ISRN AlgebraVolume 2011 (2011), Article ID 856709, 17 pagesdoi:10.5402/2011/856709

Research Article

¹-Algebra of a Locally Compact Groupoid

Massoud Amini,¹ Alireza Medghalchi,² and Ahmad Shirinkalam²

¹Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran²Faculty of Mathematical Sciences and Computer, Tarbiat Moalem University, 50 Taleghani Avenue, Tehran, Iran

Received 12 May 2011; Accepted 6 June 2011

Academic Editor: B. Rangipour

Copyright © 2011 Massoud Amini et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For a locally compact groupoid with a fixed Haar system and quasi-invariant measure , we introduce the notion of -measurability and construct the space ¹(, , ) of absolutely integrable functions on and show that it is a Banach -algebra and a two-sided ideal in the algebra () of complex Radon measures on . We find correspondences between representations of on Hilbert bundles and certain class of nondegenerate representations of ¹(, , ).

1. Introduction and Preliminaries

For a locally compact group with a Haar measure , the Banach algebra plays a central role in harmonic analysis on [1]. This motivated us to define a similar notion in the case where is a locally compact groupoid with a (fixed) Haar system and quasi-invariant measure . This paper is devoted to the study of such a groupoid -algebra . One may expect that as the group case, there is a full interaction between the properties of and that of . This is not completely true. For instance, unlike the group case, not every nondegenerate representation of is integrated form a representation of . In Section 2, we introduce the appropriate measurability notion used to define . Sections 3 and 4 are devoted to the algebra structure of and its embedding into as a closed ideal. In Section 5, we find the class of nondegenerate representations of which could be obtained by integrating a representation of .

We start with some basic definitions. Our main reference for groupoids is the Renault's book [2]. In this paper, we frequently use the following version of Fubini's theorem for (not necessarily -finite) Radon measures [1, Theorem  B.3.3].

Lemma 1.1. Let and be Radon measures on the Borel sets of the locally compact spaces and , respectively. Then there exists a unique Radon measure on such that (i)if is -integrable, then the partial integrals define integrable functions such that Fubini’s formula holds (ii)if is measurable such that is -finite and if one of the iterated integrals or is finite, then is integrable and the Fubini formula holds.

A groupoid is a set endowed with a subset of , called the set of composable pairs, a product map: , and an inverse map: , such that for each ,(i), (ii)if then and ,(iii) and if then ,(iv) and if then .

If , is called the source of and is called the range of . The pair is composable if and only if . The set is the unit space of , and its elements are called units in the sense that and .

For , , , , and . The latter is called the isotropy group at . We define if . It is checked that ~ is an equivalence relation on the unit space . The equivalence class of is denoted by and is called the orbit of .

A topological groupoid consists of a groupoid and a topology compatible with the groupoid structure, such that the composition map is continuous on in the induced product topology, and the inversion map is continuous on . A locally compact groupoid is a topological groupoid which satisfies the following conditions:(i) is locally compact and Hausdorff in the relative topology.(ii)There is a countable family of compact Hausdorff subsets of whose interiors form a basis for the topology of .(iii)Every is locally compact Hausdorff in the relative topology.

A locally compact groupoid is -discrete if its unit space is an open subset. Let be a locally compact groupoid. The support of a function is the complement of the union of all open, Hausdorff subsets of on which vanishes. The space consists of all continuous functions on with compact support. A left Haar system for consists of measures on such that(i)the support of is ,(ii)(continuity) for each is continuous,(iii)(left invariance) for any and any ,

Let be a measure on . The measure on induced by is , defined by , for . The measure on induced by is The inversion of is . Note that the measures are Radon. A measure on is said to be quasi invariant if its induced measure is equivalent to its inverse . Let be a quasi-invariant measure on . The Radon-Nikodym derivative is called the modular function of . An equivalent definition of modular function on is given in [3, Definition  2.3], defining the modular function as a strictly positive continuous homomorphism on such that is modular function for .

A subset of is called a -set if the restrictions of and to it are one to one. Equivalently, is a -set if and only if and are contained in .

We introduce some notations from [4, 5] which is related to the representations of . Let be a fixed Haar system on . Let be a quasi-invariant measure, its modular function, be the measure induced by on , and . Let where is defined by If and are two equivalent quasi-invariant measures, then because for each quasi-invariant measure , where is the one-dimensional trivial representation on . Now define The supremum is taken over the class of quasi-invariant measures.

Let where Under the canonical convolution and involution, becomes a Banach -algebra [5, page  4]. Here . If we consider and put , then .

2. -Measurability

For the rest of the paper, is a locally compact, Hausdorff, second countable groupoid which admits a left Haar system .

Definition 2.1. A system of measures is said to be complete if for each , is complete on its -algebra . A Borel measurable set is called -measurable if for each , . A function is -measurable if for every and every open set .

Proposition 2.2. If is the completion of and is -measureable, then there is a -measurable function such that on -a.e).

Proof. Since is -measurable for each , is -measurable on , and since is the completion of , there exists a -measurable function such that . Now define on and zero, elsewhere. Since for every and every open set , is -measurable, and since , we have on (-a.e).

From now on, we assume that the Haar system is complete.

Lemma 2.3. For each , -measurability of is equivalent to -measurability of .

Proof. We have and Since is complete and is -null, is in . Thus, for each and open set Now for , we have , hence is -measurable if and only if is -measurable.

If is -measurable and is continuous, then is -measurable. Also, if is -measurable, is continuous, and , then is -measurable. If then is -measurable if and only if are -measurable. If are -measurable, so are and . Also, if is a sequence of -valued -measurable functions, then the functions , and are all -measurable. If exists for every , then is -measurable. Thus if is a sequence of complex-valued -measurable functions and -a.e, then is -measurable.

3. The Algebra

In this section, we define the space of integrable functions on with respect to a fixed Haar system and quasi-invariant measure and show that it is a Banach *-algebra under canonical convolution and involution.

Definition 3.1. Suppose is a quasi-invariant probability measure on , and is Radon measure induced by , then we define

It is clear that is a Banach space. We show that the Banach space is a Banach *-algebra under the following convolution product.

Lemma 3.2. If , then and .

Proof. For each , The measurability of follows from -measurability of .

Next, we define an involution on . We say that the assertion holds for -a.e. if for , . Clearly an assertion holds -almost everywhere if and only if it holds -almost everywhere.

Lemma 3.3. Suppose with . Then on .

Proof. Suppose . We have Also from we have Thus Now, let . If then But -a.e.), hence -a.e.). Thus . Therefore, -a.e.). A similar argument leads to -a.e.).

Proposition 3.4. The map , where , is an isometric involution on .

Note that from [5, page  9], we have Hence the space of is in general bigger than and with respect to -norm and -norm, indeed .

According to [2, Lemma  1.4], -norm topology is coarser than the inductive limit topology. Also [5, page  15] shows that has a two-sided bounded approximate identity with respect to the inductive limit topology with the following properties:(i) for all ,(ii) for all , where and 's are compact,(iii) for all .

An argument in [5, page  15] shows that there is such that for all . Since is dense in , thus has a two-sided bounded approximate identity.

For each define when the multiplications on the right hand sides are defined. It is easy to check that the maps are homomorphisms.

Proposition 3.5. Let be a closed subspace of . Then is a left ideal if and only if it is closed under left translation, and is a right ideal if and only if it is closed under right translation.

Proof. Note that since , Now suppose is a bounded approximate identity for . For the first assertion, if and and is a left ideal, then we have Conversely, if is closed under left translation and and , is in the closed linear span of the functions and hence in . The other assertion is proved similarly.

4. as Banach -Algebra

Let be the space of complex Radon measures on . If , then the map on defined by is a linear functional on satisfying , so by Riesz representation theorem, it is given by a measure shown as called the convolution of with . If we define , then is an involution on , and is a Banach -algebra. In this section, we show that the space is a closed two-sided ideal of .

Proposition 4.1. The map with , for ; is an isometric embedding.

Proof. If , then is -measurable so the integral exists, and it is easy to check that is a measure on . We show that is Radon. If , then , so is Radon if and only if and are Radon. Since is LCH and second countable, we have , for each compact set , thus is Radon. Similarly is Radon, and so is .By definition, , so for each , there exists a partition of such that Thus, . Conversely, suppose , then and for every partition of we have, If , where then . Hence and equality holds.

Corollary 4.2. is a closed subspace of .

Lemma 4.3. If , then .

Proof. For each compact set , we have Since and are regular measures, the equality holds for each open set and then for each measurable set.

If and , we will define such that . Suppose , we put On the other hand,

Comparing these equalities, If , then it is easy to check that Thus Similarly, we want to define in such a way that the equality holds. Again suppose . We have On the other hand, Comparing the above equalities, we have

Lemma 4.4. is a two-sided ideal of .

Proof. Suppose and . Then we have Thus . Also Hence, .

5. Representation Theory of Locally Compact Groupoids

Let us briefly review some basic facts of representation theory on locally compact and Hausdorff groupoids. Recall that the measure on defined by is a Radon measure. Let be the measure on given by where is modular function of . Then for each Borel subset of , we have Hence, is symmetric under inversion.

Definition 5.1. A representation of the locally compact groupoid is a triple consisting of a Hilbert bundle , where is a quasi-invariant measure on (with associated Radon measures ) and for each , a unitary element such that(i) is the identity map on for all ,(ii) for -a.e. ,(iii) for -a.e. ,(iv)for any , the map is -measurable on .

Definition 5.2. A representation of on a Hilbert space, is a -homomorphism It is called nondegenerate if is dense in .

Continuity of automatically holds, because each -homomorphism from a Banach -algebra to a -algebra is norm decreasing, namely, , for each .

Our main aim here is to find a correspondence between unitary representations of and nondegenerate representations of . Unfortunately, this is impossible in general. Such a correspondence exists between representations of and those of , when is separable, is second countable and admits sufficiently many nonsingular -sets [2, Theorem  1.21].

Proposition 5.3. Let be a second countable locally compact groupoid with Haar system and with sufficiently many nonsingular Borel -sets. Then, every representation of on a separable Hilbert space is the integrated form of a representation of .

These assumptions satisfied in the case of -discrete groupoids and transformation groups. The main problem is that a continuous representation of is not necessarily continuous in the -norm. To get a partial result, we use the following result [2, Proposition  1.7].

Proposition 5.4. Suppose is a representation of . For and , defines a bounded nondegenerate -representation of on such that two equivalent representations of give two equivalent representations of .

The equation above is called the integrated form of a unitary representation. If is a representation of , the above proposition says that should be of the form For each and .

Next, we define a representation of , denoted by as for each and

Lemma 5.5. is a bounded representation of on .

Proof. We have Thus, we may define for -a.e. , Now, we have Therefore, the map is -integrable, and is a bounded operator of norm . We have to check that is a -homomorphism. For this, we define Then Hence, . Also The nondegeneracy follows from

Now let us turn to the problem that a continuous representation of is not necessarily continuous in -norm. Let and put Observe that is a nontrivial subspace, as is a probability measure and if in for each , then in and the calculations in the proof of Lemma 5.5 shows that , hence . On the other hand, for each , because the map is continuous. Therefore, . Hence, we have for each .

Put and define , for . Then, it follows from continuity of and polarization identity that extends to a continuous representation of on , still denoted by .

Next we focus on the notion of irreducibility which plays an important role in the theory of representations. We show that if is an irreducible representation of , the integrated representation of on is irreducible. Basic materials come from [6].

Definition 5.6. Let be a Hilbert bundle. A family , where is a closed subspace of for each , is called a subbundle. A subbundle is called nontrivial if for some . For a representation of a locally compact groupoid associated with the Hilbert bundle , a subbundle is called invariant if for each . Note that if is an invariant subbundle, and for some , then for every .

The following lemma is proved in [6, Lemma  3.4].

Lemma 5.7. Let be a representation of a locally compact groupoid associated with the Hilbert bundle . If is an invariant subbundle, then so is .

Definition 5.8. A representation of a locally compact groupoid is called reducible if admits a nontrivial invariant subbundle , otherwise is called irreducible. In this case, it is easy to check that with is called a subrepresentation of . If and are two representations of a locally compact groupoid associated with two Hilbert bundles and , respectively. Then, we put and write .

Two representations and are called (unitarily) equivalent if , and there is such that is a unitary operator for every . Note that if and denotes the adjoint operator to , then hence, . We observe that is a unital -algebra, where the operations are defined pointwise.

Following [6], for a representation of a locally compact groupoid associated with the Hilbert bundle , we set If , then whenever that is . Therefore, if , then there exists with not in . It is also obvious that .

We need the following version of Schur's lemma [6, Lemma  3.11].

Lemma 5.9. A representation of a locally compact groupoid is irreducible if and only if . In particular, in the case where is transitive, then is irreducible if and only if .

Lemma 5.10. If is closed subbundle of and is an orthogonal projection onto , then is invariant under if and only if .

Proof. If and , then Thus, so is invariant. Conversely, if is invariant, then for and for Hence, .

We show that if a representation of a locally compact groupoid associated with the Hilbert bundle is irreducible and is the corresponding integrated representation of on , then is irreducible. If , then for each subbundle , is closed subspace of with orthogonal complement . A map with is called an orthogonal projection of onto .

Let be a representation of on the Hilbert bundle and be the bounded representation of on constructed above.

Lemma 5.11. is invariant under if and only if for each , .

Proof. Suppose and , then Hence, is invariant. Conversely, if is invariant, then for , and for , Thus, .

Proposition 5.12. If is an irreducible representation of , then is an irreducible representation of .

Proof. If is reducible, then there exists a nontrivial invariant closed subspace of , and, hence, there is an orthogonal projection . It follows that for each and . Therefore, , for each , hence . Now by the Schur's lemma, is reducible.

Theorem 5.13. If is an irreducible representation of , then is an irreducible representation of on .

Proof. If is reducible, then there exists a nontrivial invariant closed subspace of . By the calculations after Lemma 5.5, , where and are the corresponding representations of on and . Therefore, is a nontrivial invariant closed subspace , and is reducible. Hence, is reducible by the above proposition.

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