Dedicated to Professor Ion Colojoară on his 80th birthday
CONVOLUTION ALGEBRAS
FOR TOPOLOGICAL GROUPOIDS
WITH LOCALLY COMPACT FIBRES
Mădălina Roxana Buneci
Abstract. The aim of this paper is to introduce various convolution algebras associated with a topological groupoid with locally compact fibres. Instead of working with continuous functions onG, we consider functions having a uniformly continuity property on fibres. We assume that the groupoid is endowed with a system of measures (supported on its fibres) subject to the “left invariance” condition in the groupoid sense.
Keywords: convolution algebra, groupoid, uniform continuity, Haar system. Mathematics Subject Classification: 22A22, 43A10.
1. INTRODUCTION
A groupoid is the natural extension of a group. The shortest way to define a groupoid is to say it means a small category where each morphism is invertible. If a groupoid has only one object, then it is a group. Much of the rich theory associated with locally compact second countable groups can be expressed in terms of algebra and measure theory. In fact, ifGis an analytic Borel group andµa quasi-invariant Borel measure onG, then there is a topology onGrelative to whichGis a locally compact group. This topology generates the given Borel structure and is uniquely determined by this fact. Moreover,µis equivalent to Haar measure onG[3].
Several generalizations of the Haar measure to the setting of groupoids were taken into considerations in the literature (see [2, 6, 8, 10, 14]). For instance, starting from the case of a locally compact second countable group Gacting on a standard Borel space X, Mackey [4] took into consideration the analytic Borel groupoids that can be endowed with a measure class C (a generalization of the class obtained in the group action case by a product of quasi-invariant measuresµonX with measures in the Haar measure class on G) such that each measure in the class C has a decom-position satisfying a kind of quasi-invariance condition. Hahn proved in [6] that the
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measure class C of a general measure groupoid contains aσ-finite measure ν which is translate-invariant in the groupoid sense. The measure v, or more precisely, the measures in the decomposition of ν with respect to the range map, share many of the properties of the Haar measure for groups, to which it reduces if the groupoid is a group. According to a result of Ramsay, Mackey’s groupoids may be assumed to have locally compact topologies. More precisely, a Mackey’s groupoid Ghas an inessential reduction G0 which has a locally compact metric topology in which it
is a topological groupoid [7]. On the other hand, there are situations in which it is not natural to fix a certain measure class C on a groupoid. In this case, in the setting of locally compact groupoids, the analogue of the Haar measure associated with a locally compact group is a system of measures, called a Haar system, subject to suitable invariance and smoothness conditions called respectively “left invariance” and “continuity”. The construction of theC∗-algebra of a locally compact Hausdorff
groupoid G endowed with a Haar system (due to Renault [8]) extends the case of a group. The space Cc(G) of continuous functions with compact support is made
into a∗-algebra and endowed with the smallestC∗-norm making its representations
continuous. The multiplication inCc(G)is a convolution and the convolution of two
functions is defined using the Haar system. The C∗-algebra of Gis the completion
ofCc(G).
However unlike the case for locally compact groups:
1. Haar systems on groupoids need not exist. Also, when a Haar system does exist, it need not be unique. The continuity assumption has topological consequences forG(it entails that the range map is open [13]). In the same time, according to a result of Seda [11], the continuity assumption is essential for construction of the C∗-algebra ofGin the sense of Renault [8].
2. The result of Ramsay [7] is only a partial generalization to groupoids of the Mackey result [3].
3. IfGis a topological group such that the singleton sets are closed, thenGis Haus-dorff. However for general topological groupoids G this is no longer true. If G is a locally compact groupoid, not necessarily Hausdorff, then as pointed out by A. Connes [2], one has to modify the choice of Cc(G) (because Cc(G) it is too
small to capture the topological or differential structure ofG). In this caseCc(G)
is replaced with Cc(G), the space of complex valued functions on G spanned by
functionsf which vanishes outside a compact setKcontained in an open Hausdorff subsetU ofGand being continuous onU. Since in a non-Hausdorff space a compact set may not be closed, the functions inCc(G)are not necessarily continuous onG.
According [5], if(Ui)i is a covering ofGby open Hausdorff subsets, then the
func-tions inCc(G)are finite sumsPifi, wherefiis a continuous compactly supported
function onUi. In the Hausdorff caseCc(G)andCc(G)coincide.
continuity property on fibres and instead of the family of compact subsets, we will work with a family of subsets having similar properties as conditionally compact subsets in the sense of [9]. We will endow these groupoids with pre-Haar systems meaning systems of measures subject to the “left invariance” condition (which do not necessarily satisfy the “continuity” property). The aim of the paper is to intro-duce various convolution algebras in this framework. In the case when G is locally compact andG can be endowed with a Haar system, the usual convolution algebra Cc(G)(respectively,Cc(G)in the non-Hausdorff case) can be recovered as a particular
∗-subalgebra of the∗-algebras constructed here.
2. SPACES OF FUNCTIONS ON GROUPOIDS WITH LOCALLY COMPACT FIBRES
Agroupoid is a setGendowed with aproduct map
(x, y)7→xy h:G(2)→Gi,
where G(2) is a subset of G×G called the set of composable pairs, and an inverse
map
x7→x−1 [:G→G]
such that the following conditions hold:
(1) If(x, y)∈G(2) and(y, z)∈G(2), then(xy, z)∈G(2),(x, yz)∈G(2) and(xy)z=
x(yz).
(2) x−1−1=xfor allx∈G.
(3) For allx∈G, x, x−1
∈G(2), and if(z, x)∈G(2), then(zx)x−1=z.
(4) For allx∈G, x−1, x
∈G(2), and if(x, y)∈G(2), thenx−1(xy) =y.
The mapsr anddonG, defined by the formulaer(x) =xx−1 andd(x) =x−1x,
are called the range map, respectively the source (domain) map. It follows easily from the definition that they have a common image called theunit space ofG, which is denoted G(0). Its elements areunits in the sense thatxd(x) =r(x)x=x.
The fibres of the range and the source maps are denoted Gu = r−1({u}) and
Gv=d−1({v}), respectively. Also foru, v∈G(0),Guv =Gu∩Gv.
IfA andB are subsets ofG, one may form the following subsets ofG:
A−1=
x∈G: x−1∈A
AB=nxy: (x, y)∈G(2)∩(A×B)o.
A topological groupoid consists of a groupoid G and a topology compatible with the groupoid structure. This means that the inverse map x 7→ x−1 [:G→G] is
continuous, as well as the product map (x, y) 7→ xy
:G(2)→G
In the following a topological spaceX is said to belocally compact if every point x∈X has a compact Hausdorff neighborhood. IfX,Y andZare sets andp:X→Z andq:Y →Z are maps, then the fibred product they determine, denotedXp×qY,
is defined to be
Xp×qY ={(x, y)∈X×Y : p(x) =q(y)}.
It is well known that any continuous function with compact support defined on a topological group is uniformly continuous. Let us show that in a suitable sense (intro-duced in [1]) the property of topological groups generalizes to topological groupoids.
Definition 2.1 ([1, Definition 3.1, p. 39]). Let Gbe a topological groupoid andE be a Banach space. The functionh:G→E is said to beleft “uniformly continuous on fibres” if and only if for each ε >0 there is a neighborhoodW of the unit space G(0) such that:
kh(x)−h(y)k< ε for all (x, y)∈Gr×rG, x−1y∈W.
The functionh:G→Eis said to beright “uniformly continuous on fibres”if and only if for eachε >0there is a neighborhoodW of the unit space such that:
kh(x)−h(y)k< ε for all (x, y)∈Gd×dG, yx−1∈W.
It is easy to see that his left “uniformly continuous on fibres” if and only if for eachε >0there is a neighborhoodW of the unit space such that:
kh(x)−h(xs)k< ε for alls∈W andx∈Gr(s)
and thathis right “uniformly continuous on fibres” if and only if for eachε >0there is a neighborhoodW of the unit space such that:
kh(x)−h(sx)k< ε for all s∈W andx∈Gd(s).
Lemma 2.2. Let X,Y, Z and S be four topological spaces and let p: X →Z and
q : Y → Z be two continuous maps. LetK be a subset of Y with the property that
K∩q−1({z}) is a compact set for all z ∈Z and p−1(q(K\V)) is a closed set for
every open neighborhood V of q−1({z})∩K and for every z ∈Z. IfW is an open subset of S andf :Xp×q K→S is a continuous function, where Xp×q K has the induced topology from X×K, then
L=
x∈X:f(x, y)∈W for all y∈K∩q−1({p(x)})
is an open subset ofX.
Proof. Let x0 ∈ L and y ∈ K with q(y) = p(x0). Since f(x0, y) ∈ W and f is
continuous in (x0, y), it follows that there are two open sets Uy and Vy such that
(x0, y)∈Uy×Vy andf((Uy×Vy)∩(Xp×qK))⊂W.
Since{Vy}y∈K∩q−1({p(x0)})is an open covering of the compact setK∩q−1({p(x0)}),
it follows that there arey1, y2, . . . , yn∈Ksuch thatSi=1,nVyi⊃K∩q
Let us denote U = T
i=1,nUyi and V = S
i=1,nVyi. Then U is an open
neigh-borhood of x0 and V is an open neighborhood of q−1({p(x0)}) ∩ K.
Fur-thermore U0 = U ∩ X\p−1(q(K\V)) is an open neighborhood of x0 and
f((U0×K)∩(Xp×q(K)))⊂W. Consequently,x0∈U0⊂L.
Proposition 2.3. Let Gbe a topological groupoid and E be a Banach space. Let W
andW0be two symmetric open neighborhoods ofG(0) and letK be a subset ofGwith
the property that there is aK0 such that:
1. KW ⊂K0, respectivelyW K ⊂K0.
2. For allu∈G(0),K
0∩Gu, respectively K0∩Gu, are compact sets for allu∈G(0).
3. For all u ∈ G(0), r(K
0\V), respectively d(K0\V), are closed sets for all open
neighborhoodsV of Gu∩K0, respectivelyGu∩K0.
Then any functionh:G→E which vanishes outsideK and is continuous onK0W0,
respectivelyW0K0, is left, respectively right, “uniformly continuous on fibres”.
Proof. If we set
f :G(2)→E, f(s, y) =h(ys)−h(y),
then f is a continuous function in every point of (W0r×dK0). Let ε > 0. By
Lemma 2.2,
L=
s∈W0:|h(ys)−h(y)|< εfor all y∈K0∩Gr(s)
is an open subset ofW0, and consequently, ofG. Let us note thatG(0)⊂L. Ifs∈L
andy∈Gr(s), theny∈K∩Gr(s)and|h(ys)−h(y)|< ε, ory∈Gr(s)\ K∩Gr(s)
and in this case y, ys /∈K, therefore|h(sy)−h(y)|= 0. Hencehis left “uniformly continuous on fibres”.
Similarly, his right “uniformly continuous on fibres”.
Definition 2.4. LetGbe a topological groupoid. By afamily of “abstractly compact” subsets ofGwe mean a familyK of subsets ofGsatisfying the following conditions:
1. For everyK∈ K,K−1∈ K.
2. For everyK1, K2∈ K, there isK3∈ Ksuch thatK1K2⊂K3.
3. For everyK1, K2∈ K,K1∪K2∈ K.
4. For everyu∈G(0) and everyK∈ K,K∩Gu andK∩G
u are compact.
AnyK∈ Kwill be called a“abstractly compact” subset ofG.
Definition 2.5. LetGbe a topological groupoid such that:
1. The points are closed inG(or equivalently,Gis aT1-space).
2. For everyu∈G(0),Gu andG
uare Hausdorff.
Remark 2.6. LetGbe a topological groupoid with the property that the points are closed inG. IfG(0) is Hausdorff, then for everyu∈G(0), Guand G
u are Hausdorff.
As in [12], the set
Z ={(x, y)∈Gu×Gu: d(x) =d(y)}
is closed inGu×Gu, being the set where two continuous maps (to a Hausdorff space)
coincide. Letϕ: Z →Gdefined by ϕ(x, y) =xy−1 for all(x, y)∈Z. Since{u} is
closed in G, ϕ−1({u})is closed in Z. Furthermore, Z being closed in G, it follows
thatϕ−1({u}), which is the diagonal ofGu×Gu, is closed inG.
If Gis a “locally abstractly compact” groupoid with respect toK, then for every u ∈ G(0), Gu and G
u are locally compact Hausdorff subspaces of G. Indeed, each
point xin Gu (respectively, in G
u) has a neighborhoodV ∈ K in G. Then V ∩Gu
(respectively,V ∩Gu) is a compact neighborhood ofxinGu (respectively, inGu).
Lemma 2.7. Let Gbe a topological groupoid,K a subset ofGandF a closed subset ofG. IfF K andKF are closed inG, thenr K∩F−1
andd K∩F−1
are closed subsets ofG(0).
Proof. It suffices to note that
r K∩F−1
=KF ∩G(0) andd K∩F−1
=F K∩G(0)
and thereforer K∩F−1
andd K∩F−1
are closed subsets ofG(0).
Proposition 2.8. Let Gbe a topological groupoid such that:
1. The points are closed in G(or equivalently,Gis aT1-space).
2. For every u∈G(0),Gu andGu are Hausdorff.
3. Each point has a neighborhood basis of sets K with the properties:
(i) for every closed subsetF of G,F K andKF are closed inG,
(ii) for everyu∈G(0),K∩Gu andK∩G
u are compact.
If
K={K: for every closed setF ⊂G,F K andKF are closed and
for every u∈G(0),K∩Gu and K∩G
u are compact}, thenGis a “locally abstractly compact” groupoid with respect to K.
Proof. We have to prove that forK1, K2∈ K, there isK3∈ Ksuch thatK1K2⊂K3.
Let us note that for every open setV andK∈ K,
r(K\V) =KF−1∩G(0) andd(K\V) =F−1K∩G(0)
withF =G\V a closed set. Thusr(K\V)andd(K\V)are closed subsets ofG(0).
Letu∈G(0) andx∈Gu. For everyy∈Gd(x)there is a neighborhoodK
y∈ Kof
xyand there are two open setsUyandVysuch that(x, y)∈Uy×Vyandxy∈UyVy⊂Ky.
Since {Vy}y∈K2∩r−1({d(x)}) is an open covering of the compact set K2∩G
d(x), it
follows that there are y1, y2, . . . , yn ∈ K2 such that Si=1,nVyi ⊃ K2∩G
d(x). Let
us denote Ux =Ti=1,nUyi, Vx = S
i=1,nVyi and Kx = S
open neighborhood of x, Vx is an open neighborhood of Gd(x)∩K2 and Kx ⊂ K.
Furthermore Lx =Ux∩ G\d−1(r(K2\Vx)) is an open neighborhood of x and
LxK2⊂Kx. Since{Lx}x∈K1∩r−1({u})is an open covering of the compact setK1∩G
u,
it follows that there arex1, x2, . . . , xm∈K1such thatSi=1,mLxi ⊃K1∩G
u. Let us
notice thatKu=Si=1,mKxi ∈ K. Hence
Gu∩K1K2= (Gu∩K1)K2⊂ [
i=1,m
LxiK2⊂ [
i=1,m
Kxi
is relatively compact in Gu. Since Gu is closed GuK1 is closed and furthermore (GuK
1)K2 is closed. ThusGu∩K1K2= (Gu∩K1)K2 is closed and therefore
com-pact. Similarly, Gu∩K1K2is compact.
Let F be a closed subset of G. Since K2F is closed, if follows that K1K2F =
K1(K2F)is closed. Similarly,F(K1K2) = (F K1)K2 is closed.
Notation 2.9. LetGbe a “locally abstractly compact” groupoid with respect toK (Definition 2.5). Let us denote by:
1. UF(G)the space of complex valued functions onGwhich are left and right “uni-formly continuous on fibres”.
2. UFK(G)the subspace ofUF(G)consisting of functionsf :G→Cwhich vanish
outside an abstractly compact setK∈ K.
3. UFb(G)⊂ UF(G)the subspace of bounded functions.
4. UFKb(G)⊂ UFK(G)the subspace of bounded functions.
Remark 2.10. 1. Iff ∈ UF(G), then for allu∈G(0),f|
Guandf|G
uare continuous.
Indeed, letube a unit,x0 ∈Gu andε >0. Sincef is left “uniformly continuous
on fibres”, there is a neighborhood WεofG(0) such that:
|f(x)−f(y)|< ε for all (x, y)∈Gr×rG, x−1y∈Wε.
Since y 7→ x0y is a homeomorphism from Gd(x0) to Gu, it follows that x0Wε =
x0 Wε∩Gd(x0)is an open subset of Gu containingx0. But if y ∈ x0Wε, then
x−01y ∈ Wε and therefore |f(x0)−f(y)| < ε. Thus f|Gu is continuous in x0.
Similarly, f|Gu is continuous.
2. If f ∈ UFK(G), then for all u∈ G(0), f|Gu and f|G
u are compactly supported
continuous functions (because f :G→Cvanishes outside an abstractly compact
setK∈ KandK∩Gu, respectivelyK∩G
u are compact).
3. Iff ∈ UF(G), respectivelyUFb(G),UFK(G),UFKb(G),UFI(G), then|f|, f ∈
UF(G), respectively UFb(G), UFK(G),UFKb(G), UFI(G). If f, g ∈ UFb(G),
respectively UFKb(G), thenf g∈ UFb(G), respectivelyUFKb(G).
Examples. A topological groupoid G is said to be locally compact if it is locally compact as a topological space (this means that every point x∈ G has a compact Hausdorff neighborhood). Thus any locally compact groupoidG(in the above sense) is locally Hausdorff, and therefore a T1-space. Aconditionally compact subset of a
topological groupoid Gis a subset K such that for every compact subset Lof G(0),
1. Let G be a locally compact groupoid having a Hausdorff unit space. Then G endowed with the family of compact subsets and satisfies the conditions of the Definition 2.5.
2. IfGis a locally conditionally compact groupoid in the sense of [9], thenGendowed with the family of conditionally compact subsets (in the sense of [9]) is a “locally abstractly compact” groupoid in the sense of Definition 2.5.
3. CONVOLUTION ALGEBRAS FOR GROUPOIDS WITH LOCALLY COMPACT FIBRES
The analogue notion of the Haar measure on locally compact groups that we consider in this section is a system of measures, called pre-Haar system, subject to a suitable left invariance condition.
Definition 3.1. Let G be a “locally abstractly compact” groupoid with respect to K (Definition 2.5). A left pre-Haar system on G is a family of positive measures,
νu, u∈ G(0) , with the following properties:
(1) is a positive Radon measure onGufor allu∈G(0),
(2) R
f(y)dνr(x)(y) = R
f(xy)dνd(x)(y)for allx∈ Gand allf ∈ UF K(G).
Definition 3.2. LetGbe a “locally abstractly compact” groupoid with respect toK (Definition 2.5). The pre-Haar system
νu, u∈ G(0) onGis said to bebounded on
abstractly compact sets if
supnνu(K), u∈ G(0)o < ∞
for each K∈ K.
Notation 3.3. LetGbe a “locally abstractly compact” groupoid with respect toK (Definition 2.5) and
νu, u∈ G(0) be a pre-Haar system onG. For eachf ∈ UF(G),
let us denote by
kfkI,r= sup Z
|f(y)|dνu(y), u∈ G(0)
,
kfkI,d= sup Z
f y−1
dνu(y), u∈ G(0)
,
kfkI = maxnkfkI,r, kfkI,do
and let us considerUFI(G) ={f ∈ UF(G) : kfkI <∞}.
Forf ∈ UF(G), theinvolution is defined by
f∗(x) =f(x−1), x∈G.
Obviously, if f ∈ UF(G), respectively UFb(G), UFK(G), UFKb(G),UFI(G),
thenf∗∈ UF(G), respectivelyUF
Forf,g∈ UFI(G)theconvolution is defined by:
f∗g(x) = Z
f(xy)g y−1
dνd(x)(y) =
= Z
f(y)g y−1x
dνr(x)(y), x∈G.
Theorem 3.4. Let G be a “locally abstractly compact” groupoid with respect to K (Definition 2.5)and
νu, u∈ G(0) be a pre-Haar system onG. ThenUF
I(G)and
UFK(G) are ∗-algebras. If νu, u∈ G(0) is bounded on abstractly compact sets,
then alsoUFKb(G)is closed under involution and convolution, and consequently it is a∗-algebra.
Proof. Let us prove if f, g∈ UFI(G), thenf ∗g ∈ UFI(G). Let ε >0. Since g is
left “uniformly continuous on fibres”, there is a neighborhoodWεofG(0) such that:
|g(x)−g(y)|< ε for all (x, y)∈Gr×rG, x−1y∈Wε.
Hence
|f∗g(x)−f∗g(y)|= Z
f(z)g z−1x
dνr(x)(z)−Z f(z)g z−1y
dνr(y)(z) ≤ ≤ Z
|f(z)| g z−1x
−g z−1y
dνr(y)(z)≤
≤εkfkI,r,
and thereforef∗gis left “uniformly continuous on fibres”. Sincef is right “uniformly continuous on fibres”, there is a neighborhoodWε ofG(0) such that:
|f(x)−f(y)|< ε for all (x, y)∈Gd×dG, xy−1∈Wε.
Hence
|f∗g(x)−f ∗g(y)|= Z
f(xz)g z−1
dνd(x)(z)−Z f(yz)g z−1
dνd(y)(z) ≤ ≤ Z
|f(xz)−f(yz)| g z−1
dνd(y)(z)≤
≤εkgkI,d,
and thereforef∗g is right “uniformly continuous on fibres”. Foru∈G(0) we have
Z
|f∗g(x)|dνu(x)≤
Z Z
f(y)g y−1x
dνr(x)(y)dνu(x) =
= Z Z
f(y)g y−1x
dνu(x)dνu(y) =
= Z Z
|f(y)g(x)|dνd(y)(x)dνu(y) =
= Z
|f(y)|
Z
and
Z
f∗g x−1
dνu(x)≤ Z Z
f x−1y
g y−1
dνr(x)(y)dνu(x) =
= Z Z
f x−1y
g y−1
dνu(x)dνu(y) =
= Z Z
f x−1
g y−1
dνd(y)(x)dνu(y) =
= Z
g y−1
Z
f x−1
dνd(y)(x)dνu(y).
Thus kf∗gkI,r ≤ kfkI,rkgkI,r and kf∗gkI,d ≤ kfkI,dkgkI,d. Therefore under the involution and convolution defined above,UFI(G)becomes a∗-algebra.
It is also easy to see that UFK(G) is closed under involution and convolution
(if f vanishes outside K1 and g vanishes outside K2, then f ∗g vanishes outside
K1K2⊂K3∈ K).
Theorem 3.5. Let G be a “locally abstractly compact” groupoid with respect to K (Definition 2.5) and
νu, u∈ G(0) be a pre-Haar system on G. Let C be a class of complex valued functions on G(0) closed to addition and scalar multiplication (for instance, the class of functions continuous, or the class of Borel functions, etc.). Let
ACK=
f ∈ UFK(G), u7→ Z
f(y)ϕ(d(y))dνu(y)∈ C for allϕ∈ C and
u7→
Z
f(y−1)ϕ(d(y))dνu(y)∈ C
for allϕ∈ C
,
ACI =
f ∈ UFI(G), u7→
Z
f(y)ϕ(d(y))dνu(y)∈ C for allϕ∈ C and
u7→
Z
f(y−1)ϕ(d(y))dνu(y)∈ C for allϕ∈ C
.
ThenAC
I andACK are ∗-algebras.
Proof. It suffices to prove that AC
K and ACI are closed to convolution. But this is
obvious, since forf, g∈ AK (respectively,ACI ) we have
Z
f∗g(x)ϕ(d(x))dνu(x) = Z Z
f(y)g y−1x
ϕ(d(x))dνr(x)(y)dνu(x) =
= Z Z
f(y)g y−1x
ϕ(d(x))dνu(x)dνu(y) =
= Z Z
f(y)g(x)ϕ(d(x))dνd(y)(x)dνu(y) =
= Z
f(y) Z
g(x)ϕ(d(x))dνd(y)(x)dνu(y) =
= Z
whereϕg(u) =Rg(x)ϕ(d(x))dνu(x), and
Z
f ∗g(x−1)ϕ(d(x))dνu(x) =Z (f ∗g)∗
(x)ϕ(d(x))dνu(x) =
= Z
g∗∗f∗(x)ϕ(d(x))dνu(x) =
= Z
g∗(y)ϕ
f∗(d(y))dνu(y).
Theorem 3.6. Let G be a “locally abstractly compact” groupoid with respect to K (Definition 2.5) and
νu, u∈ G(0) be a pre-Haar system on G. Let C
K(G)be the
space of complex valued functions onGspanned by functionsf with the property that there are two symmetric open neighborhoods W and W0 of G(0) and two sets K,
K0∈ Ksuch that:
1. KW∪W K⊂K0.
2. For allu∈G(0),r(K
0\V)andd(K0\V)are closed sets for all open setsV.
3. f vanishes outside K and is continuous on K0W0∪W0K0.
If we assume that for everyf,g∈ CK(G),f∗g∈ CK(G), thenCK(G)is a ∗-subalgebra
of UFK(G).
Proof. According to Proposition 2.3 each f ∈ CK(G) is left and right “uniformly
continuous on fibres”, thereforef ∈ UFK(G).
Remark 3.7. LetGbe a topological groupoid such that:
1. The points are closed inG(or equivalently,Gis aT1-space).
2. For everyu∈G(0),Gu andG
uare Hausdorff.
3. Each point has a neighborhood basis of sets K with the property that for every closed subsetF ofG,F K andKF are closed inGand for everyu∈G(0),K∩Gu
andK∩Gu are compact.
Let
K={K:for every closed setF ⊂G,F K andKF are closed and
for everyu∈G(0),K∩Gu andK∩Gu are compact}.
ThenGis a “locally abstractly compact” groupoid with respect toK(Proposition 2.8). In this case the space CK(G) from the above theorem is the space of complex
valued functions on Gspanned by functions f with the property that there are two symmetric open neighborhoods W and W0 of G(0) and two sets K, K0 ∈ K such
that:
1. KW∪W K⊂K0.
2. f vanishes outsideK and is continuous onK0W0∪W0K0.
In particular let us consider a locally compact topological groupoid Gwhose unit spaceG(0) is paracompact. LetK be the family of compact subsets of G. LetC
be the space of complex valued functions onGspanned by functionsf which vanishe outside a compact set K contained in an open Hausdorff subset U of Gand being continuous onU. Let
νu, u∈ G(0) be a left Haar system onG. This means that
νu, u∈ G(0) is a left pre-Haar system onGand the map
u7→
Z
f(x)dνu(x) h:G(0)→Ci
is continuous is for all f ∈ Cc(G). ThenCc(G)is a ∗-subalgebra ofUFK(G).
Examples.
Let us consider the following special cases in Theorem 3.5:
(i) C(τ) =
ϕ:G0→C,ϕcontinuous with respect toτ , where τ is a topology
possible weaker than the topology induced onG(0) fromG.
(ii) C(B) =
ϕ:G0→C,ϕmeasurable with respect toσ-algebra B - we can work
with a Borel structure weaker than the Borel structure generated by the open sets.
Let us consider now two classes of groupoids: the groups and the sets (as co-trivial groupoids):
1. Groups: If G is a locally compact Hausdorff group, then G (as a groupoid)
admits an essentially unique (left) pre-Haar system{λ}whereλis a Haar measure onG. LetCc(G)be space of complex valued continuous functions with compact
support onGand letU(G)be the space of complex valued left and right uniformly continuous functions on G. The convolution of f, g ∈ Cc(G) (G is seen as a
groupoid) is given by:
f∗g(x) = Z
f(xy)g y−1
dλ(y) (usual convolution onG)
and the involution by
f∗(x) =f(x−1).
The involution defined above is slightly different from the usual involution on groups. In fact if∆is the modular function of the Haar measureλ, thenf 7→∆1/2f
is ∗-isomorphism from Cc(G) forGseen as a groupoid toCc(G)for Gseen as a
group.
It is easy to see that K is a family of “abstractly compact” subsets (in the sense of Definition 2.4) such that each point of G has a neighborhood basis of sets K belonging toK if and only ifKis the family of compact subsets ofG. Therefore,
ACK(τ)=A C(B)
K =UFK(G) =Cc(G),
ACI(τ)=A
C(B)
I =UI(G),
where UI(G)is the space of complex valued left and right uniformly continuous
2. Sets: IfX is a topological space which is a T1-space, then X seen as a co-trivial
groupoid is a“locally abstractly compact” groupoid with respect toK if a) K is a family of subsets ofX closed at union.
b) each point of X has a neighborhood basis of setsK belonging toK.
The system of measures {α(x)εx, x∈X} is a pre-Haar system on X
(α(x)εx(f) =α(x)f(x)), whereαis a positive function on X. The convolution
off,g∈Cc(G)is given by:
f∗g(x) = Z
f(xy)g y−1
α(x)dεx(y) =
=f(xx)g x−1
α(x) = =f(x)g(x)α(x)
and the involution by
f∗(x) =f(x).
In this framework:
a) UF(X)is the space of all complex valued functions onX.
b) UFb(X)is the space of all complex valued bounded functions onX.
c) UFK(X)is the space of functions which vanish outside a setK∈ K.
d) UFKb(X)is the space of functions which vanish outside a setK ∈ Kand are
bounded onX.
e) UFI(X)is the space of functionsf such thatx7→f(x)α(x)is bounded onX.
f) ACK(τ) is the space of functions f which vanish outside a set K ∈ K and such thatx7→f(x)α(x)isτ-continuous.
g) ACK(B)is the space of functions f which vanish outside a setK ∈ K and such thatx7→f(x)α(x)is measurable with respect to the σ-algebraB.
h) ACI(τ) is the space of functionsf such that x7→f(x)α(x) is bounded and is τ-continuous onX.
i) ACI(B) is the space of functions f such that x7→ f(x)α(x) is bounded and measurable with respect to theσ-algebra BonX.
WhenX is a locally compact Hausdorff space,τ is the topology ofX,α≡1 and K is the the family of compact subsets ofX, then AKC(τ) = Cc(X), the space of
continuous functions with compact support onX and ACI(τ) =Cb(X), the space
of continuous bounded functions onX .
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Mădălina Roxana Buneci [email protected]
Constantin Brâncuşi University of Târgu-Jiu Str. Geneva, Nr. 3, 210136 Târgu-Jiu, Romania
