Table of Contents
ISRN Algebra
VolumeΒ 2011, Article IDΒ 856709, 17 pages
http://dx.doi.org/10.5402/2011/856709
Research Article

𝐿1-Algebra of a Locally Compact Groupoid

1Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran
2Faculty of Mathematical Sciences and Computer, Tarbiat Moalem University, 50 Taleghani Avenue, Tehran, Iran

Received 12 May 2011; Accepted 6 June 2011

Academic Editor: B.Β Rangipour

Copyright Β© 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 𝐺 with a fixed Haar system πœ† and quasi-invariant measure πœ‡, we introduce the notion of πœ†-measurability and construct the space 𝐿1(𝐺, πœ†, πœ‡) of absolutely integrable functions on 𝐺 and show that it is a Banach βˆ—-algebra and a two-sided ideal in the algebra 𝑀(𝐺) of complex Radon measures on 𝐺. We find correspondences between representations of 𝐺 on Hilbert bundles and certain class of nondegenerate representations of 𝐿1(𝐺, πœ†, πœ‡).

1. Introduction and Preliminaries

For a locally compact group 𝐺 with a Haar measure πœ†, the Banach algebra 𝐿1(𝐺,πœ†) plays a central role in harmonic analysis on 𝐺 [1]. This motivated us to define a similar notion in the case where 𝐺 is a locally compact groupoid with a (fixed) Haar system πœ† and quasi-invariant measure πœ‡. This paper is devoted to the study of such a groupoid 𝐿1-algebra 𝐿1(𝐺,πœ†,πœ‡). One may expect that as the group case, there is a full interaction between the properties of 𝐺 and that of 𝐿1(𝐺,πœ†,πœ‡). This is not completely true. For instance, unlike the group case, not every nondegenerate representation of 𝐿1(𝐺,πœ†,πœ‡) is integrated form a representation of 𝐺. In Section 2, we introduce the appropriate measurability notion used to define 𝐿1(𝐺,πœ†,πœ‡). Sections 3 and 4 are devoted to the algebra structure of 𝐿1(𝐺,πœ†,πœ‡) and its embedding into 𝑀(𝐺) as a closed ideal. In Section 5, we find the class of nondegenerate representations of 𝐿1(𝐺,πœ†,πœ‡) which could be obtained by integrating a representation of 𝐺.

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 𝜎-finite) Radon measures [1, Theorem  B.3.3].

Lemma 1.1. Let 𝜈 and 𝜌 be Radon measures on the Borel sets of the locally compact spaces 𝑋 and π‘Œ, respectively. Then there exists a unique Radon measure πœˆβ‹…πœŒ on π‘‹Γ—π‘Œ such that (i)if π‘“βˆΆπ‘‹Γ—π‘Œβ†’πΆ is πœˆβ‹…πœŒ-integrable, then the partial integrals ξ€œπ‘‹ξ€œπ‘“(π‘₯,𝑦)π‘‘πœˆ(π‘₯),π‘Œπ‘“(π‘₯,𝑦)π‘‘πœŒ(𝑦)(1.1) define integrable functions such that Fubini’s formula holds ξ€œπ‘‹Γ—π‘Œξ€œπ‘“π‘‘πœˆβ‹…πœŒ=π‘‹ξ€œπ‘Œξ€œπ‘“π‘‘πœˆπ‘‘πœŒ=π‘Œξ€œπ‘‹π‘“π‘‘πœŒπ‘‘πœˆ,(1.2)(ii)if 𝑓 is measurable such that 𝐴={(π‘₯,𝑦)βˆˆπ‘‹Γ—π‘ŒβˆΆπ‘“(π‘₯,𝑦)β‰ 0} is 𝜎-finite and if one of the iterated integrals βˆ«π‘‹βˆ«π‘Œ|𝑓|π‘‘πœˆπ‘‘πœŒ or βˆ«π‘Œβˆ«π‘‹|𝑓|π‘‘πœŒπ‘‘πœˆ 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 composable pairs, a product map: 𝐺2→𝐺;(π‘₯,𝑦)↦π‘₯𝑦, and an inverse map: 𝐺→𝐺;π‘₯↦π‘₯βˆ’1, such that for each π‘₯,𝑦,π‘§βˆˆπΊ,(i)(π‘₯βˆ’1)βˆ’1=π‘₯, (ii)if (π‘₯,𝑦),(𝑦,𝑧)∈𝐺2 then (π‘₯𝑦,𝑧),(π‘₯,𝑦𝑧)∈𝐺2 and (π‘₯𝑦)𝑧=π‘₯(𝑦𝑧),(iii)(π‘₯βˆ’1,π‘₯)∈𝐺2 and if (π‘₯,𝑦)∈𝐺2 then π‘₯βˆ’1(π‘₯𝑦)=𝑦,(iv)(π‘₯,π‘₯βˆ’1)∈𝐺2 and if (𝑧,π‘₯)∈𝐺2 then (𝑧π‘₯)π‘₯βˆ’1=𝑧.

If π‘₯∈𝐺, 𝑠(π‘₯)=π‘₯βˆ’1π‘₯ is called the source of π‘₯ and π‘Ÿ(π‘₯)=π‘₯π‘₯βˆ’1 is called the range of π‘₯. The pair (π‘₯,𝑦) is composable if and only if 𝑠(π‘₯)=π‘Ÿ(𝑦). The set 𝐺0=𝑠(𝐺)=π‘Ÿ(𝐺) is the unit space of 𝐺, and its elements are called units in the sense that π‘₯𝑠(π‘₯)=π‘₯ and π‘Ÿ(π‘₯)π‘₯=π‘₯.

For 𝑒,π‘£βˆˆπΊ0, 𝐺𝑒=π‘Ÿβˆ’1{𝑒}, 𝐺𝑣=π‘ βˆ’1{𝑣}, 𝐺𝑒𝑣=πΊπ‘’βˆ©πΊπ‘£, and 𝐺{𝑒}=𝐺𝑒𝑒. The latter is called the isotropy group at 𝑒. We define π‘’βˆΌπ‘£ if πΊπ‘’π‘£β‰ βˆ…. It is checked that ~ is an equivalence relation on the unit space 𝐺0. The equivalence class of 𝑒 is denoted by [𝑒] and is called the orbit of 𝑒.

A topological groupoid consists of a groupoid 𝐺 and a topology compatible with the groupoid structure, such that the composition map is continuous on 𝐺2 in the induced product topology, and the inversion map is continuous on 𝐺. A locally compact groupoid is a topological groupoid 𝐺 which satisfies the following conditions:(i)𝐺0 is locally compact and Hausdorff in the relative topology.(ii)There is a countable family 𝐢 of compact Hausdorff subsets of 𝐺 whose interiors form a basis for the topology of 𝐺.(iii)Every 𝐺𝑒 is locally compact Hausdorff in the relative topology.

A locally compact groupoid is π‘Ÿ-discrete if its unit space is an open subset. Let 𝐺 be a locally compact groupoid. The support of a function π‘“βˆΆπΊβ†’β„‚ is the complement of the union of all open, Hausdorff subsets of 𝐺 on which 𝑓 vanishes. The space 𝐢𝑐(𝐺) consists of all continuous functions on 𝐺 with compact support. A left Haar system for 𝐺 consists of measures {πœ†π‘’,π‘’βˆˆπΊ0} on 𝐺 such that(i)the support of πœ†π‘’ is 𝐺𝑒,(ii)(continuity) for each π‘“βˆˆπΆπ‘βˆ«(𝐺),π‘’β†¦πœ†(𝑓)(𝑒)=π‘“π‘‘πœ†π‘’ is continuous,(iii)(left invariance) for any π‘₯∈𝐺 and any π‘“βˆˆπΆπ‘(𝐺), ξ€œπ‘“(π‘₯𝑦)π‘‘πœ†π‘ (π‘₯)(ξ€œπ‘¦)=𝑓(𝑦)π‘‘πœ†π‘Ÿ(π‘₯)(𝑦).(1.3)

Let πœ‡ be a measure on 𝐺0. The measure on 𝐺 induced by πœ‡ is βˆ«πœ†πœˆ=π‘’π‘‘πœ‡, defined by βˆ«πΊβˆ«π‘“π‘‘πœˆ=𝐺0βˆ«π‘‘πœ‡(𝑒)πΊπ‘“π‘‘πœ†π‘’, for π‘“βˆˆπΆπ‘(𝐺). The measure on 𝐺2 induced by πœ‡ is 𝜈2=βˆ«πœ†π‘’Γ—πœ†π‘’π‘‘πœ‡(𝑒). The inversion of 𝜈 is πœˆβˆ’1=βˆ«πœ†π‘’π‘‘πœ‡(𝑒). Note that the measures 𝜈,𝜈2,πœˆβˆ’1 are Radon. A measure πœ‡ on 𝐺0 is said to be quasi invariant if its induced measure 𝜈 is equivalent to its inverse πœˆβˆ’1. Let πœ‡ be a quasi-invariant measure on 𝐺0. The Radon-Nikodym derivative 𝐷=π‘‘πœˆ/π‘‘πœˆβˆ’1 is called the modular function of πœ‡. An equivalent definition of modular function on 𝐺 is given in [3, Definition  2.3], defining the modular function Ξ” as a strictly positive continuous homomorphism on 𝐺 such that Ξ”|𝐺𝑒𝑒 is modular function for 𝐺𝑒𝑒.

A subset 𝐴 of 𝐺 is called a 𝐺-set if the restrictions of π‘Ÿ and 𝑠 to it are one to one. Equivalently, 𝐴 is a 𝐺-set if and only if π΄π΄βˆ’1 and π΄βˆ’1𝐴 are contained in 𝐺0.

We introduce some notations from [4, 5] which is related to the representations of 𝐿1(𝐺,πœ†,πœ‡). Let {πœ†π‘’}𝑒 be a fixed Haar system on 𝐺. Let πœ‡ be a quasi-invariant measure, Ξ” its modular function, 𝜈 be the measure induced by πœ‡ on 𝐺, and 𝜈0=Ξ”βˆ’1/2𝜈. Let πΌπΌπœ‡ξ€½(𝐺,𝜈,πœ‡)=π‘“βˆˆπΏ1𝐺,𝜈0ξ€ΈβˆΆβ€–π‘“β€–πΌπΌ,πœ‡ξ€Ύ,<∞(1.4) where ‖𝑓‖𝐼𝐼,πœ‡ is defined by ‖𝑓‖𝐼𝐼,πœ‡ξ‚»ξ€œ||||=sup𝑓(π‘₯)𝑗(𝑠(π‘₯))π‘˜(π‘Ÿ(π‘₯))π‘‘πœˆ0ξ€œ||𝑗||(π‘₯);2ξ€œ||π‘˜||π‘‘πœ‡=2ξ‚Ό.π‘‘πœ‡=1(1.5) If πœ‡1 and πœ‡2 are two equivalent quasi-invariant measures, then ‖𝑓‖𝐼𝐼,πœ‡1=‖𝑓‖𝐼𝐼,πœ‡2,(1.6) because ‖𝑓‖𝐼𝐼,πœ‡=β€–πΌπΌπœ‡(|𝑓|)β€– for each quasi-invariant measure πœ‡, where πΌπΌπœ‡ is the one-dimensional trivial representation on πœ‡. Now define ‖𝑓‖𝐼𝐼=supπœ‡β€–π‘“β€–πΌπΌ,πœ‡.(1.7) The supremum is taken over the class of quasi-invariant measures.

Let 𝐼(𝐺,𝜈,πœ‡)=π‘“βˆˆπΏ1𝐺,𝜈0ξ€ΈβˆΆβ€–π‘“β€–πΌ,πœ‡ξ€Ύ,<∞(1.8) where ‖𝑓‖𝐼,πœ‡ξ‚»β€–β€–β€–ξ€œ||𝑓||=maxπ‘’βŸΆπ‘‘πœ†π‘’β€–β€–β€–βˆž,β€–β€–β€–ξ€œ||π‘“π‘’βŸΆβˆ—||π‘‘πœ†π‘’β€–β€–β€–βˆžξ‚Ό.(1.9) Under the canonical convolution and involution, 𝐼(𝐺,𝜈,πœ‡) becomes a Banach βˆ—-algebra [5, page  4]. Here π‘“βˆ—(π‘₯)=𝑓(π‘₯βˆ’1). If we consider ‖𝑓‖𝐼,π‘Ÿ=supπ‘’ξ‚΅ξ€œ||𝑓||π‘‘πœ†π‘’ξ‚Ά,‖𝑓‖𝐼,𝑠=supπ‘’ξ‚΅ξ€œ||𝑓||π‘‘πœ†π‘’ξ‚Ά,(1.10) and put ‖𝑓‖𝐼=max(‖𝑓‖𝐼,π‘Ÿ,‖𝑓‖𝐼,𝑠), then ‖𝑓‖𝐼=‖𝑓‖𝐼,πœ‡.

2. πœ†-Measurability

For the rest of the paper, 𝐺 is a locally compact, Hausdorff, second countable groupoid which admits a left Haar system πœ†={πœ†π‘’}.

Definition 2.1. A system of measures {πœ†π‘’}π‘’βˆˆπΊ0 is said to be complete if for each π‘’βˆˆπΊ0, πœ†π‘’ is complete on its 𝜎-algebra π”πœ†π‘’. A Borel measurable set πΈβŠ†πΊ is called πœ†-measurable if for each π‘’βˆˆπΊ0, πΈβˆ©πΊπ‘’βˆˆπ”πœ†π‘’. A function π‘“βˆΆπΊβ†’β„‚ is πœ†-measurable if for every π‘’βˆˆπΊ0 and every open set π‘‚βŠ†β„‚,π‘“βˆ’1(𝑂)βˆ©πΊπ‘’βˆˆπ”πœ†π‘’.

Proposition 2.2. If πœ†={πœ†π‘’} is the completion of πœ†={πœ†π‘’} and π‘“βˆΆπΊβ†’[0,∞] is πœ†-measureable, then there is a πœ†-measurable function 𝑔 such that 𝑓=𝑔 on 𝐺𝑒(πœ†π‘’-a.e).

Proof. Since 𝑓 is πœ†-measurable for each π‘’βˆˆπΊ0, 𝑓 is πœ†π‘’-measurable on 𝐺𝑒, and since πœ†π‘’ is the completion of πœ†π‘’, there exists a πœ†π‘’-measurable function 𝑔𝑒 such that 𝑓=𝑔𝑒(πœ†π‘’-a.e). Now define 𝑔=𝑔𝑒 on 𝐺𝑒 and zero, elsewhere. Since for every π‘’βˆˆπΊ0 and every open set π‘‚βŠ†β„‚,π‘”βˆ’1(𝑂)βˆ©πΊπ‘’=π‘”π‘’βˆ’1(𝑂)βˆˆπ”πœ†π‘’, 𝑔 is πœ†-measurable, and since πœ†π‘’(𝐺𝑒)𝑐=0, we have 𝑓=𝑔 on 𝐺𝑒 (πœ†π‘’-a.e).

From now on, we assume that the Haar system πœ† is complete.

Lemma 2.3. For each π‘“βˆΆπΊβ†’β„‚, πœ†-measurability of 𝑓 is equivalent to 𝜈-measurability of 𝑓.

Proof. We have suppπœ†π‘’=𝐺𝑒 and π‘“βˆ’1𝑓(𝑂)=βˆ’1(𝑂)βˆ©πΊπ‘’ξ€Έβˆͺξ€·π‘“βˆ’1(𝑂)∩(𝐺𝑒)𝑐.(2.1) Since πœ†π‘’ is complete and (𝐺𝑒)𝑐 is πœ†π‘’-null, (π‘“βˆ’1(𝑂)∩(𝐺𝑒)𝑐)βŠ†(𝐺𝑒)𝑐 is in π”πœ†π‘’. Thus, for each π‘’βˆˆπΊ0 and open set π‘‚βŠ†β„‚π‘“βˆ’1(𝑂)βˆˆπ”πœ†π‘’βŸΊπ‘“βˆ’1(𝑂)βˆ©πΊπ‘’βˆˆπ”πœ†π‘’.(2.2) Now for βˆ«πœ†πœˆ=π‘’π‘‘πœ‡(𝑒), we have π”πœˆ=β‹‚π‘’βˆˆπΊ0π”πœ†π‘’, hence 𝑓 is 𝜈-measurable if and only if 𝑓 is πœ†-measurable.

If π‘“βˆΆπΊβ†’β„‚ is πœ†-measurable and π‘”βˆΆβ„‚β†’β„‚ is continuous, then π‘”βˆ˜π‘“βˆΆπΊβ†’β„‚ is πœ†-measurable. Also, if 𝑓,π‘”βˆΆπΊβ†’β„ is πœ†-measurable, Ξ¦βˆΆβ„‚β†’π‘Œ is continuous, and β„Ž(π‘₯)=Ξ¦(𝑓(π‘₯),𝑔(π‘₯)), then β„Ž is πœ†-measurable. If 𝑓=𝑒+𝑖𝑣 then 𝑓 is πœ†-measurable if and only if 𝑒,𝑣 are πœ†-measurable. If 𝑓,π‘”βˆΆπΊβ†’β„ are πœ†-measurable, so are 𝑓+𝑔 and 𝑓𝑔. Also, if {𝑓𝑗}∞1 is a sequence of ℝ-valued πœ†-measurable functions, then the functions 𝑔1(π‘₯)=sup𝑗𝑓𝑗(π‘₯),𝑔2(π‘₯)=inf𝑗𝑓𝑗(π‘₯),𝑔3(π‘₯)=limsup𝑗𝑓𝑗(π‘₯), and 𝑔4(π‘₯)=liminf𝑗𝑓𝑗(π‘₯) are all πœ†-measurable. If 𝑓(π‘₯)=lim𝑗𝑓𝑗(π‘₯) exists for every π‘₯∈𝐺, then 𝑓 is πœ†-measurable. Thus if {𝑓𝑗}∞1 is a sequence of complex-valued πœ†-measurable functions and π‘“π‘—β†’π‘“πœ†π‘’-a.e, then 𝑓 is πœ†-measurable.

3. The Algebra 𝐿1(𝐺,πœ†,πœ‡)

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

Definition 3.1. Suppose πœ‡ is a quasi-invariant probability measure on 𝐺0, and 𝜈 is Radon measure induced by πœ‡, then we define 𝐿1(𝐺,𝜈)=𝐿1ξ‚»(𝐺,πœ†,πœ‡)=π‘“βˆΆπΊβŸΆβ„‚βˆΆπ‘“isπœ†-measurable,‖𝑓‖1=ξ€œπΊ||||ξ‚Ό.𝑓(π‘₯)π‘‘πœˆ(π‘₯)<∞(3.1)

It is clear that 𝐿1(𝐺,πœ†,πœ‡) is a Banach space. We show that the Banach space 𝐿1(𝐺,πœ†,πœ‡) is a Banach *-algebra under the following convolution product.(ξ€œπ‘“βˆ—π‘”)(π‘₯)=πΊπ‘Ÿ(π‘₯)𝑦𝑓(𝑦)π‘”βˆ’1π‘₯ξ€Έπ‘‘πœˆ(𝑦).(3.2)

Lemma 3.2. If 𝑓,π‘”βˆˆπΏ1(𝐺,πœ†,πœ‡), then π‘“βˆ—π‘”βˆˆπΏ1(𝐺,πœ†,πœ‡) and β€–π‘“βˆ—π‘”β€–1≀‖𝑓‖1‖𝑔‖1.

Proof. For each 𝑓,𝑔, β€–π‘“βˆ—π‘”β€–1=ξ€œπΊ||||β‰€ξ€œπ‘“βˆ—π‘”(π‘₯)π‘‘πœˆ(π‘₯)πΊξ€œπΊπ‘Ÿ(π‘₯)||||||𝑔𝑦𝑓(𝑦)βˆ’1π‘₯ξ€Έ||β‰€ξ€π‘‘πœˆ(𝑦)π‘‘πœˆ(π‘₯)𝐺||||||𝑔𝑦𝑓(𝑦)βˆ’1π‘₯ξ€Έ||=ξ€π‘‘πœˆ(𝑦)π‘‘πœˆ(π‘₯)𝐺||𝑓||||𝑔||=ξ€œ(𝑦)(π‘₯)π‘‘πœˆ(𝑦)π‘‘πœˆ(𝑦π‘₯)𝐺0ξ€œπΊ||||ξ€œπ‘”(π‘₯)𝐺0ξ€œπΊ||||𝑓(𝑦)π‘‘πœ†π‘’(𝑦)π‘‘πœ‡(𝑒)π‘‘πœ†π‘’=ξ€œ(𝑦π‘₯)π‘‘πœ‡(𝑒)𝐺||||ξ‚΅ξ€œπ‘”(π‘₯)𝐺||||𝑓(𝑦)π‘‘πœˆ(𝑦)π‘‘πœˆ(π‘₯)=‖𝑓‖1‖𝑔‖1.(3.3) The measurability of π‘“βˆ—π‘” follows from πœ†-measurability of 𝑓,𝑔.

Next, we define an involution on 𝐿1(𝐺,πœ†,πœ‡). We say that the assertion 𝑃(π‘₯) holds for πœ†-a.e. π‘₯ if for 𝐸={π‘₯βˆΆΒ¬π‘ƒ(π‘₯)}, πœ‡{π‘’βˆΆπœ†π‘’(𝐸)>0}=0. Clearly an assertion holds πœ†-almost everywhere if and only if it holds 𝜈-almost everywhere.

Lemma 3.3. Suppose π·π‘’βˆΆπΊπ‘’β†’β„+ with 𝐷𝑒(π‘₯)=π‘‘πœ†π‘’(π‘₯)/π‘‘πœ†π‘’(π‘₯)(π‘₯∈𝐺). Then 𝐷=𝐷𝑒 on 𝐺𝑒(π‘Ž.𝑒.).

Proof. Suppose πΈβŠ†πΊπ‘’. We have ξ€œπœˆ(𝐸)=𝐺0πœ†π‘’(ξ€œπΈ)π‘‘πœ‡(𝑒)=𝐺0ξ€œπΈπ‘‘πœ†π‘’(ξ€œπ‘₯)π‘‘πœ‡(𝑒)=𝐺0ξ€œπΈπ·π‘’(π‘₯)π‘‘πœ†π‘’(π‘₯)π‘‘πœ‡(𝑒).(3.4) Also from π‘‘πœˆ=π·π‘‘πœˆβˆ’1 we have ξ€œπœˆ(𝐸)=𝐸𝐷(π‘₯)π‘‘πœˆβˆ’1(ξ€œπ‘₯)=πΈξ€œπ·(π‘₯)𝐺0π‘‘πœ†π‘’(ξ€œπ‘₯)π‘‘πœ‡(𝑒)=𝐺0ξ€œπΈπ·(π‘₯)π‘‘πœ†π‘’(π‘₯)π‘‘πœ‡(𝑒).(3.5) Thus ξ€œπΊ0ξ€œπΈξ€·π·π‘’(ξ€Έπ‘₯)βˆ’π·(π‘₯)π‘‘πœ†π‘’(π‘₯)π‘‘πœ‡(𝑒)=0.(3.6) Now, let 𝐸𝑒={π‘₯βˆˆπΊπ‘’βˆΆπ·π‘’(π‘₯)>𝐷(π‘₯)}. If πœ†π‘’(𝐸𝑒)>0 then ξ€œπΈπ‘’ξ€·π·π‘’(ξ€Έπ‘₯)βˆ’π·(π‘₯)π‘‘πœ†π‘’(π‘₯)>0.(3.7) But βˆ«πΈπ‘’(𝐷𝑒(π‘₯)βˆ’π·(π‘₯))π‘‘πœ†π‘’(π‘₯)=0(πœ‡-a.e.), hence πœ†π‘’(𝐸𝑒)=0(πœ‡-a.e.). Thus πœ‡{π‘’βˆΆπœ†π‘’(𝐸𝑒)>0}=0. Therefore, 𝐷𝑒(π‘₯)≀𝐷(π‘₯)(πœ†-a.e.). A similar argument leads to 𝐷𝑒(π‘₯)β‰₯𝐷(π‘₯)(πœ†-a.e.).

Proposition 3.4. The map βˆ—βˆΆπΏ1(𝐺,πœ†,πœ‡)→𝐿1(𝐺,πœ†,πœ‡);π‘“β†¦π‘“βˆ—, where π‘“βˆ—(π‘₯)=𝑓(π‘₯βˆ’1)𝐷(π‘₯βˆ’1), is an isometric involution on 𝐿1(𝐺,πœ†,πœ‡).

Note that from [5, page  9], we have ‖𝑓‖𝐿1(𝐺,πœ†,πœ‡)=β€–π‘“βˆ—β€–πΏ1(𝐺,πœ†,πœ‡)=‖𝑓‖𝐿1(𝐺,𝜈0)=β€–π‘“βˆ—β€–πΏ1(𝐺,𝜈0)≀‖𝑓‖𝐼𝐼,πœ‡=β€–π‘“βˆ—β€–πΌπΌ,πœ‡β‰€β€–π‘“β€–πΌ,πœ‡=β€–π‘“βˆ—β€–πΌ,πœ‡.(3.8) Hence the space of 𝐿1(𝐺,πœ†,πœ‡) is in general bigger than 𝐼(𝐺,𝜈,πœ‡) and πΌπΌπœ‡(𝐺,𝜈,πœ‡) with respect to 𝐼-norm and 𝐼𝐼-norm, indeed 𝐼(𝐺,𝜈,πœ‡)βŠ†πΌπΌπœ‡(𝐺)βŠ†πΏ1(𝐺,πœ†,πœ‡).

According to [2, Lemma  1.4], 𝐼-norm topology is coarser than the inductive limit topology. Also [5, page  15] shows that 𝐢𝑐(𝐺) has a two-sided bounded approximate identity (𝑒𝑛)βˆžπ‘›=1 with respect to the inductive limit topology with the following properties:(i)𝑒𝑛(π‘₯)β‰₯0 for all π‘₯∈𝐺,(ii)|βˆ«π‘’π‘›(π‘₯)π‘‘πœ†π‘’(π‘₯)βˆ’1|<1/𝑛 for all π‘’βˆˆπΎπ‘›, where βˆͺ𝑛𝐾𝑛=𝐺0 and 𝐾𝑛's are compact,(iii)𝑒𝑛(π‘₯)=𝑒𝑛(π‘₯βˆ’1) for all π‘₯∈𝐺.

An argument in [5, page  15] shows that there is 𝑀>0 such that ‖𝑒𝑛‖𝐼𝐼,πœ‡β‰€π‘€ for all 𝑛. Since 𝐢𝑐(𝐺) is dense in 𝐿1(𝐺,πœ†,πœ‡), thus 𝐿1(𝐺,πœ†,πœ‡) has a two-sided bounded approximate identity.

For each π‘“βˆˆπΏ1(𝐺,πœ†,πœ‡) define 𝐿π‘₯ξ€·π‘₯𝑓(𝑦)=π‘“βˆ’1𝑦,𝑅π‘₯𝑓(𝑦)=𝑓(𝑦π‘₯),(3.9) when the multiplications on the right hand sides are defined. It is easy to check that the maps 𝐿π‘₯,𝑅π‘₯ are homomorphisms.

Proposition 3.5. Let 𝐼 be a closed subspace of 𝐿1(𝐺,πœ†,πœ‡). Then 𝐼 is a left ideal if and only if it is closed under left translation, and 𝐼 is a right ideal if and only if it is closed under right translation.

Proof. Note that since βˆ«π‘“βˆ—π‘”=πΊπ‘Ÿ(𝑦)𝑓(𝑦)πΏπ‘¦π‘”π‘‘πœˆ(𝑦), 𝐿π‘₯(ξ€œπ‘“βˆ—π‘”)=πΊπ‘Ÿ(𝑦)𝑓(𝑦)𝐿π‘₯πΏπ‘¦ξ€œπ‘”π‘‘πœˆ(𝑦)=πΊπ‘Ÿ(𝑦)𝑓(𝑦)𝐿π‘₯𝑦=ξ€œπ‘”π‘‘πœˆ(𝑦)πΊπ‘Ÿ(𝑦)𝑓π‘₯βˆ’1π‘¦ξ€ΈπΏπ‘¦ξ€œπ‘”π‘‘πœˆ(𝑦)=πΊπ‘Ÿ(𝑦)𝐿π‘₯𝑓(𝑦)πΏπ‘¦ξ€·πΏπ‘”π‘‘πœˆ(𝑦)=π‘₯π‘“ξ€Έβˆ—π‘”.(3.10) Now suppose (𝑒𝑛)𝑛 is a bounded approximate identity for 𝐿1(𝐺,πœ†,πœ‡). For the first assertion, if π‘“βˆˆπΏ1(𝐺,πœ†,πœ‡) and π‘”βˆˆπΌ and 𝐼 is a left ideal, then we have 𝐿π‘₯ξ€·π‘’π‘›ξ€Έβˆ—π‘“=𝐿π‘₯ξ€·π‘’π‘›ξ€Έβˆ—π‘“βŸΆπΏπ‘₯𝑓.(3.11) Conversely, if 𝐼 is closed under left translation and π‘“βˆˆπΏ1(𝐺,πœ†,πœ‡) and π‘”βˆˆπΌ, ξ€œπ‘“βˆ—π‘”=πΊπ‘Ÿ(𝑦)𝑓(𝑦)πΏπ‘¦π‘”π‘‘πœˆ(𝑦)(3.12) is in the closed linear span of the functions 𝐿𝑦𝑔 and hence in 𝐼. The other assertion is proved similarly.

4. 𝑀(𝐺) as Banach βˆ—-Algebra

Let 𝑀(𝐺) be the space of complex Radon measures on 𝐺. If πœ‚,πœƒβˆˆπ‘€(𝐺), then the map πœ“β†¦πΌ(πœ“) on 𝐢0(𝐺) defined by ∫𝐼(πœ“)=πΊβˆ«πΊπ‘Ÿ(𝑦)πœ“(π‘₯𝑦)π‘‘πœ‚(π‘₯)π‘‘πœƒ(𝑦) is a linear functional on 𝐢0(𝐺) satisfying |𝐼(πœ“)|β‰€β€–πœ“β€–supβ€–πœ‚β€–β€–πœƒβ€–, so by Riesz representation theorem, it is given by a measure shown as πœ‚βˆ—πœƒ called the convolution of πœ‚,πœƒ with β€–πœ‚βˆ—πœƒβ€–β‰€β€–πœ‚β€–β€–πœƒβ€–. If we define πœ‚βˆ—(𝐸)=πœ‚(πΈβˆ’1), then πœ‚β†¦πœ‚βˆ— is an involution on 𝑀(𝐺), and 𝑀(𝐺) is a Banach βˆ—-algebra. In this section, we show that the space 𝐿1(𝐺,πœ†,πœ‡) is a closed two-sided ideal of 𝑀(𝐺).

Proposition 4.1. The map 𝐿1(𝐺,πœ†,πœ‡)β†ͺ𝑀(𝐺) with πœˆπ‘“βˆ«(𝐸)=πΈπ‘“πœ’πΈπ‘‘πœˆ, for (πΈβŠ†πΊ); π‘“β†¦πœˆπ‘“ is an isometric embedding.

Proof. If π‘“βˆˆπΏ1(𝐺,πœ†,πœ‡), then 𝑓 is πœ†-measurable so the integral exists, and it is easy to check that πœˆπ‘“ is a measure on 𝐺. We show that πœˆπ‘“ is Radon. If 𝑓=𝑒+𝑖𝑣, then πœˆπ‘“=πœˆπ‘’+π‘–πœˆπ‘£, so πœˆπ‘“ is Radon if and only if πœˆπ‘’ and πœˆπ‘£ are Radon. Since 𝐺 is LCH and second countable, we have πœˆπ‘’βˆ«(𝐾)=πΎβˆ«π‘’π‘‘πœˆβ‰€πΊ|𝑒|π‘‘πœˆ=‖𝑒‖1<∞, for each compact set 𝐾, thus πœˆπ‘’ is Radon. Similarly πœˆπ‘£ is Radon, and so is πœˆπ‘“.
By definition, β€–πœˆπ‘“βˆ‘β€–=sup{𝑛1|πœˆπ‘“(πΈπ‘–βˆ)|βˆΆπ‘›βˆˆπ‘,𝐺=𝑛1𝐸𝑖}, so for each πœ–>0, there exists a partition {𝐸𝑖}𝑛1 of 𝐺 such that β€–β€–πœˆπ‘“β€–β€–βˆ’πœ–<𝑛1||πœˆπ‘“ξ€·πΈπ‘–ξ€Έ||=𝑛1||||ξ€œπΊπ‘“πœ’πΈπ‘–||||β‰€ξ€œπ‘‘πœˆπΊξƒ©||𝑓||𝑛1πœ’πΈπ‘–ξƒͺπ‘‘πœˆ=‖𝑓‖1.(4.1) Thus, β€–πœˆπ‘“β€–β‰€β€–π‘“β€–1. Conversely, suppose 𝑓β‰₯0, then πœˆπ‘“β‰₯0 and for every partition {𝐸𝑖}𝑛1 of 𝐺 we have, β€–β€–πœˆπ‘“β€–β€–β‰₯𝑛1πœˆπ‘“ξ€·πΈπ‘–ξ€Έ=𝑛1ξ€œπΊπ‘“πœ’πΈπ‘–ξ€œπ‘‘πœˆ=πΊπ‘“π‘‘πœˆ=‖𝑓‖1.(4.2) If 𝑓=𝑒+𝑖𝑣=(𝑓1βˆ’π‘“2)+𝑖(𝑓3βˆ’π‘“4), where 𝑓𝑖β‰₯0 then β€–πœˆπ‘“β€–=β€–πœˆπ‘“1β€–+β€–πœˆπ‘“2β€–+β€–πœˆπ‘“3β€–+β€–πœˆπ‘“4β€–β‰₯‖𝑓1β€–1+‖𝑓2β€–1+‖𝑓3β€–1+‖𝑓4β€–1β‰₯‖𝑓‖1. Hence β€–πœˆπ‘“β€–β‰₯‖𝑓‖1 and equality holds.

Corollary 4.2. 𝐿1(𝐺,πœ†,πœ‡) is a closed subspace of 𝑀(𝐺).

Lemma 4.3. If 𝑓,π‘”βˆˆπΏ1(𝐺,πœ†,πœ‡), then 𝜈(π‘“βˆ—π‘”)=πœˆπ‘“βˆ—πœˆπ‘”.

Proof. For each compact set 𝐾, we have πœˆπ‘“βˆ—πœˆπ‘”(ξ€œπΎ)=πΊπœ’πΎ(ξ€·πœˆπ‘₯)π‘‘π‘“βˆ—πœˆπ‘”ξ€Έ(ξ€œπ‘₯)=πΊξ€œπΊπ‘Ÿ(π‘₯)πœ’πΎ(𝑦π‘₯)π‘‘πœˆπ‘“(𝑦)π‘‘πœˆπ‘”(=ξ€œπ‘₯)πΊξ€œπ‘“(𝑦)𝐺𝑠(𝑦)πœ’πΎ(𝑦π‘₯)π‘‘πœˆπ‘”ξ€œ(π‘₯)π‘‘πœˆ(𝑦)=πΊξ€œπΊπ‘ (𝑦)𝑦𝑓(𝑦)π‘”βˆ’1π‘₯ξ€Έπœ’πΎ=ξ€œ(π‘₯)π‘‘πœˆ(𝑦)π‘‘πœˆ(π‘₯)πΊπœ’πΎξ€œ(π‘₯)πΊπ‘Ÿ(π‘₯)𝑦𝑓(𝑦)π‘”βˆ’1π‘₯ξ€Έξ€œπ‘‘πœˆ(𝑦)π‘‘πœˆ(π‘₯)=𝐺(π‘“βˆ—π‘”)(π‘₯)πœ’πΎ(π‘₯)π‘‘πœˆ(π‘₯)=πœˆπ‘“βˆ—π‘”(𝐾).(4.3) Since πœˆπ‘“βˆ—π‘” and πœˆπ‘“βˆ—πœˆπ‘” are regular measures, the equality holds for each open set and then for each measurable set.

If π‘“βˆˆπΏ1(𝐺,πœ†,πœ‡) and πœ‚βˆˆπ‘€(𝐺), we will define πœ‚βˆ—π‘“ such that πœˆπœ‚βˆ—π‘“=πœ‚βˆ—πœˆπ‘“. Suppose πœ‘βˆˆπΆ0(𝐺), we put ξ€œπœ‚βˆ—π‘“(πœ‘)=πΊπœ‘(π‘₯)π‘‘πœˆπœ‚βˆ—π‘“(ξ€œπ‘₯)=πΊπœ‘(π‘₯)(πœ‚βˆ—π‘“)(π‘₯)π‘‘πœˆ(π‘₯).(4.4) On the other hand, πœ‚βˆ—πœˆπ‘“(ξ€œπœ‘)=πΊξ€œπΊπ‘Ÿ(π‘₯)πœ‘(𝑦π‘₯)π‘‘πœ‚(𝑦)π‘‘πœˆπ‘“(ξ€œπ‘₯)=πΊξ€œπΊπ‘Ÿ(π‘₯)=ξ€œπœ‘(𝑦π‘₯)π‘‘πœ‚(𝑦)𝑓(π‘₯)π‘‘πœˆ(π‘₯)πΊξ€œπΊπ‘ (𝑦)π‘“ξ€·π‘¦βˆ’1π‘₯ξ€Έξ€·π‘¦πœ‘(π‘₯)π‘‘πœ‚(𝑦)π‘‘πœˆβˆ’1π‘₯ξ€Έ=ξ€œπΊξ€œπœ‘(π‘₯)𝐺𝑠(𝑦)π‘“ξ€·π‘¦βˆ’1π‘₯ξ€Έπ‘‘πœ‚(𝑦)π‘‘πœˆ(π‘₯).(4.5)

Comparing these equalities, (ξ€œπœ‚βˆ—π‘“)(π‘₯)=πΊπ‘Ÿ(π‘₯)π‘“ξ€·π‘¦βˆ’1π‘₯ξ€Έπ‘‘πœ‚(𝑦).(4.6) If π‘“βˆˆπΏ1(𝐺,πœ†,πœ‡), then it is easy to check that ξ€œπΊπ‘’π‘…π‘¦π‘“(π‘₯)π‘‘πœ†π‘’(ξ€œπ‘₯)=𝐺𝑒𝑓(π‘₯)π‘‘πœ†π‘’ξ€·π‘₯π‘¦βˆ’1𝑦=π·βˆ’1ξ€Έξ€œπΊπ‘’π‘“(π‘₯)π‘‘πœ†π‘’(π‘₯).(4.7) Thus ξ€œπΊπ‘…π‘¦ξ€·π‘¦π‘“(π‘₯)π‘‘πœˆ(π‘₯)=π·βˆ’1ξ€Έξ€œπΊπ‘“(π‘₯)π‘‘πœˆ(π‘₯).(4.8) Similarly, we want to define π‘“βˆ—πœ‚ in such a way that the equality 𝜈(π‘“βˆ—πœ‚)=πœˆπ‘“βˆ—πœ‚ holds. Again suppose πœ‘βˆˆπΆ0(𝐺). We have 𝜈(π‘“βˆ—πœ‚)(ξ€œπœ‘)=πΊπœ‘(π‘₯)π‘‘πœˆπ‘“βˆ—πœ‚(ξ€œπ‘₯)=πΊξ€œπœ‘(π‘₯)(π‘“βˆ—πœ‚)(π‘₯)π‘‘πœˆ(π‘₯)=𝐺0ξ€œπΊπœ‘(π‘₯)(π‘“βˆ—πœ‚)(π‘₯)π‘‘πœ†π‘’(=ξ€œπ‘₯)π‘‘πœ‡(𝑒)𝐺0ξ€œπΊπ‘’πœ‘(π‘₯)(π‘“βˆ—πœ‚)(π‘₯)π‘‘πœ†π‘’ξ€œ(π‘₯)π‘‘πœ‡(𝑒)=πΊπ‘’πœ‘(π‘₯)(π‘“βˆ—πœ‚)(π‘₯)π‘‘πœˆ(π‘₯).(4.9) On the other hand, ξ€·πœˆπ‘“ξ€Έ(ξ€œβˆ—πœ‚πœ‘)=πΊξ€œπΊπ‘Ÿ(𝑦)πœ‘(π‘₯𝑦)π‘‘πœˆπ‘“(=ξ€œπ‘₯)π‘‘πœ‚(𝑦)πΊξ€œπΊπ‘Ÿ(𝑦)=ξ€œπœ‘(π‘₯𝑦)𝑓(π‘₯)π‘‘πœˆ(π‘₯)π‘‘πœ‚(𝑦)πΊξ€œπΊπ‘Ÿ(𝑦)ξ€·πœ‘(π‘₯)𝑓π‘₯π‘¦βˆ’1ξ€Έξ€·π‘‘πœˆπ‘₯π‘¦βˆ’1ξ€Έ=ξ€œπ‘‘πœ‚(𝑦)πΊξ€œπΊπ‘Ÿ(𝑦)πœ‘ξ€·(π‘₯)𝑓π‘₯π‘¦βˆ’1ξ€Έπ·ξ€·π‘¦βˆ’1ξ€Έ=ξ€œπ‘‘πœˆ(π‘₯)π‘‘πœ‚(𝑦)𝐺𝑠(π‘₯)ξ€œπœ‘(π‘₯)𝐺𝑓π‘₯π‘¦βˆ’1ξ€Έπ·ξ€·π‘¦βˆ’1ξ€Έπ‘‘πœ‚(𝑦)π‘‘πœˆ(π‘₯).(4.10) Comparing the above equalities, we have (ξ€œπ‘“βˆ—πœ‚)(π‘₯)=𝐺𝑓π‘₯π‘¦βˆ’1ξ€Έπ·ξ€·π‘¦βˆ’1ξ€Έπ‘‘πœ‚(𝑦).(4.11)

Lemma 4.4. 𝐿1(𝐺,πœ†,πœ‡) is a two-sided ideal of 𝑀(𝐺).

Proof. Suppose π‘“βˆˆπΏ1(𝐺,πœ†,πœ‡) and πœ‚βˆˆπ‘€(𝐺). Then we have β€–πœ‚βˆ—π‘“β€–1=ξ€œπΊ||||ξ€œπœ‚βˆ—π‘“(π‘₯)π‘‘πœˆ(π‘₯)β‰€πΊξ€œπΊπ‘Ÿ(π‘₯)||π‘“ξ€·π‘¦βˆ’1π‘₯ξ€Έ||𝑑||πœ‚||(=ξ€œπ‘¦)π‘‘πœˆ(π‘₯)πΊξ€œπΊπ‘Ÿ(π‘₯)||||𝑑||πœ‚||ξ€œπ‘“(π‘₯)(𝑦)π‘‘πœˆ(𝑦π‘₯)=𝐺||||ξ€œπ‘“(π‘₯)πΊπ‘Ÿ(π‘₯)𝑑||πœ‚||β€–(𝑦)π‘‘πœˆ(π‘₯)β‰€πœ‚β€–β€–π‘“β€–1<∞.(4.12) Thus πœ‚βˆ—π‘“βˆˆπΏ1(𝐺,πœ†,πœ‡). Also β€–π‘“βˆ—πœ‚β€–1=ξ€œπΊ||(||ξ€π‘“βˆ—πœ‚)(π‘₯)π‘‘πœˆ(π‘₯)≀𝐺||𝑓π‘₯π‘¦βˆ’1ξ€Έπ·ξ€·π‘¦βˆ’1ξ€Έ||𝑑||πœ‚||(=𝑦)π‘‘πœˆ(π‘₯)𝐺||||𝐷𝑦𝑓(π‘₯)βˆ’1ξ€Έ||πœ‚||=𝐷(𝑦)π‘‘πœˆ(π‘₯)𝑑(𝑦)𝐺||𝑓||||πœ‚||π‘‘πœˆπ‘‘=‖𝑓‖1β€–πœ‚β€–<∞.(4.13) Hence, π‘“βˆ—πœ‚βˆˆπΏ1(𝐺,πœ†,πœ‡).

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 𝐺 defined by βˆ«πœ†πœˆ=π‘’π‘‘πœ‡(𝑒) is a Radon measure. Let 𝜈0 be the measure on 𝐺 given by π‘‘πœˆ0=π·βˆ’1/2π‘‘πœˆ,(5.1) where 𝐷 is modular function of πœ‡. Then for each Borel subset 𝐸 of 𝐺, we have 𝜈0(ξ€œπœ’πΈ)=𝐸(π‘₯)π·βˆ’1/2(ξ€œπœ’π‘₯)π‘‘πœˆ(π‘₯)=𝐸π‘₯βˆ’1ξ€Έπ·βˆ’1/2ξ€·π‘₯βˆ’1ξ€Έπ‘‘πœˆβˆ’1(=ξ€œπœ’π‘₯)πΈβˆ’1(π‘₯)𝐷1/2(π‘₯)π·βˆ’1(π‘₯)π‘‘πœˆ(π‘₯)=𝜈0ξ€·πΈβˆ’1ξ€Έ.(5.2) Hence, 𝜈0 is symmetric under inversion.

Definition 5.1. A representation of the locally compact groupoid 𝐺 is a triple (πœ‡,{𝐻𝑒}𝑒,πœ‹) consisting of a Hilbert bundle (𝐺0,{𝐻𝑒}π‘’βˆˆπΊ0,πœ‡), where πœ‡ is a quasi-invariant measure on 𝐺0 (with associated Radon measures 𝜈,πœˆβˆ’1,𝜈2,𝜈0) and for each π‘₯∈𝐺, a unitary element πœ‹(π‘₯)βˆˆβ„¬(𝐻𝑠(π‘₯),π»π‘Ÿ(π‘₯)) such that(i)πœ‹(𝑒) is the identity map on 𝐻𝑒 for all 𝑒,(ii)πœ‹(π‘₯𝑦)=πœ‹(π‘₯)πœ‹(𝑦) for 𝜈2-a.e. (π‘₯,𝑦)∈𝐺2,(iii)πœ‹(π‘₯βˆ’1)=πœ‹βˆ’1(π‘₯) for 𝜈-a.e. π‘₯∈𝐺,(iv)for any πœ‰,πœ‚βˆˆπΏ2(𝐺0,{𝐻𝑒}𝑒,πœ‡), the mapπ‘₯βŸΌβŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))⟩(5.3) is 𝜈-measurable on 𝐺.

Definition 5.2. A representation Ξ  of 𝐿1(𝐺,πœ†,πœ‡) on a Hilbert space, 𝐻 is a βˆ—-homomorphism Π∢𝐿1(𝐺,πœ†,πœ‡)βŸΆβ„¬(𝐻).(5.4) It is called nondegenerate if ⟨Π(𝑓)πœ‰βˆΆπ‘“βˆˆπΏ1,πœ‰βˆˆπ»βŸ© is dense in 𝐻.

Continuity of Ξ  automatically holds, because each βˆ—-homomorphism from a Banach βˆ—-algebra to a πΆβˆ—-algebra is norm decreasing, namely, β€–Ξ (𝑓)‖≀‖𝑓‖1, for each π‘“βˆˆπΏ1.

Our main aim here is to find a correspondence between unitary representations of 𝐺 and nondegenerate representations of 𝐿1(𝐺,πœ†,πœ‡). Unfortunately, this is impossible in general. Such a correspondence exists between representations of 𝐺 and those of 𝐢𝑐(𝐺), when 𝐻 is 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 with sufficiently many nonsingular Borel 𝐺-sets. Then, every representation of 𝐢𝑐(𝐺) on a separable Hilbert space is the integrated form of a representation of 𝐺.

These assumptions satisfied in the case of π‘Ÿ-discrete groupoids and transformation groups. The main problem is that a continuous representation of (𝐿1(𝐺,πœ†,πœ‡),‖⋅‖𝐼) is not necessarily continuous in the 𝐿1-norm. To get a partial result, we use the following result [2, Proposition  1.7].

Proposition 5.4. Suppose (πœ‹,{𝐻𝑒}𝑒,πœ‡) is a representation of 𝐺. For πœ‰,πœ‚βˆˆπΏ2(𝐺0,{𝐻𝑒}𝑒,πœ‡) and π‘“βˆˆπΆπ‘(𝐺), ξ€œβŸ¨Ξ (𝑓)πœ‰,πœ‚βŸ©=𝐺𝑓(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0(π‘₯)(5.5) defines a bounded nondegenerate βˆ—-representation of 𝐢𝑐(𝐺) on 𝐿2(𝐺0,{𝐻𝑒}𝑒,πœ‡) such that two equivalent representations of 𝐺 give two equivalent representations of 𝐢𝑐(𝐺).

The equation above is called the integrated form of a unitary representation. If Ξ  is a representation of 𝐢𝑐(𝐺), the above proposition says that Ξ  should be of the form ξ€œβŸ¨Ξ (𝑓)πœ‰,πœ‚βŸ©=𝐺𝑓(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0(π‘₯).(5.6) For each πœ‰,πœ‚βˆˆπΏ2(𝐺0,{𝐻𝑒}𝑒,πœ‡) and π‘“βˆˆπΆπ‘(𝐺).

Next, we define a representation of 𝐼(𝐺,πœ†,πœ‡), denoted by Π𝐼 as ⟨Π𝐼(ξ€œπ‘“)πœ‰,πœ‚βŸ©=𝐺𝑓(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0(π‘₯)(5.7) for each πœ‰,πœ‚βˆˆπΏ2(𝐺0,{𝐻𝑒}𝑒,πœ‡) and π‘“βˆˆπΌ(𝐺,πœ†,πœ‡)

Lemma 5.5. Π𝐼 is a bounded representation of 𝐼(𝐺,πœ†,πœ‡) on ℬ(𝐿2(𝐺0,{𝐻𝑒}𝑒,πœ‡)).

Proof. We have ⟨Π𝐼(ξ€œπ‘“)πœ‰,πœ‚βŸ©=𝐺𝑓(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0(=ξ€œπ‘₯)𝐺0ξ€œπΊπ‘’π‘“(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘βˆ’1/2(π‘₯)π‘‘πœ†π‘’=ξ€œ(π‘₯)π‘‘πœ‡(𝑒)𝐺0ξƒ‘ξ€œπΊπ‘’π‘“(π‘₯)πœ‹(π‘₯)πœ‰(𝑠(π‘₯))π·βˆ’1/2(π‘₯)π‘‘πœ†π‘’ξƒ’(π‘₯),πœ‚(𝑒)π‘‘πœ‡(𝑒).(5.8) Thus, we may define for πœ‡-a.e. π‘’βˆˆπΊ0, Π𝐼(ξ€œπ‘“)πœ‰(𝑒)=𝐺𝑒𝑓(π‘₯)πœ‹(π‘₯)πœ‰(𝑠(π‘₯))π·βˆ’1/2(π‘₯)π‘‘πœ†π‘’(π‘₯).(5.9) Now, we have ||⟨Π𝐼||=||||ξ€œ(𝑓)πœ‰,πœ‚βŸ©πΊπ‘“(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0||||β‰€ξ€œ(π‘₯)𝐺||𝑓||(π‘₯)β€–πœ‰βˆ˜π‘ (π‘₯)β€–β€–πœ‚βˆ˜π‘Ÿ(π‘₯)β€–π‘‘πœˆ0β‰€ξ‚Έξ€œ(π‘₯)𝐺||||𝑓(π‘₯)β€–πœ‰βˆ˜π‘ (π‘₯)β€–2π‘‘πœˆβˆ’1ξ‚Ή(π‘₯)1/2ξ‚Έξ€œπΊ||||𝑓(π‘₯)β€–πœ‚βˆ˜π‘Ÿ(π‘₯)β€–2ξ‚Ήπ‘‘πœˆ(π‘₯)1/2β‰€ξ‚Έξ€œπΊ0ξ€œπΊπ‘’||||𝑓(π‘₯)π‘‘πœ†π‘’(π‘₯)β€–πœ‰(𝑒)β€–2ξ‚Ήπ‘‘πœ‡(𝑒)1/2Γ—ξ‚Έξ€œπΊ0ξ€œπΊπ‘’||||𝑓(π‘₯)π‘‘πœ†π‘’(π‘₯)β€–πœ‚(𝑒)β€–2ξ‚Ήπ‘‘πœ‡(𝑒)1/2≀‖𝑓‖1/2𝐼,π‘ β€–πœ‰β€–β€–π‘“β€–1/2𝐼,π‘Ÿβ‰€β€–πœ‚β€–β€–π‘“β€–πΌβ€–πœ‰β€–β€–πœ‚β€–.(5.10) Therefore, the map π‘₯↦𝑓(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))⟩ is 𝜈0-integrable, and Π𝐼(𝑓) is a bounded operator of norm ‖Π𝐼(𝑓)‖≀‖𝑓‖𝐼. We have to check that Π𝐼 is a βˆ—-homomorphism. For this, we define 𝐹𝑑(π‘₯,𝑑)=𝑓(π‘₯𝑑)π‘”βˆ’1ξ€ΈβŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘1/2(π‘₯).(5.11) Then ⟨Π𝐼(ξ€œπ‘“βˆ—π‘”)πœ‰,πœ‚βŸ©=πΊπ‘“βˆ—π‘”(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0(=ξ€œπ‘₯)πΊξ‚΅ξ€œπΊξ€·π‘‘π‘“(π‘₯𝑑)π‘”βˆ’1ξ€Έπ‘‘πœ†π‘ (π‘₯)ξ‚Ά(𝑑)βŸ¨πœ‹(π‘₯)πœ‰βˆ˜π‘ (π‘₯),πœ‚βˆ˜π‘Ÿ(π‘₯)βŸ©π‘‘βˆ’1/2(π‘₯)𝐷(π‘₯)π‘‘πœˆβˆ’1=ξ€œ(π‘₯)𝐺2𝐹(π‘₯,𝑑)π‘‘πœˆ2=ξ€œ(π‘₯,𝑑)𝐺2𝐹π‘₯𝑑,π‘‘βˆ’1ξ€ΈπœŒβˆ’1ξ€·π‘₯𝑑,π‘‘βˆ’1ξ€Έπ‘‘πœˆ2=ξ€œ(π‘₯,𝑑)𝐺0ξ€œπΊπ‘’ξƒ‘ξ‚΅ξ€œπ‘“(π‘₯)πœ‹(π‘₯)𝐺𝑠(π‘₯)𝑔(𝑑)πœ‹(𝑑)πœ‰βˆ˜π‘ (𝑑)π·βˆ’1/2(𝑑)π‘‘πœ†π‘ (π‘₯)ξ‚Άξƒ’(𝑑),πœ‚(𝑒)Γ—π·βˆ’1/2(π‘₯)π‘‘πœ†π‘’=ξ€œ(π‘₯)π‘‘πœ‡(𝑒)𝐺𝑓(π‘₯)βŸ¨πœ‹(π‘₯)Ξ (𝑔)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0(π‘₯)=⟨Π𝐼(𝑓)Π𝐼(𝑔)πœ‰,πœ‚βŸ©.(5.12) Hence, Π𝐼(π‘“βˆ—π‘”)=Π𝐼(𝑓)Π𝐼(𝑔). Also βŸ¨Ξ πΌξ€·π‘“βˆ—ξ€Έξ€œπœ‰,πœ‚βŸ©=πΊπ‘“βˆ—(π‘₯)βŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0(=ξ€œπ‘₯)𝐺𝑓π‘₯βˆ’1𝐷π‘₯βˆ’1ξ€ΈβŸ¨πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))βŸ©π‘‘βˆ’1/2=ξ€œ(π‘₯)π‘‘πœˆ(π‘₯)𝐺π‘₯𝑓(π‘₯)π·βˆ’1ξ€ΈβŸ¨πœ‹βˆ—(π‘₯)πœ‰(π‘Ÿ(π‘₯)),πœ‚(𝑠(π‘₯))βŸ©π‘‘1/2ξ€·π‘₯(π‘₯)π·βˆ’1ξ€Έ=π‘‘πœˆ(π‘₯)ξ€œπΊπ‘“(π‘₯)βŸ¨πœ‚(𝑠(π‘₯)),πœ‹βˆ—(π‘₯)πœ‰(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0=(π‘₯)ξ€œπΊπ‘“(π‘₯)βŸ¨πœ‹(π‘₯)πœ‚(𝑠(π‘₯)),πœ‰(π‘Ÿ(π‘₯))βŸ©π‘‘πœˆ0=(π‘₯)⟨Π𝐼(𝑓)πœ‰,πœ‚βŸ©=βŸ¨πœ‰,Π𝐼Π(𝑓)πœ‚βŸ©=βˆ—πΌξ¬.(𝑓)πœ‰,πœ‚(5.13) The nondegeneracy follows from ⟨Π𝐼(𝑓)πœ‰βˆ£π‘“βˆˆπΆπ‘ξ«Ξ (𝐺),πœ‰βˆˆπ»βŸ©βŠ†πΌ(𝑓)πœ‰βˆ£π‘“βˆˆπΏ1𝐺,‖⋅‖𝐼,πœ‰βˆˆπ»βŠ†π».(5.14)

Now let us turn to the problem that a continuous representation of 𝐼(𝐺,πœ†,πœ‡) is not necessarily continuous in 𝐿1-norm. Let 𝐻=𝐿2(𝐺0,{𝐻𝑒}𝑒,πœ‡) and put 𝐻Π1=ξ«πœ‰βˆˆπ»βˆΆThemapπ‘“βŸΌβŸ¨Ξ πΌ(𝑓)πœ‰,πœ‰βŸ©iscontinuousin𝐿1-norm.(5.15) Observe that 𝐻Π1 is a nontrivial subspace, as πœ‡ is a probability measure and if β€–πœ‰(𝑒)β€–=1 in 𝐻𝑒 for each π‘’βˆˆπΊ0, then β€–πœ‰β€–=1 in 𝐻 and the calculations in the proof of Lemma 5.5 shows that |Π𝐼(𝑓)πœ‰,πœ‰βŸ©|≀‖𝑓‖1, hence πœ‰βˆˆπ»1. On the other hand, Π𝐼(𝑓)𝐻Π1βŠ†π»Ξ 1 for each π‘“βˆˆπΌ(𝐺,πœ†,πœ‡), because the map π‘“βŸΌβŸ¨Ξ πΌ(𝑓)Π𝐼(𝑔)πœ‰,Π𝐼Π(𝑔)πœ‰βŸ©=πΌξ€·π‘”βˆ—ξ€Έξ¬βˆ—π‘“βˆ—π‘”πœ‰,πœ‰(5.16) is continuous. Therefore, Π𝐼(𝑓)π»βŸ‚Ξ 1βŠ†π»βŸ‚Ξ 1. Hence, we have Π𝐼(𝑓)=Π𝐼(𝑓)|𝐻Π1βŠ•Ξ πΌ(𝑓)|π»βŸ‚Ξ 1 for each π‘“βˆˆπΌ(𝐺,πœ†,πœ‡).

Put 𝐻1=𝐻Π1 and define Ξ 1(𝑓)∢=Π𝐼(𝑓)|𝐻1, for π‘“βˆˆπΌ(𝐺,πœ†,πœ‡). Then, it follows from continuity of π‘“β†¦βŸ¨Ξ πΌ(𝑓)πœ‰,πœ‰βŸ© and polarization identity that Ξ 1 extends to a continuous representation of 𝐿1(𝐺,πœ†,πœ‡) on 𝐻1, still denoted by Ξ 1.

Next we focus on the notion of irreducibility which plays an important role in the theory of representations. We show that if πœ‹ is an irreducible representation of 𝐺, the integrated representation Ξ 1 of 𝐿1(𝐺,πœ†,πœ‡) on 𝐻1 is irreducible. Basic materials come from [6].

Definition 5.6. Let (𝐺0,{𝐻𝑒}π‘’βˆˆπΊ0,πœ‡) be a Hilbert bundle. A family 𝑀={𝑀𝑒}π‘’βˆˆπΊ0, where 𝑀𝑒 is a closed subspace of 𝐻𝑒 for each π‘’βˆˆπΊ0, is called a subbundle. A subbundle {𝑀𝑒}π‘’βˆˆπΊ0 is called nontrivial if 0≠𝑀𝑒≠𝐻𝑒 for some π‘’βˆˆπΊ0. For a representation πœ‹ of a locally compact groupoid 𝐺 associated with the Hilbert bundle (𝐺0,{𝐻𝑒}π‘’βˆˆπΊ0,πœ‡), a subbundle {𝑀𝑒}π‘’βˆˆπΊ0 is called invariant if πœ‹(π‘₯)𝑀𝑠(π‘₯)βŠ†π‘€π‘Ÿ(π‘₯) for each π‘₯∈𝐺. Note that if 𝑀 is an invariant subbundle, and 0≠𝑀𝑒≠𝐻𝑒 for some π‘’βˆˆπΊ0, then 0≠𝑀𝑀≠𝐻𝑀 for every π‘€βˆˆ[𝑒].

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

Lemma 5.7. Let πœ‹ be a representation of a locally compact groupoid 𝐺 associated with the Hilbert bundle (𝐺0,{𝐻𝑒}π‘’βˆˆπΊ0,πœ‡). If 𝑀={𝑀𝑒}π‘’βˆˆπΊ0 is an invariant subbundle, then so is π‘€βŸ‚={π‘€βŸ‚π‘’}π‘’βˆˆπΊ0.

Definition 5.8. A representation πœ‹ of a locally compact groupoid 𝐺 is called reducible if πœ‹ admits a nontrivial invariant subbundle 𝑀={𝑀𝑒}π‘’βˆˆπΊ0, otherwise πœ‹ is called irreducible. In this case, it is easy to check that πœ‹π‘€ with πœ‹π‘€(π‘₯)=πœ‹(π‘₯)|𝑀𝑠(π‘₯)βˆΆπ‘€π‘ (π‘₯)β†’π‘€π‘Ÿ(π‘₯) is called a subrepresentation of πœ‹. If πœ‹ and πœ‹ξ…ž are two representations of a locally compact groupoid 𝐺 associated with two Hilbert bundles (𝐺0,{𝐻𝑒}π‘’βˆˆπΊ0,πœ‡) and (𝐺0,{π»ξ…žπ‘’}π‘’βˆˆπΊ0,πœ‡ξ…ž), respectively. Then, we put π’žξ€·πœ‹,πœ‹ξ…žξ€Έ=⎧βŽͺ⎨βŽͺβŽ©ξ€·π‘‡π‘’ξ€Έπ‘’βˆˆξ‘π‘’βˆˆπΊ0ℬ𝐻𝑒,π»ξ…žπ‘’ξ€ΈβˆΆπœ‹ξ…ž(π‘₯)𝑇𝑠(π‘₯)=π‘‡π‘Ÿ(π‘₯)⎫βŽͺ⎬βŽͺβŽ­πœ‹(π‘₯)(π‘₯∈𝐺)(5.17) and write π’ž(πœ‹,πœ‹)=π’ž(πœ‹).

Two representations πœ‹ and πœ‹ξ…ž are called (unitarily) equivalent if πœ‡βˆΌπœ‡ξ…ž, and there is (𝑇𝑒)π‘’βˆˆπΊ0βˆˆπ’ž(πœ‹,πœ‹ξ…ž) such that 𝑇𝑒 is a unitary operator for every π‘’βˆˆπΊ0. Note that if (𝑇𝑒)π‘’βˆˆπΊ0βˆˆπ’ž(πœ‹) and π‘‡βˆ—π‘’ denotes the adjoint operator to 𝑇𝑒, then πœ‹(π‘₯)π‘‡βˆ—π‘ (π‘₯)=𝑇𝑠(π‘₯)πœ‹ξ€·π‘₯βˆ’1ξ€Έξ€»βˆ—=ξ€Ίπœ‹ξ€·π‘₯βˆ’1ξ€Έπ‘‡π‘Ÿ(π‘₯)ξ€»βˆ—=π‘‡βˆ—π‘Ÿ(π‘₯)πœ‹(π‘₯),(5.18) hence, (π‘‡βˆ—π‘’)π‘’βˆˆπΊ0βˆˆπ’ž(πœ‹). We observe that π’ž(πœ‹) is a unital βˆ—-algebra, where the operations are defined pointwise.

Following [6], for a representation πœ‹ of a locally compact groupoid 𝐺 associated with the Hilbert bundle (𝐺0,{𝐻𝑒}π‘’βˆˆπΊ0,πœ‡), we set ⎧βŽͺ⎨βŽͺβŽ©ξ€·πœ†Ξ›=π‘’ξ€Έπœ‹(𝑒)π‘’βˆˆξ‘π‘’βˆˆπΊ0β„¬ξ€·π»π‘’ξ€ΈβˆΆπœ†π‘’βˆˆβ„‚,πœ†π‘’=πœ†π‘€whenever⎫βŽͺ⎬βŽͺ⎭.π‘’βˆΌπ‘€(5.19) If (πœ†π‘’πœ‹(𝑒))π‘’βˆˆπ’ž(πœ‹), then πœ†π‘’=πœ†π‘€ whenever π‘’βˆΌπ‘€ that is (πœ†π‘’πœ‹(𝑒))π‘’βˆˆΞ›. Therefore, if (𝑇𝑒)π‘’βˆˆπΊ0βˆˆπ’ž(πœ‹)⧡Λ, then there exists π‘’βˆˆπΊ0 with 𝑇𝑒 not in β„‚πœ‹(𝑒). It is also obvious that Ξ›βŠ†π’ž(πœ‹).

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

Lemma 5.9. A representation πœ‹ of a locally compact groupoid 𝐺 is irreducible if and only if Ξ›=π’ž(πœ‹). In particular, in the case where 𝐺 is transitive, then πœ‹ is irreducible if and only if π’ž(πœ‹)=β„‚(πœ‹(𝑒))π‘’βˆˆπΊ0.

Lemma 5.10. If β„³=(𝐺0,{𝑀𝑒}𝑒,πœ‡) is closed subbundle of β„‹=(𝐺0,{𝐻𝑒}𝑒,πœ‡) and 𝑃𝑒 is an orthogonal projection 𝐻𝑒 onto 𝑀𝑒, then β„³ is invariant under πœ‹ if and only if (𝑃𝑒)π‘’βˆˆπ’ž(πœ‹).

Proof. If (𝑃𝑒)π‘’βˆˆπ’ž(πœ‹) and π‘šπ‘’βˆˆπ‘€π‘’, then πœ‹(π‘₯)π‘šπ‘ (π‘₯)=πœ‹(π‘₯)𝑃𝑠(π‘₯)π‘šπ‘ (π‘₯)=π‘ƒπ‘Ÿ(π‘₯)πœ‹(π‘₯)π‘šπ‘ (π‘₯)βˆˆπ‘€π‘Ÿ(π‘₯).(5.20) Thus, πœ‹(π‘₯)𝑀𝑠(π‘₯)βŠ†π‘€π‘Ÿ(π‘₯) so β„³ is invariant. Conversely, if β„³ is invariant, then for π‘šβˆˆπ‘€π‘ (π‘₯)πœ‹(π‘₯)𝑃𝑠(π‘₯)π‘š=πœ‹(π‘₯)π‘š=π‘ƒπ‘Ÿ(π‘₯)πœ‹(π‘₯)π‘š(5.21) and for π‘šβˆˆπ‘€βŸ‚π‘ (π‘₯)πœ‹(π‘₯)𝑃𝑠(π‘₯)π‘š=0=π‘ƒπ‘Ÿ(π‘₯)πœ‹(π‘₯)π‘š.(5.22) Hence, πœ‹(π‘₯)𝑃𝑠(π‘₯)=π‘ƒπ‘Ÿ(π‘₯)πœ‹(π‘₯).

We show that if a representation πœ‹ of a locally compact groupoid 𝐺 associated with the Hilbert bundle (𝐺0,{𝐻𝑒}π‘’βˆˆπΊ0,πœ‡) is irreducible and Ξ 1 is the corresponding integrated representation of 𝐿1(𝐺,πœ†,πœ‡) on 𝐻1βŠ†πΏ2(𝐺0,{𝐻𝑒}𝑒,πœ‡), then Ξ 1 is irreducible. If 𝐻=𝐿2(𝐺0,{𝐻𝑒}𝑒,πœ‡), then for each subbundle {𝑀𝑒}𝑒, 𝑀=𝐿2(𝐺0,{𝑀𝑒}𝑒,πœ‡) is closed subspace of 𝐻 with orthogonal complement π‘€βŸ‚=𝐿2(𝐺0,{π‘€βŸ‚π‘’}𝑒,πœ‡). A map Θ∢𝐿2(𝐺0,{𝐻𝑒}𝑒,πœ‡)→𝐿2(𝐺0,{𝑀𝑒}𝑒,πœ‡);πœ‰β†¦Ξ˜πœ‰ with Ξ˜πœ‰(𝑠(π‘₯))βˆˆπ‘€π‘Ÿ(π‘₯) is called an orthogonal projection of 𝐻 onto 𝑀.

Let πœ‹ be a representation of 𝐺 on the Hilbert bundle (𝐺0,{𝐻𝑒}π‘’βˆˆπΊ0,πœ‡) and Π𝐼 be the bounded representation of 𝐼(𝐺,πœ†,πœ‡) on ℬ(𝐿2(𝐺0,{𝐻𝑒}𝑒,πœ‡)) constructed above.

Lemma 5.11. 𝑀 is invariant under Π𝐼 if and only if for each π‘“βˆˆπΌ(𝐺,πœ†,πœ‡), Π𝐼(𝑓)Θ=ΘΠ𝐼(𝑓).

Proof. Suppose Π𝐼(𝑓)Θ=ΘΠ𝐼(𝑓) and πœ‰βˆˆπ‘€, then Π𝐼(𝑓)πœ‰=Π𝐼(𝑓)Ξ˜πœ‰=ΘΠ𝐼(𝑓)πœ‰βˆˆπ‘€.(5.23) Hence, 𝑀 is invariant. Conversely, if 𝑀 is invariant, then for πœ‰βˆˆπ‘€, Π𝐼(𝑓)Ξ˜πœ‰=Π𝐼(𝑓)πœ‰=ΘΠ𝐼(𝑓)πœ‰(5.24) and for πœ‚βˆˆπ‘€βŸ‚, Π𝐼(𝑓)Ξ˜πœ‚=0=ΘΠ𝐼(𝑓)πœ‚.(5.25) Thus, Π𝐼(𝑓)Θ=ΘΠ𝐼(𝑓).

Proposition 5.12. If πœ‹ is an irreducible representation of 𝐺, then Π𝐼 is an irreducible representation of 𝐼(𝐺,πœ†,πœ‡).

Proof. If Π𝐼 is reducible, then there exists a nontrivial invariant closed subspace 𝑀 of 𝐿2(𝐺0,{𝐻𝑒}𝑒,πœ‡), and, hence, there is an orthogonal projection Ξ˜βˆˆπ’ž(Π𝐼). It follows that ξ€œπΊξ«π‘“(π‘₯)πœ‹(π‘₯)𝑃𝑠(π‘₯)ξ¬πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))π‘‘πœˆ0(π‘₯)=⟨Π𝐼(𝑓)Ξ˜πœ‰,πœ‚βŸ©=⟨ΘΠ𝐼=ξ€œ(𝑓)πœ‰,πœ‚βŸ©πΊξ«π‘ƒπ‘“(π‘₯)π‘Ÿ(π‘₯)ξ¬πœ‹(π‘₯)πœ‰(𝑠(π‘₯)),πœ‚(π‘Ÿ(π‘₯))π‘‘πœˆ0(π‘₯),(5.26) for each π‘“βˆˆπΆπ‘(𝐺) and πœ‰,πœ‚βˆˆπΏ2(𝐺0,{𝐻𝑒}𝑒,πœ‡). Therefore, πœ‹(π‘₯)𝑃𝑠(π‘₯)=π‘ƒπ‘Ÿ(π‘₯)πœ‹(π‘₯), for each π‘₯∈𝐺, hence (𝑃𝑒)π‘’βˆˆπ’ž(πœ‹). Now by the Schur's lemma, πœ‹ is reducible.

Theorem 5.13. If πœ‹ is an irreducible representation of 𝐺, then Ξ 1 is an irreducible representation of 𝐿1(𝐺,πœ†,πœ‡) on 𝐻1.

Proof. If Ξ 1 is reducible, then there exists a nontrivial invariant closed subspace 𝑀1 of 𝐻1βŠ†π»=𝐿2(𝐺0,{𝐻𝑒}𝑒,πœ‡). By the calculations after Lemma 5.5, Π𝐼=Π1βŠ•ξ‚Ξ 2, where Π1 and Π2 are the corresponding representations of 𝐼(𝐺,πœ†,πœ‡) on 𝐻1 and π»βŸ‚1. Therefore, 𝑀=𝑀1βŠ•π»βŸ‚1 is a nontrivial invariant closed subspace 𝐻, and Π𝐼 is reducible. Hence, πœ‹ is reducible by the above proposition.

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