)−

g(y)dβ_d(x^d(xi)

i)(y) +

g(y)dβ^d(x_d(xⁱ⁾

i)(y)−

f(x_iy)dβ_d(x^d(xi)

i)(y)

≤

g(y)dβ_d(x)^d(x)(y)−

g(y)dβ_d(x^d(xi)

i)(y) + sup

y∈G^d(_d(^xi_xi⁾₎

|g(y)−f(x_iy)|β_d(x^d(xi)

i)(K).

A compactness argument shows that sup

y∈G^d(_d(^xi_xi⁾₎|g(y)−f(x_iy)| converges to 0. Also

g(y)dβ_d(x)^d(x)(y)−

g(y)dβ_d(x^d(xi)

i)(y) converges to 0 because the function u→

f(y)dβ_u^u(y) is continuous. Hence

f(y)dβ_d(x)^r(x)(y)−

f(y)dβ_d(x^r(xi)

i)(y)

converges to 0.

Proposition 5. Let Gbe a second countable locally compact groupoid with proper orbit space. Let {ν^u, u ∈ G⁽⁰⁾} be a Haar system on G and let ({β_v^u},{μ^u˙}) be its decomposition over the principal groupoid associated to G. Then for each g ∈ C_c(G⁽⁰⁾), the map

u→

g(v)dμ^π(u)(v) is continuous.

Proof. Let g ∈ C_c(G⁽⁰⁾) andu₀ ∈G⁽⁰⁾. Let K₁ be a compact neighborhood of u₀ and K₂ be the support of g. Since G is locally compact and (r, d) is open fromGto (r, d)(G), there is a compact subsetK ofGsuch that (r, d)(K) contains (K₁×K₂)∩(r, d)(G). Let F₁∈C_c(G) be a nonnegative function equal to 1 on a compact neighborhoodU ofK. LetF₂ ∈C_c(G) be a function which extends toG

the functionx→F₁(x)/

F₁(y)dβ_d(x)^r(x)(y),x∈U. We have

F₂(y)dβ_v^u(y) = 1 for all (u, v)∈(r, d)(K). Since for allu∈K₁,

g(v)dμ^π(u)(v) =

g(v)

F₂(y)dβ_v^u(y)dμ^π(u)(v)

=

g(d(y))F₂(y)dν^u(y), it follows thatu→

g(v)dμ^π(u)(v) is continuous atu₀. Remark 6. Let G be a locally compact second countable groupoid with proper orbit space. Let {ν^u, u ∈ G⁽⁰⁾}be a Haar system on G and ({β_v^u},{μ^u˙}) be its decomposition over the associated principal groupoid. If μ is a quasi-invariant probability measure for the Haar system, thenμ₁=

μ^π(u)dμ(u) is a Radon mea- sure which is equivalent toμ. Indeed, letf ≥0 Borel onG⁽⁰⁾ such thatμ(f) = 0.

Sinceμis quasi-invariant, it follows that forμa.a.u,ν^u(f◦d) = 0, and sinceμ^π(u) is equivalent to d_∗(v^u), it follows that μ^π(u)(f) = 0 for μ a.a. u. Conversely, if μ₁(f) = 0, then μ^π(u)(f) = 0 for μa.a. u, and thereforeν^u(f ◦d) = 0. Thus the quasi-invariance ofμimplies μ(f) = 0. Thus each Radon quasi-invariant measure is equivalent to a Radon measure of the form

μ^u˙dμ( ˙u), whereμis a probability measure on the orbit spaceG/G⁽⁰⁾.

4. A C

-algebra associated to a locally compact groupoid with proper orbit space

LetGbe a locally compact second countable groupoid with proper orbit space.

Let

π:G⁽⁰⁾ →G⁽⁰⁾/G

be the quotient map. Since the quotient space is proper,G⁽⁰⁾/G is Hausdorff.

As we mentioned at the outset, our standing hypothesis that G has a Haar system guarantees thatr is open. Consequently, so is the mapπ.

Applying Lemma 1.1 of [3] to the locally compact second countable spacesG⁽⁰⁾ and G⁽⁰⁾/G and to the continuous open surjection π: G⁽⁰⁾ → G⁽⁰⁾/G, it follows that there is a Borel setF inG⁽⁰⁾ such that:

(1) F contains exactly one element in each orbit [u] =π⁻¹(π(u)).

(2) For each compact subsetKofG⁽⁰⁾,F∩[K] =F∩π⁻¹(π(K)) has a compact closure.

For each unit u let us define e(u) to be the unique element in the orbit of u that is contained in F, i.e., {e(u)} =F ∩[u]. For each Borel subset B of G⁽⁰⁾, πis continuous and one-to-one on B∩F and hence π(B∩F) is Borel in G⁽⁰⁾/G.

Therefore the map e : G⁽⁰⁾ → G⁽⁰⁾ is Borel (for each Borel subset B of G⁽⁰⁾, e⁻¹(B) = [B∩F] =π⁻¹(π(B∩F)) is Borel inG⁽⁰⁾). Also for each compact subset K ofG⁽⁰⁾,e(K) has a compact closure becausee(K)⊂F∩[K].

Since the orbit spaceG⁽⁰⁾/Gis proper the map

(r, d) :G→R,(r, d)(x) = (r(x), d(x))

is open andR is closed inG⁽⁰⁾×G⁽⁰⁾. Applying Lemma 1.1 of [3] to the locally compact second countable spaces Gand R and to the continuous open surjection

(r, d) :G→ R, it follows that there is a regular cross section σ₀ : R → G. This means that σ₀ is Borel, (r, d)(σ₀(u, v)) = (u, v) for all (u, v) ∈ R, and σ₀(K) is relatively compact inGfor each compact subsetK ofR.

Let us define σ:G⁽⁰⁾ →G^F by σ(u) =σ₀(e(u), u) for allu. It is easy to note thatσis a cross section ford:G^F →G⁽⁰⁾andσ(K) is relatively compact inGfor all compactK⊂G⁽⁰⁾. IfF is closed, then σis regular.

Replacingσby

v→σ(e(v))⁻¹σ(v)

we may assume thatσ(e(v)) =e(v) for allv. Let us defineq:G→G^F_F by q(x) =σ(r(x))xσ(d(x))⁻¹, x∈G.

Let ν = {ν^u : u∈ G⁽⁰⁾} be a Haar system on G and let ({β_v^u},{μ^u˙}) be its decompositions over the principal groupoid . Letδ be the 1-cocycle associated to the decomposition.

Let us denote byBσ(G) the linear span of the functions of the form x→g₁(r(x))g(q(x))g₂(d(x))

where g₁, g₂ are compactly supported bounded Borel functions onG⁽⁰⁾ andg is a bounded Borel function onG^F_F such that ifSis the support ofg, then the closure of S is compact inG. Bσ(G) is a subspace ofBc(G), the space of compactly supported bounded Borel functions onG.

Iff₁,f₂∈ Bσ(G) are defined by

f₁(x) =g₁(r(x))g(q(x))g₂(d(x)) f₂(x) =h₁(r(x))h(q(x))h₂(d(x)) then

f₁∗f₂(x) =g∗h(q(x))g₁(r(x))h₂(d(x))g₂, h_1π(r(x)) f₁^∗(x) =g₂(r(x))g(q(x)⁻¹)g₁(d(x)).

ThusBσ(G) is closed under convolution and involution.

Letωbe the universal representation ofC^∗(G, ν) the usualC^∗-algebra associated to a Haar system ν ={ν^u, u ∈ G⁽⁰⁾} (constructed as in [9]). Since every cyclic representation ofC^∗(G, ν) is the integrated form of a representation ofG, it follows that ω can be also regarded as a representation of B_c(G), the space of compactly supported bounded Borel functions on G. Arlan Ramsay and Martin E. Walter have used the notationM^∗(G, ν) for the operator norm closure ofω(B_c(G)). Since ω is an∗-isomorphism on C^∗(G, ν), we can regarded C^∗(G, ν) as a subalgebra of M^∗(G, ν).

Definition 7. We denote byM_σ^∗(G, ν) the operator norm closure ofω(Bσ(G)).

Lemma 8. Let {μ^u₁^˙}u˙ and{μ^u₂^˙}u˙ be two systems of measures on G⁽⁰⁾ satisfying:

(1) supp(μ^u_i^˙) = [u] for allu,˙ i= 1,2.

(2) For all compactly supported bounded Borel functionsf onG⁽⁰⁾ the function u→

f(v)μ^π(u)_i (v) is bounded and Borel.

Then there is a family{U_u˙}u˙ of unitary operators with the following properties:

(1) U_u˙ :L²(μ^u₁^˙)→L²(μ^u₂^˙)is a unitary operator for eachu˙ ∈G⁽⁰⁾/G.

(2) For all bounded Borel functionsf onG⁽⁰⁾, u→U_π(u)(f)

is a bounded Borel function with compact support.

(3) For all bounded Borel functionsf onG⁽⁰⁾, U_π(u)(f) =U_π(u)(f).

Proof. Using the same argument as in [7] (p. 323) we can construct a sequence f₁, f₂, . . . of real valued bounded Borel function onG⁽⁰⁾such that dim(L²(μ^u₁^˙)) =∞ if and only iff_n2= 1 in L²(μ^u₁^˙) for n= 1,2, . . . and then{f₁, f₂, . . .} gives an orthonormal basis ofL²(μ^u₁^˙), while dim(L²(μ^u₁^˙)) =k <∞if and only iff_n2= 1 forn≤k, and f_n2= 0 for n > kand then{f₁, f₂, . . . , f_k} gives an orthonormal basis ofL²(μ^u₁^˙). Letg₁, g₂, . . . be a sequence with the same properties asf₁, f₂, . . . corresponding to{μ^u₂^˙}_u˙. Let us defineU_u˙ :L²(μ^u₁^˙)→L²(μ^u₂^˙) by

U_u˙(f_n) =g_n for alln

Then the family{U_u˙}u˙ has the required properties.

Theorem 9. Let G be a locally compact second countable groupoid with proper orbit space. Let {ν_i^u, u ∈ G⁽⁰⁾}, i = 1,2 be two Haar systems on G. Let F be a Borel subset of G⁽⁰⁾ containing only one element e(u) in each orbit [u]. Let σ : G⁽⁰⁾ →G^F be a cross section for d: G^F → G⁽⁰⁾ with σ(e(v)) = e(v) for all v∈G⁽⁰⁾ and such thatσ(K)is relatively compact in Gfor all compactK ⊂G⁽⁰⁾. Then the C^∗-algebrasM_σ^∗(G, ν₁)andM_σ^∗(G, ν₂)are ∗-isomorphic.

Proof. Let ({β_v^u},{μ^u_i^˙}) be the decompositions of the Haar systems over the prin- cipal groupoid. Letδbe the 1-cocycle associated to the decompositions,i= 1,2.

We shall denote by·,·i,u˙ the inner product of (L²(G⁽⁰⁾, δ(σ(·))μ^u_i^˙)),i= 1,2.

Let us defineq:G→G^F_F by

q(x) =σ(r(x))xσ(d(x))⁻¹, x∈G.

We shall define a∗-homomorphism Φ fromB_σ(G) toB_σ(G). It suffices to define Φ on the set of functions onGof the form

x→g₁(r(x))g(q(x))g₂(d(x))

Let{U_u˙}u˙ be the family of unitary operators with the properties stated in Lemma8, associated to the systems of measures{δ(σ(·))μ^u_i^˙}u˙,i= 1,2.

Let us define Φ by

Φ(f) = (x→U_π(r(x))(g₁)(r(x))g(q(x))U_π(d(x))(g₂)(d(x))) wheref is defined by

f(x) =g₁(r(x))g(q(x))g₂(d(x)).

Iff₁ andf₂ are defined by

f₁(x) =g₁(r(x))g(q(x))g₂(d(x)) f₂(x) =h₁(r(x))h(q(x))h₂(d(x))

then

f₁∗f₂(x) =g∗h(q(x))g₁(r(x))h₂(d(x))g₂, h_11,π(r(x)) and consequently

Φ(f₁∗f₂) =g∗h(q(x))U_π(r(x))(g₁)(r(x))U_π(r(x))(h₂)(d(x))g₂, h_11,π(r(x))

= Φ(f₁)∗Φ(f₂).

Letη be a probability measure on G⁽⁰⁾/Gand η_i =

μ^u_i^˙dη( ˙u), i= 1,2. LetL₁ be the integrated form of a representation (L,H∗G⁽⁰⁾, η₁) andL₂be the integrated form of (L,H ∗G⁽⁰⁾, η₂). Let B be the Borel function defined by

B(u) =L(σ(u)) andW :_⊕

G⁽⁰⁾H(u)dη₁(u)→_⊕

G⁽⁰⁾ H(e(u))dη₁(u) be defined by W(ζ) = (u→B(u)(ζ(u))).

Since every element ofL²(G⁽⁰⁾, δ(σ(·))μ^w₁^˙,H(e(w))) is a limit of linear combina- tions of elementsu→a(u)ξwitha∈L²(G⁽⁰⁾, δ(σ(·))μ^w₁^˙) andξ∈ H(e(w)), we can define a unitary operator

V_w˙ :L²(G⁽⁰⁾, δ(σ(·))μ^w₁^˙,H(e(w)))→L²(G⁽⁰⁾, δ(σ(·))μ^w₂^˙,H(e(w))) by

V_w˙(u→a(u)ξ) =U_w˙(a)ξ.

LetV :_⊕

G⁽⁰⁾H(e(u))dη₁(u)→_⊕

G⁽⁰⁾ H(e(u))dη₂(u) be defined by V(ζ) = (u→V_u˙(ζ(u))).

Ifζ₁, ζ₂∈_⊕

G⁽⁰⁾ H(e(u))dη₁(u) andf is of the form f(x) =g₁(r(x))g(q(x))g₂(d(x)), we have

W L₁(f)W^∗ζ₁, ζ₂= g(x)δ(x)⁻¹² L(x)A₁( ˙w), B₁( ˙w)dβ^e(w)_e(w)(x)dη( ˙w) where

A₁( ˙w) =

g₂(v)ζ₁(v)δ(σ(v))¹²dμ^w₁^˙(v) B₁( ˙w) =

g₁(u)ζ₂(u)δ(σ(u))¹²dμ^w₁^˙(u).

Moreover, if f is of the form f(x) = g₁(r(x))g(q(x)g₂(d(x)) and ζ₁, ζ₂ ∈ _⊕

G⁽⁰⁾

H(e(u))dη₂(u), then

V W L₁(f)W^∗V^∗ζ₁, ζ₂= g(x)δ(x)⁻¹² L(x)A₂( ˙w), B₂( ˙w)dβ_e(u)^e(u)(x)dη( ˙w)

=W L₂(Φ(f))W^∗ζ₁, ζ₂

where

A₂( ˙w) =

g₂(v)V^∗ζ₁(v)δ(σ(v))¹²dμ^w₁^˙(v)

=

U_v˙(g₂)(v)ζ₁(v)δ(σ(v))¹²dμ^w₂^˙(v) B₂( ˙w) =

g₁(v)V^∗ζ₂(v)δ(σ(v))¹²dμ^w₁^˙(v)

=

U_u˙(g₁)(u)ζ₂(u)δ(σ(u))¹²dμ^w₂^˙(u).

Therefore L₁(f) = L₂(Φ(f)). Consequently we can extend Φ to a ∗- homomorphism between the M_σ^∗(G, ν₁) and M_σ^∗(G, ν₂). It is not hard to see that Φ is in fact a∗-isomorphism:

Φ⁻¹(f) = (x→U_π(r(x))^∗ (g₁)(r(x))g(q(x))U_π(d(x))^∗ (g₂)(d(x))) for eachf of the form

f(x) =g₁(r(x))g(q(x))g₂(d(x)).

5. The case of locally transitive groupoids

A locally compactlocally transitive groupoidGis a groupoid for which all orbits [u] are open inG⁽⁰⁾. We shall prove that ifGis a locally compact second countable locally transitive groupoid endowed with a Haar systemν, then

C^∗(G, ν) =M^∗(G, ν) =M_σ^∗(G, ν) for any regular cross sectionσ.

Notation 10. Let{ν^u, u∈G⁽⁰⁾}be a fixed Haar system onG. Letμbe a quasi- invariant measure, Δ its modular function,ν₁ be the measure induced by μonG andν₀= Δ⁻¹²ν₁. Let

II_μ(G) ={f ∈L¹(G, ν₀) : fII,μ<∞}, wherefII,μ is defined by

fII,μ= sup

|f(x)j(d(x))k(r(x))|dν₀(x),

|j|²dμ=

|k|²dμ= 1

. Ifμ₁ andμ₂are two equivalent quasi-invariant measures, then

fII,μ1 =fII,μ2,

because fII,μ =II_μ(|f|)for each quasi-invariant measureμ, whereII_μ is the one-dimensional trivial representation onμ.

Define

fII = sup

fII,μ:μquasi-invariant Radon measure onG⁽⁰⁾

. The supremum can be taken over the classes of quasi-invariant measures.

If·is the fullC^∗-norm onC_c(G), then (see [8]) f ≤ fII for allf.

Lemma 11. LetGbe a locally compact second countable groupoid with proper orbit space. Let{ν^u, u∈G⁽⁰⁾}be a Haar system onG, let({β_v^u},{μ^u˙})its decomposition over the principal groupoid associated to Gand letδ the associated 1-cocycle. If f is a universally measurable function onG, then

fII ≤sup

˙ w

|f(x)|δ(x)⁻¹²dβ^u_v(x) ₂

dμ^w˙(v)dμ^w˙(u) ¹₂

.

Proof. Each Radon quasi-invariant measure is equivalent with a Radon measure of the form

μ^u˙dμ( ˙u), whereμis a probability measure on the orbit spaceG/G⁽⁰⁾. Therefore for the computation of .II it is enough to consider only the quasi- invariant measures of the formμ=

μ^u˙dμ( ˙u), whereμis a probability measure on G⁽⁰⁾/G. It is easy to see that the modular function of

μ^u˙dμ( ˙u) is Δ =δ.

Letj, k∈L²(G⁽⁰⁾, μ) with

|j|²dμ=

|k|²dμ= 1. We have

|f(x)|δ(x)⁻¹²dβ_v^u(x)|j(v)||k(u)|dμ^w˙(v)dμ^w˙(u)dμ( ˙w)

≤

|f(x)|δ(x)⁻¹²dβ_v^u(x) ₂

dμ^w˙(v)dμ^w˙(u) ¹₂

· |j(v)|²|k(u)|²dμ^w˙(v)dμ^w˙(u) ¹₂

dμ( ˙w)

≤sup

˙ w

|f(x)|δ(x)⁻¹²dβ_v^u(x) ₂

dμ^w˙(v)dμ^w˙(u) ¹₂

· |j(v)|²dμ^w˙(v))¹²(|k(u)|²dμ^w˙(u) ¹₂

dμ( ˙w)

≤sup

˙ w

|f(x)|δ(x)⁻¹²dβ_v^u(x) ₂

dμ^w˙(v)dμ^w˙(u) ¹₂

.

Consequently, fII ≤sup

˙ w

|f(x)|δ(x)⁻¹²dβ^u_v(x) ₂

dμ^w˙(v)dμ^w˙(u) ¹₂

.

If G is locally transitive, each orbit [u] is open in G⁽⁰⁾. Each measure μ^u˙ is supported on [u]. Since ([u]) is a partition ofG⁽⁰⁾ into open sets, it follows that there is a unique Radon measuremonG⁽⁰⁾such that the restriction ofmatC_c([u]) isμ^u˙ for each [u].

Corollary 12. LetGbe a locally compact second countable locally transitive group- oid endowed with a Haar system ν ={ν^u, u∈G⁽⁰⁾}. Let f be a universally mea- surable function such that fII <∞.

(1) If(f_n)_nis a uniformly bounded sequence of universally measurable functions supported on a compact set, and if(f_n)_nconverges pointwise tof, then(f_n)_n converges tof in the norm of C^∗(G, ν).