Full text of DOI:10.1007/s002330010074

Semigroup Forum Vol. 63 (2001) 449–456
c
ꢀ 2001 Springer-Verlag New York Inc.
DOI: 10.1007/s002330010074
RESEARCH ARTICLE
Ideals, Idempotents and Right Cancelable
Elements in the Uniform Compactification
S. Ferri and D. Strauss
Communicated byJohn Pym
Introduction
In this note, we discuss the number of left ideals, idempotents and right can-
celable elements in the uniform compactification of a topological group.These
are related to covering numbers defined for the group.
G will denote a topological group with identity e.We assume that
G
has the right uniform structure in which a basis for the vicinities is provided
by the sets of the form {(x, y) ∈ G × G: xy−1 ∈ U}, where U denotes a
neighbourhood of e in G. Cu(G) will denote the set of real-valued bounded
uniformly continuous functions defined on G, and uG will denote the uniform
compactification of G. G can be topologically embedded in uG, and so we
shall regard G as a subspace of uG.Furthermore, a semigroup operation can
be defined on uG which extends that defined on G.This has the property
that, for every x ∈ uG, the map ρx: y → yx is a continuous map from uG to
itself.Furthermore, the semigroup operation as a map from uG×uG to uG, is
jointly continuous at every point of G × uG.A real-valued bounded continuous
function defined on G has a continuous extension to uG if and only if it is
uniformly continuous.If f ∈ Cu(G), we shall use f to denote the continuous
extension of f to uG.
The reader is referred to [1] or [4] for proofs of these statements.We
observe that uG is often denoted by GLUC or LC(G) in the literature.
We shall use Gd to denote G with the discrete topology, and βGd ∼ uGd
ˇ
to denote its Stone-Cech compactification.There is a continuous surjective
homomorphism π: βGd → uG.This has the property that, for every x, y ∈
βGd , π(x) = π(y) if and only if fβ(x) = fβ(y) for every f ∈ Cu(G), where
fβ: βGd → R denotes the continuous extension of f .
If U is a neighbourhood of e in G, we define the cardinal κU (G) to be
the smallest number of sets of the form Us (s ∈ G) required to cover G.We
put
κ(G) = sup{κU (G): U is a neighbourhood of e in G}.
κ(G)
We show that, under certain conditions, uG has 22
and that uG \ G has 22
minimal left ideals
κ(G)
elements which are right cancelable in uG.These

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Ferri and Strauss
conditions are always satisfied if G is locally compact.We show that the number
of left invariant means on Cu(G) is at least equal to the number of minimal left
ideals in uG, assuming that G is left amenable.We also give a result about
the number of idempotents in uG in terms of another covering number.
Our results about the number of disjoint left ideals in uG and the number
of left invariant means on uG are not new in the case in which G is locally
compact.They were proved in [6] and in [5].However, we believe that our
results are new for many topological groups which are not locally compact.
M.Filali has previously obtained a result about right cancelable elements
in uG.It was shown in [2] that uG \ G contains a dense open set of right
cancelable elements in the special case in which G is the product of Rn and a
topological group which contains a compact open normal subgroup.M.Filali
and J.S.Pym have recently proved that uG \ G contains a dense set of right
cancelable elements if G is any locally compact topological group [3].
We wish to thank J.S.Pym for very helpful comments. In particular,
Example 1.7 is due to him.
1. Disjoint left ideals, idempotents
and right cancelable points in uG
Lemma 1.1.
that xꢀ ∈/ Ux whenever x and xꢀ are distinct points of X , then π: βGd → uG
Let X, Y ⊆ G. If there is a neighbourhood U of e in G such
is injective on clβG (X). If Y ⊆ G and V X ∩ Y = ∅ for some neighbourhood
d
V of e in G, then cluG(X) ∩ cluG(Y ) = ∅.
Proof.
Let p, q ∈ clβG (X), with p = q.We can choose A, B ⊆ X such
d
that A ∈ p, B ∈ q and A ∩ B = ∅.Since UA ∩ B = ∅, there is a function
f ∈ Cu(G) such that f = 0 on A and f = 1 on B (see [4], ex. 21.5.3). Thus
fβ(p) = 0 and fβ(q) = 1, and so π(p) = π(q).
There is a function g ∈ Cu(G) such that g = 0 on X and g = 1 on Y .
Since g = 0 on cluG(X) and g = 1 on cluG(Y ), cluG(X) ∩ cluG(Y ) = ∅.
Theorem 1.2.
is infinite, uG has at least 22
Let U be a symmetric neighbourhood of e in G. If κU (G)
κ (G)
U
points.
Proof.
whenever x and xꢀ are distinct elements of X .Since
Let X ⊆ G be maximal subject to the condition that xꢀ ∈/ Ux
ꢀ
x∈X Ux covers G,
κ
(G)
U
|X| ≥ κU (G).So |clβG(X)| ≥ 22
([4], Theorem 3.58). Our claim therefore
follows from Lemma 1.1.
Our next theorem concerns the number of disjoint left ideals and the
number of right cancelable elements of uG.
Theorem 1.3.
Let κ(G) be infinite.
Suppose that there exists a neighbourhood U of e in G such that G cannot
be covered by less than κ(G) sets of the form sUt (s, t ∈ G).

Ferri and Strauss
451
Let A denote the family of subsets A ⊆ G which have the property that
G \ A can be covered by less than κ(G) sets of the form sUt (s, t ∈ G) and
ꢁ
define C := A∈A cluGA. If p ∈ C and N is a neighbourhood of p in uG,
then there is a set Q ⊆ N ∩ (uG \ G) such that |Q| = 22κ(G) , the left ideals of
the form (uG)q (q ∈ Q) are pairwise disjoint and the elements of Q are right
cancelable in uG.
Proof.
Suppose N to be closed and consider a symmetric neighbourhood
V of e in G such that V 3 ⊆ U .By hypothesis we can cover G with κ(G)
sets V sα (α < κ) where each sα ∈ G.We may also suppose that s0 = e.
Define then by induction a κ-sequence (tα)α<κ in N in such a way that
tβ ∈/ s−γ 1Usδtα whenever α, γ, δ < β .Let K be the set of κ-uniform ultrafilters
κ(G)
on {tα: α < κ}.The cardinality of K is 22
(see [4], Theorem 3.58). Since
tβ ∈/ Utα whenever α, β < κ are distinct, it follows from Lemma 1.1 that
π: βGd → uG is injective on K .We put Q := π(K) and consider two distinct
elements q1, q2 ∈ Q with q1 = π(x1) and q2 = π(x2), (x1, x2 ∈ K ).Let
˜
X1 ∈ x1 and X2 ∈ x2 be disjoint.We define sets Xi := {vsαtβ: v ∈ V, α <
˜
β, tβ ∈ Xi}, (i = 1, 2).We note that, for every x ∈ uG, xqi ∈ cluGXi ,
because xqi = limvs →x limt
vsαtβ .Using the property which defines
(tα)α<κ and the fact that V 3 ⊆ U , we have V X1 ∩ X2 = ∅.So, by Lemma 11.,
β →qi
α
˜
˜
˜
˜
cluG(X1) ∩ cluG(X2) = ∅ and hence (uG)q1 ∩ (uG)q2 = ∅, as required.Notice
also that Q ⊆ uG \ G, because (uG)a = uG if a ∈ G.
It remains to show that all q ∈ Q are right cancelable in uG.Let
x, y ∈ uG be distinct and suppose, by contradiction, that xq = yq with q ∈ Q.
¯
¯
Since x = y there exists h ∈ Cu(G) such that h(x) = 0 and h(y) = 1.Since
h is uniformly continuous there exists a neighbourhood W of e in G (that we
1
3
may suppose contained in V ) such that |h(s) − h(t)| <
whenever s, t ∈ G
1
are such that st−1 ∈ W .Let A := {vsαtβ: v ∈ V, α < β, h(vsα) < 3 } and
2
B := {vsαtβ: v ∈ V, α < β, h(vsα) > 3 }.Since, as before, xq ∈ cluG(A) and
yq ∈ cluG(B), we have WA ∩ B = ∅.So wvsαtβ = vꢀsγtδ for some w ∈ W ,
v, vꢀ ∈ V , α, β < κ, where α < β , γ < δ, h(vsα) < and h(vꢀsγ) > 23 .Using
1
3
once more the property of (tα)α<κ we get β = δ and hence wvsα = vꢀsγ .So
|h(vsα) − h(vꢀsγ)| < 31 , which is a contradiction.
When G is locally compact and non-compact, κ(G) is the cardinality
of the smallest number of compact subsets required to cover G.Therefore
hypothesis † of Theorem 1.3 holds for any compact neighbourhood U of e.
So the conclusions of Theorem 1.3 hold for all locally compact, non-compact
groups.In particular, if G is locally compact and σ-compact, there is a dense
subset of uG \ G whose elements are all right cancelable in G.
Hypothesis † also holds for many other groups.For example, suppose
that G is a SIN-group.(This means that e has a basis of neighbourhoods U
for which tUt−1 = U for every t ∈ G. ) Ifκ(G) = ω and if G is not totally
bounded, then hypothesis † of Theorem 1.3 holds. It is not hard to prove, in
this case, that we again have a dense subset of uG \ G whose elements are all

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Ferri and Strauss
right cancelable in uG.In particular, these statements hold for all separable
topological vector spaces.
We note that † also holds for any SIN-group G for which κ(G) = κU (G)
for some neighbourhood U of e in G.
In some cases, † is necessary, as well as sufficient, for the existence of
κ(G)
22
minimal left ideals.For example, if G is a SIN-group and κ(G) = ω,
then † fails to hold if and only if G is totally bounded.In this case, uG is
the completion of G, a compact topological group, which has precisely one left
ideal.In the following theorem, we see that † is necessary for the existence of 2c
disjoint left ideals in uG if G is a metrisable topological group with κ(G) = ω.
Theorem 1.4.
Let G be a metrisable group. Suppose that, for every neigh-
bourhood of e, G is covered by a finite number of sets of the form xUy
(x, y ∈ G). Then uG cannot have more than c disjoint left ideals.
Proof.
be a finite subset of G×G such that G =
Let (Un) be a basis of neighbourhoods of e.For every n ∈ N let Fn
ꢀ
ꢂ
xUny.Let Φ :=
Fn .
(x,y)∈Fn
n∈N
xUny, and so
ꢀ
Then |Φ| ≤ c.Let p ∈ uG.For every n ∈ N, G =
(x,y)∈Fn
there exists (x, y) ∈ Fn such that p ∈ cluG(xUny).We can choose f ∈ Φ such
that f(n) = (xn, yn) implies that p ∈ cluG(xnUnyn).We shall prove that if
p1 and p2 correspond to the same function f , then the principal left ideals
corresponding to p1 and p2 intersect.
We have x−n 1p1yn−1 ∈ cluGUn for every n ∈ N. So x−n 1p1yn−1 −→ e in
uG.Let ( z, y) be a limit point of (x−n 1, yn) in uG × uG and let (x−1, yn ) be
nι
ι
a net in G × G converging to (z, y) in uG.The semigroup operation of uG
is jointly continuous at every point of G × uG (see [4], Theorem 21.44). So
(xn−1p1y−1)yn = x−n 1p1 converges both to y and zp1 .Thus zp1 = y.We also
nι
ι
ι
ι
have zp2 = y, and so (uG)p1 ∩ (uG)p2 = ∅.
Theorem 1.5.
Suppose that κU (G) < κ(G) for every neighbourhood U of e
in G and that there is a basis of cardinality at most κ(G) for the neighbourhoods
of e in G. Then, assuming the generalised continuum hypothesis, uG contains
at most 2κ(G) points.
Proof.
Let (Uα)α<κ(G) be a basis for the neighbourhoods of e in G.For
each α < κ(G), we can choose Xα ⊆ G such that G = UαXα and |Xα| =
κU (G).For each p ∈ uG and each α < κ(G), let φ(α, p) := {Y ∈ P(Xα): p ∈
α
κ
|φ(α, p)| ≤ 22 (G) .Hence, assuming the
U
α
cluG(UαY )}.We observe that
generalised continuum hypothesis, |φ(α, p)| < κ(G).
We shall show that, if p, q ∈ u(G) have the property that ϕ(α, p) =
ϕ(α, q) for every α < κ(G), then p = q.
To see this, assume that p = q.We can choose f ∈ CU (G) such that
f(p) = 0 and f(q) = 1.We can then choose α < κ(G) such that |f(x)−f(y)| <
1
8
1
whenever x, y ∈ G satisfy xy−1 ∈ Uα .Then, if Y := {x ∈ Xα: f(x) < 4 },
3
8
we have Y ∈ φ(p, α) and hence Y ∈ φ(q, α).However, f < on UαY and so
q ∈/ cluG(UαY )—a contradiction.

Ferri and Strauss
453
Thus we have shown that we have an injective mapping from uG into
κ(G)κ(G) , defined by p → (φ(p, α))α<κ(G) .Our claim now follows from the fact
that κ(G)κ(G) = 2κ(G)
.
Theorem 1.6.
Let U be a symmetric neighbourhood of e in G and let
λU (G) denote the least number of sets of the form sUt (s, t ∈ G) required
λ
(G)
U
to cover G. If λU (G) is infinite, uG has at least 22
idempotents.
Proof.
Put λ = λU (G).We can inductively choose a λ-sequence (xα)α<λ
in G such that x0 = e and, whenever F and Fꢀ are finite subsets of [0, β)
ꢂ
ꢂ
arranged in increasing order, xβ ∈/ (
xα)−1U(
ꢀ xα).
α∈F
α∈F
Let X = {xα: α < β} and let FP(X) denote the set of all finite
products of the form xα xα · · · xα with α1 < α2 < · · · < αm < λ.If
1
2
m
β < λ, let FPβ(X) denote the set of products of this form with α1 > β .Let
ꢁ
C =
clβG (FPβ(X)).
d
β<λ
We observe that C is a subsemigroup of βGd (see [4], Theorem 4.20). We
shall show that Cp ∩ Cq = ∅ whenever p and q are distinct uniform ultrafilters
on X .To see this, choose disjoint subsets P and Q of X with P ∈ p and
Q ∈ q.Put
Y = {xα xα · · · xα : α1 < α2 < · · · < αm, αm ∈ P} and
m
1
2
m
Z = {xα xα · · · xα : α1 < α2 < · · · < αm, αm ∈ Q}.Then Y ∩ Z = ∅ and so
1
2
clβG (Y ) ∩ clβG (Z) = ∅. However, Cp ⊆ clβG (Y ) and Cq ⊆ clβG (Z).
d
d
d
d
λ
λ
Since there are 22 uniform ultrafilters on X , there are 22 disjoint left
ideals in C .Each of these contains an idempotent (see [4], Corollary 26.).
We claim that π: βGd → uG is injective on clβG (FP(X)).To see this,
d
we shall apply Lemma 1.1 and show that, if y and z are distinct elements of
FP(X), then z ∈/ Uy.We suppose the contrary.Let y = xα xα · · · xα and
1
2
m
z = xβ xβ · · · xβ with α1 < α2 < · · · < αm < λ and β1 < β2 < · · · < βn < λ.
1
2
n
We assume that m+n has been chosen to be as small as possible subject to the
condition that z ∈ Uy.Our choice of {xα: α < λ} implies that αm = βn and
hence that xα xα · · · xα
∈ Uxβ xβ · · · xβn−1 , contradicting the minimality
1
2
m−1
1
2
of m + n.
λ
Since π is injective on clβG (FP(X)), uG has at least 22 idempot-
d
ents.
For each neighbourhood U of e in G, let λU (G) denote the smallest
number of sets of the form xUy (x, y ∈ G) required to cover G, and let
λ(G) = sup{λU (G): U a neighbourhood of e in G}.We remark that it follows
from Theorem 1.6 that uG has at least λ(G) idempotents, unless λU (G) is
finite for every neighbourhood U of e in G.
We are indebted to J.S.Pym for the following example of a group
G
for which λ(G) < κ(G).This example also allows us to show that † is not, in
general, necessary for the conclusions of Theorem 1.3 to hold.
Example 1.7.
Let η > 1 be any cardinal.We shall define a group G for
which κ(G) = max(c, η) and λ(G) = ω.
Let F be a group with identity 1F and |F| = η, and let Fi = F for each

454
Ferri and Strauss
ꢃ
i ∈ (0, 1).We put H := i∈(0,1) Fi , with the topology defined by choosing as a
ꢃ
base of neighbourhoods of the identity 1H the sets of the form Uꢀ = i∈(0,1) Vi ,
where 5 ∈ (0, 1), Vi = F if i < 5 and Vi = {1F } if i ≥ 5.
We define Φ to be the group of functions t → tr defined on (0,1), where
r denotes a positive rational number, with composition as the group operation.
We denote the identity of Φ by 1Φ .We give Φ the discrete topology.We define
an action of Φ on H by putting φ(xi) = (xφ(i)), where φ ∈ Φ and (xi) ∈ H .
We take G to be the semidirect product of H and Φ.So G = H × Φ
as a topological space, with the group operation of G given by (x, φ)(y, ψ) =
(xφ(y), φψ).
ꢄ
ꢄ
ꢄ
Let H = H × {1Φ} and Uꢀ = Uꢀ × {1Φ}.Note that H is an open
ꢄ
subsemigroup of G and that {Uꢀ: 5 ∈ (0, 1)} is a base for the neighbourhoods
of e = (1H, 1Φ) in G.
ꢄ
ꢄ
A set of the form Uꢀ(x, φ) meets H if and only if φ = 1H .Since
ꢄ
κ
(H) = κU (H) = max(c, η), we have κ(G) = max(c, η).
ꢀ
ꢄ
Uꢀ
On the other hand, we have (1H, φ)Uꢀ(1H, φ−1) = Uφ(ꢀ) × {1Φ} for every
ꢄ
ꢀ
1
n
−1
ꢄ
ꢄ
φ ∈ Φ.Let φn = t .Since φn ꢃ 1,
G/ H is countable and so λ(G) = ω.
(1H, φn)Uꢀ(1H, φn ) = H .Now
n∈N
ꢄ
We now claim that, with a suitable choice of η, G nevertheless satisfies
the conclusions of Theorem 1.3. Our proof is similar to that of Theorem 1.3.
We choose η > c, with the property that η = sup{ηn: n ∈ N} for some
ꢄ
increasing sequence (ηn) of distinct cardinals.We enumerate H as (sα)α<η
with s0 = e, and Φ as (φn)n∈N, with φ0 = 1Φ .We observe that every element
,
of G can be expressed uniquely in the form (1H, φn)sα for some n ∈ N and
some α < η.
ꢄ
We inductively choose (tβ)β<η in H , with t0 = e and
−1
tδ ∈/ s−β 1(1H, φn)−1U (1H, φn)sαtγ = sβ Uφ
−1
(
n
)sαtγ,
ꢄ
ꢄ
1
2
1
2
whenever α, β, γ, ηn < δ.This is possible, because, for a given δ < η, {n ∈
N: ηn < δ} is finite.
As in the proof of Theorem 1.3, we choose K to be the set of η-uniform
η
ultrafilters on {tβ: β < η}, and we put Q = π(K).We note that |K| = 22
η
and hence that |Q| = 22 , because π is injective on K , by Lemma 1.1.
We claim that, if q1 and q2 are distinct elements of Q, then (uG)q1 ∩
(uG)q2 = ∅.To see this, we choose x1 and x2 in K with π(x1) = q1 and
π(x2) = q2 .We then choose disjoint subsets X1 and X2 of {tβ: β < η} with
X1 ∈ x1 and X2 ∈ x2 .For i ∈ {1, 2}, we put
ꢄ
Xi := {(1H, φn)sαtβ: ηn < β, α < β, tβ ∈ Xi}.
ꢀ
ꢀ
ꢀ
We note that an equation of the form u(1H, φn)sαtβ = (1H, φn )sα tβ , with
ꢀ
ꢄ
ꢄ ꢄ
ꢄ
ꢀ
ꢀ
1
2
u, sα, sα , tβ, tβ ∈ H , can only hold if n = n .It follows that U X1 ∩ X2 = ∅.

Ferri and Strauss
455
ꢄ
ꢄ
ꢄ
So cluGX1 ∩ cluGX2 = ∅, by Lemma 1.1. Now, if i ∈ {1, 2}, (uG)qi ⊆ cluGXi ,
and so our claim follows.
It is also true that each element of Q is right cancelable in uG.We
omit the proof, which is essentially the same as the proof of the corresponding
statement in Theorem 1.3.
2. Consequences
The main consequence of Theorem 1.3 concerns the number of left invariant
means on a left amenable group.In the following theorem we use the terminol-
ogy of [1].
A mean on Cu(G) is a real linear functional µ on Cu(G) with the
property that
inf f(s) ≤ µ(f) ≤ sup f(s).
s∈G
s∈G
for every f ∈ Cu(G).If ν is a mean on Cu(G), we define Tν: Cu(G) → Cu(G)
by Tν(f)(s) = ν(Lsf), where Lsf ∈ Cu(G) is given by Lsf(t) = f(st)
(∀s, t ∈ G).If µ, ν are means on Cu(G) a mean µν on Cu(G) is defined
by µν(f) = µ(Tν(f)).
A mean µ on uG is said to be multiplicative if µ(fg) = µ(f)µ(g)
for every f, g ∈ Cu(G).We note that uG can be identified with the set of
multiplicative means on Cu(G), endowed with the w∗ -topology of Cu∗(G).If µ
is a multiplicative mean on Cu(G) and f ∈ Cu , then f(µ) = µ(f).
A mean µ on Cu(G) is said to be left invariant if µ(Lsf) = µ(f) for
every f ∈ Cu(G) and every s ∈ G. Cu(G) is said to be left amenable if a left
invariant mean on Cu(G) exists.
We shall use M(Cu(G)), MM (Cu(G)) and LIM (Cu(G)) respectively for
the set of means, the set of multiplicative means and the set of left invariant
means on Cu(G).We shall use the fact that MM (Cu(G)) is the set of extreme
points of M(Cu(G)).We refer the reader to [1], Chapter 2, for the proofs of
these statements.
Theorem 2.1.
Let G be left amenable. Suppose that uG has η minimal left
ideals, then there are at least η means in LIM (Cu(G)).
Proof.
Let µ ∈ LIM (Cu(G)).
By Proposition 3.5, p. 81 of [1], LIM (G) is a right ideal in M(Cu(G)).We
shall prove that, if L and Lꢀ are disjoint closed left ideals in uG, then µν = µνꢀ
if ν ∈ L and νꢀ ∈ Lꢀ .Notice that we are regarding uG as MM (Cu(G)).Since
µ ∈ M(Cu(G)), µ is in the closed convex hull of MM (Cu(G)), by the Krein-
Milman Theorem.Let f ∈ Cu(G) satisfy f(L) = {0} and f(Lꢀ) = {1}, where
f: uG → R denotes the continuous extension of f .If λ is a convex combination
of elements in MM (Cu(G)), then (λν)(f) = 0 and (λνꢀ)(f) = 1, because λν
is a convex combination of elements in L and λνꢀ is a convex combination of
elements in Lꢀ .So ( µν)(f) = 0 and (µνꢀ)(f) = 1, and therefore µν = µνꢀ .

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Ferri and Strauss
Corollary 2.2.
Theorem 1.1, then |LIM (Cu(G))| ≥ 22
If Cu(G) is left amenable and G satisfies hypothesis † of
κ(G)
.
References
[1] Berglund, J.F,. H.D.Junghenn and P.Milnes, “Analysis on Semigroups”,
Wiley-Interscience, New York 1989.
[2] Filali, M., On some semigroup compactifications, In Proc.of the 12th Sum-
mer Conference on General Topology and its Applications (North Bay, ON,
1997), Topology Proc. 22 (Summer 1997), 111–123.
[3] Filali, M,. and J.S.Pym, manuscript.
ˇ
[4] Hindman, N., and D. Strauss, “Algebra in the Stone-Cech Compactifica-
tion”, Walter de Gruyter, Berlin, New York, 1998.
[5] Lau, A.T,. P.Milnes and J.S.Pym,
Locally compact groups, invariant
means and the centres of compactifications, J.London Math.Soc. 56, No.2
(1997), 77–90.
[6] Lau, A.T., and A.L.Paterson, The exact cardinality of the set of topological
left invariant means on an amenable locally compact group, Proc.Amer.
Math.Soc. 98 (1986), 75–80.
Department of Mathematics
Universityof Hull
Cottingham Road
Kingston upon Hull HU6 7RX, UK
s.ferri@maths.hull.ac.uk
d.strauss@maths.hull.ac.uk
Received November 6, 2000
and in final form November 27, 2000
Online publication June 29, 2001