𝐿¹-Algebra of a Locally Compact Groupoid

Abstract

For a locally compact groupoid 𝐺 with a fixed Haar system 𝜆 and quasi-invariant measure 𝜇, we introduce the notion of 𝜆-measurability and construct the space 𝐿¹(𝐺, 𝜆, 𝜇) of absolutely integrable functions on 𝐺 and show that it is a Banach ∗-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, 𝜆, 𝜇).

1. Introduction and Preliminaries

For a locally compact group 𝐺 with a Haar measure 𝜆, the Banach algebra 𝐿¹(𝐺,𝜆) 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 𝜆 and quasi-invariant measure 𝜇. This paper is devoted to the study of such a groupoid 𝐿¹-algebra 𝐿¹(𝐺,𝜆,𝜇). One may expect that as the group case, there is a full interaction between the properties of 𝐺 and that of 𝐿¹(𝐺,𝜆,𝜇). This is not completely true. For instance, unlike the group case, not every 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 into 𝑀(𝐺) as a closed ideal. In Section 5, we find the class of nondegenerate representations of 𝐿¹(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'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 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 _𝑋𝑓(𝑥,𝑦)𝑑𝜈(𝑥),_𝑌𝑓(𝑥,𝑦)𝑑𝜌(𝑦)(1.1) define integrable functions such that Fubini’s formula holds _𝑋×𝑌𝑓𝑑𝜈⋅𝜌=_𝑋_𝑌𝑓𝑑𝜈𝑑𝜌=_𝑌_𝑋𝑓𝑑𝜌𝑑𝜈,(1.2)(ii)if 𝑓 is measurable such that 𝐴={(𝑥,𝑦)∈𝑋×𝑌∶𝑓(𝑥,𝑦)≠0} is 𝜎-finite and if one of the iterated integrals ∫_𝑋∫_𝑌|𝑓|𝑑𝜈𝑑𝜌 or ∫_𝑌∫_𝑋|𝑓|𝑑𝜌𝑑𝜈 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;(𝑥,𝑦)↦𝑥𝑦, and an inverse map: 𝐺→𝐺;𝑥↦𝑥⁻¹, such that 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 unit space of 𝐺, and its elements are called units in the sense that 𝑥𝑠(𝑥)=𝑥 and 𝑟(𝑥)𝑥=𝑥.

For 𝑢,𝑣∈𝐺⁰, 𝐺^𝑢=𝑟⁻¹{𝑢}, 𝐺_𝑣=𝑠⁻¹{𝑣}, 𝐺^𝑢𝑣=𝐺^𝑢∩𝐺_𝑣, and 𝐺{𝑢}=𝐺^𝑢𝑢. The latter is called the isotropy group at 𝑢. We define 𝑢∼𝑣 if 𝐺^𝑢𝑣≠∅. It is checked that ~ is an equivalence relation on the unit space 𝐺⁰. The equivalence class of 𝑢 is denoted by [𝑢] and is called the orbit of 𝑢.

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 𝐶 of compact Hausdorff subsets of 𝐺 whose interiors form a basis for the topology of 𝐺.(iii)Every 𝐺^𝑢 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 be a locally compact groupoid. The support of a function 𝑓∶𝐺→ℂ is the complement of the union of all open, Hausdorff subsets of 𝐺 on which 𝑓 vanishes. The space 𝐶_𝑐(𝐺) consists of all continuous functions on 𝐺 with compact support. A left Haar system for 𝐺 consists of measures {𝜆^𝑢,𝑢∈𝐺⁰} on 𝐺 such that(i)the support of 𝜆^𝑢 is 𝐺^𝑢,(ii)(continuity) for each 𝑓∈𝐶_𝑐(𝐺),𝑢↦𝜆(𝑓)(𝑢)=∫𝑓𝑑𝜆^𝑢 is continuous,(iii)(left invariance) for any 𝑥∈𝐺 and any 𝑓∈𝐶_𝑐(𝐺), 𝑓(𝑥𝑦)𝑑𝜆^𝑠(𝑥)(𝑦)=𝑓(𝑦)𝑑𝜆^𝑟(𝑥)(𝑦).(1.3)

Let 𝜇 be a measure on 𝐺⁰. The measure on 𝐺 induced by 𝜇 is 𝜈=∫𝜆^𝑢𝑑𝜇, defined by ∫_𝐺𝑓𝑑𝜈=∫_𝐺0𝑑𝜇(𝑢)∫_𝐺𝑓𝑑𝜆^𝑢, for 𝑓∈𝐶_𝑐(𝐺). The measure on 𝐺² induced by 𝜇 is 𝜈²=∫𝜆_𝑢×𝜆^𝑢𝑑𝜇(𝑢). The inversion of 𝜈 is 𝜈⁻¹=∫𝜆_𝑢𝑑𝜇(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 its inverse 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 function on 𝐺 is given in [3, Definition  2.3], defining the modular function Δ as a strictly positive continuous homomorphism on 𝐺 such that Δ|_𝐺^𝑢𝑢 is modular function for 𝐺^𝑢𝑢.

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 of 𝐿¹(𝐺,𝜆,𝜇). Let {𝜆^𝑢}_𝑢 be a fixed Haar system on 𝐺. Let 𝜇 be a quasi-invariant measure, Δ its modular function, 𝜈 be the measure induced by 𝜇 on 𝐺, and 𝜈₀=Δ^−1/2𝜈. Let 𝐼𝐼_𝜇(𝐺,𝜈,𝜇)=𝑓∈𝐿¹𝐺,𝜈₀∶‖𝑓‖_{𝐼𝐼,𝜇}<∞,(1.4) where ‖𝑓‖_{𝐼𝐼,𝜇} is defined by ‖𝑓‖_{𝐼𝐼,𝜇}=sup||𝑓(𝑥)𝑗(𝑠(𝑥))𝑘(𝑟(𝑥))||𝑑𝜈₀(𝑥);||𝑗||²𝑑𝜇=||𝑘||²𝑑𝜇=1.(1.5) If 𝜇₁ and 𝜇₂ are two equivalent quasi-invariant measures, then ‖𝑓‖_{𝐼𝐼,𝜇1}=‖𝑓‖_{𝐼𝐼,𝜇2},(1.6) because ‖𝑓‖_{𝐼𝐼,𝜇}=‖𝐼𝐼_𝜇(|𝑓|)‖ for each quasi-invariant measure 𝜇, where 𝐼𝐼_𝜇 is the one-dimensional trivial representation on 𝜇. Now define ‖𝑓‖_𝐼𝐼=sup_𝜇‖𝑓‖_{𝐼𝐼,𝜇}.(1.7) The supremum is taken over the class of quasi-invariant measures.

Let 𝐼(𝐺,𝜈,𝜇)=𝑓∈𝐿¹𝐺,𝜈₀∶‖𝑓‖_𝐼,𝜇<∞,(1.8) where ‖𝑓‖_𝐼,𝜇=max‖‖‖𝑢⟶||𝑓||𝑑𝜆^𝑢‖‖‖_∞,‖‖‖𝑢⟶||𝑓^∗||𝑑𝜆^𝑢‖‖‖_∞.(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=sup_𝑢||𝑓||𝑑𝜆^𝑢,‖𝑓‖^𝐼,𝑠=sup_𝑢||𝑓||𝑑𝜆^𝑢,(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 which admits a left Haar system 𝜆={𝜆^𝑢}.

Definition 2.1. A system of measures {𝜆^𝑢}_𝑢∈𝐺⁰ is said to be complete if for each 𝑢∈𝐺⁰, 𝜆^𝑢 is complete on its 𝜎-algebra 𝔐_𝜆^𝑢. A Borel measurable set 𝐸⊆𝐺 is called 𝜆-measurable if for each 𝑢∈𝐺⁰, 𝐸∩𝐺^𝑢∈𝔐_𝜆^𝑢. A function 𝑓∶𝐺→ℂ is 𝜆-measurable if for every 𝑢∈𝐺⁰ and every open set 𝑂⊆ℂ,𝑓⁻¹(𝑂)∩𝐺^𝑢∈𝔐_𝜆^𝑢.

Proposition 2.2. If 𝜆={𝜆^𝑢} is the completion of 𝜆={𝜆^𝑢} and 𝑓∶𝐺→[0,∞] is 𝜆-measureable, then there is a 𝜆-measurable function 𝑔 such that 𝑓=𝑔 on 𝐺^𝑢(𝜆^𝑢-a.e).

Proof. Since 𝑓 is 𝜆-measurable for each 𝑢∈𝐺⁰, 𝑓 is 𝜆^𝑢-measurable on 𝐺^𝑢, and since 𝜆^𝑢 is the completion of 𝜆^𝑢, there exists a 𝜆^𝑢-measurable function 𝑔_𝑢 such that 𝑓=𝑔_𝑢(𝜆^𝑢-a.e). Now define 𝑔=𝑔_𝑢 on 𝐺^𝑢 and zero, elsewhere. Since for every 𝑢∈𝐺⁰ and every open set 𝑂⊆ℂ,𝑔⁻¹(𝑂)∩𝐺^𝑢=𝑔^𝑢−1(𝑂)∈𝔐_𝜆^𝑢, 𝑔 is 𝜆-measurable, and since 𝜆^𝑢(𝐺^𝑢)^𝑐=0, we have 𝑓=𝑔 on 𝐺^𝑢 (𝜆^𝑢-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 supp𝜆^𝑢=𝐺^𝑢 and 𝑓⁻¹(𝑂)=𝑓⁻¹(𝑂)∩𝐺^𝑢∪𝑓⁻¹(𝑂)∩(𝐺^𝑢)^𝑐.(2.1) Since 𝜆^𝑢 is complete and (𝐺^𝑢)^𝑐 is 𝜆^𝑢-null, (𝑓⁻¹(𝑂)∩(𝐺^𝑢)^𝑐)⊆(𝐺^𝑢)^𝑐 is in 𝔐_𝜆^𝑢. Thus, for each 𝑢∈𝐺⁰ and open set 𝑂⊆ℂ𝑓⁻¹(𝑂)∈𝔐_𝜆^𝑢⟺𝑓⁻¹(𝑂)∩𝐺^𝑢∈𝔐_𝜆^𝑢.(2.2) Now for 𝜈=∫𝜆^𝑢𝑑𝜇(𝑢), we have 𝔐_𝜈=⋂_𝑢∈𝐺0𝔐_𝜆^𝑢, hence 𝑓 is 𝜈-measurable if and only if 𝑓 is 𝜆-measurable.

If 𝑓∶𝐺→ℂ is 𝜆-measurable and 𝑔∶ℂ→ℂ is continuous, then 𝑔∘𝑓∶𝐺→ℂ is 𝜆-measurable. Also, if 𝑓,𝑔∶𝐺→ℝ is 𝜆-measurable, Φ∶ℂ→𝑌 is continuous, and ℎ(𝑥)=Φ(𝑓(𝑥),𝑔(𝑥)), then ℎ is 𝜆-measurable. If 𝑓=𝑢+𝑖𝑣 then 𝑓 is 𝜆-measurable if and only if 𝑢,𝑣 are 𝜆-measurable. If 𝑓,𝑔∶𝐺→ℝ are 𝜆-measurable, so are 𝑓+𝑔 and 𝑓𝑔. Also, if {𝑓_𝑗}^∞1 is a sequence of ℝ-valued 𝜆-measurable functions, then the functions 𝑔₁(𝑥)=sup_𝑗𝑓_𝑗(𝑥),𝑔₂(𝑥)=inf_𝑗𝑓_𝑗(𝑥),𝑔₃(𝑥)=limsup_𝑗𝑓_𝑗(𝑥), and 𝑔₄(𝑥)=liminf_𝑗𝑓_𝑗(𝑥) are all 𝜆-measurable. If 𝑓(𝑥)=lim_𝑗𝑓_𝑗(𝑥) exists for every 𝑥∈𝐺, then 𝑓 is 𝜆-measurable. Thus if {𝑓_𝑗}^∞1 is a sequence of complex-valued 𝜆-measurable functions and 𝑓_𝑗→𝑓𝜆^𝑢-a.e, then 𝑓 is 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 fixed Haar system 𝜆 and quasi-invariant measure 𝜇 and show that it is a Banach *-algebra under canonical convolution and involution.

Definition 3.1. Suppose 𝜇 is a quasi-invariant probability measure on 𝐺⁰, and 𝜈 is Radon measure induced by 𝜇, then we define 𝐿¹(𝐺,𝜈)=𝐿¹(𝐺,𝜆,𝜇)=𝑓∶𝐺⟶ℂ∶𝑓is𝜆-measurable,‖𝑓‖₁=_𝐺||𝑓(𝑥)||𝑑𝜈(𝑥)<∞.(3.1)

It is clear that 𝐿¹(𝐺,𝜆,𝜇) is a Banach space. We show that the Banach space 𝐿¹(𝐺,𝜆,𝜇) is a Banach *-algebra under the following convolution product.(𝑓∗𝑔)(𝑥)=_{𝐺𝑟(𝑥)}𝑓(𝑦)𝑔𝑦⁻¹𝑥𝑑𝜈(𝑦).(3.2)

Lemma 3.2. If 𝑓,𝑔∈𝐿¹(𝐺,𝜆,𝜇), then 𝑓∗𝑔∈𝐿¹(𝐺,𝜆,𝜇) and ‖𝑓∗𝑔‖₁≤‖𝑓‖₁‖𝑔‖₁.

Proof. For each 𝑓,𝑔, ‖𝑓∗𝑔‖₁=≤_𝐺||𝑓∗𝑔(𝑥)||𝑑𝜈(𝑥)_𝐺_{𝐺𝑟(𝑥)}≤||𝑓(𝑦)||||𝑔𝑦⁻¹𝑥||𝑑𝜈(𝑦)𝑑𝜈(𝑥)_𝐺=||𝑓(𝑦)||||𝑔𝑦⁻¹𝑥||𝑑𝜈(𝑦)𝑑𝜈(𝑥)_𝐺=||𝑓(𝑦)||||𝑔(𝑥)||𝑑𝜈(𝑦)𝑑𝜈(𝑦𝑥)_𝐺0_𝐺||𝑔(𝑥)||_𝐺0_𝐺=||𝑓(𝑦)||𝑑𝜆^𝑢(𝑦)𝑑𝜇(𝑢)𝑑𝜆^𝑢(𝑦𝑥)𝑑𝜇(𝑢)_𝐺=‖𝑓||‖𝑔₁(𝑥)||_𝐺||𝑓(𝑦)||𝑑𝜈(𝑦)𝑑𝜈(𝑥)‖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 for 𝜆-a.e. 𝑥 if for 𝐸={𝑥∶¬𝑃(𝑥)}, 𝜇{𝑢∶𝜆_𝑢(𝐸)>0}=0. Clearly an assertion holds 𝜆-almost everywhere if and only if it holds 𝜈-almost everywhere.

Lemma 3.3. Suppose 𝐷_𝑢∶𝐺^𝑢→ℝ⁺ with 𝐷_𝑢(𝑥)=𝑑𝜆^𝑢(𝑥)/𝑑𝜆_𝑢(𝑥)(𝑥∈𝐺). Then 𝐷=𝐷_𝑢 on 𝐺^𝑢(𝑎.𝑒.).

Proof. Suppose 𝐸⊆𝐺^𝑢. We have 𝜈(𝐸)=_𝐺0𝜆^𝑢(𝐸)𝑑𝜇(𝑢)=_𝐺0_𝐸𝑑𝜆^𝑢(𝑥)𝑑𝜇(𝑢)=_𝐺0_𝐸𝐷_𝑢(𝑥)𝑑𝜆_𝑢(𝑥)𝑑𝜇(𝑢).(3.4) Also from 𝑑𝜈=𝐷𝑑𝜈⁻¹ we have 𝜈(𝐸)=_𝐸𝐷(𝑥)𝑑𝜈⁻¹(𝑥)=_𝐸𝐷(𝑥)_𝐺0𝑑𝜆_𝑢(𝑥)𝑑𝜇(𝑢)=_𝐺0_𝐸𝐷(𝑥)𝑑𝜆_𝑢(𝑥)𝑑𝜇(𝑢).(3.5) Thus _𝐺0_𝐸𝐷_𝑢(𝑥)−𝐷(𝑥)𝑑𝜆_𝑢(𝑥)𝑑𝜇(𝑢)=0.(3.6) Now, let 𝐸^𝑢={𝑥∈𝐺^𝑢∶𝐷_𝑢(𝑥)>𝐷(𝑥)}. If 𝜆_𝑢(𝐸^𝑢)>0 then _𝐸𝑢𝐷_𝑢(𝑥)−𝐷(𝑥)𝑑𝜆_𝑢(𝑥)>0.(3.7) But ∫_𝐸𝑢(𝐷_𝑢(𝑥)−𝐷(𝑥))𝑑𝜆_𝑢(𝑥)=0(𝜇-a.e.), hence 𝜆_𝑢(𝐸^𝑢)=0(𝜇-a.e.). Thus 𝜇{𝑢∶𝜆_𝑢(𝐸^𝑢)>0}=0. Therefore, 𝐷_𝑢(𝑥)≤𝐷(𝑥)(𝜆-a.e.). A similar argument leads to 𝐷_𝑢(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 𝐼𝐼_𝜇(𝐺,𝜈,𝜇) with respect to 𝐼-norm and 𝐼𝐼-norm, indeed 𝐼(𝐺,𝜈,𝜇)⊆𝐼𝐼_𝜇(𝐺)⊆𝐿¹(𝐺,𝜆,𝜇).

According to [2, Lemma  1.4], 𝐼-norm topology is coarser than the inductive limit topology. Also [5, page  15] shows that 𝐶_𝑐(𝐺) has a two-sided bounded approximate identity (𝑒_𝑛)^∞𝑛=1 with respect to the inductive limit topology with the following properties:(i)𝑒_𝑛(𝑥)≥0 for all 𝑥∈𝐺,(ii)|∫𝑒_𝑛(𝑥)𝑑𝜆^𝑢(𝑥)−1|<1/𝑛 for all 𝑢∈𝐾_𝑛, where ∪_𝑛𝐾_𝑛=𝐺⁰ and 𝐾_𝑛's are compact,(iii)𝑒_𝑛(𝑥)=𝑒_𝑛(𝑥⁻¹) for all 𝑥∈𝐺.

An argument in [5, page  15] shows that there is 𝑀>0 such that ‖𝑒_𝑛‖_{𝐼𝐼,𝜇}≤𝑀 for all 𝑛. Since 𝐶_𝑐(𝐺) is dense in 𝐿¹(𝐺,𝜆,𝜇), thus 𝐿¹(𝐺,𝜆,𝜇) has a two-sided bounded approximate identity.

For each 𝑓∈𝐿¹(𝐺,𝜆,𝜇) define 𝐿_𝑥𝑓(𝑦)=𝑓𝑥⁻¹𝑦,𝑅_𝑥𝑓(𝑦)=𝑓(𝑦𝑥),(3.9) when the multiplications on the right hand sides are defined. It is easy to check that the maps 𝐿_𝑥,𝑅_𝑥 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 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 𝑓∗𝑔=∫_{𝐺𝑟(𝑦)}𝑓(𝑦)𝐿_𝑦𝑔𝑑𝜈(𝑦), 𝐿_𝑥(𝑓∗𝑔)==_{𝐺𝑟(𝑦)}𝑓(𝑦)𝐿_𝑥𝐿_𝑦𝑔𝑑𝜈(𝑦)=_{𝐺𝑟(𝑦)}𝑓(𝑦)𝐿_𝑥𝑦𝑔𝑑𝜈(𝑦)_{𝐺𝑟(𝑦)}𝑓𝑥⁻¹𝑦𝐿_𝑦𝑔𝑑𝜈(𝑦)=_{𝐺𝑟(𝑦)}𝐿_𝑥𝑓(𝑦)𝐿_𝑦𝑔𝑑𝜈(𝑦)=𝐿_𝑥𝑓∗𝑔.(3.10) Now suppose (𝑒_𝑛)_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 𝐿_𝑥𝑒_𝑛∗𝑓=𝐿_𝑥𝑒_𝑛∗𝑓⟶𝐿_𝑥𝑓.(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)𝐿_𝑦𝑔𝑑𝜈(𝑦)(3.12) is in the closed linear span of the functions 𝐿_𝑦𝑔 and hence in 𝐼. The other assertion is proved similarly.

4. 𝑀(𝐺) as Banach ∗-Algebra

Let 𝑀(𝐺) be the space of complex Radon measures on 𝐺. If 𝜂,𝜃∈𝑀(𝐺), then the map 𝜓↦𝐼(𝜓) on 𝐶₀(𝐺) defined by 𝐼(𝜓)=∫_𝐺∫_{𝐺𝑟(𝑦)}𝜓(𝑥𝑦)𝑑𝜂(𝑥)𝑑𝜃(𝑦) is a linear functional on 𝐶₀(𝐺) satisfying |𝐼(𝜓)|≤‖𝜓‖_sup‖𝜂‖‖𝜃‖, so by Riesz representation theorem, it is given by a measure shown as 𝜂∗𝜃 called the convolution of 𝜂,𝜃 with ‖𝜂∗𝜃‖≤‖𝜂‖‖𝜃‖. If we define 𝜂^∗(𝐸)=𝜂(𝐸⁻¹), then 𝜂↦𝜂^∗ is an involution on 𝑀(𝐺), and 𝑀(𝐺) is a Banach ∗-algebra. In this section, we show that the space 𝐿¹(𝐺,𝜆,𝜇) is a closed two-sided ideal of 𝑀(𝐺).

Proposition 4.1. The map 𝐿¹(𝐺,𝜆,𝜇)↪𝑀(𝐺) with 𝜈_𝑓(𝐸)=∫_𝐸𝑓𝜒_𝐸𝑑𝜈, for (𝐸⊆𝐺); 𝑓↦𝜈_𝑓 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 check that 𝜈_𝑓 is a measure on 𝐺. We show that 𝜈_𝑓 is Radon. If 𝑓=𝑢+𝑖𝑣, then 𝜈_𝑓=𝜈_𝑢+𝑖𝜈_𝑣, so 𝜈_𝑓 is Radon if and only if 𝜈_𝑢 and 𝜈_𝑣 are Radon. Since 𝐺 is LCH and second countable, we have 𝜈_𝑢(𝐾)=∫_𝐾𝑢𝑑𝜈≤∫_𝐺|𝑢|𝑑𝜈=‖𝑢‖₁<∞, for each compact set 𝐾, thus 𝜈_𝑢 is Radon. Similarly 𝜈_𝑣 is Radon, and so is 𝜈_𝑓.
By definition, ‖𝜈_𝑓‖=sup{∑^𝑛1|𝜈_𝑓(𝐸_𝑖)|∶𝑛∈𝑁,𝐺=∐^𝑛1𝐸_𝑖}, so for each 𝜖>0, there exists a partition {𝐸_𝑖}^𝑛1 of 𝐺 such that ‖‖𝜈_𝑓‖‖−𝜖<^𝑛₁||𝜈_𝑓𝐸_𝑖||=^𝑛₁||||_𝐺𝑓𝜒_𝐸𝑖𝑑𝜈||||≤_𝐺||𝑓||^𝑛₁𝜒_𝐸𝑖𝑑𝜈=‖𝑓‖¹.(4.1) Thus, ‖𝜈_𝑓‖≤‖𝑓‖₁. Conversely, suppose 𝑓≥0, then 𝜈_𝑓≥0 and for every partition {𝐸_𝑖}^𝑛1 of 𝐺 we have, ‖‖𝜈_𝑓‖‖≥^𝑛₁𝜈_𝑓𝐸_𝑖=^𝑛₁_𝐺𝑓𝜒_𝐸𝑖𝑑𝜈=_𝐺𝑓𝑑𝜈=‖𝑓‖₁.(4.2) If 𝑓=𝑢+𝑖𝑣=(𝑓₁−𝑓₂)+𝑖(𝑓₃−𝑓₄), where 𝑓_𝑖≥0 then ‖𝜈_𝑓‖=‖𝜈_𝑓1‖+‖𝜈_𝑓2‖+‖𝜈_𝑓3‖+‖𝜈_𝑓4‖≥‖𝑓₁‖₁+‖𝑓₂‖₁+‖𝑓₃‖₁+‖𝑓₄‖₁≥‖𝑓‖₁. Hence ‖𝜈_𝑓‖≥‖𝑓‖₁ and equality holds.

Corollary 4.2. 𝐿¹(𝐺,𝜆,𝜇) is a closed subspace of 𝑀(𝐺).

Lemma 4.3. If 𝑓,𝑔∈𝐿¹(𝐺,𝜆,𝜇), then 𝜈_{(𝑓∗𝑔)}=𝜈_𝑓∗𝜈_𝑔.

Proof. For each compact set 𝐾, we have 𝜈_𝑓∗𝜈_𝑔(𝐾)=_𝐺𝜒_𝐾(𝑥)𝑑𝜈_𝑓∗𝜈_𝑔(𝑥)=_𝐺_{𝐺𝑟(𝑥)}=𝜒_𝐾(𝑦𝑥)𝑑𝜈_𝑓(𝑦)𝑑𝜈_𝑔(𝑥)_𝐺𝑓(𝑦)_{𝐺𝑠(𝑦)}𝜒_𝐾(𝑦𝑥)𝑑𝜈_𝑔(𝑥)𝑑𝜈(𝑦)=_𝐺_{𝐺𝑠(𝑦)}=𝑓(𝑦)𝑔𝑦⁻¹𝑥𝜒_𝐾(𝑥)𝑑𝜈(𝑦)𝑑𝜈(𝑥)_𝐺𝜒_𝐾(𝑥)_{𝐺𝑟(𝑥)}𝑓(𝑦)𝑔𝑦⁻¹𝑥𝑑𝜈(𝑦)𝑑𝜈(𝑥)=_𝐺(𝑓∗𝑔)(𝑥)𝜒_𝐾(𝑥)𝑑𝜈(𝑥)=𝜈_𝑓∗𝑔(𝐾).(4.3) Since 𝜈_𝑓∗𝑔 and 𝜈_𝑓∗𝜈_𝑔 are regular measures, the equality holds for each open set and then for each measurable set.

If 𝑓∈𝐿¹(𝐺,𝜆,𝜇) and 𝜂∈𝑀(𝐺), we will define 𝜂∗𝑓 such that 𝜈_𝜂∗𝑓=𝜂∗𝜈_𝑓. Suppose 𝜑∈𝐶₀(𝐺), we put 𝜂∗𝑓(𝜑)=_𝐺𝜑(𝑥)𝑑𝜈_𝜂∗𝑓(𝑥)=_𝐺𝜑(𝑥)(𝜂∗𝑓)(𝑥)𝑑𝜈(𝑥).(4.4) On the other hand, 𝜂∗𝜈_𝑓(𝜑)=_𝐺_{𝐺𝑟(𝑥)}𝜑(𝑦𝑥)𝑑𝜂(𝑦)𝑑𝜈_𝑓(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=_𝐺𝜑(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 _𝐺𝑢𝑅_𝑦𝑓(𝑥)𝑑𝜆^𝑢(𝑥)=_𝐺𝑢𝑓(𝑥)𝑑𝜆^𝑢𝑥𝑦⁻¹=𝐷𝑦⁻¹_𝐺𝑢𝑓(𝑥)𝑑𝜆^𝑢(𝑥).(4.7) Thus _𝐺𝑅_𝑦𝑓(𝑥)𝑑𝜈(𝑥)=𝐷𝑦⁻¹_𝐺𝑓(𝑥)𝑑𝜈(𝑥).(4.8) Similarly, we want to define 𝑓∗𝜂 in such a way that the equality 𝜈_{(𝑓∗𝜂)}=𝜈_𝑓∗𝜂 holds. Again suppose 𝜑∈𝐶₀(𝐺). We have 𝜈_{(𝑓∗𝜂)}(𝜑)=_𝐺𝜑(𝑥)𝑑𝜈_𝑓∗𝜂(𝑥)=_𝐺𝜑(𝑥)(𝑓∗𝜂)(𝑥)𝑑𝜈(𝑥)=_𝐺0_𝐺=𝜑(𝑥)(𝑓∗𝜂)(𝑥)𝑑𝜆^𝑢(𝑥)𝑑𝜇(𝑢)_𝐺0_𝐺𝑢𝜑(𝑥)(𝑓∗𝜂)(𝑥)𝑑𝜆^𝑢(𝑥)𝑑𝜇(𝑢)=_𝐺𝑢𝜑(𝑥)(𝑓∗𝜂)(𝑥)𝑑𝜈(𝑥).(4.9) On the other hand, 𝜈_𝑓∗𝜂(𝜑)==_𝐺_{𝐺𝑟(𝑦)}𝜑(𝑥𝑦)𝑑𝜈_𝑓(𝑥)𝑑𝜂(𝑦)_𝐺_{𝐺𝑟(𝑦)}=𝜑(𝑥𝑦)𝑓(𝑥)𝑑𝜈(𝑥)𝑑𝜂(𝑦)_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)_𝐺𝑓𝑥𝑦⁻¹𝐷𝑦⁻¹𝑑𝜂(𝑦)𝑑𝜈(𝑥).(4.10) Comparing the above equalities, we have (𝑓∗𝜂)(𝑥)=_𝐺𝑓𝑥𝑦⁻¹𝐷𝑦⁻¹𝑑𝜂(𝑦).(4.11)

Lemma 4.4. 𝐿¹(𝐺,𝜆,𝜇) is a two-sided ideal of 𝑀(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<∞.(4.12) Thus 𝜂∗𝑓∈𝐿¹(𝐺,𝜆,𝜇). Also ‖𝑓∗𝜂‖₁==_𝐺||(𝑓∗𝜂)(𝑥)||𝑑𝜈(𝑥)≤_𝐺||𝑓𝑥𝑦⁻¹𝐷𝑦⁻¹||𝑑||𝜂||(𝑦)𝑑𝜈(𝑥)_𝐺=||𝑓(𝑥)||𝐷𝑦⁻¹𝐷(𝑦)𝑑𝜈(𝑥)𝑑||𝜂||(𝑦)_𝐺||𝑓||𝑑𝜈𝑑||𝜂||=‖𝑓‖₁‖𝜂‖<∞.(4.13) Hence, 𝑓∗𝜂∈𝐿¹(𝐺,𝜆,𝜇).

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 𝜈=∫𝜆^𝑢𝑑𝜇(𝑢) is a Radon measure. Let 𝜈₀ be the measure on 𝐺 given by 𝑑𝜈₀=𝐷^−1/2𝑑𝜈,(5.1) where 𝐷 is modular function of 𝜇. Then for each Borel subset 𝐸 of 𝐺, we have 𝜈₀(𝐸)==𝜒𝜒_𝐸(𝑥)𝐷^−1/2(𝑥)𝑑𝜈(𝑥)=𝜒_𝐸𝑥⁻¹𝐷^−1/2𝑥⁻¹𝑑𝜈⁻¹(𝑥)_𝐸−1(𝑥)𝐷^1/2(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,{𝐻_𝑢}_𝑢,𝜋) consisting of a Hilbert bundle (𝐺⁰,{𝐻_𝑢}_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(𝐻_𝑠(𝑥),𝐻_𝑟(𝑥)) such that(i)𝜋(𝑢) is the identity map on 𝐻_𝑢 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,{𝐻_𝑢}_𝑢,𝜇), the map𝑥⟼⟨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 in 𝐻.

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 𝐺 and nondegenerate representations of 𝐿¹(𝐺,𝜆,𝜇). Unfortunately, this is impossible in general. Such a correspondence exists between representations of 𝐺 and those of 𝐶_𝑐(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 𝐶_𝑐(𝐺) on a separable Hilbert space is the integrated form of a representation of 𝐺.

These assumptions satisfied in the case of 𝑟-discrete groupoids and transformation groups. The main problem is that a continuous representation of (𝐿¹(𝐺,𝜆,𝜇),‖⋅‖_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,{𝐻_𝑢}_𝑢,𝜇) is a representation of 𝐺. For 𝜉,𝜂∈𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇) and 𝑓∈𝐶_𝑐(𝐺), ⟨Π(𝑓)𝜉,𝜂⟩=_𝐺𝑓(𝑥)⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑𝜈₀(𝑥)(5.5) defines a bounded nondegenerate ∗-representation of 𝐶_𝑐(𝐺) on 𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇) such that two equivalent representations of 𝐺 give two equivalent representations of 𝐶_𝑐(𝐺).

The equation above is called the integrated form of a unitary representation. If Π is a representation of 𝐶_𝑐(𝐺), the above proposition says that Π should be of the form ⟨Π(𝑓)𝜉,𝜂⟩=_𝐺𝑓(𝑥)⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑𝜈₀(𝑥).(5.6) For each 𝜉,𝜂∈𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇) and 𝑓∈𝐶_𝑐(𝐺).

Next, we define a representation of 𝐼(𝐺,𝜆,𝜇), denoted by Π_𝐼 as ⟨Π_𝐼(𝑓)𝜉,𝜂⟩=_𝐺𝑓(𝑥)⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑𝜈₀(𝑥)(5.7) for each 𝜉,𝜂∈𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇) and 𝑓∈𝐼(𝐺,𝜆,𝜇)

Lemma 5.5. Π_𝐼 is a bounded representation of 𝐼(𝐺,𝜆,𝜇) on ℬ(𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇)).

Proof. We have ⟨Π_𝐼(𝑓)𝜉,𝜂⟩==_𝐺𝑓(𝑥)⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑𝜈₀(𝑥)_𝐺0_𝐺𝑢=𝑓(𝑥)⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑^−1/2(𝑥)𝑑𝜆^𝑢(𝑥)𝑑𝜇(𝑢)_𝐺0_𝐺𝑢𝑓(𝑥)𝜋(𝑥)𝜉(𝑠(𝑥))𝐷^−1/2(𝑥)𝑑𝜆^𝑢(𝑥),𝜂(𝑢)𝑑𝜇(𝑢).(5.8) Thus, we may define for 𝜇-a.e. 𝑢∈𝐺⁰, Π_𝐼(𝑓)𝜉(𝑢)=_𝐺𝑢𝑓(𝑥)𝜋(𝑥)𝜉(𝑠(𝑥))𝐷^−1/2(𝑥)𝑑𝜆^𝑢(𝑥).(5.9) Now, we have ||⟨Π_𝐼(𝑓)𝜉,𝜂⟩||=≤||||_𝐺𝑓(𝑥)⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑𝜈₀(𝑥)_𝐺||||≤||𝑓(𝑥)||‖𝜉∘𝑠(𝑥)‖‖𝜂∘𝑟(𝑥)‖𝑑𝜈₀(𝑥)_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)||𝑑𝜆_𝑢(𝑥)‖𝜉(𝑢)‖²𝑑𝜇(𝑢)^1/2×_𝐺0_𝐺𝑢||𝑓(𝑥)||𝑑𝜆^𝑢(𝑥)‖𝜂(𝑢)‖²𝑑𝜇(𝑢)^1/2≤≤‖‖𝑓𝑓‖‖^1/2𝐼,𝑠_𝐼‖𝜉‖‖𝑓‖^1/2𝐼,𝑟‖𝜂‖‖𝜉‖‖𝜂‖.(5.10) Therefore, the map 𝑥↦𝑓(𝑥)⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩ is 𝜈₀-integrable, and Π_𝐼(𝑓) is a bounded operator of norm ‖Π_𝐼(𝑓)‖≤‖𝑓‖_𝐼. We have to check that Π_𝐼 is a ∗-homomorphism. For this, we define 𝐹(𝑥,𝑡)=𝑓(𝑥𝑡)𝑔𝑡⁻¹⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑^1/2(𝑥).(5.11) Then ⟨Π_𝐼(𝑓∗𝑔)𝜉,𝜂⟩==_𝐺𝑓∗𝑔(𝑥)⟨𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑𝜈₀(𝑥)_𝐺_𝐺=𝑓(𝑥𝑡)𝑔𝑡⁻¹𝑑𝜆^𝑠(𝑥)(𝑡)⟨𝜋(𝑥)𝜉∘𝑠(𝑥),𝜂∘𝑟(𝑥)⟩𝑑^−1/2(𝑥)𝐷(𝑥)𝑑𝜈⁻¹(𝑥)_𝐺2=𝐹(𝑥,𝑡)𝑑𝜈²(𝑥,𝑡)_𝐺2=𝐹𝑥𝑡,𝑡⁻¹𝜌⁻¹𝑥𝑡,𝑡⁻¹𝑑𝜈²(𝑥,𝑡)_𝐺0_𝐺𝑢×𝑓𝐷(𝑥)_−1/2𝜋(𝑥)_{𝐺𝑠(𝑥)}𝑔(𝑡)𝜋(𝑡)𝜉∘𝑠(𝑡)𝐷^−1/2(𝑡)𝑑𝜆^𝑠(𝑥)(𝑡),𝜂(𝑢)=(𝑥)𝑑𝜆^𝑢(𝑥)𝑑𝜇(𝑢)_𝐺=⟨Π_𝐼𝑓(𝑥)⟨𝜋(𝑥)Π(𝑔)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))⟩𝑑𝜈₀(𝑥)(𝑓)Π_𝐼(𝑔)𝜉,𝜂⟩.(5.12) Hence, Π_𝐼(𝑓∗𝑔)=Π_𝐼(𝑓)Π_𝐼(𝑔). Also ⟨Π_𝐼𝑓^∗𝜉,𝜂⟩==_𝐺𝑓^∗(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)_𝐺=𝑓(𝑥)𝐷𝑥⁻¹⟨𝜋^∗(𝑥)𝜉(𝑟(𝑥)),𝜂(𝑠(𝑥))⟩𝑑^1/2(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)⟨Π_𝐼(𝑓)𝜉,𝜂⟩=⟨𝜉,Π_𝐼(𝑓)𝜂⟩=Π^∗𝐼(𝑓)𝜉,𝜂.(5.13) The nondegeneracy follows from ⟨Π_𝐼(𝑓)𝜉∣𝑓∈𝐶_𝑐(𝐺),𝜉∈𝐻⟩⊆Π_𝐼(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 necessarily continuous in 𝐿¹-norm. Let 𝐻=𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇) and put 𝐻_Π1=𝜉∈𝐻∶Themap𝑓⟼⟨Π_𝐼(𝑓)𝜉,𝜉⟩iscontinuousin𝐿¹-norm.(5.15) Observe that 𝐻_Π1 is a nontrivial subspace, as 𝜇 is a probability measure and if ‖𝜉(𝑢)‖=1 in 𝐻_𝑢 for each 𝑢∈𝐺⁰, then ‖𝜉‖=1 in 𝐻 and the calculations in the proof of Lemma 5.5 shows that |Π_𝐼(𝑓)𝜉,𝜉⟩|≤‖𝑓‖₁, hence 𝜉∈𝐻₁. On the other hand, Π_𝐼(𝑓)𝐻_Π1⊆𝐻_Π1 for each 𝑓∈𝐼(𝐺,𝜆,𝜇), because the map 𝑓⟼⟨Π_𝐼(𝑓)Π_𝐼(𝑔)𝜉,Π_𝐼(𝑔)𝜉⟩=Π_𝐼𝑔^∗∗𝑓∗𝑔𝜉,𝜉(5.16) is continuous. Therefore, Π_𝐼(𝑓)𝐻^⟂Π₁⊆𝐻^⟂Π₁. Hence, we have Π_𝐼(𝑓)=Π_𝐼(𝑓)|_𝐻Π1⊕Π_𝐼(𝑓)|_𝐻^⟂Π1 for each 𝑓∈𝐼(𝐺,𝜆,𝜇).

Put 𝐻₁=𝐻_Π1 and define Π₁(𝑓)∶=Π_𝐼(𝑓)|_𝐻1, for 𝑓∈𝐼(𝐺,𝜆,𝜇). Then, it follows from continuity of 𝑓↦⟨Π_𝐼(𝑓)𝜉,𝜉⟩ and polarization identity that Π₁ extends to a continuous representation of 𝐿¹(𝐺,𝜆,𝜇) on 𝐻₁, still denoted by Π₁.

Next we focus on the notion of irreducibility which plays an important role in the theory of representations. We show that if 𝜋 is an irreducible representation of 𝐺, the integrated representation Π₁ of 𝐿¹(𝐺,𝜆,𝜇) on 𝐻₁ is irreducible. Basic materials come from [6].

Definition 5.6. Let (𝐺⁰,{𝐻_𝑢}_𝑢∈𝐺0,𝜇) be a Hilbert bundle. A family 𝑀={𝑀_𝑢}_𝑢∈𝐺0, where 𝑀_𝑢 is a closed subspace of 𝐻_𝑢 for each 𝑢∈𝐺⁰, is called a subbundle. A subbundle {𝑀_𝑢}_𝑢∈𝐺0 is called nontrivial if 0≠𝑀_𝑢≠𝐻_𝑢 for some 𝑢∈𝐺⁰. For a representation 𝜋 of a locally compact groupoid 𝐺 associated with the Hilbert bundle (𝐺⁰,{𝐻_𝑢}_𝑢∈𝐺0,𝜇), a subbundle {𝑀_𝑢}_𝑢∈𝐺0 is called invariant if 𝜋(𝑥)𝑀_𝑠(𝑥)⊆𝑀_𝑟(𝑥) for each 𝑥∈𝐺. Note that if 𝑀 is an invariant subbundle, and 0≠𝑀_𝑢≠𝐻_𝑢 for some 𝑢∈𝐺⁰, then 0≠𝑀_𝑤≠𝐻_𝑤 for every 𝑤∈[𝑢].

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

Lemma 5.7. Let 𝜋 be a representation of a locally compact groupoid 𝐺 associated with the Hilbert bundle (𝐺⁰,{𝐻_𝑢}_𝑢∈𝐺0,𝜇). If 𝑀={𝑀_𝑢}_𝑢∈𝐺0 is an invariant subbundle, then so is 𝑀^⟂={𝑀^⟂𝑢}_𝑢∈𝐺0.

Definition 5.8. A representation 𝜋 of a locally compact groupoid 𝐺 is called reducible if 𝜋 admits a nontrivial invariant subbundle 𝑀={𝑀_𝑢}_𝑢∈𝐺0, otherwise 𝜋 is called irreducible. In this case, it is easy to check that 𝜋^𝑀 with 𝜋^𝑀(𝑥)=𝜋(𝑥)|_{𝑀𝑠(𝑥)}∶𝑀_𝑠(𝑥)→𝑀_𝑟(𝑥) is called a subrepresentation of 𝜋. If 𝜋 and 𝜋^ are two representations of a locally compact groupoid 𝐺 associated with two Hilbert bundles (𝐺⁰,{𝐻_𝑢}_𝑢∈𝐺0,𝜇) and (𝐺⁰,{𝐻^𝑢}_𝑢∈𝐺0,𝜇^), respectively. Then, we put 𝒞𝜋,𝜋^=⎧⎪⎨⎪⎩𝑇_𝑢_𝑢∈_𝑢∈𝐺0ℬ𝐻_𝑢,𝐻^𝑢∶𝜋^(𝑥)𝑇_𝑠(𝑥)=𝑇_𝑟(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 (𝑇_𝑢)_𝑢∈𝐺⁰∈𝒞(𝜋,𝜋^) such that 𝑇_𝑢 is a unitary operator for every 𝑢∈𝐺⁰. Note that if (𝑇_𝑢)_𝑢∈𝐺⁰∈𝒞(𝜋) and 𝑇^∗𝑢 denotes the adjoint operator to 𝑇_𝑢, then 𝜋(𝑥)𝑇^{∗𝑠(𝑥)}=𝑇_𝑠(𝑥)𝜋𝑥⁻¹^∗=𝜋𝑥⁻¹𝑇_𝑟(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 are defined pointwise.

Following [6], for a representation 𝜋 of a locally compact groupoid 𝐺 associated with the Hilbert bundle (𝐺⁰,{𝐻_𝑢}_𝑢∈𝐺0,𝜇), we set Λ=⎧⎪⎨⎪⎩𝜆_𝑢𝜋(𝑢)_𝑢∈_𝑢∈𝐺0ℬ𝐻_𝑢∶𝜆_𝑢∈ℂ,𝜆_𝑢=𝜆_𝑤whenever𝑢∼𝑤⎫⎪⎬⎪⎭.(5.19) If (𝜆_𝑢𝜋(𝑢))_𝑢∈𝒞(𝜋), then 𝜆_𝑢=𝜆_𝑤 whenever 𝑢∼𝑤 that is (𝜆_𝑢𝜋(𝑢))_𝑢∈Λ. Therefore, if (𝑇_𝑢)_𝑢∈𝐺⁰∈𝒞(𝜋)⧵Λ, then there exists 𝑢∈𝐺⁰ with 𝑇_𝑢 not in ℂ𝜋(𝑢). It is also obvious that Λ⊆𝒞(𝜋).

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

Lemma 5.9. A representation 𝜋 of a locally compact groupoid 𝐺 is irreducible if and only if Λ=𝒞(𝜋). In particular, in the case where 𝐺 is transitive, then 𝜋 is irreducible if and only if 𝒞(𝜋)=ℂ(𝜋(𝑢))_𝑢∈𝐺⁰.

Lemma 5.10. If ℳ=(𝐺⁰,{𝑀_𝑢}_𝑢,𝜇) is closed subbundle of ℋ=(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇) and 𝑃_𝑢 is an orthogonal projection 𝐻_𝑢 onto 𝑀_𝑢, then ℳ is invariant under 𝜋 if and only if (𝑃_𝑢)_𝑢∈𝒞(𝜋).

Proof. If (𝑃_𝑢)_𝑢∈𝒞(𝜋) and 𝑚_𝑢∈𝑀_𝑢, then 𝜋(𝑥)𝑚_𝑠(𝑥)=𝜋(𝑥)𝑃_𝑠(𝑥)𝑚_𝑠(𝑥)=𝑃_𝑟(𝑥)𝜋(𝑥)𝑚_𝑠(𝑥)∈𝑀_𝑟(𝑥).(5.20) Thus, 𝜋(𝑥)𝑀_𝑠(𝑥)⊆𝑀_𝑟(𝑥) so ℳ is invariant. Conversely, if ℳ is invariant, then for 𝑚∈𝑀_𝑠(𝑥)𝜋(𝑥)𝑃_𝑠(𝑥)𝑚=𝜋(𝑥)𝑚=𝑃_𝑟(𝑥)𝜋(𝑥)𝑚(5.21) and for 𝑚∈𝑀^{⟂𝑠(𝑥)}𝜋(𝑥)𝑃_𝑠(𝑥)𝑚=0=𝑃_𝑟(𝑥)𝜋(𝑥)𝑚.(5.22) Hence, 𝜋(𝑥)𝑃_𝑠(𝑥)=𝑃_𝑟(𝑥)𝜋(𝑥).

We show that if a representation 𝜋 of a locally compact groupoid 𝐺 associated with the Hilbert bundle (𝐺⁰,{𝐻_𝑢}_𝑢∈𝐺0,𝜇) is irreducible and Π₁ is the corresponding integrated representation of 𝐿¹(𝐺,𝜆,𝜇) on 𝐻₁⊆𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇), then Π₁ is irreducible. If 𝐻=𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇), then for each subbundle {𝑀_𝑢}_𝑢, 𝑀=𝐿²(𝐺⁰,{𝑀_𝑢}_𝑢,𝜇) is closed subspace of 𝐻 with orthogonal complement 𝑀^⟂=𝐿²(𝐺⁰,{𝑀^⟂𝑢}_𝑢,𝜇). A map Θ∶𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇)→𝐿²(𝐺⁰,{𝑀_𝑢}_𝑢,𝜇);𝜉↦Θ𝜉 with Θ𝜉(𝑠(𝑥))∈𝑀_𝑟(𝑥) is called an orthogonal projection of 𝐻 onto 𝑀.

Let 𝜋 be a representation of 𝐺 on the Hilbert bundle (𝐺⁰,{𝐻_𝑢}_𝑢∈𝐺0,𝜇) and Π_𝐼 be the bounded representation of 𝐼(𝐺,𝜆,𝜇) on ℬ(𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇)) constructed above.

Lemma 5.11. 𝑀 is invariant under Π_𝐼 if and only if for each 𝑓∈𝐼(𝐺,𝜆,𝜇), Π_𝐼(𝑓)Θ=ΘΠ_𝐼(𝑓).

Proof. Suppose Π_𝐼(𝑓)Θ=ΘΠ_𝐼(𝑓) and 𝜉∈𝑀, then Π_𝐼(𝑓)𝜉=Π_𝐼(𝑓)Θ𝜉=ΘΠ_𝐼(𝑓)𝜉∈𝑀.(5.23) Hence, 𝑀 is invariant. Conversely, if 𝑀 is invariant, then for 𝜉∈𝑀, Π_𝐼(𝑓)Θ𝜉=Π_𝐼(𝑓)𝜉=ΘΠ_𝐼(𝑓)𝜉(5.24) and for 𝜂∈𝑀^⟂, Π_𝐼(𝑓)Θ𝜂=0=ΘΠ_𝐼(𝑓)𝜂.(5.25) Thus, Π_𝐼(𝑓)Θ=ΘΠ_𝐼(𝑓).

Proposition 5.12. If 𝜋 is an irreducible representation of 𝐺, then Π_𝐼 is an irreducible representation of 𝐼(𝐺,𝜆,𝜇).

Proof. If Π_𝐼 is reducible, then there exists a nontrivial invariant closed subspace 𝑀 of 𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇), and, hence, there is an orthogonal projection Θ∈𝒞(Π_𝐼). It follows that _𝐺𝑓=(𝑥⟨)Π_𝐼𝜋(𝑥)𝑃_𝑠(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))𝑑𝜈₀(𝑥)=(𝑓)Θ𝜉,𝜂⟩=⟨ΘΠ_𝐼(𝑓)𝜉,𝜂⟩_𝐺𝑓(𝑥)𝑃_𝑟(𝑥)𝜋(𝑥)𝜉(𝑠(𝑥)),𝜂(𝑟(𝑥))𝑑𝜈₀(𝑥),(5.26) for each 𝑓∈𝐶_𝑐(𝐺) and 𝜉,𝜂∈𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇). Therefore, 𝜋(𝑥)𝑃_𝑠(𝑥)=𝑃_𝑟(𝑥)𝜋(𝑥), for each 𝑥∈𝐺, hence (𝑃_𝑢)_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 of 𝐿¹(𝐺,𝜆,𝜇) on 𝐻₁.

Proof. If Π₁ is reducible, then there exists a nontrivial invariant closed subspace 𝑀₁ of 𝐻₁⊆𝐻=𝐿²(𝐺⁰,{𝐻_𝑢}_𝑢,𝜇). By the calculations after Lemma 5.5, Π_𝐼=Π₁⊕Π₂, where Π₁ and Π₂ are the corresponding representations of 𝐼(𝐺,𝜆,𝜇) on 𝐻₁ and 𝐻^⟂1. Therefore, 𝑀=𝑀₁⊕𝐻^⟂1 is a nontrivial invariant closed subspace 𝐻, and Π_𝐼 is reducible. Hence, 𝜋 is reducible by the above proposition.