The Stone–Čech Compactification of Groupoids
Let $G$ be a discrete groupoid. Consider the Stone–Čech compactification $βG$ of $G$ . We extend the operation on the set of composable elements $G^{(2)}$ of $G$ to the operation * on a subset $(βG)^{(2)}$ of $βG×βG$ such that the triple $(βG, (βG)^{(2)}$, *) is a compact right topological semigroup...
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| author | Behrouzi, F. Бехрузі, Ф. |
| author_facet | Behrouzi, F. Бехрузі, Ф. |
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| description | Let $G$ be a discrete groupoid. Consider the Stone–Čech compactification $βG$ of $G$ . We extend the operation on the set of composable elements $G^{(2)}$ of $G$ to the operation * on a subset $(βG)^{(2)}$ of $βG×βG$ such that the triple $(βG, (βG)^{(2)}$, *) is a compact right topological semigroupoid. |
| first_indexed | 2026-03-24T02:16:41Z |
| format | Article |
| fulltext |
UDC 512.5
F. Behrouzi ( Alzahra Univ. and School Math., Inst. Res. Fundam. Sci. (IPM), Tehran, Iran)
THE STONE – ČECH COMPACTIFICATION OF GROUPOIDS*
КОМПАКТИФIКАЦIЯ СТОУНА – ЧЕХА ДЛЯ ГРУПОЇДIВ
LetG be a discrete groupoid and consider the Stone – Čech compactification βG ofG.We extend the operation on the set of
composable elements G(2) of G to the operation “∗” on a subset (βG)(2) of βG×βG such that the triple (βG, (βG)(2), ∗)
is a compact right topological semigroupoid.
Нехай G — дискретний групоїд. Розглянемо компактифiкацiю Стоуна – Чexa βG групоїда G. Розширимо операцiю
на множинi G(2) елементiв G, що компонуються, до операцiї “∗” на пiдмножинi (βG)(2) множини βG× βG такої,
що трiйка (βG, (βG)(2), ∗) є компактним топологiчним напiвгрупоїдом.
1. Introduction. A compactification of a topological space X is a compact space K together with
an embedding e : X −→ K with e(X) dense in K. We usually identify X with e(X) and consider
X as a subspace of K. There exists a very special type of compactification of X in which X is
embedded in such a way that every bounded, real-valued (complex-valued) continuous function on
X will extend continuously to the compactification. Such a compactification of X is called the
Stone – Čech compactification and denoted by βX.
As known, the Stone – Čech compactification βG of an infinite discrete group G can be turned
into a (compact) semigroup by an operation, extended from G [1, 4]. This operation can be taken
in many ways depending on how we regard βG. We can regard βG as the maximal ideal space of
B(G), the C*-algebra of all bounded complex-valued functions on G. In this case, the product of two
elements θ, η ∈ βG, is described by the following steps:
Lg(f)(h) = f(gh), Tη,f (g) = η(Lgf), θ ∗ η(f) = θ(Tη,f ).
Let g ∈ G. By using the universal property of βG (see S1 below), one can extend the continuous map
h 7→ gh : G −→ βG to a continuous map η 7→ g ∗ η : βG −→ βG. Then the mappings g 7→ g ∗ η :
G −→ βG are in turn continuously extended to βG leading to a binary operation in βG. This operation
in βG is associative, so βG is a compact right topological semigroup, that is, the map θ 7→ θ ∗ η :
βG −→ βG is continuous for every η ∈ βG. More generally, for any topological group, there are
many compactifications. Each compactification can be described as the maximal ideal space of a
function algebra.
In this paper, we deal with groupoids instead of groups. Unlike groups, in a groupoid G, the
product is not defined for each two elements of G. But, the product defined on a subset of G × G,
the set of composable pairs. The product on composable elements is associative (see Definition 2.1
below). We will show that, like the group case, the operation of any groupoid G can be extend to
βG such that this operation is still associative.
* This paper was partially supported by a grant from IPM (No. 89470014).
c© F. BEHROUZI, 2015
456 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
THE STONE – ČECH COMPACTIFICATION OF GROUPOIDS 457
2. Preliminaries. The Stone – Čech compactification. For the convenience of the reader we
repeat the relevant material about βX, the Stone – Čech compactification of X, from [3, 8] without
proofs, thus making our exposition self-contained.
Let X be a topological space. Then Cb(X) stands for the algebra of all bounded continuous
complex-valued functions on the topological space X. Also, a subset E of X is called C∗-embedded
if every function in Cb(E) can be extended to a function in Cb(X). A subset E is called zero-set if
there exists a continuous function f in Cb(X) such that E = {x ∈ X : f(x) = 0}. Trivially, every
subset E of a discrete space X is C∗-embedded and also is zero-set. Every (completely regular)
space X has a compactification βX, with the following properties:
S1. (Stone) Every continuous mapping T from X into any compact space Y has a continuous
extension T̃ from βX into Y.
S2. (Stone – Čech) Every function f in Cb(X) has an extension to a function f̃ in C(βX).
S3. (Čech) Any two disjoint zero-sets in X have disjoint closures in βX.
S4. For any two zero-sets Z1 and Z2 in X,
Z1 ∩ Z2 = Z1 ∩ Z2.
S5. A subset S of X is C∗-embedded in X if and only if βS = S.
S6. If S is open-and-closed in X, then S is open-and-closed in βX.
Let E be a subset of a discrete space X. Then, by applying S1 and S5, we can deduce that every
map f from E into compact space Y can be extended to a continuous map f̃ from Ẽ = βE into
Y (S1, S5).
Let X be a discrete space. It is customary to write B(X) rather than Cb(X). So, B(X) by
pointwise operations and the norm
||f ||X = sup
x∈X
|f(x)|
is a commutative unital C*-algebra. Since B(X) is isometrically isomorphism to C(βX) (by S2),
we can identify βX with the maximal ideal space of B(X). So, the topology of βX coincides with
the Gelfand topology. Thus a net {θi}i∈I converges to θ in βX if and only if for every f ∈ B(X)
the net {θi(f)}i∈I converges to θ(f).
Groupoids. Here is some elementary definitions in groupoid literatures. For more details we
refer the reader to [5 – 7].
Definition 2.1. A groupoid is a set G endowed with a product map (g, h) 7→ gh : G(2) −→ G
where G(2) is a subset of G × G called the set of composable pairs and an inverse map g 7→ g−1 :
G→ G such that the following relation are satisfied:
(1) (g−1)−1 = g;
(2) if (g, h) ∈ G(2) and (h, k) ∈ G(2), then (gh, k), (g, hk) ∈ G(2) and we have
(gh)k = g(hk);
(3) (g−1, g) ∈ G(2) and if (g, h) ∈ G(2), then g−1(gh) = h;
(4) (g, g−1) ∈ G(2) and if (h, g) ∈ G(2), then (hg)g−1 = h.
The unit space G0 is the subset of elements gg−1 where g ranges over G. The rang map r :
G −→ G0 and the source map d : G −→ G0 is defined by r(g) = gg−1 and d(g) = g−1g. The pair
(g, h) belongs to the set G(2) if and only if d(g) = r(h). For each u ∈ G0, the subsets Gu and Gu
are given by Gu = d−1({u}), Gu = r−1({u}).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
458 F. BEHROUZI
Definition 2.2. A topological groupoid consists of a groupoid G and a topology compatible with
the groupoid structure:
(1) (x, y) 7→ xy : G(2) −→ G is continuous where G(2) has the induced topology from G×G;
(2) g 7→ g−1 : G −→ G is continuous.
If G is a topological groupoid, then the maps r, d are continuous. In addition, if G0 is Hausdorff
in the relative topology, then G(2) is closed in G×G.
3. Discrete groupoids. Let G be a groupoid and g ∈ G. For any f ∈ B(G), we define the left
g-translation and the right g-translation of f, respectively, by
Lgf(x) =
{
f(gx), x ∈ Gd(g),
0, x /∈ Gd(g),
Rgf(x) =
{
f(xg), x ∈ Gr(g),
0, x /∈ Gr(g).
Since f is bounded, so are Lgf and Rgf. Therefore, for any θ ∈ βG and f ∈ B(G), we can consider
two new functions
Tθ,f : G −→ C, Sθ,f : G −→ C
given by
Tθ,f (g) = θ(Lgf), Sθ,f (g) = θ(Rgf).
It is clear that Tθ,f and Sθ,f are bounded. Next, we collect some elementary properties of these
functions.
Lemma 3.1. Let f be a bounded function on a groupoid G. Then for any g, h ∈ G and any
θ ∈ βG:
(1) If (g, h) ∈ G(2), then Lh(Lgf) = Lghf. Also, if (g, h) /∈ G(2), then Lh(Lgf) = 0.
(2) Lg(Tθ,f ) = Tθ,Lgf and Rg(Sθ,f ) = Sθ,Rgf .
Proof. (1) Let x ∈ G and (g, h) ∈ G(2). Suppose that x /∈ Gd(h) = Gd(gh). By the definition,
Lh(Lgf)(x) = 0 = Lghf(x). Let x ∈ Gd(h) = Gd(gh). Accordingly,
Lh(Lgf)(x) = Lgf(hx) = f(ghx) = Lghf(x).
Therefore, Lghf = Lh(Lgf). Now, suppose that (g, h) /∈ G(2). In this case, if x /∈ Gd(h), then
Lh(Lgf)(x) = 0. If x ∈ Gd(h), then Lh(Lgf)(x) = Lgf(hx). Since g /∈ Gd(h) = Gd(hx), we have
Lgf(hx) = 0.
(2) We only prove the first identity, the proof of the second one is similar. Let h be any element
in G with h /∈ Gd(g). Then Lg(Tθ,f )(h) = 0. On the other hand, Tθ,Lgf (h) = θ(Lh(Lgf)) = 0. Now,
suppose that h ∈ Gd(g). So,
LgTθ,f (h) = Tθ,f (gh) = θ(Lghf) = θ(Lh(Lgf)) = Tθ,Lgf (h).
Lemma 3.1 is proved.
According to G, we define the set of composable elements of βG by
(βG)(2) =
⋃
u∈G0
Gu ×Gu =
⋃
u∈G0
βGu × βGu.
It is trivial that G(2) ⊆ (βG)(2). In the following result, we extend the operation of G(2) to
(βG)(2).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
THE STONE – ČECH COMPACTIFICATION OF GROUPOIDS 459
Theorem 3.1. Let G be a discrete groupoid. There is a unique operation ∗ on (βG)(2) satisfying
the following conditions:
(1) For every (g, h) ∈ G(2), g ∗ h = gh.
(2) For every u ∈ G0 and g ∈ Gu, the map η 7→ g ∗ η : Gu −→ βG is continuous.
(3) For every u ∈ G0 and η ∈ Gu, the map θ 7→ θ ∗ η : Gu −→ βG is continuous.
Proof. Let u ∈ G0. Given any g ∈ Gu, define `ug : Gu −→ G ⊆ βG by `ug (x) = gx. By S1, there
is a continuous map ˜̀ug : βGu = Gu −→ βG such that ˜̀ug |Gu = `ug . Now, let η ∈ Gu and define ruη :
Gu −→ βG by ruη (g) = ˜̀u
g (η). Then there is a continuous map r̃uη : βGu = Gu −→ βG such that
r̃uη |Gu = ruη . For any (θ, η) ∈ Gu ×Gu, set
θ ∗ η = r̃uη (θ).
For (1), suppose that (g, h) ∈ G(2). Then there is a u ∈ G0 such that (g, h) ∈ Gu ×Gu. Therefore,
g ∗ h = r̃uh(g) = ruh(g) = ˜̀u
g (h) = `ug (h) = gh.
The map η 7→ g ∗ η : Gu −→ βG is just the map ˜̀ug and the map θ 7→ θ ∗ η : Gu −→ βG, is just the
map r̃uη and the continuity of these maps follow from the continuity of r̃uη and ˜̀ug .
Theorem 3.1 is proved.
Theorem 3.2. Suppose that G is a discrete groupoid and (θ, η) ∈ (βG)(2).
(1) If {gi}i∈I and {hj}j∈J are nets in G such that limi gi = θ and limj hj = η, then θ ∗ η =
= limi limj gihj .
(2) θ ∗ η(f) = θ(Tη,f ).
Proof. Since (θ, η) ∈ (βG)(2), there is a u ∈ G0 such that (θ, η) ∈ Gu × Gu. As Gu and Gu
are open, we can suppose that {gi}i∈I is a net in Gu and {hj}j∈J is a net in Gu. We have
θ ∗ η = r̃uη (θ) = lim
i
r̃uη (gi) = lim
i
ruη (gi) = lim
i
˜̀u
gi(η) =
= lim
i
lim
j
˜̀u
gi(hj) = lim
i
lim
j
`ugi(hj) = lim
i
lim
j
gihj .
For (2), suppose that {gi}i∈I is a net in Gu and {hj}j∈J is a net in Gu such that limi gi = θ and
limj hj = η. Then for any f in B(G), we have
θ ∗ η(f) = lim
i
lim
j
f(gihj) = Lgif(hj) =
= lim
i
η(Lgif) = lim
i
Tη,f (gi) = θ(Tη,f ).
Theorem 3.2 is proved.
One can consider the inversion map defined by g 7→ g−1 : G −→ G. By S2, this map has a
continuous extension ĩnv : βG −→ βG. We denote again the ĩnv(θ) by θ−1. By the continuity, if
{gi}i∈I is any net in G converging to θ in βG, then {g−1i }i∈I converges to θ−1. Consequently,
(θ−1)−1 = θ and if θ ∈ Gu, then θ−1 ∈ Gu. Let f ∈ B(G) and define the transformation f̂ on
G by f̂(g) = f(g−1). This relation can be extended to βG, that is, for any θ ∈ βG, we have
θ−1(f) = θ(f̂). If G is a groupoid, then (g, g−1) ∈ G(2) for all g ∈ G. But this property does not
hold for the Stone – Čech compactification βG, unless Go is finite.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
460 F. BEHROUZI
Theorem 3.3. Let G be a discrete groupoid. For every θ ∈ βG, (θ, θ−1) ∈ (βG)(2) if and only
if G0 is finite.
Proof. Assume that (θ, θ−1) ∈ (βG)(2) for all θ ∈ βG. Then, it follows that βG =
⋃
u∈G0 Gu.
Since Gu is open in βG, by compactness of βG, there is u1, u2 . . . , un ∈ G0 such that βG =
=
⋃n
k=1Guk . Thus G0 = {u1, u2, . . . , un}. Conversely, suppose that G0 is finite. Then, for every
θ ∈ βG, there exists u ∈ G0 with θ ∈ Gu. Let {gi}i∈I be a net in Gu converging to θ. So, {g−1i }i∈I
is a net in Gu which converges to θ−1. Thus (θ, θ−1) ∈ (βG)(2).
Theorem 3.3 is proved.
Lemma 3.2. Let G be a discrete groupoid, (θ, η) ∈ (βG)(2) and let v ∈ G0. Then
(1) η ∈ Gv if and only if θ ∗ η ∈ Gv;
(2) θ ∈ Gv if and only if θ ∗ η ∈ Gv.
Proof. (1) Let u be in G0 such that (θ, η) ∈ Gu×Gu. Then there exist nets {gi}i∈I and {hj}j∈J ,
respectively, in Gu and Gu such that gi −→ θ and hj −→ η. Since Gv is open set βG containing η,
we can assume that {hj}j∈J is also a net in Gv. By the Theorem 3.2, θ ∗ η = limi limj gihj . Since
for any i and j, gihj ∈ Gv, by Theorem 3.2, we deduce that θ ∗η = limi limj gihj ∈ Gv. Conversely,
suppose that θ ∗ η ∈ Gv. Let {gi}i∈I and {hj}j∈J are net respectively, in Gu and Gu such that
gi −→ θ and hj −→ η. Since Gv is open and containing θ ∗ η, we can assume that gihj ∈ Gv. Thus
hj ∈ Gv and hence θ ∈ Gv.
(2) The proof is similar to (1).
Lemma 3.2 is proved.
Definition 3.1. A semigroupoid is a triple (Λ,Λ(2), ∗) such that Λ is a set, Λ(2) is a subset of
Λ× Λ, and
∗ : Λ(2) −→ Λ
is an operation which is associative in the following sense: if f, g, h ∈ Λ are such that either
(i) (f, g) ∈ Λ(2) and (g, h) ∈ Λ(2), or
(ii) (f, g) ∈ Λ(2) and (f ∗ g, h) ∈ Λ(2), or
(iii) (g, h) ∈ Λ(2) and (f, g ∗ h) ∈ Λ(2),
then all (f, g), (g, h), (f ∗ g, h) and (f, g ∗ h) lie in Λ(2), and
(f ∗ g) ∗ h = f ∗ (g ∗ h).
Moreover, for f ∈ Λ, we will set
Λf = {g ∈ Λ : (f, g) ∈ Λ(2)}, Λf = {g ∈ Λ : (g, f) ∈ Λ(2)}.
Let (Λ,Λ(2), ∗) and (Λ′,Λ′(2), ∗′) be semigroupoids. A map T : Λ −→ Λ′ is called homomorphism
if (f, g) ∈ Λ(2), then (T (f), T (g)) ∈ Λ′(2) and T (f ∗ g) = T (f) ∗′ T (g).
Definition 3.2. Let (Λ,Λ(2), ∗) be a semigroupoid and a topological space. Then
(i) Λ is called left topological semigroupoid if for every f ∈ Λ the map g 7→ f ∗ g : Λf −→ Λ is
continuous.
(ii) Λ is called right topological semigroupoid if for every f ∈ Λ the map g 7→ g ∗ f : Λf −→ Λ
is continuous.
Let Λ be a right topological semigroupoid. The topological center of Λ is the set of all f ∈ Λ
such that the map g 7→ f ∗ g : Λg −→ Λ is continuous.
Theorem 3.4. If G is discrete groupoid, then (βG, (βG)(2), ∗) is a compact right topological
semigroupoid. Moreover, the topological center of βG contains G.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
THE STONE – ČECH COMPACTIFICATION OF GROUPOIDS 461
Proof. Suppose that θ, η, γ are in βG. By Lemma 3.2, each one of conditions (i) – (iii) of
Definition 3.1 implies that (f, g), (g, h), (f ∗ g, h) and (f, g ∗ h) lie in Λ(2). Therefore, it is enough
to prove that if (θ, η) ∈ (βG)(2) and (η, γ) ∈ (βG)(2), then (θ ∗ η) ∗ γ = θ ∗ (η ∗ γ). For, first we
show the following identity:
Tθ,Tη,f = Tθ∗η,f .
Suppose that g ∈ G. Then
Tθ,Tη,f (g) = θ(Lg(Tη,f )) = θ(Tη,Lgf ) = θ ∗ η(Lgf) = Tθ∗η,f (g).
Now, for any f ∈ B(G), we have
θ ∗ (η ∗ γ)(f) = θ(Tη∗γ,f ) = θ(Tη,Tγ,f ) = θ ∗ η(Tγ,f ) = (θ ∗ η) ∗ γ(f).
The above arguments show that βG is a semigroupoid. Also, by the Theorem 3.1, βG is a compact
right topological semigroupoid such that the topological center of βG contains G.
Theorem 3.4 is proved.
Theorem 3.5. Let G be a discrete groupoid and let (K,K(2), ?) be a compact right topological
semigroupoid which is such that the following properties are satisfied:
(1) there is a morphism e : G→ K such that e(G) is dense in K;
(2) the topological center of K contains e(G);
(3)
⋃
u∈G0 e(Gu)× e(Gu) ⊆ K(2).
Then there exists a continuous surjective homomorphism T : βG −→ K such that for each g ∈ G,
T (g) = e(g).
Proof. Since K is compact topological space, there exists a continuous surjective map T :
βG −→ K such that the following diagram is commutative:
G
eβ
//
e
��
βG
T
~~||
||
||
|
K.
Let (θ, η) ∈ (βG)(2). By the definition, there exist u ∈ G0 and a net {gi}i∈I in Gu and a net {hj}j∈J
in Gu such that gi −→ θ and hj −→ η. Therefore, e(gi) −→ T (θ) and e(hj) −→ T (η), and so the
property (3) yields that (T (θ), T (η)) ∈ K(2). Since K(2) is right topological semigroupoid and the
topological center of K contains e(G), we have
T (θ ∗ η) = T (lim
i
lim
j
gihj) = lim
i
lim
j
T (gihj) =
= lim
i
lim
j
e(gihj) = lim
i
lim
j
e(gi) ? e(hj) =
= lim
i
lim
j
T (gi) ? T (hj) = T (θ) ? T (η).
Theorem 3.5 is proved.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
462 F. BEHROUZI
We can start the definition of the product on (βG)(2) by extending the h-right translation map ruh :
x 7→ xh (from Gu to βG) for h ∈ Gu to the map r̃uh : Gu −→ βG . Then consider θ ∈ Gu and define
the map `uθ : Gu −→ βG by `uθ (h) = r̃uh(θ). We extend `uθ to βGu and denote it by ˜̀uθ . Now, define
θ2 η = ˜̀u
θ (η).
So, we have the following results:
(1) For every g, h ∈ G, g2h = gh.
(2) For every u ∈ G0 and θ ∈ Gu, the map η 7→ θ2η : Gu −→ βG is continuous.
(3) For every u ∈ G0 and h ∈ Gu, the map θ 7→ θ2h : Gu −→ βG is continuous.
(4) For every u ∈ G0 and (θ, η) ∈ Gu ×Gu :
θ2 η = lim
j
lim
i
gihj ,
where {gi}i∈I is a net in Gu and {hj}j∈J is a net in Gu such that gi −→ θ and hj −→ η.
(5) For every u ∈ G0, (θ, η) ∈ Gu ×Gu and f ∈ B(G) :
θ2 η(f) = η(Sθ,f ).
(6) (βG, (βG)(2),2) is a compact left topological semigroupoid.
Lemma 3.3. Suppose that G is a groupoid, g ∈ G and θ ∈ βG. Then
(1) Lgf̂ = R̂g−1f,
(2) Tθ,f̂ = Ŝθ−1,f .
Proof. (1) Assume that x ∈ Gd(g). Then x−1 ∈ Gr(g−1) and we have
Lgf̂(x) = f̂(gx) = f(x−1g−1) = Rg−1f(x−1) = R̂g−1f(x).
Also, if x /∈ Gd(g), then x−1 /∈ Gr(g−1), and so
Lgf̂(x) = 0 = Rg−1f(x−1) = R̂g−1f (x).
(2) Let x be any element in G. Then
Tθ,f̂ (x) = θ(Lxf̂) = θ(R̂x−1f) = θ−1(Rx−1f) =
= Sθ−1,f (x−1) = Ŝθ−1,f (x).
Lemma 3.3 is proved.
Theorem 3.6. Let G be a discrete groupoid and (θ, η) ∈ (βG)(2). Then (η−1, θ−1) ∈ (βG)(2)
and we have
η−1 ∗ θ−1 = (θ2 η)−1.
Proof. Let u ∈ G0 be such that θ ∈ Gu and η ∈ Gu. Then η−1 ∈ Gu and θ−1 ∈ Gu. Thus
(η−1, θ−1) ∈ (βG)(2), and so
η−1 ∗ θ−1(f) = η−1(Tθ−1,f ) = η(T̂θ−1,f ) = η(Sθ,f̂ ) =
= θ2η(f̂) = (θ2 η)−1(f).
Theorem 3.6 is proved.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
THE STONE – ČECH COMPACTIFICATION OF GROUPOIDS 463
Example 3.1. An interesting example of a groupoid is an equivalence relation R on a set X.
Here, R(2) = {((g, h), (h, k)) : (g, h), (h, k) ∈ R} and the product map and inversion map are
given by (g, h)(h, k) = (g, k) and (g, h)−1 = (h, g). So, the set of units is {(g, g) : g ∈ X}. Also,
G(g,g) = {g} × [g] and G(g,g) = [g]× {g}. Here [g] denotes the equivalence class of g. In this case,
the set of composable elements is
(βG)(2) =
⋃
g∈G
[g]× {g} × {g} × [g].
To specify the product, let us first determine [g]× {g} and {g} × [g]. Consider the bijection Π1 :
[g] × {g} −→ [g] defined by Π1((h, g)) = h. Thus, there exists homeomorphism Π̃1 : β([g] ×
× {g}) −→ β[g] which is an extension of Π1. If we repeat the argument for {g} × [g] we obtain the
homeomorphism Π2 : β({g} × [g]) −→ β[g] which is an extension of the bijection Π̃2 : {g} × [g] −→
−→ [g] defined by Π2((g, h)) = h. Let f ∈ B(G) and g ∈ X and define fg : [g] −→ C by
fg(h) = f(g, h). Also, for θ ∈ βG define
Uθ,f : X −→ C
by Uθ,f (g) = θ(fg). Let {(gi, g)}i∈I and {(g, hj)}j∈J be nets, respectively, in [g]× {g} and {g} × [g]
such that (gi, g) −→ θ′ and (g, hj) −→ η′ in βG. Since [g]× {g} and {g} × [g] are homeomorphic
to βX, we can assume that there exist θ and η in βX such that gi −→ θ and hj −→ η and Π̃1(θ
′) = θ
and Π̃2(η
′) = η. For any f ∈ B(G), we have
θ′ ∗ η′(f) = lim
i
lim
j
f((gi, g)(g, hj)) = lim
i
lim
j
f(gi, hj) =
= lim
i
lim
j
fgi(hj) = lim
i
η(fgi) =
= lim
i
Uη,f (gi) = θ(Uη,f ).
Note that, we can deduce that the composable elements (βG)(2) is homeomorphic to the disjoint
union of β[g]× β[g]’ s, that is,
◦⊔
g∈X
β[g]× β[g].
Example 3.2. Another example of a groupoid is the transformation group groupoid. Suppose that
the group S acts on a set U on the right. The image of the point u by the transformation s is denoted
u.s. We let G be U ×S and define the following groupoid structure: (u, s) and (v, t) are composable
if and if v = u.s, (u, s)(u.s, t) = (u, st), and (u, s)−1 = (u.s, s−1). Then r(u, s) = (u, e) and
d(u, s) = (u.s, e). The map (u, e) 7→ u identifies G0 with U. It is easy to check that
Gu = {u} × S, Gu = {(u.s−1, s) : s ∈ S}.
By the same argument mentioned in Example 3.1, we can identify Gu with S and obtain a home-
omorphism between βGu and βS. Also, the map (u.s−1, s) 7→ s is a bijection from Gu onto S.
So, this map has a continuous extension from βGu onto βS. Thus, the set composable elements is
homeomorphic to
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
464 F. BEHROUZI
◦⊔
u∈U
βS × βS.
Let (θ′, η′) ∈ Gu×Gu. Let {(u.s−1i , si)}i∈I and {(u, tj)}j∈J be nets in βG such that (u.s−1i , si) −→
−→ θ′ and (u, tj) −→ η′. Therefore, there exist η and θ in βS such that si −→ θ and tj −→ η. For
any f ∈ B(G), one has
θ′ ∗ η′(f) = lim
i
lim
j
(u.s−1i , si)(u, tj) =
= lim
i
lim
j
(u, sitj)(f) =
= lim
i
lim
j
fu(sitj) = θ ∗ η(fu),
where fu is a map from S into C defined by fu(s) = f(u, s).
4. Topological groupoids. Let G be a topological groupoid such that for every u ∈ G0, Gu is
C*-embedded. Let l̃ug be the extension of the map lug mentioned in the previous section. Then for
each fixed η ∈ Gu, we may consider the mapping ruη from Gu into βG. Defined by ruη (g) = l̃ug (g).
But unlike the discrete case, nothing guarantees that the mapping ruη is continuous for every η ∈ Gu.
Therefore we might not able to extend these mappings to βG leading to a continuous operation on
βG.
We can start this process by extending the mappings ruh : Gu −→ βG to mappings r̃uh : Gu −→
−→ βG, where h ∈ Gu. If for every θ ∈ Gu we define luθ : Gu −→ βG by luθ (h) = r̃uh(θ), again
nothing guarantees the continuity of the mappings lθ for each θ ∈ Gu.
Let G be a topological groupoid. As the previous section, we can define left g-translation Lgf
and right g-translation. But these map are not continuous in general. As we know we can regard βG
as the maximal ideal space of Cb(G). It seems that we can not define the extended operation on βG
in the term of elements of the maximal ideal space of Cb(G). Because the mapping g 7→ η(Lgf) is
not well-defined.
Lemma 4.1. Suppose that u ∈ G0 , η ∈ Gu and f ∈ Cb(Gu). Let F1, F2 ∈ Cb(G)such that
F1|Gu = F2|Gu = f. Then η(F1) = η(F2).
Let u ∈ G0 , g ∈ Gu and let h ∈ Gu. For f ∈ Cb(G) define Lugf : Gu −→ C by Lugf(x) = f(gx).
Also, define Ruhf : Gu −→ C by Ruhf(x) = f(xh).
Definition 4.1. Let u ∈ G0, η ∈ Gu, θ ∈ Gu. For f ∈ Cb(G), set
T uη,f : Gu −→ C T uη,f (g) = η(L̃ugf),
where L̃ugf is an extension of Lugf in Cb(G). By Lemma 4.1, T uη,f is well-defined. Also defined
Suθ,f : Gu −→ C, Suθ,f (h) = θ(R̃uhf),
where R̃uhf is an extension of R̃uhf in Cb(G).
Theorem 4.1. Let G be a topological groupoid and let f ∈ Cb(G). Then the followings are
equivalent:
(1) For every u ∈ G0 and for every θ ∈ Gu, luθ is continuous.
(2) For every u ∈ G0 and for every η ∈ Gu, ruη is continuous.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
THE STONE – ČECH COMPACTIFICATION OF GROUPOIDS 465
(3) For every u ∈ G0, for every η ∈ Gu and every f ∈ Cb(G), T uh f is continuous.
(4) For every u ∈ G0, for every θ ∈ Gu and every f ∈ Cb(G), Suθ,f is continuous.
Proof. First, we prove the following identities, for any u ∈ G0, g ∈ Gu, h ∈ Gu and f ∈ B(G),
are satisfied
f(ruη (g)) = f̂(luη−1(g−1)), (4.1)
f(ruη (g)) = T uη,f (g), (4.2)
f(luθ (h)) = Suθ,f (h). (4.3)
Let u ∈ G0 and η ∈ Gu. Let {hj}j∈J be a net in Gu such that hj −→ η. So, η−1 ∈ Gu and
hj −→ η−1. For (4.1),
f(ruη (g)) = f(l̃ug (η)) = lim
j
f(lug (hj)) = lim
j
f(ghj) =
= lim
j
f̂(h−1j g−1) = lim
j
f(rug−1(h−1j )) =
= f̂(r̃u
g−1(η−1)) = f̂(luη−1(g−1)).
On the other hand
f(ruη (g)) = f(l̃ug (η)) = lim
j
f(lug (hj)) = lim
j
f(ghj) =
= lim
j
Lugf(hj) = η(L̃ugf) = T uη,f (g).
Similarly, one can prove (4.3). The identity (4.1) implies that (1)⇔ (2) and the identity (4.2) implies
that (2)⇔ (3) and the identity (4.3) implies that (1)⇔ (4) .
Theorem 4.1 is proved.
Example 4.1. Let G be the groupoid [0,∞) × [0,∞). Consider the sequence ((1, n))∞n=1. So
there exist a subnet ((1, nk))
∞
k=1 and η ∈ βG such that (1, nk) −→ η in βG. Let f be a function in
Cb([0,∞)) such that f(n) = 1 for every n and f(t) = 0 if n+
1
n
< t < n+ 1− 1
n+ 1
. Define F :
G −→ C by F (x, y) = f(x + y). Then for every m ∈ N, Tη,F
(
1
m
, 1
)
= limk f
(
nk +
1
m
)
= 0.
But Tη,F (0, 1) = f(nk) = 1. Therefore, Tη,F is not continuous.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
466 F. BEHROUZI
1. Filali M., Protasov I. Ultrafilters and topologies on groups // Math. Stud. Monogr. Ser. – Lviv: VNTL, 2010.
2. Filali M., Vedenjuoksu T. The Stone – Čech compactification of a topological group and the β-extension property //
Houston J. Math. – 2010. – 36, № 2. – P. 477 – 488.
3. Gillman L., Jerison M. Rings of continuous functions. – Princeton: van Nostrand, 1960.
4. Hindman N., Strauss D. Algebra in the Stone – Čech compactification theory and applications. – Berlin: Water de
Gruyter, 1998.
5. Muhly P. Coordinates in operator algebra (Book in preparation).
6. Paterson A. L. T. Groupoids, inverse semigroups, and their operator algebras // Progr. Math. – Boston, MA: Birkhäuser
Boston Inc., 1999. – 170.
7. Renault J. A groupoid approach to C∗-algebras // Lect. Notes Math. – 1980. – 793.
8. Walker R. The Stone – Čech compactification. – Berlin: Springer-Verlag, 1974.
Received 27.11.12,
after revision — 28.11.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
|
| id | umjimathkievua-article-1996 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:16:41Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/c6/f1f3881c542df5d1486b3485f868b2c6.pdf |
| spelling | umjimathkievua-article-19962019-12-05T09:48:26Z The Stone–Čech Compactification of Groupoids Компактифікація Стоуна-Чеха для групоїдів Behrouzi, F. Бехрузі, Ф. Let $G$ be a discrete groupoid. Consider the Stone–Čech compactification $βG$ of $G$ . We extend the operation on the set of composable elements $G^{(2)}$ of $G$ to the operation * on a subset $(βG)^{(2)}$ of $βG×βG$ such that the triple $(βG, (βG)^{(2)}$, *) is a compact right topological semigroupoid. Нехай $G$ — дискретний групоїд. Розглянемо компактифікацію Стоуна-Чexa $βG$ групоїда $G$. Розширимо операцію на множині $G^{(2)}$ єлємєнтів $G$, що компонуються, до операції "*" на підмножині $(βG)^{(2)}$ множини $βG×βG$ такої, що трійка $(βG, (βG)^{(2)}$ є компактним топологічним напівгрупоїдом. Institute of Mathematics, NAS of Ukraine 2015-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1996 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 4 (2015); 456–466 Український математичний журнал; Том 67 № 4 (2015); 456–466 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1996/1010 https://umj.imath.kiev.ua/index.php/umj/article/view/1996/1011 Copyright (c) 2015 Behrouzi F. |
| spellingShingle | Behrouzi, F. Бехрузі, Ф. The Stone–Čech Compactification of Groupoids |
| title | The Stone–Čech Compactification of Groupoids |
| title_alt | Компактифікація Стоуна-Чеха для групоїдів |
| title_full | The Stone–Čech Compactification of Groupoids |
| title_fullStr | The Stone–Čech Compactification of Groupoids |
| title_full_unstemmed | The Stone–Čech Compactification of Groupoids |
| title_short | The Stone–Čech Compactification of Groupoids |
| title_sort | stone–čech compactification of groupoids |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1996 |
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