# L1-Algebra of a Locally Compact Groupoid

Authors: Massoud Amini; Alireza Medghalchi; Ahmad Shirinkalam
Journal: ISRN Algebra
 (2011)

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## Abstract

For a locally compact groupoidG 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).


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## Body

## 1. Introduction and Preliminaries

For a locally compact groupG 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.
Letν 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)
iff:X×Y→C is ν·ρ-integrable, then the partial integrals
(1.1)∫Xf(x,y)dν(x),∫Yf(x,y)dρ(y)
define integrable functions such that Fubini’s formula holds
(1.2)∫X×Yfdν⋅ρ=∫X∫Yfdνdρ=∫Y∫Xfdρdν,(ii)
iff 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 setG 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.Ifx∈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.Foru,v∈G0, Gu=r-1{u}, Gv=s-1{v}, Gvu=Gu∩Gv, and G{u}=Guu. The latter is called the isotropy group at u. We define u~v if Gvu≠∅. 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 groupoidG 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 familyC of compact Hausdorff subsets of G whose interiors form a basis for the topology of G.(iii)
EveryGu is locally compact Hausdorff in the relative topology.A locally compact groupoid isr-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 Cc(G) consists of all continuous functions on G with compact support. A left Haar system for G consists of measures {λu,u∈G0} on G such that(i)
the support ofλu is Gu,(ii)
(continuity) for eachf∈Cc(G),u↦λ(f)(u)=∫fdλu is continuous,(iii)
(left invariance) for anyx∈G and any f∈Cc(G),(1.3)∫f(xy)dλs(x)(y)=∫f(y)dλr(x)(y).Letμ be a measure on G0. The measure on G induced by μ is ν=∫λudμ, defined by ∫Gfdν=∫G0dμ(u)∫Gfdλu, 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 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|Guu is modular function for Guu.A subsetA 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 {λu}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 (1.4)IIμ(G,ν,μ)={f∈L1(G,ν0):‖f‖II,μ<∞},
where ∥f∥II,μ is defined by (1.5)‖f‖II,μ=sup⁡{∫|f(x)j(s(x))k(r(x))|dν0(x);∫|j|2dμ=∫|k|2dμ=1}.
If μ1 and μ2 are two equivalent quasi-invariant measures, then (1.6)‖f‖II,μ1=‖f‖II,μ2,
because ∥f∥II,μ=∥IIμ(|f|)∥ for each quasi-invariant measure μ, where IIμ is the one-dimensional trivial representation on μ. Now define (1.7)‖f‖II=sup⁡μ‖f‖II,μ.
The supremum is taken over the class of quasi-invariant measures.Let(1.8)I(G,ν,μ)={f∈L1(G,ν0):‖f‖I,μ<∞},
where (1.9)‖f‖I,μ=max⁡{‖u⟶∫|f|dλu‖∞,‖u⟶∫|f*|dλu‖∞}.
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 (1.10)‖f‖I,r=sup⁡u(∫|f|dλu),‖f‖I,s=sup⁡u(∫|f|dλu),
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={λu}.Definition 2.1.
A system of measures{λu}u∈G0 is said to be complete if for each u∈G0, λu is complete on its σ-algebra 𝔐λu. A Borel measurable set E⊆G is called λ-measurable if for each u∈G0, E∩Gu∈𝔐λu. A function f:G→ℂ is λ-measurable if for every u∈G0 and every open set O⊆ℂ,f-1(O)∩Gu∈𝔐λu.Proposition 2.2.
Ifλ̅={λ̅u} is the completion of  λ={λu} and  f:G→[0,∞] is  λ̅-measureable, then there is a  λ-measurable function  g such that  f=g on  Gu(λu-a.e).Proof.
Sincef is λ-measurable for each u∈G0, f is λ̅u-measurable on Gu, and since λ̅u is the completion of λu, there exists a λu-measurable function gu such that f=gu(λ̅u-a.e). Now define g=gu on Gu and zero, elsewhere. Since for every u∈G0 and every open set O⊆ℂ,g-1(O)∩Gu=gu-1(O)∈𝔐λu, g is λ-measurable, and since λu(Gu)c=0, we have f=g on Gu (λu-a.e).From now on, we assume that the Haar systemλ is complete.Lemma 2.3.
For eachf:G→ℂ,  λ-measurability of  f is equivalent to  ν-measurability of  f.Proof.
We havesuppλu=Gu and
(2.1)f-1(O)=(f-1(O)∩Gu)∪(f-1(O)∩(Gu)c).
Since λu is complete and (Gu)c is λu-null, (f-1(O)∩(Gu)c)⊆(Gu)c is in 𝔐λu. Thus, for each u∈G0 and open set O⊆ℂ(2.2)f-1(O)∈Mλu⟺f-1(O)∩Gu∈Mλu.
Now for ν=∫λudμ(u), we have 𝔐ν=⋂u∈G0𝔐λu, hence f is ν-measurable if and only if f is λ-measurable.Iff: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 {fj}1∞ is a sequence of ℝ¯-valued λ-measurable functions, then the functions g1(x)=supjfj(x),g2(x)=infjfj(x),g3(x)=limsupjfj(x), and g4(x)=liminfjfj(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→fλu-a.e, then f is λ-measurable.


## 3. The AlgebraL1(G,λ,μ)

In this section, we define the space of integrable functions onG 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.
Supposeμ is a quasi-invariant probability measure on G0, and ν is Radon measure induced by μ, then we define
(3.1)L1(G,ν)=L1(G,λ,μ)={f:G⟶C:fisλ-measurable,‖f‖1=∫G|f(x)|dν(x)<∞}.It is clear thatL1(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.(3.2)(f*g)(x)=∫Gr(x)f(y)g(y-1x)dν(y).Lemma 3.2.
Iff,g∈L1(G,λ,μ), then  f*g∈L1(G,λ,μ) and  ∥f*g∥1≤∥f∥1∥g∥1.Proof.
For eachf,g,
(3.3)‖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)|dλu(y)dμ(u)dλu(yx)dμ(u)=∫G|g(x)|(∫G|f(y)|dν(y))dν(x)=‖f‖1‖g‖1.
The measurability of f*g follows from λ-measurability of f,g.Next, we define an involution onL1(G,λ,μ). We say that the assertion P(x) holds for λ-a.e. x if for E={x:¬P(x)}, μ{u:λu(E)>0}=0. Clearly an assertion holds λ-almost everywhere if and only if it holds ν-almost everywhere.Lemma 3.3.
SupposeDu:Gu→ℝ+ with  Du(x)=dλu(x)/dλu(x)(x∈G). Then  D=Du on  Gu(a.e.).Proof.
SupposeE⊆Gu. We have
(3.4)ν(E)=∫G0λu(E)dμ(u)=∫G0∫Edλu(x)dμ(u)=∫G0∫EDu(x)dλu(x)dμ(u).
Also from dν=Ddν-1 we have
(3.5)ν(E)=∫ED(x)dν-1(x)=∫ED(x)∫G0dλu(x)dμ(u)=∫G0∫ED(x)dλu(x)dμ(u).
Thus
(3.6)∫G0∫E(Du(x)-D(x))dλu(x)dμ(u)=0.
Now, let Eu={x∈Gu:Du(x)>D(x)}. If λu(Eu)>0 then
(3.7)∫Eu(Du(x)-D(x))dλu(x)>0.
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)=f̅(x-1)D(x-1), is an isometric involution on  L1(G,λ,μ).Note that from [5, page  9], we have (3.8)‖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.
Hence the space of L1(G,λ,μ) is in general bigger than I(G,ν,μ) and IIμ(G,ν,μ) with respect to I-norm and II-norm, indeed I(G,ν,μ)⊆IIμ(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 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 approximate identity.For eachf∈L1(G,λ,μ) define (3.9)Lxf(y)=f(x-1y),Rxf(y)=f(yx),
when the multiplications on the right hand sides are defined. It is easy to check that the maps Lx,Rx are homomorphisms.Proposition 3.5.
LetI 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 sincef*g=∫Gr(y)f(y)Lygdν(y),
(3.10)Lx(f*g)=∫Gr(y)f(y)LxLygdν(y)=∫Gr(y)f(y)Lxygdν(y)=∫Gr(y)f(x-1y)Lygdν(y)=∫Gr(y)Lxf(y)Lygdν(y)=(Lxf)*g.
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
(3.11)Lx(en)*f=Lx(en*f)⟶Lxf.
Conversely, if I is closed under left translation and f∈L1(G,λ,μ) and g∈I,
(3.12)f*g=∫Gr(y)f(y)Lygdν(y)
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 *-Algebra

LetM(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 mapL1(G,λ,μ)↪M(G) with  νf(E)=∫EfχEdν, for  (E⊆G);  f↦νf is an isometric embedding.Proof.
Iff∈L1(G,λ,μ), then f is λ-measurable so the integral exists, and it is easy to check that νf is a measure on G. We show that νf is Radon. If f=u+iv, then νf=νu+iνv, so νf is Radon if and only if νu and νv are Radon. Since G is LCH and second countable, we have νu(K)=∫Kudν≤∫G|u|dν=∥u∥1<∞, for each compact set K, thus νu is Radon. Similarly νv is Radon, and so is νf.
By definition,∥νf∥=sup{∑1n|νf(Ei)|:n∈N,G=∐1nEi}, so for each ϵ>0, there exists a partition {Ei}1n of G such that
(4.1)‖νf‖-ϵ<∑1n|νf(Ei)|=∑1n|∫GfχEidν|≤∫G(|f|∑1nχEi)dν=‖f‖1.
Thus, ∥νf∥≤∥f∥1. Conversely, suppose f≥0, then νf≥0 and for every partition {Ei}1n of G we have,
(4.2)‖νf‖≥∑1nνf(Ei)=∑1n∫GfχEidν=∫Gfdν=‖f‖1.
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 of  M(G).Lemma 4.3.
Iff,g∈L1(G,λ,μ), then  ν(f*g)=νf*νg.Proof.
For each compact setK, we have
(4.3)νf*νg(K)=∫GχK(x)d(νf*νg)(x)=∫G∫Gr(x)χK(yx)dνf(y)dνg(x)=∫Gf(y)∫Gs(y)χK(yx)dνg(x)dν(y)=∫G∫Gs(y)f(y)g(y-1x)χK(x)dν(y)dν(x)=∫GχK(x)∫Gr(x)f(y)g(y-1x)dν(y)dν(x)=∫G(f*g)(x)χK(x)dν(x)=νf*g(K).
Since νf*g and νf*νg are regular measures, the equality holds for each open set and then for each measurable set.Iff∈L1(G,λ,μ) and η∈M(G), we will define η*f such that νη*f=η*νf. Suppose φ∈C0(G), we put (4.4)η*f(φ)=∫Gφ(x)dνη*f(x)=∫Gφ(x)(η*f)(x)dν(x).
On the other hand, (4.5)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)=∫Gφ(x)∫Gs(y)f(y-1x)dη(y)dν(x).Comparing these equalities,(4.6)(η*f)(x)=∫Gr(x)f(y-1x)dη(y).
If f∈L1(G,λ,μ), then it is easy to check that (4.7)∫GuRyf(x)dλu(x)=∫Guf(x)dλu(xy-1)=D(y-1)∫Guf(x)dλu(x).
Thus (4.8)∫GRyf(x)dν(x)=D(y-1)∫Gf(x)dν(x).
Similarly, we want to define f*η in such a way that the equality ν(f*η)=νf*η holds. Again suppose φ∈C0(G). We have (4.9)ν(f*η)(φ)=∫Gφ(x)dνf*η(x)=∫Gφ(x)(f*η)(x)dν(x)=∫G0∫Gφ(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).
On the other hand, (4.10)(νf*η)(φ)=∫G∫Gr(y)φ(xy)dνf(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).
Comparing the above equalities, we have (4.11)(f*η)(x)=∫Gf(xy-1)D(y-1)dη(y).Lemma 4.4.
L1(G,λ,μ) is a two-sided ideal of  M(G).Proof.
Supposef∈L1(G,λ,μ) and η∈M(G). Then we have
(4.12)‖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<∞.
Thus η*f∈L1(G,λ,μ). Also
(4.13)‖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‖<∞.
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 onG defined by ν=∫λudμ(u) is a Radon measure. Let ν0 be the measure on G given by (5.1)dν0=D-1/2dν,
where D is modular function of μ. Then for each Borel subset E of G, we have (5.2)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).
Hence, ν0 is symmetric under inversion.Definition 5.1.
A representation of the locally compact groupoidG 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 on Hu 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 anyξ,η∈L2(G0,{Hu}u,μ), the map(5.3)x⟼〈π(x)ξ(s(x)),η(r(x))〉
is ν-measurable on G.Definition 5.2.
A representationΠ of L1(G,λ,μ) on a Hilbert space, H is a *-homomorphism
(5.4)Π:L1(G,λ,μ)⟶B(H).
It is called nondegenerate if 〈Π(f)ξ:f∈L1,ξ∈H〉 is dense in H.Continuity ofΠ 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 ofG 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 Cc(G), when H is separable, G is second countable and admits sufficiently many nonsingular G-sets [2, Theorem  1.21].Proposition 5.3.
LetG be a second countable locally compact groupoid with Haar system and with sufficiently many nonsingular Borel  G-sets. Then, every representation of  Cc(G) on a separable Hilbert space is the integrated form of a representation of  G.These assumptions satisfied in the case ofr-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,{Hu}u,μ) is a representation of  G. For  ξ,η∈L2(G0,{Hu}u,μ) and  f∈Cc(G),
(5.5)〈Π(f)ξ,η〉=∫Gf(x)〈π(x)ξ(s(x)),η(r(x))〉dν0(x)
defines a bounded nondegenerate *-representation of Cc(G) on L2(G0,{Hu}u,μ) such that two equivalent representations of G give two equivalent representations of Cc(G).The equation above is called the integrated form of a unitary representation. IfΠ is a representation of Cc(G), the above proposition says that Π should be of the form (5.6)〈Π(f)ξ,η〉=∫Gf(x)〈π(x)ξ(s(x)),η(r(x))〉dν0(x).
For each ξ,η∈L2(G0,{Hu}u,μ) and f∈Cc(G).Next, we define a representation ofI(G,λ,μ), denoted by ΠI as (5.7)〈ΠI(f)ξ,η〉=∫Gf(x)〈π(x)ξ(s(x)),η(r(x))〉dν0(x)
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(5.8)〈ΠI(f)ξ,η〉=∫Gf(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).
Thus, we may define for μ-a.e. u∈G0,
(5.9)ΠI(f)ξ(u)=∫Guf(x)π(x)ξ(s(x))D-1/2(x)dλu(x).
Now, we have
(5.10)|〈ΠI(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)|dλu(x)‖ξ(u)‖2dμ(u)]1/2×[∫G0∫Gu|f(x)|dλu(x)‖η(u)‖2dμ(u)]1/2≤‖f‖I,s1/2‖ξ‖‖f‖I,r1/2‖η‖≤‖f‖I‖ξ‖‖η‖.
Therefore, the map x↦f(x)〈π(x)ξ(s(x)),η(r(x))〉 is ν0-integrable, and ΠI(f) is a bounded operator of norm ∥ΠI(f)∥≤∥f∥I. We have to check that ΠI is a *-homomorphism. For this, we define
(5.11)F(x,t)=f(xt)g(t-1)〈π(x)ξ(s(x)),η(r(x))〉d1/2(x).
Then
(5.12)〈ΠI(f*g)ξ,η〉=∫Gf*g(x)〈π(x)ξ(s(x)),η(r(x))〉dν0(x)=∫G(∫Gf(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∫Guf(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)=∫Gf(x)〈π(x)Π(g)ξ(s(x)),η(r(x))〉dν0(x)=〈ΠI(f)ΠI(g)ξ,η〉.
Hence, ΠI(f*g)=ΠI(f)ΠI(g). Also
(5.13)〈ΠI(f*)ξ,η〉=∫Gf*(x)〈π(x)ξ(s(x)),η(r(x))〉dν0(x)=∫Gf(x-1)¯D(x-1)〈π(x)ξ(s(x)),η(r(x))〉d-1/2(x)dν(x)=∫Gf(x)¯D(x-1)〈π*(x)ξ(r(x)),η(s(x))〉d1/2(x)D(x-1)dν(x)=∫Gf(x)〈η(s(x)),π*(x)ξ(r(x))〉¯dν0(x)=∫Gf(x)〈π(x)η(s(x)),ξ(r(x))〉¯dν0(x)=〈ΠI(f)ξ,η〉¯=〈ξ,ΠI(f)η〉=〈ΠI*(f)ξ,η〉.
The nondegeneracy follows from
(5.14)〈ΠI(f)ξ∣f∈Cc(G),ξ∈H〉⊆〈ΠI(f)ξ∣f∈L1(G,‖⋅‖I),ξ∈H〉⊆H.Now let us turn to the problem that a continuous representation ofI(G,λ,μ) is not necessarily continuous in L1-norm. Let H=L2(G0,{Hu}u,μ) and put (5.15)HΠ1=〈ξ∈H:Themapf⟼〈ΠI(f)ξ,ξ〉iscontinuousinL1-normhemapf⟼〈ΠI(f)ξ,ξ〉iscontinuousinL1〉̅.
Observe that HΠ1 is a nontrivial subspace, as μ is a probability measure and if ∥ξ(u)∥=1 in Hu for each u∈G0, then ∥ξ∥=1 in H and the calculations in the proof of Lemma 5.5 shows that |ΠI(f)ξ,ξ〉|≤∥f∥1, hence ξ∈H1. On the other hand, ΠI(f)HΠ1⊆HΠ1 for each f∈I(G,λ,μ), because the map (5.16)f⟼〈ΠI(f)ΠI(g)ξ,ΠI(g)ξ〉=〈ΠI(g**f*g)ξ,ξ〉
is continuous. Therefore, ΠI(f)HΠ1⊥⊆HΠ1⊥. Hence, we have ΠI(f)=ΠI(f)|HΠ1⊕ΠI(f)|HΠ1⊥ for each f∈I(G,λ,μ).PutH1=HΠ1 and define Π1(f):=ΠI(f)|H1, for f∈I(G,λ,μ). Then, it follows from continuity of f↦〈ΠI(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 ifπ 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,{Hu}u∈G0,μ) be a Hilbert bundle. A family M={Mu}u∈G0, where Mu is a closed subspace of Hu for each u∈G0, is called a subbundle. A subbundle {Mu}u∈G0 is called nontrivial if 0≠Mu≠Hu for some u∈G0. For a representation π of a locally compact groupoid G associated with the Hilbert bundle (G0,{Hu}u∈G0,μ), a subbundle {Mu}u∈G0 is called invariant if π(x)Ms(x)⊆Mr(x) for each x∈G. Note that if M 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].Lemma 5.7.
Letπ be a representation of a locally compact groupoid  G associated with the Hilbert bundle  (G0,{Hu}u∈G0,μ). If  M={Mu}u∈G0 is an invariant subbundle, then so is  M⊥={Mu⊥}u∈G0.Definition 5.8.
A representationπ of a locally compact groupoid G is called reducible if π admits a nontrivial invariant subbundle M={Mu}u∈G0, otherwise π is called irreducible. In this case, it is easy to check that πM with πM(x)=π(x)|Ms(x):Ms(x)→Mr(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,{Hu}u∈G0,μ) and (G0,{Hu′}u∈G0,μ′), respectively. Then, we put
(5.17)C(π,π′)={(Tu)u∈∏u∈G0B(Hu,Hu′):π′(x)Ts(x)=Tr(x)π(x)(x∈G)}
and write 𝒞(π,π)=𝒞(π).Two representationsπ 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 Tu* denotes the adjoint operator to Tu, then (5.18)π(x)Ts(x)*=[Ts(x)π(x-1)]*=[π(x-1)Tr(x)]*=Tr(x)*π(x),
hence, (Tu*)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,{Hu}u∈G0,μ), we set (5.19)Λ={(λuπ(u))u∈∏u∈G0B(Hu):λu∈C,λu=λwwheneveru~w}.
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 that Λ⊆𝒞(π).We need the following version of Schur's lemma [6, Lemma  3.11].Lemma 5.9.
A representationπ 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.
Ifℳ=(G0,{Mu}u,μ) is closed subbundle of  ℋ=(G0,{Hu}u,μ) and  Pu is an orthogonal projection  Hu onto  Mu, then  ℳ is invariant under  π if and only if  (Pu)u∈𝒞(π).Proof.
If(Pu)u∈𝒞(π) and mu∈Mu, then
(5.20)π(x)ms(x)=π(x)Ps(x)ms(x)=Pr(x)π(x)ms(x)∈Mr(x).
Thus, π(x)Ms(x)⊆Mr(x) so ℳ is invariant. Conversely, if ℳ is invariant, then for m∈Ms(x)(5.21)π(x)Ps(x)m=π(x)m=Pr(x)π(x)m
and for m∈Ms(x)⊥(5.22)π(x)Ps(x)m=0=Pr(x)π(x)m.
Hence, π(x)Ps(x)=Pr(x)π(x).We show that if a representationπ of a locally compact groupoid G associated with the Hilbert bundle (G0,{Hu}u∈G0,μ) is irreducible and Π1 is the corresponding integrated representation 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 subspace of H with orthogonal complement M⊥=L2(G0,{Mu⊥}u,μ). A map Θ:L2(G0,{Hu}u,μ)→L2(G0,{Mu}u,μ);ξ↦Θξ with Θξ(s(x))∈Mr(x) is called an orthogonal projection of H onto M.Letπ be a representation of G on the Hilbert bundle (G0,{Hu}u∈G0,μ) and ΠI be the bounded 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.
SupposeΠI(f)Θ=ΘΠI(f) and ξ∈M, then
(5.23)ΠI(f)ξ=ΠI(f)Θξ=ΘΠI(f)ξ∈M.
Hence, M is invariant. Conversely, if M is invariant, then for ξ∈M,
(5.24)ΠI(f)Θξ=ΠI(f)ξ=ΘΠI(f)ξ
and for η∈M⊥,
(5.25)ΠI(f)Θη=0=ΘΠI(f)η.
Thus, ΠI(f)Θ=ΘΠI(f).Proposition 5.12.
Ifπ is an irreducible representation of  G, then  ΠI is an irreducible representation of  I(G,λ,μ).Proof.
IfΠI is reducible, then there exists a nontrivial invariant closed subspace M of L2(G0,{Hu}u,μ), and, hence, there is an orthogonal projection Θ∈𝒞(ΠI). It follows that
(5.26)∫Gf(x)〈π(x)Ps(x)ξ(s(x)),η(r(x))〉dν0(x)=〈ΠI(f)Θξ,η〉=〈ΘΠI(f)ξ,η〉=∫Gf(x)〈Pr(x)π(x)ξ(s(x)),η(r(x))〉dν0(x),
for each f∈Cc(G) and ξ,η∈L2(G0,{Hu}u,μ). Therefore, π(x)Ps(x)=Pr(x)π(x), for each x∈G, hence (Pu)u∈𝒞(π). Now by the Schur's lemma, π is reducible.Theorem 5.13.
Ifπ is an irreducible representation of  G, then  Π1 is an irreducible representation of  L1(G,λ,μ) on  H1.Proof.
IfΠ1 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 Π̃2 are the corresponding representations of I(G,λ,μ) on H1 and H1⊥. Therefore, M=M1⊕H1⊥ is a nontrivial invariant closed subspace H, and ΠI is reducible. Hence, π is reducible by the above proposition.



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*Source: 856709-2011-06-29.xml*