International Scholarly Research NetworkISRN AlgebraVolume 2011, Article ID 856709, 17 pagesdoi:10.5402/2011/856709

Research ArticleL1-Algebra of a Locally Compact Groupoid

Massoud Amini,1 Alireza Medghalchi,2 and Ahmad Shirinkalam2

1 Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University,Tehran 14115-134, Iran

2 Faculty of Mathematical Sciences and Computer, Tarbiat Moalem University, 50 Taleghani Avenue,Tehran, Iran

Correspondence should be addressed to Massoud Amini, mamini@modares.ac.ir

Received 12 May 2011; Accepted 6 June 2011

Academic Editor: B. Rangipour

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

For a locally compact groupoid G with a fixed Haar system λ and quasi-invariant measure μ, weintroduce the notion of λ-measurability and construct the space L1�G, λ, μ� of absolutely integrablefunctions onG and show that it is a Banach ∗-algebra and a two-sided ideal in the algebraM�G� ofcomplex Radon measures on G. We find correspondences between representations of G on Hilbertbundles and certain class of nondegenerate representations of L1�G, λ, μ�.

1. Introduction and Preliminaries

For a locally compact group G with a Haar measure λ, the Banach algebra L1�G, λ� plays acentral role in harmonic analysis on G �1�. This motivated us to define a similar notion in thecase where G is a locally compact groupoid with a �fixed�Haar system λ and quasi-invariantmeasure μ. This paper is devoted to the study of such a groupoid L1-algebra L1�G, λ, μ�. Onemay expect that as the group case, there is a full interaction between the properties of Gand that of L1�G, λ, μ�. This is not completely true. For instance, unlike the group case, notevery nondegenerate representation of L1�G, λ, μ� is integrated form a representation of G.In Section 2, we introduce the appropriate measurability notion used to define L1�G, λ, μ�.Sections 3 and 4 are devoted to the algebra structure of L1�G, λ, μ� and its embedding intoM�G� as a closed ideal. In Section 5, we find the class of nondegenerate representations ofL1�G, λ, μ� which could be obtained by integrating a representation of G.

We start with some basic definitions. Our main reference for groupoids is the Renault’sbook �2�. In this paper, we frequently use the following version of Fubini’s theorem for �notnecessarily σ-finite� Radon measures �1, Theorem B.3.3�.

2 ISRN Algebra

Lemma 1.1. Let ν and ρ be Radon measures on the Borel sets of the locally compact spaces X and Y ,respectively. Then there exists a unique Radon measure ν · ρ on X × Y such that

�i� if f : X × Y → C is ν · ρ-integrable, then the partial integrals∫X

f(x, y

)dν�x�,

∫Y

f(x, y

)dρ(y)

�1.1�

define integrable functions such that Fubini’s formula holds

∫X×Y

f dν · ρ �∫X

∫Y

f dν dρ �∫Y

∫X

f dρ dν, �1.2�

�ii� if f is measurable such that A � {�x, y� ∈ X × Y : f�x, y�/� 0} is σ-finite and if one ofthe iterated integrals

∫X

∫Y |f |dν dρ or

∫Y

∫X |f |dρ dν is finite, then f is integrable and the

Fubini formula holds.

A groupoid is a set G endowed with a subset G2 of G×G, called the set of composablepairs, a product map: G2 → G; �x, y� �→ xy, and an inverse map: G → G; x �→ x−1, suchthat for each x, y, z ∈ G,

�i� �x−1�−1 � x,

�ii� if �x, y�, �y, z� ∈ G2 then �xy, z�, �x, yz� ∈ G2 and �xy�z � x�yz�,�iii� �x−1, x� ∈ G2 and if �x, y� ∈ G2 then x−1�xy� � y,�iv� �x, x−1� ∈ G2 and if �z, x� ∈ G2 then �zx�x−1 � z.If x ∈ G, s�x� � x−1x is called the source of x and r�x� � xx−1 is called the range of x.

The pair �x, y� is composable if and only if s�x� � r�y�. The set G0 � s�G� � r�G� is the unitspace of G, and its elements are called units in the sense that xs�x� � x and r�x�x � x.

For u, v ∈ G0, Gu � r−1{u}, Gv � s−1{v}, Guv � Gu ∩ Gv, and G{u} � Guu. The latter iscalled the isotropy group at u. We define u ∼ v if Guv /� ∅. It is checked that ∼ is an equivalencerelation on the unit space G0. The equivalence class of u is denoted by �u� and is called theorbit of u.

A topological groupoid consists of a groupoid G and a topology compatible with thegroupoid structure, such that the composition map is continuous on G2 in the induced pro-duct topology, and the inversion map is continuous on G. A locally compact groupoid is atopological groupoid Gwhich satisfies the following conditions:

�i� G0 is locally compact and Hausdorff in the relative topology.

�ii� There is a countable family C of compact Hausdorff subsets of G whose interiorsform a basis for the topology of G.

�iii� Every Gu is locally compact Hausdorff in the relative topology.

A locally compact groupoid is r-discrete if its unit space is an open subset. Let G bea locally compact groupoid. The support of a function f : G → C is the complement of theunion of all open, Hausdorff subsets of G on which f vanishes. The space Cc�G� consists ofall continuous functions on G with compact support. A left Haar system for G consists of

ISRN Algebra 3

measures {λu, u ∈ G0} on G such that�i� the support of λu is Gu,

�ii� �continuity� for each f ∈ Cc�G�, u �→ λ�f��u� �∫f dλu is continuous,

�iii� �left invariance� for any x ∈ G and any f ∈ Cc�G�,

∫f(xy)dλs�x�

(y)�∫f(y)dλr�x�

(y). �1.3�

Let μ be a measure on G0. The measure on G induced by μ is ν �∫λudμ, defined by∫

G f dν �∫G0dμ�u�

∫G f dλ

u, for f ∈ Cc�G�. The measure on G2 induced by μ is ν2 �∫λu ×

λudμ�u�. The inversion of ν is ν−1 �∫λudμ�u�. Note that the measures ν, ν2, ν−1 are Radon.

A measure μ on G0 is said to be quasi invariant if its induced measure ν is equivalent to itsinverse ν−1. Let μ be a quasi-invariant measure on G0. The Radon-Nikodym derivative D �dν/dν−1 is called the modular function of μ. An equivalent definition of modular functionon G is given in �3, Definition 2.3�, defining the modular function Δ as a strictly positivecontinuous homomorphism on G such that Δ|Guu is modular function for Guu.

A subset A of G is called a G-set if the restrictions of r and s to it are one to one.Equivalently, A is a G-set if and only if AA−1 and A−1A are contained in G0.

We introduce some notations from �4, 5� which is related to the representations ofL1�G, λ, μ�. Let {λu}u be a fixed Haar system on G. Let μ be a quasi-invariant measure, Δ itsmodular function, ν be the measure induced by μ on G, and ν0 � Δ−1/2ν. Let

IIμ(G, ν, μ

)�{f ∈ L1�G, ν0� :

∥∥f∥∥II,μ

4 ISRN Algebra

where

∥∥f∥∥I,μ � max{∥∥∥∥u −→

∫∣∣f∣∣dλu∥∥∥∥∞,

∥∥∥∥u −→∫∣∣f∗∣∣dλu

∥∥∥∥∞

}. �1.9�

Under the canonical convolution and involution, I�G, ν, μ� becomes a Banach ∗-algebra �5,page 4�. Here f∗�x� � f�x−1�. If we consider

∥∥f∥∥I,r � supu

(∫∣∣f∣∣dλu),

∥∥f∥∥I,s � supu

(∫∣∣f∣∣dλu), �1.10�

and put ‖f‖I � max�‖f‖I,r , ‖f‖I,s�, then ‖f‖I � ‖f‖I,μ.

2. λ-Measurability

For the rest of the paper,G is a locally compact, Hausdorff, second countable groupoid whichadmits a left Haar system λ � {λu}.

Definition 2.1. A system of measures {λu}u∈G0 is said to be complete if for each u ∈ G0, λu iscomplete on its σ-algebra Mλu . A Borel measurable set E ⊆ G is called λ-measurable if foreach u ∈ G0, E ∩ Gu ∈ Mλu . A function f : G → C is λ-measurable if for every u ∈ G0 andevery open set O ⊆ C, f−1�O� ∩Gu ∈ Mλu .

Proposition 2.2. If λ � {λu} is the completion of λ � {λu} and f : G → �0,∞� is λ-measureable,then there is a λ-measurable function g such that f � g on Gu �λu-a.e).

Proof. Since f is λ-measurable for each u ∈ G0, f is λu-measurable on Gu, and since λu isthe completion of λu, there exists a λu-measurable function gu such that f � gu �λu-a.e�.Now define g � gu on Gu and zero, elsewhere. Since for every u ∈ G0 and every open setO ⊆ C, g−1�O�∩Gu � g−1u �O� ∈ Mλu , g is λ-measurable, and since λu�Gu�c � 0, we have f � gon Gu �λu-a.e�.

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

Lemma 2.3. For each f : G → C, λ-measurability of f is equivalent to ν-measurability of f .

Proof. We have suppλu � Gu and

f−1�O� �(f−1�O� ∩Gu

)∪(f−1�O� ∩ �Gu�c

). �2.1�

Since λu is complete and �Gu�c is λu-null, �f−1�O� ∩ �Gu�c� ⊆ �Gu�c is in Mλu . Thus, for eachu ∈ G0 and open set O ⊆ C

f−1�O� ∈ Mλu ⇐⇒ f−1�O� ∩Gu ∈ Mλu . �2.2�

Now for ν �∫λudμ�u�, we have Mν �

⋂u∈G0 Mλu , hence f is ν-measurable if and only if f is

λ-measurable.

ISRN Algebra 5

If f : G → C is λ-measurable and g : C → C is continuous, then g ◦ f : G → Cis λ-measurable. Also, if f, g : G → R is λ-measurable, Φ : C → Y is continuous, andh�x� � Φ�f�x�, g�x��, then h is λ-measurable. If f � u iv then f is λ-measurable if andonly if u, v are λ-measurable. If f, g : G → R are λ-measurable, so are f g and fg.Also, if {fj}∞1 is a sequence of R-valued λ-measurable functions, then the functions g1�x� �supjfj�x�, g2�x� � infjfj�x�, g3�x� � lim supjfj�x�, and g4�x� � lim infjfj�x� are all λ-measurable. If f�x� � limjfj�x� exists for every x ∈ G, then f is λ-measurable. Thus if{fj}∞1 is a sequence of complex-valued λ-measurable functions and fj → f λu-a.e, then fis λ-measurable.

3. The Algebra L1�G, λ, μ�

In this section, we define the space of integrable functions on G with respect to a fixedHaar system λ and quasi-invariant measure μ and show that it is a Banach ∗-algebra undercanonical convolution and involution.

Definition 3.1. Suppose μ is a quasi-invariant probabilitymeasure onG0, and ν is Radonmeas-ure induced by μ, then we define

L1�G, ν� � L1(G, λ, μ

)�{f : G −→ C : f is λ-measurable, ∥∥f∥∥1 �

∫G

∣∣f�x�∣∣dν�x�

6 ISRN Algebra

�∫G

∣∣g�x�∣∣(∫

G

∣∣f(y)∣∣dν(y))dν�x�

�∥∥f∥∥1

∥∥g∥∥1.�3.3�

The measurability of f ∗ g follows from λ-measurability of f, g.

Next, we define an involution on L1�G, λ, μ�. We say that the assertion P�x� holds forλ-a.e. x if for E � {x : ¬P�x�}, μ{u : λu�E� > 0} � 0. Clearly an assertion holds λ-almosteverywhere if and only if it holds ν-almost everywhere.

Lemma 3.3. Suppose Du : Gu → R with Du�x� � dλu�x�/dλu�x� �x ∈ G�. Then D � Du onGu �a.e.�.

Proof. Suppose E ⊆ Gu. We have

ν�E� �∫G0λu�E�dμ�u� �

∫G0

∫E

dλu�x�dμ�u� �∫G0

∫E

Du�x�dλu�x�dμ�u�. �3.4�

Also from dν � Ddν−1 we have

ν�E� �∫E

D�x�dν−1�x� �∫E

D�x�∫G0dλu�x�dμ�u� �

∫G0

∫E

D�x�dλu�x�dμ�u�. �3.5�

Thus∫G0

∫E

�Du�x� −D�x��dλu�x�dμ�u� � 0. �3.6�

Now, let Eu � {x ∈ Gu : Du�x� > D�x�}. If λu�Eu� > 0 then∫Eu�Du�x� −D�x��dλu�x� > 0. �3.7�

But∫Eu�Du�x� −D�x��dλu�x� � 0 �μ-a.e.�, hence λu�Eu� � 0 �μ-a.e.�. Thus μ{u : λu�Eu� > 0} �

0. Therefore, Du�x� ≤ D�x� �λ-a.e.�. A similar argument leads to Du�x� ≥ D�x� �λ-a.e.�.

Proposition 3.4. The map ∗ : L1�G, λ, μ� → L1�G, λ, μ�; f �→ f∗, where f∗�x� � f�x−1�D�x−1�,is an isometric involution on L1�G, λ, μ�.

Note that from �5, page 9�, we have

∥∥f∥∥L1�G,λ,μ� �∥∥f∗∥∥L1�G,λ,μ� �

∥∥f∥∥L1�G,ν0� �∥∥f∗∥∥L1�G,ν0� ≤

∥∥f∥∥II,μ �∥∥f∗∥∥II,μ ≤

∥∥f∥∥I,μ �∥∥f∗∥∥I,μ.

�3.8�

Hence the space of L1�G, λ, μ� is in general bigger than I�G, ν, μ� and IIμ�G, ν, μ�with respectto I-norm and II-norm, indeed I�G, ν, μ� ⊆ IIμ�G� ⊆ L1�G, λ, μ�.

ISRN Algebra 7

According to �2, Lemma 1.4�, I-norm topology is coarser than the inductive limittopology. Also �5, page 15� shows that Cc�G� has a two-sided bounded approximate identity�en�

∞n�1 with respect to the inductive limit topology with the following properties:

�i� en�x� ≥ 0 for all x ∈ G,�ii� | ∫ en�x�dλu�x� − 1| < 1/n for all u ∈ Kn, where ∪nKn � G0 and Kn’s are compact,�iii� en�x� � en�x−1� for all x ∈ G.An argument in �5, page 15� shows that there is M > 0 such that ‖en‖II,μ ≤ M for all

n. Since Cc�G� is dense in L1�G, λ, μ�, thus L1�G, λ, μ� has a two-sided bounded approximateidentity.

For each f ∈ L1�G, λ, μ� define

Lxf(y)� f(x−1y

), Rxf

(y)� f(yx), �3.9�

when the multiplications on the right hand sides are defined. It is easy to check that the mapsLx, Rx are homomorphisms.

Proposition 3.5. Let I be a closed subspace of L1�G, λ, μ�. Then I is a left ideal if and only if it isclosed under left translation, and I is a right ideal if and only if it is closed under right translation.

Proof. Note that since f ∗ g � ∫Gr�y� f�y�Lyg dν�y�,

Lx(f ∗ g) �

∫Gr�y�

f(y)LxLyg dν

(y)�∫Gr�y�

f(y)Lxyg dν

(y)

�∫Gr�y�

f(x−1y

)Lyg dν

(y)�∫Gr�y�

Lxf(y)Lyg dν

(y)�(Lxf

) ∗ g.�3.10�

Now suppose �en�n is a bounded approximate identity for L1�G, λ, μ�. For the first assertion,

if f ∈ L1�G, λ, μ� and g ∈ I and I is a left ideal, then we have

Lx�en� ∗ f � Lx(en ∗ f

) −→ Lxf. �3.11�

Conversely, if I is closed under left translation and f ∈ L1�G, λ, μ� and g ∈ I,

f ∗ g �∫Gr�y�

f(y)Lyg dν

(y)

�3.12�

is in the closed linear span of the functions Lyg and hence in I. The other assertion is provedsimilarly.

4. M�G� as Banach ∗-AlgebraLet M�G� be the space of complex Radon measures on G. If η, θ ∈ M�G�, then the mapψ �→ I�ψ� on C0�G� defined by I�ψ� �

∫G

∫Gr�y� ψ�xy�dη�x�dθ�y� is a linear functional on

8 ISRN Algebra

C0�G� satisfying |I�ψ�| ≤ ‖ψ‖sup‖η‖‖θ‖, so by Riesz representation theorem, it is given by ameasure shown as η ∗ θ called the convolution of η, θ with ‖η ∗ θ‖ ≤ ‖η‖‖θ‖. If we defineη∗�E� � η�E−1�, then η �→ η∗ is an involution on M�G�, and M�G� is a Banach ∗-algebra. Inthis section, we show that the space L1�G, λ, μ� is a closed two-sided ideal ofM�G�.

Proposition 4.1. The map L1�G, λ, μ� ↪→ M�G� with νf�E� �∫E fχEdν, for �E ⊆ G�; f �→ νf is

an isometric embedding.

Proof. If f ∈ L1�G, λ, μ�, then f is λ-measurable so the integral exists, and it is easy to checkthat νf is a measure on G. We show that νf is Radon. If f � u iv, then νf � νu iνv, so νfis Radon if and only if νu and νv are Radon. Since G is LCH and second countable, we haveνu�K� �

∫K udν ≤

∫G |u|dν � ‖u‖1 0, there

exists a partition {Ei}n1 of G such that

∥∥νf∥∥ − � <n∑1

∣∣νf�Ei�∣∣ �n∑1

∣∣∣∣∫G

fχEidν

∣∣∣∣ ≤∫G

(∣∣f∣∣ n∑1

χEi

)dν �

∥∥f∥∥1. �4.1�

Thus, ‖νf‖ ≤ ‖f‖1. Conversely, suppose f ≥ 0, then νf ≥ 0 and for every partition {Ei}n1 of Gwe have,

∥∥νf∥∥ ≥n∑1

νf�Ei� �n∑1

∫G

fχEidν �∫G

f dν �∥∥f∥∥1. �4.2�

If f � u iv � �f1 − f2� i�f3 − f4�, where fi ≥ 0 then ‖νf‖ � ‖νf1‖ ‖νf2‖ ‖νf3‖ ‖νf4‖ ≥‖f1‖1 ‖f2‖1 ‖f3‖1 ‖f4‖1 ≥ ‖f‖1. Hence ‖νf‖ ≥ ‖f‖1 and equality holds.

Corollary 4.2. L1�G, λ, μ� is a closed subspace ofM�G�.

Lemma 4.3. If f, g ∈ L1�G, λ, μ�, then ν�f∗g� � νf ∗ νg .

Proof. For each compact set K, we have

νf ∗ νg�K� �∫G

χK�x�d(νf ∗ νg

)�x� �

∫G

∫Gr�x�

χK(yx)dνf

(y)dνg�x�

�∫G

f(y) ∫

Gs�y�χK(yx)dνg�x�dν

(y)�∫G

∫Gs�y�

f(y)g(y−1x

)χK�x�dν

(y)dν�x�

�∫G

χK�x�∫Gr�x�

f(y)g(y−1x

)dν(y)dν�x� �

∫G

(f ∗ g)�x�χK�x�dν�x� � νf∗g�K�.

�4.3�

Since νf∗g and νf ∗ νg are regular measures, the equality holds for each open set and then foreach measurable set.

ISRN Algebra 9

If f ∈ L1�G, λ, μ� and η ∈ M�G�, we will define η ∗ f such that νη∗f � η ∗ νf . Supposeϕ ∈ C0�G�, we put

η ∗ f(ϕ) �∫G

ϕ�x�dνη∗f�x� �∫G

ϕ�x�(η ∗ f)�x�dν�x�. �4.4�

On the other hand,

η ∗ νf(ϕ)�∫G

∫Gr�x�

ϕ(yx)dη(y)dνf�x� �

∫G

∫Gr�x�

ϕ(yx)dη(y)f�x�dν�x�

�∫G

∫Gs�y�

f(y−1x

)ϕ�x�dη

(y)dν(y−1x

)�∫G

ϕ�x�∫Gs�y�

f(y−1x

)dη(y)dν�x�.

�4.5�

Comparing these equalities,

(η ∗ f)�x� �

∫Gr�x�

f(y−1x

)dη(y). �4.6�

If f ∈ L1�G, λ, μ�, then it is easy to check that∫GuRyf�x�dλu�x� �

∫Guf�x�dλu

(xy−1

)� D

(y−1)∫

Guf�x�dλu�x�. �4.7�

Thus

∫G

Ryf�x�dν�x� � D(y−1)∫

G

f�x�dν�x�. �4.8�

Similarly, we want to define f ∗ η in such a way that the equality ν�f∗η� � νf ∗ η holds. Againsuppose ϕ ∈ C0�G�. We have

ν�f∗η�(ϕ)�∫G

ϕ�x�dνf∗η�x� �∫G

ϕ�x�(f ∗ η)�x�dν�x� �

∫G0

∫G

ϕ�x�(f ∗ η)�x�dλu�x�dμ�u�

�∫G0

∫Guϕ�x�

(f ∗ η)�x�dλu�x�dμ�u� �

∫Guϕ�x�

(f ∗ η)�x�dν�x�.

�4.9�

On the other hand,

(νf ∗ η

)(ϕ)�∫G

∫Gr�y�

ϕ(xy)dνf�x�dη

(y)

�∫G

∫Gr�y�

ϕ(xy)f�x�dν�x�dη

(y)

10 ISRN Algebra

�∫G

∫Gr�y�

ϕ�x�f(xy−1

)dν(xy−1

)dη(y)

�∫G

∫Gr�y�

ϕ�x�f(xy−1

)D(y−1)dν�x�dη

(y)

�∫Gs�x�

ϕ�x�∫G

f(xy−1

)D(y−1)dη(y)dν�x�.

�4.10�

Comparing the above equalities, we have

(f ∗ η)�x� �

∫G

f(xy−1

)D(y−1)dη(y). �4.11�

Lemma 4.4. L1�G, λ, μ� is a two-sided ideal ofM�G�.

Proof. Suppose f ∈ L1�G, λ, μ� and η ∈M�G�. Then we have

∥∥η ∗ f∥∥1 �∫G

∣∣η ∗ f�x�∣∣dν�x� ≤∫G

∫Gr�x�

∣∣∣f(y−1x)∣∣∣d∣∣η∣∣(y)dν�x�

�∫G

∫Gr�x�

∣∣f�x�∣∣d∣∣η∣∣(y)dν(yx) �∫G

∣∣f�x�∣∣∫Gr�x�

d∣∣η∣∣(y)dν�x� ≤ ∥∥η∥∥∥∥f∥∥1

ISRN Algebra 11

where D is modular function of μ. Then for each Borel subset E of G, we have

ν0�E� �∫χE�x�D−1/2�x�dν�x� �

∫χE(x−1)D−1/2

(x−1)dν−1�x�

�∫χE−1�x�D1/2�x�D−1�x�dν�x� � ν0

(E−1).

�5.2�

Hence, ν0 is symmetric under inversion.

Definition 5.1. A representation of the locally compact groupoid G is a triple �μ, {Hu}u, π�consisting of a Hilbert bundle �G0, {Hu}u∈G0 , μ�, where μ is a quasi-invariant measure on G0�with associated Radon measures ν, ν−1, ν2, ν0� and for each x ∈ G, a unitary element π�x� ∈B�Hs�x�,Hr�x�� such that

�i� π�u� is the identity map onHu for all u,

�ii� π�xy� � π�x�π�y� for ν2-a.e. �x, y� ∈ G2,�iii� π�x−1� � π−1�x� for ν-a.e. x ∈ G,�iv� for any ξ, η ∈ L2�G0, {Hu}u, μ�, the map

x �−→ 〈π�x�ξ�s�x��, η�r�x��〉 �5.3�

is ν-measurable on G.

Definition 5.2. A representation Π of L1�G, λ, μ� on a Hilbert space,H is a ∗-homomorphism

Π : L1(G, λ, μ

) −→ B�H�. �5.4�

It is called nondegenerate if 〈Π�f�ξ : f ∈ L1, ξ ∈ H〉 is dense inH.

Continuity of Π automatically holds, because each ∗-homomorphism from a Banach∗-algebra to a C∗-algebra is norm decreasing, namely, ‖Π�f�‖ ≤ ‖f‖1, for each f ∈ L1.

Our main aim here is to find a correspondence between unitary representations of Gand nondegenerate representations of L1�G, λ, μ�. Unfortunately, this is impossible in general.Such a correspondence exists between representations of G and those of Cc�G�, when His separable, G is second countable and admits sufficiently many nonsingular G-sets �2,Theorem 1.21�.

Proposition 5.3. Let G be a second countable locally compact groupoid with Haar system and withsufficiently many nonsingular BorelG-sets. Then, every representation ofCc�G� on a separable Hilbertspace is the integrated form of a representation of G.

These assumptions satisfied in the case of r-discrete groupoids and transformationgroups. The main problem is that a continuous representation of �L1�G, λ, μ�, ‖ · ‖I� is notnecessarily continuous in the L1-norm. To get a partial result, we use the following result �2,Proposition 1.7�.

12 ISRN Algebra

Proposition 5.4. Suppose �π, {Hu}u, μ� is a representation of G. For ξ, η ∈ L2�G0, {Hu}u, μ� andf ∈ Cc�G�,

〈Π(f)ξ, η〉�∫G

f�x�〈π�x�ξ�s�x��, η�r�x��

〉dν0�x� �5.5�

defines a bounded nondegenerate ∗-representation of Cc�G� on L2�G0, {Hu}u, μ� such that two equiv-alent representations of G give two equivalent representations of Cc�G�.

The equation above is called the integrated form of a unitary representation. If Π is arepresentation of Cc�G�, the above proposition says that Π should be of the form

〈Π(f)ξ, η〉�∫G

f�x�〈π�x�ξ�s�x��, η�r�x��

〉dν0�x�. �5.6�

For each ξ, η ∈ L2�G0, {Hu}u, μ� and f ∈ Cc�G�.Next, we define a representation of I�G, λ, μ�, denoted by ΠI as

〈ΠI(f)ξ, η〉�∫G

f�x�〈π�x�ξ�s�x��, η�r�x��

〉dν0�x� �5.7�

for each ξ, η ∈ L2�G0, {Hu}u, μ� and f ∈ I�G, λ, μ�

Lemma 5.5. ΠI is a bounded representation of I�G, λ, μ� on B�L2�G0, {Hu}u, μ��.

Proof. We have

〈ΠI(f)ξ, η〉�∫G

f�x�〈π�x�ξ�s�x��, η�r�x��

〉dν0�x�

�∫G0

∫Guf�x�

〈π�x�ξ�s�x��, η�r�x��

〉d−1/2�x�dλu�x�dμ�u�

�∫G0

〈∫Guf�x�π�x�ξ�s�x��D−1/2�x�dλu�x�, η�u�

〉dμ�u�.

�5.8�

Thus, we may define for μ-a.e. u ∈ G0,

ΠI(f)ξ�u� �

∫Guf�x�π�x�ξ�s�x��D−1/2�x�dλu�x�. �5.9�

Now, we have

∣∣〈ΠI(f)ξ, η〉∣∣ �∣∣∣∣∫G

f�x�〈π�x�ξ�s�x��, η�r�x��

〉dν0�x�

∣∣∣∣

≤∫G

∣∣f�x�∣∣‖ξ ◦ s�x�‖∥∥η ◦ r�x�∥∥dν0�x�

ISRN Algebra 13

≤[∫

G

∣∣f�x�∣∣‖ξ ◦ s�x�‖2dν−1�x�]1/2[∫

G

∣∣f�x�∣∣∥∥η ◦ r�x�∥∥2dν�x�]1/2

≤[∫

G0

∫Gu

∣∣f�x�∣∣dλu�x�‖ξ�u�‖2dμ�u�]1/2

×[∫

G0

∫Gu

∣∣f�x�∣∣dλu�x�∥∥η�u�∥∥2dμ�u�]1/2

≤ ∥∥f∥∥1/2I,s ‖ξ‖∥∥f∥∥1/2I,r

∥∥η∥∥≤ ∥∥f∥∥I‖ξ‖

∥∥η∥∥.�5.10�

Therefore, the map x �→ f�x�〈π�x�ξ�s�x��, η�r�x��〉 is ν0-integrable, and ΠI�f� is a boundedoperator of norm ‖ΠI�f�‖ ≤ ‖f‖I . We have to check that ΠI is a ∗-homomorphism. For this,we define

F�x, t� � f�xt�g(t−1)〈π�x�ξ�s�x��, η�r�x��

〉d1/2�x�. �5.11�

Then

〈ΠI(f ∗ g)ξ, η〉 �

∫G

f ∗ g�x�〈π�x�ξ�s�x��, η�r�x��〉dν0�x��∫G

(∫G

f�xt�g(t−1)dλs�x��t�

)〈π�x�ξ ◦ s�x�, η ◦ r�x�〉d−1/2�x�D�x�dν−1�x�

�∫G2F�x, t�dν2�x, t�

�∫G2F(xt, t−1

)ρ−1(xt, t−1

)dν2�x, t�

�∫G0

∫Gu

f�x�〈π�x�

(∫Gs�x�

g�t�π�t�ξ ◦ s�t�D−1/2�t�dλs�x��t�), η�u�

〉

×D−1/2�x�dλu�x�dμ�u�

�∫G

f�x�〈π�x�Π

(g)ξ�s�x��, η�r�x��

〉dν0�x�

�〈ΠI(f)ΠI(g)ξ, η〉.

�5.12�

Hence, ΠI�f ∗ g� � ΠI�f�ΠI�g�. Also

〈ΠI(f∗)ξ, η〉 �

∫G

f∗�x�〈π�x�ξ�s�x��, η�r�x��

〉dν0�x�

�∫G

f(x−1)D(x−1)〈π�x�ξ�s�x��, η�r�x��

〉d−1/2�x�dν�x�

14 ISRN Algebra

�∫G

f�x�D(x−1)〈π∗�x�ξ�r�x��, η�s�x��

〉d1/2�x�D

(x−1)dν�x�

�∫G

f�x�〈η�s�x��, π∗�x�ξ�r�x��

〉dν0�x�

�∫G

f�x�〈π�x�η�s�x��, ξ�r�x��

〉dν0�x�

�〈ΠI(f)ξ, η〉�〈ξ,ΠI

(f)η〉�〈Π∗I(f)ξ, η〉.

�5.13�

The nondegeneracy follows from

〈ΠI(f)ξ | f ∈ Cc�G�, ξ ∈ H

〉 ⊆ 〈ΠI(f)ξ | f ∈ L1�G, ‖·‖I�, ξ ∈ H〉⊆ H. �5.14�

Now let us turn to the problem that a continuous representation of I�G, λ, μ� is not nec-essarily continuous in L1-norm. LetH � L2�G0, {Hu}u, μ� and put

HΠ1 �〈ξ ∈ H : The map f �−→ 〈ΠI(f)ξ, ξ〉 is continuous in L1-norm

〉. �5.15�

Observe that HΠ1 is a nontrivial subspace, as μ is a probability measure and if ‖ξ�u�‖ � 1in Hu for each u ∈ G0, then ‖ξ‖ � 1 in H and the calculations in the proof of Lemma 5.5shows that |ΠI�f�ξ, ξ〉| ≤ ‖f‖1, hence ξ ∈ H1. On the other hand, ΠI�f�HΠ1 ⊆ HΠ1 for eachf ∈ I�G, λ, μ�, because the map

f �−→ 〈ΠI(f)ΠI(g)ξ,ΠI(g)ξ〉 � 〈ΠI(g∗ ∗ f ∗ g)ξ, ξ〉 �5.16�

is continuous. Therefore,ΠI�f�H⊥Π1 ⊆ H⊥Π1 . Hence, we haveΠI�f� � ΠI�f�|HΠ1 ⊕ΠI�f�|H⊥Π1 foreach f ∈ I�G, λ, μ�.

Put H1 � HΠ1 and define Π1�f� :� ΠI�f�|H1 , for f ∈ I�G, λ, μ�. Then, it follows fromcontinuity of f �→ 〈ΠI�f�ξ, ξ〉 and polarization identity that Π1 extends to a continuous rep-resentation of L1�G, λ, μ� onH1, still denoted byΠ1.

Next we focus on the notion of irreducibility which plays an important role in thetheory of representations. We show that if π is an irreducible representation of G, the inte-grated representation Π1 of L1�G, λ, μ� onH1 is irreducible. Basic materials come from �6�.

Definition 5.6. Let �G0, {Hu}u∈G0 , μ� be a Hilbert bundle. A familyM � {Mu}u∈G0 , whereMuis a closed subspace of Hu for each u ∈ G0, is called a subbundle. A subbundle {Mu}u∈G0 iscalled nontrivial if 0/�Mu /�Hu for some u ∈ G0. For a representation π of a locally compactgroupoid G associated with the Hilbert bundle �G0, {Hu}u∈G0 , μ�, a subbundle {Mu}u∈G0 iscalled invariant ifπ�x�Ms�x� ⊆Mr�x� for each x ∈ G. Note that ifM is an invariant subbundle,and 0/�Mu /�Hu for some u ∈ G0, then 0/�Mw /�Hw for every w ∈ �u�.

The following lemma is proved in �6, Lemma 3.4�.

ISRN Algebra 15

Lemma 5.7. Let π be a representation of a locally compact groupoid G associated with the Hilbertbundle �G0, {Hu}u∈G0 , μ�. IfM � {Mu}u∈G0 is an invariant subbundle, then so isM⊥ � {M⊥u}u∈G0 .

Definition 5.8. A representation π of a locally compact groupoid G is called reducible if πadmits a nontrivial invariant subbundleM � {Mu}u∈G0 , otherwise π is called irreducible. Inthis case, it is easy to check that πM with πM�x� � π�x�|Ms�x� : Ms�x� → Mr�x� is called asubrepresentation of π . If π and π ′ are two representations of a locally compact groupoidG associated with two Hilbert bundles �G0, {Hu}u∈G0 , μ� and �G0, {H ′u}u∈G0 , μ′�, respectively.Then, we put

C(π,π ′) �{�Tu�u ∈

∏u∈G0

B(Hu,H ′u) : π ′�x�Ts�x� � Tr�x�π�x� �x ∈ G�}

�5.17�

and write C�π,π� � C�π�.

Two representations π and π ′ are called �unitarily� equivalent if μ ∼ μ′, and there is�Tu�u∈G0 ∈ C�π,π ′� such that Tu is a unitary operator for every u ∈ G0. Note that if �Tu�u∈G0 ∈C�π� and T ∗u denotes the adjoint operator to Tu, then

π�x�T ∗s�x� �[Ts�x�π

(x−1)]∗

�[π(x−1)Tr�x�

]∗� T ∗r�x�π�x�, �5.18�

hence, �T ∗u�u∈G0 ∈ C�π�. We observe that C�π� is a unital ∗-algebra, where the operations aredefined pointwise.

Following �6�, for a representation π of a locally compact groupoid G associated withthe Hilbert bundle �G0, {Hu}u∈G0 , μ�, we set

Λ �

{�λuπ�u��u ∈

∏u∈G0

B�Hu� : λu ∈ C, λu � λw whenever u ∼ w}. �5.19�

If �λuπ�u��u ∈ C�π�, then λu � λw whenever u ∼ w that is �λuπ�u��u ∈ Λ. Therefore, if�Tu�u∈G0 ∈ C�π� \ Λ, then there exists u ∈ G0 with Tu not in Cπ�u�. It is also obvious thatΛ ⊆ C�π�.

We need the following version of Schur’s lemma �6, Lemma 3.11�.

Lemma 5.9. A representationπ of a locally compact groupoidG is irreducible if and only ifΛ � C�π�.In particular, in the case whereG is transitive, thenπ is irreducible if and only ifC�π� � C�π�u��u∈G0 .

Lemma 5.10. If M � �G0, {Mu}u, μ� is closed subbundle of H � �G0, {Hu}u, μ� and Pu is anorthogonal projectionHu ontoMu, thenM is invariant under π if and only if �Pu�u ∈ C�π�.

Proof. If �Pu�u ∈ C�π� andmu ∈Mu, then

π�x�ms�x� � π�x�Ps�x�ms�x� � Pr�x�π�x�ms�x� ∈Mr�x�. �5.20�

16 ISRN Algebra

Thus, π�x�Ms�x� ⊆Mr�x� soM is invariant. Conversely, ifM is invariant, then form ∈Ms�x�

π�x�Ps�x�m � π�x�m � Pr�x�π�x�m �5.21�

and form ∈M⊥s�x�

π�x�Ps�x�m � 0 � Pr�x�π�x�m. �5.22�

Hence, π�x�Ps�x� � Pr�x�π�x�.

We show that if a representation π of a locally compact groupoid G associated withthe Hilbert bundle �G0, {Hu}u∈G0 , μ� is irreducible and Π1 is the corresponding integratedrepresentation of L1�G, λ, μ� on H1 ⊆ L2�G0, {Hu}u, μ�, then Π1 is irreducible. If H �L2�G0, {Hu}u, μ�, then for each subbundle {Mu}u, M � L2�G0, {Mu}u, μ� is closed subspaceof H with orthogonal complement M⊥ � L2�G0, {M⊥u}u, μ�. A map Θ : L2�G0, {Hu}u, μ� →L2�G0, {Mu}u, μ�; ξ �→ Θξ withΘξ�s�x�� ∈Mr�x� is called an orthogonal projection ofH ontoM.

Let π be a representation of G on the Hilbert bundle �G0, {Hu}u∈G0 , μ� and ΠI be thebounded representation of I�G, λ, μ� on B�L2�G0, {Hu}u, μ�� constructed above.

Lemma 5.11. M is invariant under ΠI if and only if for each f ∈ I�G, λ, μ�, ΠI�f�Θ � ΘΠI�f�.

Proof. Suppose ΠI�f�Θ � ΘΠI�f� and ξ ∈M, then

ΠI(f)ξ � ΠI

(f)Θξ � ΘΠI

(f)ξ ∈M. �5.23�

Hence,M is invariant. Conversely, ifM is invariant, then for ξ ∈M,

ΠI(f)Θξ � ΠI

(f)ξ � ΘΠI

(f)ξ �5.24�

and for η ∈M⊥,

ΠI(f)Θη � 0 � ΘΠI

(f)η. �5.25�

Thus, ΠI�f�Θ � ΘΠI�f�.

Proposition 5.12. If π is an irreducible representation of G, then ΠI is an irreducible representationof I�G, λ, μ�.

ISRN Algebra 17

Proof. If ΠI is reducible, then there exists a nontrivial invariant closed subspace M ofL2�G0, {Hu}u, μ�, and, hence, there is an orthogonal projection Θ ∈ C�ΠI�. It follows that

∫G

f�x�〈π�x�Ps�x�ξ�s�x��, η�r�x��

〉dν0�x�

�〈ΠI(f)Θξ, η

〉�〈ΘΠI

(f)ξ, η〉

�∫G

f�x�〈Pr�x�π�x�ξ�s�x��, η�r�x��

〉dν0�x�,

�5.26�

for each f ∈ Cc�G� and ξ, η ∈ L2�G0, {Hu}u, μ�. Therefore, π�x�Ps�x� � Pr�x�π�x�, for eachx ∈ G, hence �Pu�u ∈ C�π�. Now by the Schur’s lemma, π is reducible.

Theorem 5.13. If π is an irreducible representation of G, then Π1 is an irreducible representation ofL1�G, λ, μ� onH1.

Proof. If Π1 is reducible, then there exists a nontrivial invariant closed subspace M1 of H1 ⊆H � L2�G0, {Hu}u, μ�. By the calculations after Lemma 5.5, ΠI � Π̃1 ⊕ Π̃2, where Π̃1 and Π̃2are the corresponding representations of I�G, λ, μ� on H1 and H⊥1 . Therefore, M � M1 ⊕H⊥1is a nontrivial invariant closed subspaceH, andΠI is reducible. Hence, π is reducible by theabove proposition.

References

�1� A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, Springer, New York, NY, USA, 2009.�2� J. Renault, A Groupoid Approach to C∗-Algebras, vol. 793 of Lecture Notes in Mathematics, Springer, Berlin,

Germany, 1980.�3� J. Westman, “Harmonic analysis on groupoids,” Pacific Journal of Mathematics, vol. 27, pp. 621–632,

1968.�4� M. R. Buneci, “Isomorphic groupoid C∗-algebras associated with different Haar systems,” New York

Journal of Mathematics, vol. 11, pp. 225–245, 2005.�5� M. Buneci, “The equality of the reduced and the full C∗-algebras and the amenability of a topological

groupoid,” in Recent Advances in Operator Theory and Their Applications, vol. 153, pp. 61–78, Birkhäuser,Basel, Switzerland, 2005.

�6� H. Amiri and M. L. Bami, “Square integrable representation of groupoids,” Acta Mathematica Sinica,vol. 23, no. 2, pp. 327–340, 2007.

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